On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes
The lack of a positive-definite and conserved energy is a serious obstacle in the black hole stability problem. In this work, we will show that there exists a positive-definite and conserved Hamiltonian energy for axially symmetric linear perturbations of the exterior of Kerr black hole spacetimes....
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2025 |
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Інститут математики НАН України
2025
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| Цитувати: | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes. Nishanth Gudapati. SIGMA 21 (2025), 054, 72 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860277685948776448 |
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| author | Gudapati, Nishanth |
| author_facet | Gudapati, Nishanth |
| citation_txt | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes. Nishanth Gudapati. SIGMA 21 (2025), 054, 72 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The lack of a positive-definite and conserved energy is a serious obstacle in the black hole stability problem. In this work, we will show that there exists a positive-definite and conserved Hamiltonian energy for axially symmetric linear perturbations of the exterior of Kerr black hole spacetimes. In the first part, based on the Hamiltonian dimensional reduction of 3 + 1 axially symmetric, Ricci-flat Lorentzian spacetimes to a 2 + 1 Einstein-wave map system with the negatively curved hyperbolic 2-plane target, we construct a positive-definite, spacetime gauge-invariant energy functional for linear axially symmetric perturbations in the exterior of Kerr black holes, in a manner that is also gauge-independent on the target manifold. In the construction of the positive-definite energy, various dynamical terms at the boundary of the orbit space occur critically. In the second part, after setting up the initial value problem in harmonic coordinates, we prove that the positive energy for the axially symmetric linear perturbative theory of Kerr black holes is strictly conserved in time by establishing that all the boundary terms dynamically vanish for all times. This result implies a form of dynamical linear stability of the exterior of Kerr black hole spacetimes.
|
| first_indexed | 2026-03-14T22:05:08Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 054, 72 pages
On Axially Symmetric Perturbations
of Kerr Black Hole Spacetimes
Nishanth GUDAPATI
Department of Mathematics, College of the Holy Cross,
1 College Street, Worcester, MA-01610, USA
E-mail: ngudapati@holycross.edu
Received February 13, 2024, in final form June 17, 2025; Published online July 11, 2025
https://doi.org/10.3842/SIGMA.2025.054
Abstract. The lack of a positive-definite and conserved energy is a serious obstacle in
the black hole stability problem. In this work, we will show that there exists a positive-
definite and conserved Hamiltonian energy for axially symmetric linear perturbations of
the exterior of Kerr black hole spacetimes. In the first part, based on the Hamiltonian
dimensional reduction of 3+1 axially symmetric, Ricci-flat Lorentzian spacetimes to a 2 + 1
Einstein-wave map system with the negatively curved hyperbolic 2-plane target, we construct
a positive-definite, spacetime gauge-invariant energy functional for linear axially symmetric
perturbations in the exterior of Kerr black holes, in a manner that is also gauge-independent
on the target manifold. In the construction of the positive-definite energy, various dynamical
terms at the boundary of the orbit space occur critically. In the second part, after setting up
the initial value problem in harmonic coordinates, we prove that the positive energy for the
axially symmetric linear perturbative theory of Kerr black holes is strictly conserved in time,
by establishing that all the boundary terms dynamically vanish for all times. This result
implies a form of dynamical linear stability of the exterior of Kerr black hole spacetimes.
Key words: Kerr black holes; black hole stability problem; ergo-region; Hamiltonian me-
chanics; wave maps; Poisson equation
2020 Mathematics Subject Classification: 83C57; 58E20; 58E30; 35C15; 35Q75
Contents
1 Geometric mass-energy and perturbations of black holes 2
2 A Hamiltonian formalism for axially symmetric spacetimes 10
3 A Hamiltonian formalism for axially symmetric metric perturbations 17
4 A positive-definite Hamiltonian energy from negative curvature of the target
and the Hamiltonian dynamics 28
5 Boundary behaviour of the dynamics in the orbit space 32
6 Global existence and propagation of regularity 36
7 Canonical phase space variables and Lagrange multipliers
in the Weyl–Papapetrou gauge 40
8 Strict conservation of the regularized Hamiltonian HReg 56
References 69
mailto:ngudapati@holycross.edu
https://doi.org/10.3842/SIGMA.2025.054
2 N. Gudapati
1 Geometric mass-energy and perturbations of black holes
The stability of stationary solutions of a physical law serves as an impetus to the validity of
the law. In the context of Einstein’s equations for general relativity, an important stationary
solution is the Kerr family of black holes which is also an asymptotically flat, axially symmetric
family of solutions of the (3 + 1)-dimensional vacuum Einstein equations for general relativity
R̄µν = 0 for a Lorentzian manifold
(
M̄, ḡ
)
. (1.1)
In part due to the physical relevance and the mathematical beauty arising from its multiple
miraculous properties (see, e.g., [11, 84]), the problem of stability of Kerr black hole spacetimes
within the class of Einstein’s equations (1.1) has been a subject of active research interest since
their discovery by R. Kerr in 1963. However, geometric properties of Kerr black hole spacetimes
such as stationarity (as opposed to staticity), trapping of null geodesics and the general issue
of gauge dependence of metric perturbations cause significant obstacles in the resolution of
this ‘black hole stability’ problem. In this work, we focus on the serious issue caused by the
stationarity of the Kerr metric:
(P1) The problem of the ergo-region, the lack of a positive-definite and conserved energy caused
by the shift vector of the Kerr metric.
It may be noted that an asymptotically flat spacelike Riemannian hypersurface
(
Σ, q̄
)
such
that M̄ = Σ× R satisfies the Einstein’s equations for general relativity (1.1), has a positive-
definite total (ADM) mass mADM:
mADM := lim
r→∞
∫
S2(r)
3∑
i,j,k=1
(∂kq̄iℓ − ∂iq̄ℓk)
xi
r
µ̄S2 , q̄ is asymptotically Euclidean,
from the celebrated positive-mass theorems of Schoen–Yau and Witten [66, 67, 86]. However,
it is not necessary that the positivity of energy carries forward to the perturbative theory of
the Einstein equations. In a general asymptotically flat manifold, it is a priori not determinate
whether the mass-energy at infinity increases or decreases for small perturbations. This outcome
can be seen in the energy of a (linear) scalar wave equation propagating in the exterior of Kerr
black holes – an illustrative, albeit a special ‘test’ case of the perturbations of the Kerr metric.
However, as we already alluded to, the difficulty of constructing a positive-definite energy
for the perturbative theory is not only due to the shift vector (or the ergo-region) of the Kerr
metric. Even if one considers the Schwarzschild metric (the special case of vanishing angular
momentum of Kerr),
ḡ = −fdt2 + f−1dr2 + r2dω2
S2 , (1.2)
where f :=
(
1− 2mr−1
)
, it is not immediate that there exists a positive-definite energy for the
perturbative theory of (1.2). In 1974, Moncrief had devised a ‘Hamiltonian’ for the perturbative
theory of Schwarzschild based on the ADM formalism of Einstein’s equations [53]. Suppose the
Lorentzian spacetime
(
M̄, ḡ
)
admits a 3 + 1 ADM decomposition
ḡ = −N2dt2 + q̄ij
(
dxi +N idt
)
⊗
(
dxj +N jdt
)
,
then the ADM constraint and evolution equations are given by the variational principle for the
phase space XADM :=
{(
π̄ij , q̄ij
)
, i, j = 1, 2, 3
}
:
IADM :=
∫ (
π̄ij∂tq̄ij −NH −N iHi
)
d4x, (1.3a)
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 3
where π̄ij is the momentum conjugate to q̄ij ,
H := µ̄−1
q̄
(
∥π̄∥2q̄ −
1
2
Trq̄(π̄)
2
)
− µ̄q̄Rq̄, (1.3b)
Hi := −2(q̄)∇jπ̄
j
i , (1.3c)
Rq̄ is the scalar curvature of
(
Σ, q̄
)
, and {N,N i} are the Lagrange multipliers. We say that(
Σ, q̄, π̄
)
are asymptotically flat if diffeomorphic to R3 \BR(0), where BR(0) is a closed ball of
radius R centered at the origin and for α > 0
q̄ij =
(
1 +
M
r
)
δ̄ij +O
(
r−1−α
)
,
π̄ij = O
(
r−2−α
)
,
as r = |x| → 0. As a consequence, we have M = mADM. It follows from the variational
principle (1.3a) that the evolution equations are given by
∂tq̄ij = 2N̄ µ̄q̄
(
π̄ij −
1
2
qijTrq̄(π̄)
)
+ (q̄)∇jN̄i +
(q̄)∇iN̄j ,
∂tπ̄
ij = −N̄ µ̄q̄
(
(q̄)Rij − 1
2
q̄ijRq̄
)
+
1
2
N̄ µ̄−1
q̄ qij
(
Trq̄
(
π̄2
)
− 1
2
Trq̄(π̄)
2
)
− 2N̄ µ̄−1
q̄
(
π̄imπ̄j
m − 1
2
π̄ijTrq̄(π̄)
)
+ µ̄q
(
(q̄)∇i(q̄)∇jN̄ − q̄ij(q̄)∇m(q̄)∇mN̄
)
+ (q̄)∇m
(
π̄ijN̄m
)
− (q̄)∇mN̄ iπ̄mj − (q̄)∇mN̄ jπ̄mi.
Suppose we consider the small perturbations of the initial data of Schwarzschild black hole
spacetimes: q̄ = q̄s + ϵq̄′ and π̄ = π̄s + ϵπ̄′, Moncrief’s Hamiltonian energy formula is
Hpert :=
∫
Σ
{
N̄ µ̄−1
q̄
(
∥π̄′∥2q̄ −
1
2
Trq̄
(
π̄′)2) (1.4)
+
1
2
N̄ µ̄q̄
(
1
2
(q̄)∇kq̄
′
ij
(q̄)∇kq̄′ij − (q̄)∇kq̄
′
ij
(q̄)∇j q̄′ik − 1
2
(q̄)∇iTr
(
q̄′
)
(q̄)∇iTr
(
q̄′
)
+ 2(q̄)∇iTr
(
q̄′
)
(q̄)∇j q̄
′ij +Tr
(
q̄′
)
(q̄)∇2
ij q̄
′ij − Tr
(
q̄′
)
q̄′ijR̄
ij
q̄
)}
d3x,
which is a volume integral on the hypersurface Σ. Moncrief used the Hamiltonian formulation to
decompose the metric perturbations into gauge-dependent, gauge-independent and constraints;
and ultimately reconciled with the Regge–Wheeler–Zerilli results [63, 87]. An important feature
of these results is that the energy functional (1.4) can be realized to be positive-definite for both
odd and even parity perturbations. Using tensor harmonics, positive-definite energy functionals
for both odd and even parity perturbations of Schwarzschild black holes were constructed in [53].
In this spirit, a number of pioneering articles on the perturbations of static black holes were
written by Moncrief [54, 55, 56].
The subject of this article is to focus on axially symmetric perturbations of the Kerr metric.
In precise terms, the Kerr metric
(
M̄, ḡ
)
can be represented in Boyer–Lindquist coordinates
(t, r, θ, ϕ) as
ḡ = −
(
∆− a2 sin2 θ
Σ
)
dt2 −
2a sin2 θ
(
r2 + a2 −∆
)
Σ
dtdϕ
+
((
r2 + a2
)2 −∆a2 sin2 θ
Σ
)
sin2 θdϕ2 +
Σ
∆
dr2 +Σdθ2, (1.5)
4 N. Gudapati
where
Σ := r2 + a2 cos2 θ,
∆ := r2 − 2Mr + a2 with the real roots {r−, r+}, r+ := M +
√
M2 − a2 > r−
and
θ ∈ [0, π], r ∈ (r+,∞), ϕ ∈ [0, 2π).
It is well known (cf. [13, 14, 33]) that the Einstein equations (1.1) on spacetimes
(
M̄, ḡ
)
with
one isometry
(
∂
∂ϕ
)
, represented in Weyl–Papapetrou coordinates,
ḡ = e−2γg + e2γ(dϕ+Aνdx
ν)2, ν = 0, 1, 2,
where e2γ is the square of the norm of the rotational Killing vector ∂ϕ, A is a 1-form and
g is the induced metric on orbit space M := M̄/SO(2), admit a dimensional reduction to a
(2 + 1)-dimensional Einstein wave map system
Eµν = Tµν , (1.6a)
□gU
A + (h)ΓA
BCg
µν∂µU
B∂νU
C = 0 on (M, g), (1.6b)
where □g is the covariant wave operator, Eµν is the Einstein tensor in the interior of the quotient
(M, g) :=
(
M̄, ḡ
)
/SO(2) and T is the stress energy tensor of the wave map U : (M, g) → (N, h),
N is the negatively curved hyperbolic 2-plane,
Tµν = ⟨∂µU, ∂νU⟩h(U) −
1
2
gµν⟨∂σU, ∂σU⟩h(U).
Introducing the coordinates (ρ, z) such that ρ = R sin θ, z = R cos θ, where R := 1
2
(
r−M+
√
∆
)
,
the Kerr metric (1.5) can be represented in the Weyl–Papapetrou form as
ḡ = Σζ−1
(
−∆dt2 + ζR−2
(
dρ2 + dz2
))
+ sin2 θΣ−1ζ
(
dϕ− 2aMrζ−1dt
))2
,
where ζ =
(
r2 + a2
)2 − a2∆sin2 θ. Furthermore, the Kerr metric can also be represented in the
Weyl–Papapetrou form using functions (ρ̄, z̄) such that
ρ̄ = ρ−
(
M2 − a2
)
4R2
ρ, and z̄ = z +
(
M2 − a2
)
4R2
z
(cf. [61, Appendix A] for details). Now we shall turn to the axially symmetric perturbation
theory of the Kerr metric.
In view of the peculiar behaviour of the 2+1 Einstein-wave map system, a detailed discussion
of our methods is relevant for our article and perhaps also interesting to the reader. Consider
the Hamiltonian energy of an axially symmetric linear wave equation propagating on the Kerr
metric (□gu = 0),
HLW :=
∫
Σ
(
1
2
N̄ µ̄−1
q v2 + vN i∂iu+
1
2
Nµ̄q q̄
ij∂iu∂ju
)
d3x,
where v is the conjugate momentum of u, the energy is directly positive-definite. However, this
simplification does not carry forward to the Maxwell equations on the Kerr metric
HMax :=
∫ (
1
2
Nq̄ijµ̄q(E
iEj +BiBj) +N iϵijkE
jBk
)
d3x,
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 5
where
Ei :=
1
2
ϵijk∗Fjk, Bi :=
1
2
ϵijkFjk
are the electric and magnetic field densities respectively and F is the Faraday tensor which
satisfies the Maxwell equations
d ⋆ F = 0, dF = 0.
Actually, one can construct counter examples of positivity of energy density, for instance,
using time-symmetric Maxwell fields (cf. the discussion in [61, Section 2]). In a crucial work,
Dain–de Austria [22] had arrived at a positive-definite energy for the gravitational perturbations
of extremal Kerr black holes using the Brill mass formula [21] and subsequent use of Carter’s
identity [10], originally developed for black hole uniqueness theorems. In a Weyl coordinate
system for the spacelike hypersurface
(
Σ, q
)
in extremal Kerr spacetime, their positive-definite
energy for axially symmetric perturbations is obtained from perturbations of the Brill mass
formula, which in turn is obtained from multiplying a factor with the Hamiltonian constraint
that conveniently results in a volume form (in the chosen Weyl coordinate system) that is useful
in its representation.
In order to construct a positive-definite energy for the perturbations of Kerr–Newman metric
for the full-subextremal range, we delve into the variational structure of the relevant field equa-
tions. The beautiful linearization stability framework, developed by V. Moncrief, J. Marsden and
A. Fischer [28, 29, 57], provides a natural mechanism to construct an energy-functional based
on the kernel of the adjoint of the deformations around the Kerr metric of the dimensionally
reduced constraint map. This recognition allows us to extend results to the full sub-extremal
range (|a|, |Q| < M) of the perturbations of the Kerr–Newman metric [61], which is a solution
of Einstein–Maxwell equations of general relativity.
Let us briefly outline our construction of a positive-definite energy functional for axially
symmetric perturbations of Kerr black hole spacetimes. Consider the ADM decomposition of
M̄ = Σ × R. Suppose the group SO(2) acts on Σ through isometries. Let Γ be the non-
empty fixed-point set. Suppose the norm squared of the Killing vector generating the rotational
isometry is denoted by e2γ . In the dimensional reduction ansatz, the metric ḡ is
ḡ = e−2γ
(
−N2dt2 + qab(dx
a +Nadt)⊗
(
dxb +N bdt
))
+ e2γ(dϕ+A0dt+Aadx
a)2. (1.7)
It may be noted that this metric form is combination of ADM formalism and Weyl–Papapetrou
coordinates [53]. In the dimensional reduction framework, identifying the reduced conjugate
momenta, which form the reduced phase space in (M, g); and the corresponding reduced Hamil-
tonian formalism is nontrivial. This construction was done in [58]. Define the conjugate mo-
mentum corresponding to the metric qab as follows:
πab = e−2γπ̄ab, q̄ab = e−2γqab + e2γAaAb.
As a consequence, the ADM action principle transforms to
J =
∫ t2
t1
∫
Σ
(
πab∂tqab + Ea∂tAa + p∂tγ −NH −NaHa +A0∂aEa
)
d2xdt,
where the phase-space is now
{(q,π), (Aa, Ea), (γ, p)}
with the Lagrange multipliers
{N,Na,A0}
6 N. Gudapati
and the constraints
H = µ̄−1
q
(
∥π∥2q − Trq(π)
2
)
+
1
8
p2 +
1
2
e−4γqabEaEb + µ̄q
(
−Rq + 2qab∂aγ∂bγ
)
+
1
4
e4γqabqbd∂[bAa]∂[dAc], (1.8a)
Ha = −2(q)∇bπ
b
a + p∂aγ + Eb
(
∂[aAb]
)
, (1.8b)
∂aEa = 0. (1.8c)
It may be noted that E and p are the momenta conjugate to A and γ respectively. After
applying the Poincaré lemma on E and introducing the twist potential such that Ea =: ϵab∂bω,
we transform into the phase space
XEWM =
{
(γ, p), (ω, r),
(
qab,π
ab
)}
,
where r is the momentum conjugate to ω. The variational principle reduces to
J̃ :=
∫ t2
t1
∫
Σ
(
πab∂tqab + p∂tγ + r∂tω −NH −NaHa
)
d2xdt, (1.9a)
where H and Ha are now
H = µ̄−1
q
(
∥π∥2q − Trq(π)
2 +
1
8
p2 +
1
2
e4γr2
)
+ µ̄−1
q
(
−Rq + 2qab∂aγ∂bγ +
1
2
e−4γqab∂aω∂bω
)
, (1.9b)
Ha = −2(q)∇bπ
b
a + p∂aγ + r∂aω. (1.9c)
with the Lagrange multipliers N , Na. After computing the linearized field equations involving
the perturbed phase space
X ′ :=
{
(γ′, p′), (ω′, r′),
(
q′ab,π
′ab)},
it can be noted that (N, 0)T is the element of the kernel of the adjoint of the perturbed constraint
map. This in turn provides a candidate for the energy, analogous to (1.4). The resulting
expression has the potential energy
D2 · V = µ̄qq
ab
(
4∂aγ
′∂bγ
′ + e−4γ∂aω
′∂bω
′ + 8e−4γγ′2∂aω∂bω − 8e−4γγ′∂aω∂bω
′),
where D2· is the second variational derivative. This expression is then transformed to a positive-
definite form using the Carter–Robinson identities. Firstly, it may be noted that the original
Carter–Robinson identities are not restrictive to the choice of the function ‘ρ’ (in [10] and
in [65, equation (5)]) and can thus be generalized as follows:
µ̄qq
ab
(
4∂aγ
′∂bγ
′ + e−4γ∂aω
′∂bω
′ + 8e−4γγ′2∂aω∂bω − 8e−4γγ′∂aω∂bω
′)
+ ∂b
(
Nµ̄qq
ab
(
−2e−4γ∂aγω
′ + e−4γω′ + 4e−4γγ′∂aω
))
+
1
2
µ̄qe
−4γL1
(
e−2γω′)+ µ̄qL2
(
−4γ′ω′)
= Nµ̄qq
ab
(
(1)Va
(1)Vb +
(2)Va
(2)Vb +
(3)Va
(3)Vb
)
,
where
(1)Va = 2∂aγ
′ + e−4γω′∂aω,
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 7
(2)Va = −∂a
(
e−2γω′)+ 2e−2γγ′∂aω,
(3)Va = 2∂aγω
′ − 2γ′∂aω,
and
L1 := e−2γ
(
∂b
(
Nµ̄qq
ab∂aγ
)
+Ne−4γµ̄qq
ab∂aω∂bω
)
,
L2 := −∂b
(
Nµ̄qq
abe−4γ∂aω
)
.
Upon substitution into the potential energy, this results in a positive-definite energy of the form
HReg =
∫
Σ
{
Nµ̄−1
q
(
ϱ′ba ϱ
′a
b +
1
8
p′2 +
1
2
e4γr′2
)
− 1
2
µ̄qτ
′2
+Nµ̄qq
ab
(
2
(
∂aγ
′ +
1
2
e−4γω′∂aω
)(
∂bγ
′ +
1
2
e−4γω′∂bω
)
+ 2
(
γ′e−2γ∂aω − ∂a
(
e−2γω′))(γ′e−2γ∂bω − ∂b
(
e−2γω′))
+ 2e−4γ
(
∂aγω
′ − γ′∂aω
)(
∂bγω
′ − γ′∂bω
))}
d2x
modulo a time-coordinate gauge condition τ ′ = 0. It is then shown that this energy functional
is a Hamiltonian for the dynamics of the reduced Einstein equations in the perturbative phase-
space and a spacetime divergence-free vector field density is constructed,
JReg =
(
JReg
)t
∂t +
(
JReg
)a
∂a,
where
(
JReg
)t
= eReg, eReg is the integrand of HReg, and(
JReg
)a
= N2µ̄−1
q
((
p′µ̄qq
ab∂bγ
′)+ e4γr′
(
e−4γµ̄qq
ab∂bω
′))+ γ′LN ′
(
4Nµ̄qq
ab∂bγ
)
+ ω′LN ′
(
Ne−4γµ̄qq
ab∂bω
)
+ 2LN ′N
(
µ̄qq
ab∂bν
′)+ 2LX′ν ′µ̄qq
ab∂bN
− 2Xa
(
µ̄qq
bc∂bν
′∂cN
)
+ 2Nqac0 ϱ′bc e
−2ν∂bN
′ +
(
N∂bN
′ −N ′∂bN
)
τ ′µ̄qq
ab
− 2N ′qac0 ϱ′bc e
−2γ∂bN
after the imposition of the linearly perturbed constraints. Following the use of the Carter–
Robinson identities for the sub-extremal case, there was still the lingering question of why
do these transformations magically solve the issues of the ergo-region and the positivity and
conservation of energy in the stability theory of Kerr and Kerr–Newman black holes, even
in the axially symmetric case, which, as we just discussed, is nontrivial because the energy
density can in principle be locally negative. Our results demonstrate that the reason for these
transformations holding in such a way that one can obtain positivity and yet preserve the
‘symplectic structure’, is not ‘by fluke’, but that there are well-defined geometric and variational
underlying structures, namely, the covariant (in target) nature of the dimensionally reduced
system, the negative curvature of the target and the linearization stability methods. In the
context of the black hole uniqueness theorems, the associated generalizations of Carter–Robinson
identities were constructed by Bunting and Mazur.1 We adapted these results for our present
problem of (dynamical) black hole stability.
It may be noted that the analogous transformations also resolve the positivity problem for
the energy of axially symmetric Maxwell’s equations if one does the dimensional reduction to
introduce the twist potentials λ, η corresponding to the E and B fields (cf. [61]).
1It is indeed remarkable that the rather ingenious identity of Carter (Robinson for the Einstein–Maxwell case)
that was seemingly constructed from ‘trial and error’, would later have a natural geometric interpretation. These
results also hold for higher-dimensional black holes and have been used for black hole uniqueness theorems in
higher dimensions (see [41]).
8 N. Gudapati
In the axially symmetric case, even though the original Maxwell equations are linear, a non-
linear transformation is used to reduce the 3+1 Einstein–Maxwell equations to an Einstein-wave
map system [59], which introduces nonlinear coupling within the Maxwell ‘twist’ fields. How-
ever, if we turn off the background E and B fields (e.g., restrict attention to the Kerr metric),
the Maxwell equations in twist potential variables reduce to linear hyperbolic PDE.
Somewhat interestingly, it appears that the construction of a positive-definite energy for the
axially symmetric Maxwell equations on Kerr black hole spacetimes does not easily follow from
the Carter’s identity, but can be realized a special case of the full Robinson’s identity.
In [37], a positive-definite Hamiltonian energy functional for axially symmetric Maxwell equa-
tions propagating on Kerr–de Sitter black hole spacetimes was constructed, using modified
Einstein-wave maps for the Lorentzian Einstein manifolds with one rotational isometry.
In this work, we shall extend this result and construct a positive-definite energy in a way
that is gauge-invariant on the target manifold (N, h). As we shall see, this is based on negative
curvature of the target manifold (H2, h) and the convexity of 2+1 wave maps. The construction
of an energy-functional based on the convexity of wave maps, together with our application of
the linearization stability methods, suggests why the positivity of our (global) energy for the
perturbative theory is to be expected in general, not relying on the insightful and elaborate
identity of Carter, which relies on a specific gauge on the target. In such a formulation, the
intrinsic geometry within the 2 + 1 Einstein wave map system becomes more transparent.
In the context of black hole uniqueness theorems, extensions along these lines, from the initial
Carter–Robinson results, were done by Bunting [9] and Mazur ([50] and references therein).
In the mathematics literature, convexity of harmonic maps for axially symmetric (Brill) initial
data was established by Schoen–Zhou [68], which is often used in geometric inequalities between
the area of the horizon, angular-momentum and the mass.
In general, due to the geometric nature of the construction, the linearization stability ma-
chinery provides a robust mechanism to deal with the stability problems of black holes within
a symmetry class, including the initial value problem on hypersurfaces that intersect null in-
finity. The linearization stability machinery is also equipped for dealing with projections from
higher-dimensional (n+1, n > 3) black holes with suitable symmetries
(
toroidal Tn−2 spacelike
symmetries
)
, including 5D Myers–Perry black holes, the stability of which is the main open
problem in the stability of higher-dimensional black holes (see, e.g., [23]). Indeed, most of our
current work, especially the local aspects, readily extend to perturbations within the aforemen-
tioned symmetry class of higher-dimensional black holes (see below). However, we propose to
carefully address the global aspects of this problem, based on the methods developed in the
current work and [61], in a future work.
We would like to point out there are related and independent works, based on the ‘canonical
energy’ of Hollands–Wald [42]. In [62], Prabhu–Wald have extended [42] by associating the
axisymmetric stability to the existence of a positive-definite ‘canonical energy’. A positive-
definite energy functional was constructed by Dafermos–Holzegel–Rodnianski [17] in the context
of their proof of linear stability of Schwarzschild black holes (see also [43]). Subsequently,
a positive-definite energy was constructed by Prabhu–Wald using the canonical energy methods,
that is consistent with both [17] and [53]. Their approach is based on the construction of metric
perturbations using the Teukolsky variable as the Hertz potential.
From a PDE perspective, a suitable notion of (positive-definite) energy is crucial to control
the dynamics of a given system of PDEs. In case the scaling symmetries of a nonlinear hyperbolic
PDE and its corresponding energy match, powerful techniques come into play that characterize
blow up (concentration) and scattering categorically. This problem is referred to as ‘energy
critical’. In the context of 2 + 1 critical flat-space wave maps
U : R2+1 → (N, h),
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 9
the fact that this characterization can be made was demonstrated in the Landmark works
[15, 47, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82] in the analysis of geometric wave equa-
tions. It may be noted that 3 + 1 Einstein’s equations with one translational isometry can be
reduced to the 2 + 1 Einstein-wave map system (1.6). In this case, the notion of a positive-
definite, gauge-invariant Hamiltonian mass-energy is provided by Ashtekar–Varadarajan [6] (see
also Thorne’s C-energy [85])
qab = r−mAV
(
δab +O
(
r−1
))
in the asymptotic region of asymptotically flat (Σ, q). In a previous work [33], it was noted
that the aforementioned fundamental results on flat space wave maps can be extended to
the 2 + 1 Einstein-wave map system resulting from 3 + 1 Einstein’s equations with transla-
tional symmetry using the AV-mass, which in turn is related to the energy of 2 + 1 wave maps
arising from the energy-momentum tensor and a local conservation law in the equivariant case.
We would like to point out that, even though the dimensional reduction of 3 + 1 dimensional
axially symmetric, asymptotically flat spacetimes results in the same 2 + 1 Einstein-wave map
system locally, the axisymmetric problem is not a (geometric) mass-energy-critical problem [34].
There is yet another dimensional reduction, based on the ‘time-translational’ Killing vector of
stationary class of spacetimes, to which the Kerr metric also belongs, that results in harmonic
maps. This distinction between each of the three cases, which is relevant for the applicable
methods therein, is explained in [34] for the interested reader.
Without the energy-criticality of the 2 + 1 Einstein-wave map system, a direct consideration
of the nonlinear problem, analogous to [4, 33], is infeasible. A long-standing approach that is
commonly used in the stability problems of Einstein’s equations, is to first consider the linear
perturbations and hope to control the nonlinear (higher-order perturbations) using the linear
perturbation theory. However, the problem of what is the natural notion of energy for the linear
perturbative theory, that is consistent with the dimensional reduction and wave map structure
of field equations remains open:
(P2) Is there a natural notion of mass-energy for the axially symmetric linear perturbative
theory of Kerr black hole spacetimes that is consistent with the dimensional reduction
and the wave map structure of the equations?
This question is closely related to whether there exists a natural factor that multiplies
the dimensionally reduced Hamiltonian constraint of the system (compare with the discussion
in [34, pp. 3–4]), which provides a natural notion of energy for our linearized problem. We point
out that the linearization stability methods employed in our works provide a natural mechanism
that resolves both (P1) (|a| < m) and (P2).
Nevertheless, dealing with a plethora of boundary terms that arise in the construction of the
positive-definite energy, in connection with the gauge-conditions and the dimensional reduction,
is nontrivial.2 These aspects are dealt with in detail in the latter half of this article, starting
from Section 5.
In the current work, after establishing that the constraints for our system are scleronomic,
we prove that our energy functional drives the constrained Hamiltonian dynamics of our sys-
tem and that it forms a (spacetime) divergence-free vector field density, after the imposition of
the constraints. In the process of obtaining our results, we construct several variational princi-
ples from both Lagrangian and Hamiltonian perspectives, for the nonlinear (exact) and linear
perturbative theories. These may be of interest in their own right.
As we already remarked, the black hole stability problem is a very active research area.
The decay of Maxwell equations on Schwarzschild was proved in [7]. The linear stability of
2This is in contrast with the Maxwell perturbations on Kerr black hole spacetimes, which is a (locally) gauge-
invariant problem.
10 N. Gudapati
Schwarzschild was established in [17]. Likewise, the linear stability of Schwarzschild black hole
spacetimes using the Cauchy problem for metric coefficients was established in [44]. These
results build on the classic results [53, 63, 87, 88].
The important case of linear wave perturbations of Kerr black holes was studied in several
fundamental works for small angular momentum [2, 19, 83]. Likewise, the decay of Maxwell
perturbations of Kerr was established in [3]. A uniform energy bound and Morawetz estimate
for the |s| = 1, 2 Teukolsky equations was established in [48, 49]. Boundedness and decay for
the |s| = 2 Teukolsky equation was established in [16]. A positive-definite energy for axially
symmetric NP-Maxwell scalars was constructed in [39], extending our aforementioned results
on Maxwell equations. The linear stability of Kerr black holes was established in [1, 40] for
small |a|. Nonlinear stability of Schwarzschild black holes was announced in [18]. Recently, the
full proof of nonlinear stability of slowly rotating Kerr black holes was announced in [32] (see
also [31, 46]).
The effects of the ergo-region become more subtle for rapidly rotating (but |a| < M) Kerr
black holes. The decay of the scalar wave for fixed azimuthal modes was established in [25, 26, 27]
using spectral methods. The decay of a general linear wave equation was established in the
remarkable work [20]. We would like to point out that the global behaviour, especially the decay
estimates, of Maxwell and linearized Einstein perturbations of Kerr black holes, is relatively less
understood for the large, but sub-extremal (|a| < M) case. We expect that our work will be
useful in this regard.
Let us briefly summarize the layout of the article. In Section 2, we discuss the Hamiltonian
formulation of axially symmetric spacetimes. We start with the Hamiltonian reduction of the
wave map Lagrangian and proceed to perform Hamiltonian dimensional reduction of 3 + 1
axisymmetric spacetimes into (2 + 1)-dimensional Einstein-wave map system. In Section 3,
we construct the field equations for linear perturbations of the Einstein equations in axially
symmetric spacetimes. These Hamilton field equations are also constructed in the dimensionally
reduced Einstein-wave map formalism. Section 4 contains the construction of a positive-definite
Hamiltonian energy functional for linearized perturbations of the Einstein equations in axially
symmetric spacetimes. The construction uses linearization stability methods for the (2 + 1)-
dimensional Einstein-wave map system. Section 4 also contains the proof that the linearized
constraint equations are preserved in time for this system.
For stability purposes, it is crucial to establish that this positive-definite Hamiltonian en-
ergy is strictly conserved in time. The remainder of this article is dedicated to this purpose.
The strict conservation of the positive-definite-definite energy is closely connected to the bound-
ary behaviour of the fields. Indeed, the boundary behaviour of the fields is critical for the whole
construction of this work. This is discussed in detail in Section 5. In Section 6, we formulate
the Cauchy problem of linearized Einstein equations in harmonic coordinates and discuss the
propagation of regularity and causal structure.
In Section 7, we transform the phase-space variables and the Lagrange multipliers into the
Weyl–Papapetrou gauge and discuss their boundary behaviour. Finally, in Section 8, we use this
construction to establish that the integrated fluxes of the energy vanish at the horizon, the axes
and at the spatial infinity, thereby establishing the strict conservation of the positive-definite
energy we constructed earlier. In establishing these results, we classify these boundary flux
terms into the dynamical terms, the kinematic terms, and the conformal terms. This article is
a merger of the articles [35] and [36].
2 A Hamiltonian formalism for axially symmetric spacetimes
Recall that M̄ = Σ× R is a 3 + 1 Lorentzian spacetime, such that the rotational vector field Φ
acts on Σ as an isometry with the fixed point set Γ. In the case of Kerr black hole spacetime,
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 11
Γ is a union of two disjoint sets (the ‘axes’). It follows that the quotient Σ := Σ/SO(2) and
M := Σ× R are manifolds with boundary Γ.
Consider the Einstein–Hilbert action on
(
M̄, ḡ
)
SEH :=
∫
R̄ḡ µ̄ḡ. (2.1)
Suppose the axially symmetric
(
M̄, ḡ
)
is a critical point of (2.1). In the Weyl–Papapetrou
coordinates,
ḡ = |Φ|−1ḡ + |Φ|(dϕ+Aνdx
ν)2,
|Φ| is the norm squared of the Killing vector Φ := ∂ϕ, and g is the metric on the quotient
M := M̄/SO(2). Suppose II is the second fundamental form of the embedding (M, g̃) ↪→
(
M̄, ḡ
)
,
g̃ = |Φ|−1g, then following the Gauss–Kodazzi equations and the conformal transformation
R̃g̃ = |Φ|−1
(
Rg − 4gµν∇µ∇ν log |Φ|1/2 − 2gµν∇µ log |Φ|1/2∇ν log |Φ|1/2
)
.
The Einstein–Hilbert action (2.1) can be reduced to
LEWM :=
1
2
∫ (
1
κ
Rg − hAB(U)gαβ∂αU
A∂βU
B
)
µ̄g (2.2)
for κ = 2 and U is a wave map
U : (M, g) → (N, h)
to a hyperbolic 2-plane target (N, h), whose components are associated to the norm and the
twist (potential) of the Killing vector. The tangent bundle of the configuration space of (2.2)
is now
CEWM :=
{
(g, ġ),
(
UA, U̇A
)}
,
where the dot
(
e.g., U̇
)
denotes derivative with respect to a time-coordinate function t. We would
like to perform the Hamiltonian reduction of the system (2.2). Recall the ADM decomposition
of (M, g) = (Σ, q)× R
g = −N2dt2 + qab(dx
a +Nadt)⊗
(
dxb +N bdt
)
Let us split the geometric part and the wave map part of the variational principle (2.2) as
LEWM = Lgeom + LWM. Let us now start with the Hamiltonian reduction of the wave map
Lagrangian LWM,
LWM := −1
2
∫ (
hAB(U)gαβ∂αU
A∂βU
B
)
µ̄g (2.3)
over the tangent bundle of the configuration space of wave maps, CWM =
{(
UA, U̇A
)}
.
Suppose we denote the Lagrangian density of (2.3) as L and conjugate momenta as pA,
we have
pA =
1
N
µ̄qhAB(U)∂tU
B − 1
N
µ̄qhAB(U)LNUB,
where LN is the Lie derivative with respect to the shift Na. As a consequence, we have
hAB(U)∂tU
B =
1
µ̄q
NpA + hAB(U)LNUB,
12 N. Gudapati
and the Lagrangian density LWM can be expressed in terms of the wave map phase space
XWM :=
{(
UA, pA
)}
as
LWM =
1
2
pB∂tU
B − 1
2
pBLNUB − 1
2
Nµ̄qhAB(U)qab∂aU
a∂bU
B.
Let us now define the Hamiltonian density as follows:
HWM :=
1
2
pB∂tU
B +
1
2
pBLNUB +
1
2
Nµ̄qhAB(U)qab∂aU
A∂bU
B.
As a consequence, we formulate the ADM variational principle for the Hamiltonian dynamics of
the wave map phase space XWM as
LWM[XWM] :=
1
2
∫ (
pA∂tU
A − pBLNUB −Nµ̄qhAB(U)qab∂aU
a∂bU
B
)
d3x,
which has the field equations
pA =
1
N
µ̄qhAB(U)∂tU
B − 1
N
µ̄qhAB(U)LNUB (2.4)
and the critical point with respect to
(
the first variation DUA · LWM = 0
)
Ua gives
∂tpA = −Nµ̄−1
q
∂
∂UA
hBCpBpC + hAB∂a
(
Nµ̄qq
ab∂bU
B
)
+Nµ̄qhAB
(h)ΓB
CD(U)qab∂aU
C∂bU
D + LNpA, (2.5)
where (h)Γ are the Christoffel symbols,
(h)ΓA
BC :=
1
2
hAD(U)(∂ChBD + ∂BhDC − ∂DhBC).
It is straight-forward to verify that the canonical equations
DpA ·HWM = ∂tU
A and DUA ·HWM = −∂tpA
correspond to (2.4) and (2.5), respectively, whereHWM :=
∫
HWMd2x is the (total) Hamiltonian.
Subsequently, if we use the Gauss–Kodazzi equation for the ADM 2 + 1 decomposition and
defining the (geometric) phase space
Xgeom :=
{(
qab,π
ab
)}
,
we can represent the gravitational Lagrangian density as follows:
LAlt
geom :=
(
−qab∂tπ
ab −NHgeom −NaH
a
geom
)
,
where
Hgeom := µ̄−1
q
(
∥π∥2q − Trq(π)
2
)
− µ̄qRq,
Ha
geom := −2 (q)∇bπ
ab.
It may be noted that the conjugate momentum tensor of the reduced metric and the corre-
sponding components of the conjugate momentum of the 3 metric are related as follows:
|Φ|π̄ab = πab, where πab = µ̄q
(
qabTrq(K)−Kab
)
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 13
Consequently, we have the variational principle for Hamiltonian dynamics of the reduced Einstein
wave map system
JEWM :=
∫ t2
t1
∫
Σ
(
πab∂tqab + pA∂tU
A −NH −NaHa
)
d2xdt, (2.6)
where now the reduced H and Ha are
H = µ̄−1
q
((
∥π∥2q − Trq(π)
2
)
+
1
2
pAp
A
)
+ µ̄q
(
−Rq +
1
2
hABq
ab∂aU
A∂bU
B
)
,
Ha = −2 (q)∇bπ
b
a + pA∂aU
A.
Therefore, we have proved the following theorem.
Theorem 2.1. Suppose
(
M̄, ḡ
)
is an axially symmetric, Ricci-flat, globally hyperbolic Lorentzian
spacetime and that ḡ admits the decomposition (1.7) in a local coordinate system, then the
dimensionally reduced field equations in the interior of M = Σ × R, where Σ = Σ/SO(2), are
derivable from the variational principle (2.6) for the reduced phase space
XEWM :=
{(
qab,π
ab
)
,
(
UA, pA
)}
with the Lagrange multipliers {N,Na}.
As a consequence, we have the field equations for Hamiltonian dynamics in XEWM
hAB∂tU
B = Nµ̄−1
q pA + hABLNUB, (2.7a)
∂tpA = −Nµ̄−1
q
∂
∂UA
hBC(U)pBpC + hAB∂a
(
Nµ̄qq
ab∂bU
B
)
+Nµ̄qhAB
(h)ΓB
CD(U)qab∂aU
C∂bU
D + LNpA, (2.7b)
∂tqab = 2Nµ̄−1
q (πab − qabTr(π)) +
(q)∇aNb +
(q)∇bNa, (2.7c)
∂tπ
ab =
1
2
Nµ̄−1
q qab
(
∥π∥2q − Tr(π)2
)
− 2Nµ̄−1
q
(
πacπb
c − πabTr(π)
)
+ µ̄q
(
(q)∇b (q)∇aN − qab (q)∇c
(q)∇cN
)
,
+ (q)∇c
(
πabN c
)
− (q)∇cN
aπcb − (q)∇cN
bπca
+
1
4
µ̄−1
q NqabpAp
A +
1
2
Nµ̄qhAB
(
qacqbd − 1
2
qabqcd
)
∂cU
A∂dU
B (2.7d)
and the constraint equations
H = 0, Ha = 0. (2.8)
It should be pointed out that, analogous to original ADM formulation [5], we have made a sim-
plification with the coupling constant (see also the discussion in [52, pp. 520–521]). In case the
precise coupling between the 2+ 1 Einstein’s equations and its wave map source is relevant, the
original coupling can be reinstated by simply substituting the following formulas throughout
our work:
πtrue :=
1
2κ
π =
1
2κ
µ̄q
(
qabTr(K)−Kab
)
,
Htrue :=
1
2κ
Hgeom = µ̄−1
q
1
2κ
(
∥π∥2q − Tr(π)2
)
− 1
2κ
µ̄qRq,
(Htrue)a :=
1
2κ
(Hgeom)a = −1
κ
(q)∇bπ
b
a.
14 N. Gudapati
We would like to remind the reader that, in the dimensional reduction process, we introduce the
closed 1-form G such that
|Φ|−2ϵµνβg
βαGα = Fµν ,
where F = dA. In our simply connected domain, G = dw, where ω is the gravitational twist
potential and one of the components of the wave map U .
Nonlinear conservation laws
Following Komar’s definition of angular momentum,
J =
1
16π
∫
Σ
⋆dΦ,
it follows that for the Kerr metric J = aM . In view of the well-known fact that the angular mo-
mentum is conserved for our vacuum axisymmetric problem, without effective loss of generality,
we shall assume that the perturbation of the angular-momentum is zero.
The dimensional reduction provides additional structure for the original field equations.
As noted by Geroch [30], the Lie group SL(2,R) acts on the resulting target (N, h) in the di-
mensional reduction procedure. The Möbius transformations, which are the isometries of (N, h)
provide us a Poisson algebra of nonlinear conserved quantities.
Corollary 2.2. Suppose U : Σ × (t1, t2) → (N, h) is the wave map coupled to 2 + 1 Einstein
equations as above, then there exist (spacetime) divergence-free vector fields Ji, i = 1, 2, 3, such
that if Ci is the flux of Ji at Σt, t ∈ (t1, t2) hypersurface,
{Ci, Cj} = σk
ijCk, i ̸= j ̸= k,
where {·, ·} is the Poisson bracket in the phase space XEWM and σk
ij are the structure constants
of the (Möbius) isometries {Ki,K2,K3} of (N, h).
Proof. The Möbius transformations on the target (N, h), the hyperbolic 2-pane, are isometries
corresponding to translation, dilation and inversion {K1,K2,K3}. It follows that
(h)∇A(Ki)B + (h)∇B(Ki)A = 0, ∀ i = 1, 2, 3,
Consider the quantity
∂t
(
KA
i pA
)
= ∂CK
A
i pA
(
Nµ̄qp
C + LNUC
)
+KA
i
(
−Nµ̄−1
q ∂Ah
BCpBpC + hAB∂a
(
Nµ̄qq
ab∂bU
B
)
+Nµ̄qhAB
(h)ΓB
CDq
ab∂aU
C∂bU
D + LNpA
)
.
Now consider,
∂a
(
Nµ̄qq
ab∂bU
BKA
i hAB
)
= KA
i hAB∂a
(
Nµ̄qq
ab∂bU
B
)
+Nµ̄qq
ab∂aU
C∂cK
A
i hAB∂bU
B+Nµ̄qq
ab∂bU
BKA
i ∂ChAB∂aU
C
and note that
−Nµ̄qq
ab∂aU
C∂cK
AhAB∂bU
B −Nµ̄qq
ab∂bU
BKA∂ChAB∂aU
C
+Nµ̄qhAB
(h)ΓB
CDq
ab∂aU
C∂bU
DKA
i
= −1
2
Nµ̄qq
ab∂aU
C∂bU
D∂AhCD −Nµ̄ab
q ∂aU
C∂cU
B∂CK
A
i hAB = 0
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 15
after relabeling of indices and on account of the fact that the deformation tensor of Ki in the
target h is zero. Let us now define
(Ji)
t = KA
i pA,
(Ji)
a = ∂a
(
Nµ̄qq
ab∂bU
BKA
i hAB +NaKA
i pA
)
.
It follows from above that each Ji, i = 1, 2, 3, is a spacetime divergence-free vector density.
Now then
Ci :=
∫
Σt
KA
i pAd
2x
and consider the Poisson bracket
{Ci, Cj} =
{∫
Σt
KA
i pA ,
∫
Σt
KA
j pA
}
=
∫
Σt
(
∂UAKB
i KA
j −KA
i ∂UAKB
j
)
pB
=
∫
Σt
[
Ki,Kj
]A
pA =
∫
Σt
σk
ijK
A
k pA = σk
ijCk, i ̸= j ̸= k.
This result generalizes the equivalent result in [58], where each Σ is S2, to our non-compact case
and a general gauge on the target metric h. We would like to point out that these conservation
laws are closely related to the ‘moment maps’ associated to the Möbius transformations in the
phase space. It may be noted that our arguments readily extend to the (n+1) higher-dimensional
context, where the target is SL(n− 2)/SO(n− 2). ■
In the Weyl–Papapetrou coordinates, define a quantity ν such that
q = e2νq0, (2.9)
where q0 is the flat metric and (mean curvature) scalar τ := µ̄−1
q qabπ
ab. In our work, it will be
convenient to split 2-tensors into a trace part and the conformal Killing operator [58]. In the
following lemma, we shall streamline the related discussion and results obtained in [58].
Lemma 2.3. Suppose the phase space variables
{
(q,π),
(
UA, pA
)}
∈ X are smooth in the
interior of Σ, we have
(1) Suppose a vector field Y ∈ T (Σ), then conformal Killing operator defined as
(CK(Y, q))ab := µ̄q
(
(q)∇bY a + (q)∇aY b − qab (q)∇cY
c
)
,
is invariant under a conformal transformation, i.e., CK(Y, q) = CK(Y, q0).
(2) There exists a vector field Y ∈ T (Σ), which is determined uniquely up a conformal Killing
vector, such that
πab = e−2ν(CK(Y, q))ab +
1
2
τ µ̄qq
ab.
(3) If
{
(q,π),
(
UA, pA
)}
∈ X satisfy the constraint equations, the Hamilton and momentum
constraint equations (2.8) can be represented as the elliptic equations
µ̄−1
q0
(
e−2ν∥ϱ∥2q0 −
1
2
τ2e2ν µ̄2
q0 +
1
2
pAp
A
)
+ µ̄q0
(
2∆0ν + hABq
ab
0 ∂aU
A∂bU
B
)
= 0
and
−(q0)∇bϱ
b
a −
1
2
∂aτ e
2ν µ̄q0 +
1
2
pA∂aU
A = 0, (Σ, q0),
respectively, where
ϱac = µ̄q0
(
(q0)∇cY
a + (q0)∇aYc − δac
(q0)∇bY
b
)
. (2.10)
16 N. Gudapati
Proof. Part (1) follows from the definitions and direct computations. Part (2) is based on
the fact that the transverse-traceless tensors vanish for our form of the 2-metric. Consider the
decomposition of πab into a trace part and a traceless part:
πab =
1
2
τ µ̄qq
ab +��Trπ
ab =
1
2
τ µ̄qq
ab + (πTT)
ab + e−2νCK(Y, q), (2.11)
where (πTT)
ab is such that
qab(πTT)
ab = 0 and (q)∇a(πTT)
ab = 0. (2.12)
The result (2) now follows from the fact that (2.12) is invariant under the conformal trans-
formation (2.9) and the fact that transverse-traceless tensors vanish on the flat metric q0, with
suitable boundary conditions. The existence of Y follows from the following elliptic equation
(q)∇a
(
e−2νCK(Y, q)
)
= ∇a
(
πab − 1
2
τ µ̄qq
ab
)
and Fredholm theory. It may be noted that the right-hand side is L2−orthogonal to the kernel
of the linear, self-adjoint elliptic operator on the left-hand side, which contains the conformal
Killing vector fields of q0,
(q)∇b
(
e−2ν µ̄q
(
Ya +
(q)∇aY
b − δba
(q)∇cY
c
))
= (q0)∇b
(
µ̄q0
(
(q0)∇bY
a
)
+ (q0)∇aY
b − δba
(q0)∇cY
c
)
. (2.13)
It would now be convenient to define ϱ as in (2.10). Now then, using
πa
b =
1
2
τ e2ν µ̄q0δ
a
c + e−2νϱac ,
and (2.13), the momentum constraint can now be transformed into the following elliptic operator
for Y on (Σ, q0)
Ha = −2(q0)∇bϱ
b
a − ∂aτ e
2ν µ̄q0 + pA∂aU
A, (Σ, q0), a = 1, 2.
The scalar curvature Rq of (Σ, q) and ∥π∥2q can be expressed explicitly as
Rq = −2e−2ν∆0ν, ∥π∥2q =
1
2
τ 2e4ν µ̄2
q0 + ∥ϱ∥2q0 .
The Hamiltonian constraint can now be transformed to the elliptic operator
H = µ̄−1
q0
(
e−2ν∥ϱ∥2q0 −
1
2
τ 2e2ν µ̄2
q0 +
1
2
pAp
A
)
+ µ̄q0
(
2∆0ν +
1
2
hABq
ab
0 ∂aU
A∂bU
B
)
, (Σ, q0),
where
∆0ν :=
1
µ̄q0
∂b
(
qab0 µ̄q0∂bν
)
. ■
The conditions for the dimensional reduction above are modeled along the Kerr metric (1.5).
Let us now consider the corresponding field equations for the Kerr metric. It follows that for
the Kerr wave map
U : (M, g) → (N,h)
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 17
we have p1 = p2 ≡ 0 and πab ≡ 0. As a consequence, the dimensionally reduced field equations
for the Kerr metric (1.5) are
∂a
(
Nµ̄qq
abUA
)
+Nµ̄q
(h)ΓA
BCq
ab∂aU
B∂bU
C = 0 (2.14)
and
µ̄q
(
(q)∇b (q)∇aN − qab (q)∇c
(q)∇cN
)
+
1
2
Nµ̄q
(
qacqbd − 1
2
qabqcd
)
hAB∂aU
A∂bU
B = 0.
The Hamiltonian constraint
H = µ̄q
(
−Rq +
1
2
hABq
ab∂aU
A∂bU
B
)
= 0
for a, b and A,B,C = 1, 2. The scalar τ is the mean curvature of the embedding Σ ↪→ M ,
whose evolution is governed by the equation
∂tτ = −(q)∇a
(q)∇aN +Nµ̄−1
q
(
∥π∥2q +
1
2
pAp
A
)
. (2.15)
Following the notation introduced in [58], the evolution equation (2.15) can be represented as
e2ν∂tτ = −∆0N +Nq,
where
q := µ̄−1
q e−2ν
(
∥ϱ∥2q +
1
2
τ 2e4ν µ̄q +
1
2
pAp
A
)
,
where we again used the splitting expression (2.11). It follows that for the Kerr metric (1.5)
τ = ∂tτ ≡ 0, ϱ ≡ 0 and ∆qN = 0. (2.16)
The equation (2.7c) can be decomposed as
∂t(µ̄q) = NTrq(π)−
1
2
µ̄qqab
(
(q)∇aN b + (q)∇bNa
)
and the evolution of the densitized inverse metric
∂t
(
µ̄qq
ab
)
= 2N
(
πab − 1
2
qabTrq(π)
)
+ µ̄q
(
(q)∇aN b + (q)∇bNa − qab(q)∇cN
c
)
.
3 A Hamiltonian formalism for axially symmetric
metric perturbations
In this section, we shall calculate the field equations and the Lagrangian and Hamiltonian
variational principles for linear perturbation equations of the 2 + 1 Einstein-wave map system.
Consider a smooth curve
γs : [0, 1] → CEWM
18 N. Gudapati
parametrized by s in the tangent bundle of configuration space CEWM of the Einstein-wave map
system. Like previously, we shall start with the wave map system. Let Us : (M, g) → (N,h) be
a 1-parameter family of maps generated by the flow along γs such that
U0 ≡ U,
Us ≡ U, outside a compact set Ω ⊂ M,
and U ′ := Dγs
· Us
∣∣
s=0
, where U : (M, g) → (N,h) is a given (e.g., Kerr) wave map. The de-
formations along γs can be manifested, for instance, by the exponential map Exp(sU). In the
following, with a slight abuse of notation, we shall denote the manifestations of the deformations
along γs for the wave map U : (M, g) → (N,h), by γs itself. Let us now denote the deformations
along γs of a point at s = 0 in the tangent bundle of the wave map configuration space CWM
as follows:
C ′
WM :=
{
U
′A = Dγs
· UA
s
∣∣
s=0
, U̇
′A = Dγs
· U̇A
s (s)
∣∣
s=0
}
.
Now consider the Lagrangian action of wave map
LWM(CWM) = −1
2
∫ (
gµνhAB∂µU
A∂νU
B
)
µ̄g. (3.1)
For simplicity, we shall denote LWM(γ(s)) as LWM(s). We have
Dγs
· LWM(s) =
∫
hAB
(
□gU
A + (h)ΓA
BCg
µν∂µU
B∂νU
C
)
U
′Bµ̄g, (3.2)
where we have used the identity
gµνhAB(U)∂µU
A∂νU
′B +
1
2
gµν∂ChAB∂µU
A∂νU
BU
′C
= −hABU
′B
(
□gU
A + (h)ΓA
BCg
µν∂µU
B∂νU
C
)
(3.3)
modulo boundary terms (see, e.g., [33, pp. 19–20]). The following geometric construction shall
be useful to represent our formulas compactly [51]. Firstly, let us define the notions of induced
tangent bundle and the associated ‘total’ covariant derivative on the target (N, h), under the
wave mapping U : M → N . The induced tangent bundle TUN onM consists of the 2-tuple (x, y),
where x ∈ M and y ∈ TU(x)N , with the bundle projection
P : TUN → M, (x, y) → x.
Consider the vector field V̇s ∈ TM , then the image of V̇s under the wave map U is a vector field
V̇ A
s = ∂sU
A in a local coordinate system of (N,h). As a consequence, we can define a covariant
derivative on the induced bundle
(h)∇µV̇
A
s := ∂µV̇
A
s + (h)ΓA
BC V̇
B
s ∂µU
C .
It may be verified explicitly that the induced connection is metric compatible (h)∇Ah
AB ≡ 0.
Likewise, for a ‘mixed’ tensor
Λ := ΛA
µ ∂xA ⊗ dxµ,
(h)∇νΛ
A
µ := (g)∇νΛ
A
µ + (h)ΓA
BCΛ
B
µ ∂νU
C .
In particular, for eB ∈ TN , the second covariant derivative
(h)∇µ
(h)∇νeB = ∂µ
(
ΓA
νBeA
)
− (g)Γα
µν
(h)ΓA
αBeA + (h)ΓA
µB
(h)ΓC
νAeC
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 19
provides the curvature for the induced connection
[∇µ,∇ν ]eB = RA
Bµν eA.
Now consider the ‘mixed’ second covariant derivatives
(h)∇µ
(h)∇AeB and (h)∇A
(h)∇µeB.
In view of the fact that eB and (h)∇AeB do not have components in the tangent bundle of the
domain M , the quantities
∂µU
C (h)∇C
(h)∇AeB and (h)∇A
(
∂µU
C (h)∇C
)
eB
are equivalent to
(h)∇µ
(h)∇AeB and (h)∇A
(h)∇µeB, respectively.
We have
U ′A (h)∇A
(h)∇µeB = U ′A (h)∇A
(
∂µU
C (h)∇CeB
)
(3.4a)
= U ′A∂µU
C
(
∂A
(h)ΓD
CB + (h)ΓD
AE
(h)ΓE
CB
)
+ U ′A(h)∇A∂µU
C (h)∇CeB,
likewise
∂µU
A (h)∇A
(
U ′C (h)∇CeB
)
= ∂µU
AU ′C (h)∇A
(h)∇CeB + ∂µU
A(h)∇AU
′C (h)∇CeB
= ∂µU
AU ′C(∂C (h)ΓD
AB + (h)ΓD
CE
(h)ΓE
AB
)
+ ∂µU
A(h)∇AU
′C (h)∇CeB, (3.4b)
so that we have
U ′A (h)∇A
(h)∇µeB − ∂µU
A (h)∇A
(
U ′C (h)∇CeB
)
= (h)RD
BAµeDU
′A.
This ‘mixed’ derivative construction is relevant for our wave map deformations. Let us assume
that [
∂βU,U
′] ≡ 0 (3.5a)
from which, it follows that
∂βU
A (h)∇AU
′B − U ′A (h)∇A∂βU
B ≡ 0. (3.5b)
Now consider another analogous curve γλ. The quantity D2
γλγs
· LWM involves the following
terms:
□gU
′A + ∂UD
(h)ΓA
BCg
µν∂µU
B∂νU
CU ′D + 2(h)ΓA
BCg
µν∂µU
′B∂νU
C . (3.6)
Assuming that the Kerr wave map is a critical point of (3.2) at s = 0, the expression (3.1) can
consecutively be transformed as follows:
= □gU
′A + ∂UD
(h)ΓA
BCg
µν∂µU
B∂νU
CU ′D + 2(h)ΓA
BCg
µν∂µU
′B∂νU
C
+ (h)ΓA
BCU
′B(□gU
C + (h)ΓC
DEg
αβ∂αU
D∂βU
E
)
,
20 N. Gudapati
which can be transformed to
gµνU ′C (h)∇C
(h)∇µ∂νU
A
= gµνU ′C(∂C((h)∇µ∂νU
A
)
− (h)ΓD
Cµ
(h)∇D∂νU
A + (h)ΓA
CD
(h)∇µ∂νU
A
)
.
Now consider the operator
gµν (h)∇µ
(
(h)∇C∂νU
A
)
= gµν (g)∇µ
(
(h)∇C∂νU
A
)
− gµν
(
(h)ΓD
µC∇D∂νU
A + hΓA
µD
(h)∇C∂νU
D
)
and performing the computations analogous to (3.4), we get that (3.6) is equivalent to
(h)□U ′A + (h)RA
BCDg
µν∂µU
B∂νU
DU ′C ,
where
(h)□U ′A := gµν (h)∇µ
(h)∇νU
′A
= ∂µ
(
(h)∇νU
′A)− (g)Γγ
µν
(h)∇γU
′A + (h)ΓA
µC
(
(h)∇νU
′C),
which can be represented in terms of the covariant wave operator
(
gµν (g)∇µ∂νU
′A) in the domain
metric g as
= □gU
′A + gµν
(
∂µ
(
(h)ΓA
νCU
′C)− (g)Γγ
µν
(h)ΓA
γCU
′C + (h)ΓA
µC∂νU
′C + (h)ΓA
µC
(h)ΓC
νDU
′D)
and (h)R is the induced Riemannian curvature tensor
(h)RA
BCD = ∂C
(h)ΓA
DB − ∂D
(h)ΓA
CB + (h)ΓA
CE
(h)ΓE
DB − (h)ΓA
DE
(h)ΓE
CB.
Now for the Kerr wave map critical point of Dγs
· LWM at s = 0, we then have
D2
γλγs
· LWM(s = 0) =
∫
hABU
′B((h)□U ′A + (h)RA
BCDg
µν∂µU
B∂νU
DU ′C)µ̄g (3.7)
as the Lagrangian variational principle for small linear deformations of the wave map Us : (M, g)
→ (N,h). In view of the divergence identity
(h)∇µ
(
hABU
′B (h)∇µU ′A) = hAB
(h)∇µU ′A(h)∇µU ′B + U ′B(h)∇µ
(
hAB
(h)∇µU ′A)
the variational principle (3.7) can equivalently be transformed into a self-adjoint variational form
D2
γλγs
· LWM(s = 0)
= −1
2
∫ (
gµν hAB
(h)∇µU
′A(h)∇µU
′B − hABU
′B (h)RA
BCDg
µν∂µU
B∂νU
CU ′D)µ̄g. (3.8)
Let us now calculate the Hamiltonian field equations for the linear perturbation theory, using
the ADM decomposition of the background (M, g)
g = −N2dt2 + qij
(
dxi +N idt
)
⊗
(
dxj +N jdt
)
.
Let us denote the variational principle (3.7) and (3.8) by LWM(U ′). The Legendre transformation
on C ′
WM results in the phase space
X ′
WM :=
{(
U ′A, p′A
)}
, where
(
U ′A, p′A
)
are canonical pairs,
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 21
the conjugate momenta p′A = Dγ · (pA(s))|s=0 are given by
p′A =
1
N
µ̄qhAB(U)
(
∂tU
′B + (h)ΓB
tCU
′C)− µ̄q
N
hAB(U)LNU ′B
− µ̄q
N
hAB(U)Na(h)ΓB
aC (3.9)
on account of the fact that the time derivative terms in the second term of (3.8) only occur for
background wave map U . Now then, using the quantity
hAB∂tU
′B = µ̄−1
q Np′A − hAB(U)(h)ΓB
tCU
′C + hABLNU ′B + hABN
a(h)ΓB
aCU
′C ,
the Lagrangian and Hamiltonian densities, L′
WM and H′
WM can be expressed in terms of the
phase space variables X ′
WM =
{(
U ′A, p′A
)}
in a recognizable ADM form as follows:
L′
WM(U ′) :=
1
2
p′A∂tU
′A − 1
2
p′ALNU ′A +
(
(h)ΓA
tCU
′C − (h)ΓA
aCN
aU ′C)1
2
p′A
− 1
2
hAB(U)Nµ̄qq
ab
(
(h)∇aU
A (h)∇bU
B
)
+Nµ̄qhAE(U)U ′ARE
BCDq
ab∂aU
BU ′C∂bU
D
− 1
N
µ̄qhAE(U)U ′ARE
BCD LNUBU ′CLNUD
− 1
N
µ̄qhAE(U)U ′ARE
BCD∂tU
BU ′C∂tU
D
+
2
N
µ̄qhAE(U)U ′ARE
BCDq
ab∂tU
BU ′CLNUD,
likewise the Hamiltonian energy density can be expressed as
H′
WM :=
1
2
p′A∂tU
′A +
1
2
p′ALNU ′A −
(
(h)ΓA
tCU
′C − (h)ΓA
aCN
aU ′C)1
2
p′A
+
1
2
hAB(U)Nµ̄qq
ab
(
(h)∇aU
A (h)∇bU
B
)
−Nµ̄qhAE(U)U ′ARE
BCDq
ab∂aU
BU ′C∂bU
D
+
1
N
µ̄qhAE(U)U ′ARE
BCD LNUBU ′CLNUD
+
1
N
µ̄qhAE(U)U ′ARE
BCD∂tU
BU ′C∂tU
D
− 2
N
µ̄qhAE(U)U ′ARE
BCDq
ab∂tU
BU ′CLNUD,
so that the critical point of L′
WM
L′
WM =
∫ t2
t1
∫
Σ
L′
WM d2xdt
with respect to U ′A gives the field equation
∂tp
′
A = LNp′A +
(
(h)ΓC
tA − (h)ΓC
aAN
a
)
p′C + hAB
(h)∇a
(
Nµ̄qq
ab∇bU
′B)
+Nµ̄qhAE(U)RE
BCDq
ab∂aU
BU ′C∂bU
D
− 1
N
µ̄qhAE(U)RE
BCD LNUBU ′CLNUD − 1
N
µ̄qhAE(U)RE
BCD∂tU
BU ′C∂tU
D
+
2
N
µ̄qhAE(U)RE
BCDq
ab∂tU
BU ′CLNUD. (3.10)
22 N. Gudapati
Analogously, it is straightforward to note that the field equations (3.9) and (3.10) are generated
by the Hamiltonian H ′
WM =
∫
H′
WMd2x, i.e.,
Dp′A ·H ′
WM = ∂tU
′A, DU ′A ·H ′
WM = −∂tp
′
A,
respectively. Specializing to our stationary Kerr background metric, we have
hAB(U)∂tU
′B = µ̄−1
q Np′A,
∂tp
′
A = hAB(U)(h)∇a
(
Nµ̄qq
ab∇bU
′B)+Nµ̄qhAE(U)RE
BCDq
ab∂aU
BU ′C∂bU
D.
Let us now construct the variational principle for the fully coupled Einstein-wave map per-
turbations. Now suppose
q′ab = Dγs
· (qab(s))|s=0, π′
ab = Dγs
· (πab(s))|s=0,
let us then denote the phase space corresponding to the perturbative theory of Kerr metric
as X ′
EWM:
X ′ :=
{(
U ′A, p′A
)
,
(
q′ab,π
′
ab
)}
.
Using the gauge-condition that the densitized metric µ̄−1
q qab is fixed, we can construct Dγs
·H
and Dγs
·Ha at s = 0
H ′ := Dγs
·H(s = 0) = −µ̄−1
q qabπ
′ab
−
(
µ̄qRq
)′
+
1
2
µ̄qq
ab∂UChAB(U)∂aU
A∂bU
BU ′C µ̄qq
abhAB(U)∂aU
′A∂bU
B
and
H ′
a := Dγs
·Ha(s = 0) = (q)∇bπ
′b
a + p′AU
A,
where(
µ̄qRq
)′
= µ̄q
(
−∆qq
′ + (q)∇a(q)∇bq′ab
)
, q′ := Trqq
′
ab.
Again, after imposing that the Kerr metric is a critical point at s = 0, we get
D2
γλγs
·H(s = 0) = µ̄−1
q
(
2∥π′∥2q − 2
(
qabπ
′ab)2 + p′Ap
′A)
−
(
µ̄qRq
)′′
+
1
2
µ̄qq
ab∂2
UDUChAB(U)∂aU
A∂bU
BU ′CU ′D
+ µ̄qq
ab∂UChAB(U)∂aU
′A∂bU
BU ′C
+
1
2
µ̄qq
ab∂UChAB(U)∂aU
A∂bU
BU ′′C
+ µ̄qq
ab∂UChAB(U)∂aU
′A∂bU
BU ′C
+ µ̄qq
abhAB(U)∂aU
′′A∂bU
B + µ̄qq
abhAB(U)∂aU
′A∂bU
′B, (3.11)
D2
γλγs
·Ha(s = 0) = −4(q
′)∇bπ
′b
a − 2(q)∇bπ
′′b
a + 2p′A∂aU
′A + ∂aU
Ap′′A,
where
(q′)∇bV
a := ∂bV
a +
1
2
qad
(
(q)∇bq
′
dc +
(q)∇cq
′
bd − (q)∇dq
′
bc
)
V c.
We arrive at the following theorem.
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 23
Corollary 3.1. Suppose X ′ is the first variation phase space, then the field equations for the
dynamics in X ′ are given by the variational principle
JEWM
(
X ′
EWM
)
:=
∫ (
π′ab∂tq
′
ab + p′AU
′A − 1
2
NH ′′ −N ′H ′ −N ′
aH
′
a
)
,
where H ′′, H ′, H ′
a are (3.11), Dγs
· H and D · Ha at s = 0, respectively, N ′ := Dγs
· N
∣∣
s=0
and N ′
a := Dγs
·Na
∣∣
s=0
.
The approach used above is the classical Jacobian method, as remarked by Moncrief [53].
Separately, it may be noted that the construction of the wave map field equations is analogous
to that of the geodesic deviation equations or the ‘Jacobi’ fields [51, 75]. In view of the fact
that the Hamiltonian formulation of the geodesic deviation equations is relatively uncommon,
our derivation may also be adapted for this purpose. Finally, we would like to emphasize that
our assumption that (3.5) holds, is not (effectively) a restriction in the class of perturbations.
In case this condition is relaxed, we shall also pick up the Riemann curvature of the target,
but with torsion. The fact that we pick only the curvature term of the target is crucial for our
work. We would also like to remark that the deformations which correspond to the coordinate
directional derivatives along the curves γλ are equivalent to (induced) covariant deformations
on the target, on account of the fact the Kerr wave map is a critical point of (3.1).
The variational principle in Corollary 3.1 and its field equations correspond to a general
Weyl–Papapetrou gauge. If we consider further gauge-fixing (2.9), where the densitized metric
µ̄−1
q qab or equivalently the densitized inverse metric µ̄qq
ab is fixed, we obtain
H ′ = µ̄q0
(
2∆0ν
′)+ 1
2
µ̄q0∂UChABq
ab
0 ∂aU
A∂bU
BU ′C + µ̄q0q
ab
0 hAB∂aU
′A∂bU
B,
H ′′ = µ̄−1
q0
(
2e−2ν∥ϱ′∥2q0 − τ ′2e2ν µ̄2
q0 + p′Ap
′A)
+ µ̄q0
(
2∆0ν
′′ + ∂2
UCUDhABq
ab
0 ∂aU
A∂bU
BU ′CU ′D + ∂ChABq
ab
0 ∂aU
′A∂bU
BU ′C
+
1
2
∂ChABq
ab
0 ∂aU
A∂bU
BU ′′C + 2hABq
ab
0 ∂aU
′′A∂bU
B
+ 2hABq
ab
0 ∂aU
′A∂bU
′B + 2∂Chabq
ab
0 ∂aU
′A∂bU
BU ′C).
The aim of our work is to construct an energy for the linear perturbative theory of Kerr black
hole spacetimes, for which the Hamiltonian formulation is naturally suited. In contrast with the
Lagrangian variational principles (e.g., (2.1) and (2.2)), the Hamiltonian variation principles are
not spacetime diffeomorphism invariant. In this work, we shall work in the 2+1 maximal gauge
condition. We point out that this gauge condition was also used by Dain–de Austria for the
extremal case [22]. We shall need the following statement.
Claim 3.2. Suppose N ′ ∈ C∞(Σ),
∆0N
′ = 0, in the interior of (Σ, q0), (3.12a)
N ′|∂Σ = 0, (3.12b)
then N ′ ≡ 0 on (Σ, q0).
Proof. If we multiply (3.12a) with N ′ and integrate by parts, we get
∫
|∇0N |2 = 0 in the
interior of Σ, after using (3.12b). It follows that N ′ is a constant in Σ. ■
The variational principle in Corollary 3.1 now gives the following field equations (for smooth
and compactly supported variations)
hAB(U)∂tU
′B = e2ν µ̄−1
q0 Np′A + hAB(U)LN ′UB,
24 N. Gudapati
∂tp
′
A = hAB(U)(h)∇a
(
Nµ̄qq
ab∇bU
′B)+Nµ̄qhAE(U)RE
BCDq
ab∂aU
BU ′C∂bU
D
= hAB(U)(h)∇a
(
Nµ̄q0q
ab
0 ∇bU
′B)+Nµ̄q0hAE(U)RE
BCDq
ab
0 ∂aU
BU ′C∂bU
D,
∂tq
′
ab = 2Nµ̄−1
q CKab
(
Y ′, q
)
+ (q)∇aN
′
b +
(q)∇bN
′
a
= 2Ne−2ν µ̄−1
q0 CKab
(
Y ′, q0
)
+ LN ′
(
e2ν(q0)ab
)
,
∂tπ
′ab =
(
µ̄qq
bcqad
)′(
∂2
dcN − (q)Γf
cd∂fN
)
+ µ̄qq
bcqad
(
qfl
(
(q)∇dq
′
lc +
(q)∇cq
′
ld +
(q)∇lq
′
cd
)
∂fN
)
+
(
1
2
Nµ̄q
(
qacqbd − 1
2
qabqcd
))′
hAB(U)∂aU
a∂bU
B
+
1
2
Nµ̄q
(
qacqbd − 1
2
qabqcd
)(
2hAB(U)∂aU
′A∂bU
B + ∂UChAB(U)∂aU
A∂bU
BU ′C)
together with the constraints
H ′ = 0 and H ′
a = (q)∇bπ
′b
a + p′A∂aU
A = 0,
in the 2 + 1 maximal gauge. We have
q′ = Tr q′ab, τ ′ = µ̄−1
q qabπ
′ab,
and
∂tq
′ = −2Nτ ′ + 2 (q)∇cN ′
c, ∂tτ
′ = −∆0N
′ +Nq′.
In the 2 + 1 maximal gauge (cf. Claim 3.2),
∂tq
′ = 2(q)∇cN ′
c, ∆0N
′ = 0.
Let us now formally discuss the structures associated to our Hamiltonian framework. The phase
space X ′
EWM is such that
(
q′ab, U
′A) are C∞(Σ) symmetric covariant 2-tensor and smooth vector
field respectively and
(
π′ab, p′A
)
are C∞(Σ) symmetric 2-tensor densities and scalar density (for
each A) respectively, which together form the cotangent bundle T ∗M, which we had represented
as X ′
EWM. The Hamiltonian and momentum constraint spaces CH′ , CH′
a
are defined as follows:
CH′ =
{(
q′ab,π
′ab)(U ′A, p′A
)
∈ T ∗M | H ′ = 0
}
, (3.13a)
CH′
a
=
{(
q′ab,π
′ab)(U ′A, p′A
)
∈ T ∗M | H ′
a = 0, a = 1, 2
}
. (3.13b)
Furthermore, we consider our time coordinate gauge condition to be ‘2 + 1 maximal’
Cτ ′ =
{(
q′ab,π
′ab)(U ′A, p′A
)
∈ T ∗M | τ ′ = 0
}
.
In our work, we shall be interested in the space
CH′ ∩ CH′
a
∩ Cτ ′
for our initial value framework. In general, proving local existence of Einstein equations using the
Hamiltonian initial value problem is a complex problem. We note the following statement from
the Lagrangian framework of Einstein’s equations from the classical result of Choquet-Bruhat
and Geroch [12] in 3 + 1 dimensions.
Suppose
{(
q̄′ab, π̄
′ab)}
0
∈ CH̄′ ∩ CH̄′
i
, then it follows from the classic results of Choquet-Bruhat
and Geroch, adapted to our linear perturbation problem, that there exists a unique, regular,
maximal development of
{(
q̄′ab, π̄
′ab)}
0
, ι : Σ → Σ × R, such that
{(
q̄′ab, π̄
′ab)}
t
∈ CH̄′ ∩ CH̄′
i
is
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 25
causally determined from the initial data
{(
q′ab,π
′ab), (U ′A, p′A
)}
0
in a suitable gauge; where CH̄′
and CH̄′
i
are defined analogous to (3.13).
Let us introduce the following notions from the machinery of linearization stability. Let us de-
fine the constraint map Ψ of
(
Σ, q̄
)
as a map from the cotangent bundle to a 4−tuple of scalar
densities, Ψ: T ∗M → C∞(Σ)× T Σ, such that
Ψ(q̄, π̄) =
(
H̄, H̄i
)
, i = 1, 2, 3.
Let us denote the deformation of the constraint map as D · Ψ(q̄, π̄). Then the L2-adjoint,
D† · Ψ(q̄, π̄)
(
C̄, Z̄
)
, of the deformations D · Ψ of the constraint map is a 2-tuple (an element
of a Banach space) consisting of a covariant symmetric 2-tensor and a contravariant symmetric
2-tensor density and is given by
D† ·Ψ(q̄, π̄)
(
C̄, Z̄
)
:=
(
µ̄−1
q̄
(
1
2
(
∥π∥2q̄ − Trq̄(π̄)
2
)
q̄ijC̄ − 2
(
π̄ikπ̄j
k −
1
2
πijTrq̄(π̄)
)
C̄
)
− µ̄q
(
q̄ij∆q̄C − (q̄)∇i(q̄)∇jC̄ +RijC̄ − 1
2
q̄ijRq̄C̄
)
+ (q̄)∇k
(
Z̄kπ̄ij
)
− (q̄)∇kZ̄
iπ̄kj − (q̄)∇kZ̄
iπ̄jk,
− 2µ̄−1
q̄ C̄
(
π̄ij −
1
2
Tr(π̄)q̄ij
)
− (q)∇iZ̄j − (q)∇jZ̄i
)
. (3.14)
The expression (3.14) is closely related to the L2−adjoint of the Lichnerowicz operator. Moncrief
had characterized the splitting theorem, established by Fischer–Marsden [28], of the (Banach)
spaces acted on by the constant map
kerD† ·Ψ(q̄, π̄)
(
C̄, Z̄
)
⊕ rangeD ·Ψ(q̄, π̄)
(
q̄′, π̄′),
by associating the kernel of the adjoint operator
(
kerD† · Ψ(q̄, π̄)
(
C̄, Z̄
))
to the existence of
spacetime Killing isometries. In particular, Moncrief proved that kerD† ·Ψ
(
C̄, Z̄
)
is non-empty
if and only if there exists a spacetime Killing vector. This result is crucial for our work, but in
the dimensionally reduced framework. In the following, we shall establish equivalent results in
our dimensionally reduced perturbation problem.
Lemma 3.3. Suppose (M, g) is the 2 + 1 spacetime obtained from the dimensional reduction of
the axially symmetric, Ricci-flat 3 + 1 spacetime
(
M̄, ḡ
)
and D · Ψ is the deformation around
the Kerr metric of the constraint map Ψ of the dimensionally reduced 2 + 1 Einstein-wave map
system on (M, g), then
(1) The adjoint D† ·Ψ(q′,π′)(C,Z) of the constraint map Ψ is given by
D† ·Ψ =
(
µ̄q
(
(q)∇b (q)∇aC − qab (q)∇c
(q)∇cC
)
+
1
2
Cµ̄qhAB
(
qacqbd − 1
2
qabqcd
)
∂aU
A∂bU
B − (q)∇aZb − (q)∇bZa
)
. (3.15)
(2) The kernel
(
ker
(
D†Ψ
))
of the adjoint of the constraint map Ψ is one dimensional and is
equal to (N, 0)T.
Proof. Consider the constraint map Ψ
Ψ(g,π) = (H,Hi),
26 N. Gudapati
then from the deformation of Ψ, D·Ψ(q,π)(q′,π′) := (H ′, H ′
a), around a general metric, it follows
that its 2 + 1 L2−adjoint is given by
D† ·Ψ =
(
1
2
Cµ̄−1
q qab
(
∥π∥2q − Tr(π)2
)
− 2Cµ̄−1
q
(
πacπb
c − πabTrq(π)
)
+ µ̄q
(
(q)∇b (q)∇aC − qab (q)∇c
(q)∇cC
)
+ (q)∇c
(
πabZc
)
− (q)∇cZ
aπcb − (q)∇cZ
bπca
+
1
4
µ̄−1
q CqabpAp
A +
1
2
Cµ̄qhAB
(
qacqbd − 1
2
qabqcd
)
∂aU
A∂bU
B,
− 2Cµ̄−1
q (πab − qabTrqπ)− (q)∇aZb − (q)∇bZa
)
,
analogous to (3.14), while noting that the (dimensionally reduced) wave map variables are not
constrained due to the introduction of the twist potential, after using the Poincaré Lemma
(see, e.g., (1.8) and then (1.9)). The expression (3.15) follows for the case of dimensionally
reduced Kerr metric. Now assume that (C,Z) ∈ kerD† ·Ψ. It follows from (3.15) that a vector
K = K⊥n+K∥ satisfies
(g)∇αKβ + (g)∇βKα = 0 (3.16)
with K⊥ = C and K∥ = Z which implies that K = (C,Z) is a (spacetime) Killing vector
in (M, g). Conversely, assuming that (3.16) holds it follows that the left-hand side of (3.15)
vanishes, which implies K ∈ kerD† ·Ψ. In particular, for the dimensionally reduced Kerr metric
(M, g) the only remaining linearly independent Killing vector is ∂t, so (C,Z)T ≡ (N, 0)T, which,
as will be shown later, resolves (P2). ■
In the following, we shall establish that the Hamilton vector field (H ′, H ′
a) is tangential to
the flow of the phase space variables
{
(q′, π′),
(
U ′A, p′A
)}
in CH′ ∩ CH′
a
∩ Cτ ′ .
Lemma 3.4. Suppose H ′ and H ′
a are the linearized Hamiltonian and momentum constraints of
the 2 + 1 Einstein-wave map system, then their propagation equations are
∂
∂t
H ′ = qab∂aNH ′
b + ∂b
(
NqabH ′
a
)
, (3.17a)
∂
∂t
H ′
a = ∂aNH ′ (3.17b)
and
NH ′ = ∂b
(
Nµ̄qq
abhABU
′A∂bU
B − 2µ̄qq
ab∂aNν ′). (3.17c)
Proof. The statements (3.17a) and (3.17b) follow from the linearized and background (exact)
field equations of our 2+ 1 Einstein-wave map system. For simplicity in computations, we shall
perform our computations with q′0 held fixed. Recall
∂tϱ
′a
b = Nµ̄q0
(
qac0 δdb −
1
2
qcd0 δab
)(
hAB∂cU
′A∂dU
B +
1
2
∂UChAB(U)∂cU
A∂dU
BU ′C
)
+ µ̄q0q
cd
0 δab ∂cN∂dν
′ − µ̄q0q
ac
0
(
∂bN∂cν
′),
∂tν
′ =
1
2µ̄q0
∂c
(
µ̄q0N
′c)+ 2LN ′ν.
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 27
Consider the quantities
∂t
(
2µ̄−1
q0 ∂b
(
µ̄q0q
ab
0 ∂aν
′)) = µ̄−1
q0 ∂b
(
µ̄q0q
ab
0 ∂a
(
µ̄−1
q0 ∂c
(
µ̄q0N
′c)+ 2N ′c∂cν
))
,
1
2
µ̄qq
ab∂UChAB∂aU
A∂bU
B∂tU
′C =
1
2
Nqab∂UChAB∂aU
A∂bU
Bp′C
+
1
2
µ̄qq
ab∂UChAB∂aU
A∂bU
BLN ′UC ,
µ̄qq
abhAB(U)∂a
(
∂tU
′A)∂bUB = µ̄qq
abhAB(U)∂a
(
µ̄−1
q Np′A
)
∂bU
B
+ µ̄qq
abhAB(U)∂a
(
LN ′UA
)
∂bU
B
= µ̄qq
abhAB(U)∂a
(
µ̄−1
q Np′A
)
∂bU
B
+ µ̄qq
abhAB(U)LN ′
(
∂aU
A
)
∂bU
B.
Combining the results above and noting that for our gauge
CKab(N ′, q0) = µ̄q0
(
(q0)∇aN ′b + (q0)∇bN ′a − qab0
(q0)∇cN
′c) = −2Ne−2νqbc0 ϱ′ac ,
we get
∂tH
′ = qab∂bN
(
−2(q0)∇cϱ
′c
a + p′A∂aU
A
)
− 2∂b
(
Nqab(q0)∇cϱ
′c
a
)
+ ∂b
(
Nqabp′A∂aU
A
)
= qab∂aNH ′
b + ∂b
(
NqabH ′
a
)
.
Likewise, for (3.17b) consider
(q0)∇a(∂tϱ
′a
b ) =
(q0)∇a
(
Nµ̄q0
(
qac0 δdb −
1
2
qcd0 δab
)
×
(
hAB∂cU
′A∂dU
B +
1
2
∂UChAB(U)∂cU
A∂dU
BU ′C
)
+ µ̄q0q
cd
0 δab ∂cN∂dν
′ − µ̄q0q
ac
0
(
∂bN∂cν
′)),
∂aU
A∂tp
′
A = hAB∂aU
A(h)∇c
(
Nµ̄qq
cb(h)∇bU
′B)
+Nµ̄qhAB(U)∂aU
′BRE
BCDq
ab∂aU
B∂bU
DU ′C .
Now combining all the above, we have
∂tH
′
c = ∂cNH ′
in view of the background field equations (2.14) and (2.16). For (3.17c), first note that
Nµ̄q0∂UChAB(U)qab0 ∂aU
A∂bU
CU ′B = Nµ̄q0hAB
(h)ΓA
CD(U)qab0 ∂aU
C∂bU
DU ′B
− 1
2
Nµ̄q0∂UChAB(U)qab0 ∂aU
A∂bU
BU ′C (3.18)
after a suitable relabelling of the indices. Now consider
NH ′ = 2Nµ̄q0∆0ν
′ +Nµ̄qhAB
(h)ΓA
CD(U)qab0 ∂aU
C∂bU
DU ′B
−Nµ̄q0∂UChAB(U)qab0 ∂aU
A∂bU
CU ′B + µ̄q0q
ab
0 hAB∂aU
′A∂bU
B
= 2Nµ̄q0∆0ν
′ + ∂b
(
Nµ̄q0q
ab
0 hAB∂aU
AU ′B)
= ∂b
(
−2µ̄q0q
ab
0 ∂aNν ′ + 2∂aν
′ +Nµ̄qq
abhAB∂aU
AU ′B), (3.19)
where we have used (3.18) and the background field equations (2.14) and (2.16). Fundamentally,
underlying the statement (3.19) is the fact that (N, 0)T is the kernel of the adjoint of the
constraint map of our linear perturbation theory. ■
28 N. Gudapati
4 A positive-definite Hamiltonian energy from negative
curvature of the target and the Hamiltonian dynamics
In arriving at the variational principles and their corresponding field equations, we have used
smooth compactly supported deformations. In the construction of a Hamiltonian energy function
the underlying computations are bit more subtle, in connection with the boundary terms and the
initial value problem. We impose the regularity conditions on the axis of initial hypersurface Σ
by fiat, so that the fields smoothly lift up to the original Σ. We shall assume the following
conditions on the two disjoint segments of the axes Γ = Γ1 ∪ Γ2. In the wave map U : M → N,
one of the components corresponds to the norm of the Killing vector |Φ| and the other the ‘twist’.
For the twist component, we assume
U ′A∣∣
Γ1
= U ′A∣∣
Γ2
for the corresponding A
on account of our assumption that the perturbation of the angular momentum is zero. Without
(effective) loss of generality we assume,
U ′A = 0 on Γ which implies ∂t⃗ U
′A = 0,
where ∂t⃗ is the derivative tangent to the axis. To prevent conical singularity on the axis, which,
as we remarked, allows us to smoothly lift our fields up to the original manifold Σ, we assume
|Φ|′ = 0, ∂t⃗ |Φ|
′ = 0
for the ‘norm’ component of U , and
∂n⃗U
′A = 0, p′A = ∂t⃗p
′
A = 0, ∂n⃗p
′
A = 0,
where ∂n⃗ is the derivative normal to the axes Γ. In this work, for ν ′ we shall assume
∂n⃗ν
′ = 0,
which corresponds to the preservation of the condition that inner (horizon) boundary is a min-
imal surface. Now define an ‘alternative’ Hamiltonian constraint H ′Alt
H ′Alt := µ̄q0
(
2∆0ν
′ + hAB(U)U ′B(∆0U
A + (h)ΓA
BCq
ab
0 ∂aU
B∂bU
C
))
,
where we have now used the following identity to transform from H ′:
1
2
∂UChABq
ab∂aU
A∂bU
BU ′C + hAB(U)qab∂aU
′A∂bU
B
= hABU
′B(∆qU
′A + (h)ΓA
CDq
ab∂aU
C∂bU
D
)
,
which is analogous to (3.3), but now for the q metric. Analogously define
H ′′Alt := µ̄−1
q0
(
2e−2ν∥ϱ′∥2q0 − τ ′2e2ν µ̄2
q0 + p′Ap
′A)
− µ̄q0hABU
′B((h)∆U ′A +RA
BCDq
ab
0 ∂aU
B∂bU
CU ′D).
Using a further divergence identity
(h)∇a
(
hABU
′B(h)∇aU ′A)− hABU
′B(h)∇a
(h)∇aU ′A
= hABq
ab(h)∇aU
′A(h)∇bU
′B, (Σ, q),
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 29
let us now define our ‘regularized’ Hamiltonian energy density as
eReg := Nµ̄−1
q0 e
−2ν
(
∥ϱ′∥2q0 +
1
2
p′Ap
′A
)
− 1
2
Ne2ν µ̄q0τ
′2
+
1
2
Nµ̄q0q
ab
0 hAB(U)(h)∇aU
′A(h)∇bU
′B
− 1
2
Nµ̄q0q
ab
0 hAE(U)U ′A(h)RE
BCD∂aU
B∂bU
CU ′D (4.1)
and the ‘regularized’ Hamiltonian HReg
HReg :=
∫
Σ
eReg d2x. (4.2)
It is evident that HReg is manifestly positive-definite in the maximal gauge τ ′ ≡ 0, in view
of the fact that the target is the (negatively curved) hyperbolic 2-plane. Indeed, we obtain
a similar energy expressions (4.1) and (4.2) in the higher-dimensional case where the target for
wave maps is SL(n − 2)/SO(n − 2). As we already alluded to, the purpose of distinguishing
the quantities HReg is that they are transformed, using divergence identities, from H, and
thus differ by boundary terms. In case the perturbations are compactly supported in (Σ) it is
immediate that they have the same value. In general, dealing with all the boundary terms and
their evolution in our problem is considerably subtle (see Section 5).
The aim of this work is the construction of the positive-definite energy functional HReg.
In the following, we shall establish the validity and consistency of our approach to construct
the energy using two separate methods. Firstly, we shall show that HReg serves as a Hamil-
tonian that drives the dynamics of the unconstrained or ‘independent’ phase-space variables.
Secondly, we shall establish that there exists a spacetime divergence-free vector density, whose
flux through Σ is HReg.
We would like to point out that, in our problem, the 2 + 1 geometric phase space variables(
e.g., ν ′, ϱ′ab
)
are completely determined by the constraints and gauge-conditions. Therefore,
their Hamiltonian dynamics are governed by the ‘independent’ or ‘unconstrained’ dynamical
variables
(
U ′A, p′A
)
. In the following, we shall prove that HReg drives the coupled Einstein-wave
map dynamics of
(
U ′A, p′A
)
.
Theorem 4.1. Suppose the globally regular, maximal development of the smooth, compactly
supported perturbation initial data in the domain of outer communications of the Kerr metric
is such that
{(
q′ab,π
′ab), (U ′A, p′A
)}
t
∈ CH′ ∩ CH′
a
∩ Cτ ′, then functional HReg is a Hamiltonian
for the coupled dynamics of
(
U ′A, p′A
)
,
Dp′A
·HReg = ∂tU
′A, DU ′A ·HReg = −∂tp
′
A.
Proof. The first variation Dp′A
·HReg contains the terms
ϱ′ab ϱ
′′a
b =
1
2
N−1µ̄q
(
(q)∇bN ′
a +
(q)∇aN
′b − δba
(q)∇cN
′c)
×
(
(q)∇aN ′′
b + (q)∇bN
′′a − δab
(q)∇cN
′′c)
=
1
2
N−1µ̄q
(q)∇aN
′b((q)∇aN ′′
b + (q)∇bN
′′a − δab
(q)∇cN
′′c).
We have the divergence identity
(q)∇a
(
N−1N ′bµ̄q
(
(q)∇aN ′′
b + (q)∇bN
′′a − δab
(q)∇cN
′′c))
×N−1N ′b((q)∇a
(
(q)∇aN ′′
b + (q)∇bN
′′a − δab
(q)∇cN
′′c))
30 N. Gudapati
+N−1µ̄q
(q)∇aN
′b((q)∇aN ′′
b + (q)∇bN
′′a − δab
(q)∇cN
′′c)
= −N ′b(∂bUAp′′A
)
+N−1µ̄q
(q)∇aN
′b((q)∇aN ′′
b + (q)∇bN
′′a − δab
(q)∇cN
′′c)
after using the momentum constraint, and
Dp′A
· 1
2
p′Ap
′A = p′A.
Collecting the terms above gives the Hamilton equation
Dp′A
·HReg = Nµ̄−1
q p′A +N ′b∂bU
A = ∂tU
′A.
The quantity DU ′A ·HReg contains the terms
DU ′A · 1
2
hAB(U)(h)∇aU
′A(h)∇bU
′B = Nµ̄qq
abhAB
(q)∇aU
′′A(q)∇bU
′B
note that
(q)∇a
(
Nµ̄qq
abhABU
′′A∂bU
′B) = U ′′A(q)∇a
(
Nµ̄qq
abhAB∂aU
′B)
+Nµ̄qq
abhAB
(q)∇aU
′′A(q)∇bU
′B
and
DU ′A ·
(
−1
2
Nµ̄qq
abhAE(U)U ′A(h)RE
BCD∂aU
B∂bU
CU ′D
)
= −Nµ̄qq
abhAE(U)(h)RE
BCD∂aU
B∂bU
CU ′D,
which combine to give
DU ′A ·HReg = −(q)∇a
(
Nµ̄qq
abhAB∂aU
′B)
−Nµ̄qq
abhAE(U)(h)RE
BCD∂aU
B∂bU
CU ′D
which is the Hamilton equation
= −∂tp
′
A. ■
Theorem 4.2. Suppose the variables
{(
q′ab,π
′ab), (U ′A, p′A
)}
∈ CH′ ∩ CH′
a
∩ Cτ ′, then there
exists a (spacetime) divergence-free vector field density such that its flux through t−constant
hypersurfaces is HReg (positive-definite).
Proof. In the proof we shall use the perturbation evolution equations and the background (Kerr
metric) field equations. Consider ∂te
Reg and it contains the following terms:
1. Nµ̄−1
q p′A∂tp
′
A = Nµ̄−1
q p′A
(
hAB
(h)∇a
(
Nµ̄qq
ab(h)∇bU
′B)
+Nµ̄qhABR
E
BCDq
ab∂aU
B∂bU
DU ′C), (4.3)
2. Nµ̄qq
ab(h)∇a
(
∂tU
′A)(h)∇bU
′B
= Nµ̄qq
abhAB(U)(h)∇bU
′B((h)∇a
(
µ̄−1
q Np′A + LN ′UA
))
. (4.4)
Note the divergence relation involving the terms from (4.3) and (4.4),
(h)∇a
(
N2qabhABp
′A(h)∇bU
′B) = µ̄−1
q Np′A(h)∇a
(
Nµ̄qq
ab(h)∇bU
′B)
+Nµ̄qq
abhAB
(h)∇bU
′B(h)∇a
(
Nµ̄−1
q p′A
)
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 31
3. Nµ̄qhAE(U)∂tU
′ARA
BCDq
ab∂aU
B∂bU
DU ′C
= Nµ̄qhAE
(
µ̄−1
q Np′A + LN ′UA
)
RE
BCDq
ab∂aU
B∂bU
DU ′C
4. e−2νNµ̄−1
q0 ϱ
′c
a ∂tϱ
′a
c
= N
(
(q)∇aN
′b + (q)∇aN
′b − δba
(q)∇cN
′c)
×
(
Nµ̄q
(
qacδdb −
1
2
δab q
cd
)(
hAB∂bU
′A∂dU
B +
1
2
∂UhAB∂bU
′A∂dU
B
)
+ 2µ̄qq
ac∂cN∂bν
′ − µ̄qδ
a
b q
cd∂cN∂dν
′
)
= LN ′
(
µ̄q0q
ab
0
)(
hAB∂aU
′A∂bU
B +
1
2
∂UChAB∂aU
A∂bU
BU ′C − 2∂aN∂bν
′
)
.
Consider the following divergence identities:
(q0)∇a
(
N ′b∂aN∂bν
′µ̄q0
)
= µ̄q0
(q0)∇aN ′b∂aN∂bν
′ +N ′b(q0)∇a
(
µ̄q0∂aN∂bν
′) (4.5a)
(q0)∇b
(
N ′a∂aN∂bν
′µ̄q0
)
= µ̄q0
(q0)∇bN ′a∂aN∂bν
′ +N ′a(q0)∇b
(
µ̄q0∂aN∂bν
′) (4.5b)
(q0)∇c
(
N ′cqab0 ∂aN∂bν
′µ̄q0
)
= (q0)∇cN
′c(qab0 ∂aN∂bν
′µ̄q0
)
+N ′c(q0)∇c
(
qab0 ∂aN∂bν
′µ̄q0
)
. (4.5c)
Based on the right-hand sides of the divergence identities in (4.5), we get after using the back-
ground field equation (2.16)
−2LN ′
(
µ̄qq
ab
)
∂aN∂bν
′
= −2N ′b∂bN∂a
(
µ̄qq
ab∂aν
′)+ 2(q0)∇a
(
N ′c∂cν
′µ̄q0q
ab
0 ∂bN
)
+ 2(q0)∇a
(
N ′c∂cNµ̄q0q
ab
0 ∂bν
′)− 2(q0)∇c
(
N ′c∂aν
′µ̄q0q
ab
0 ∂bN
)
= LN ′N
(
−H ′ + hAB∂aU
′A∂bU
B +
1
2
∂UChAB∂aU
A∂bU
BU ′C
)
+ 2(q0)∇b
(
LN ′ν ′µ̄q0∂
bN + LN ′Nµ̄q0∂
bν′ − µ̄q0N
′b∂aN∂aν ′).
Now let us focus on the remaining ‘shift’ terms:
Nµ̄qq
abhAB
(h)∇bU
′B(h)∇a
(
LN ′UA
)
,
−Nµ̄qhAE
(
LN ′UA
)
RE
BCDq
ab∂aU
B∂bU
DU ′C ,
LN ′
(
µ̄q0q
ab
0
)(
hAB∂aU
′A∂bU
B +
1
2
∂UhAB∂aU
′A∂bU
B
)
. (4.6)
Consider the quantity Nµ̄qq
abhAB∂aU
′A(LN ′
(
∂bU
B
))
that occurs in (4.6), we have
Nµ̄qq
abhAB∂aU
′A(LN ′
(
∂bU
B
))
+ ∂UChABLN ′UCNµ̄qq
ab∂aU
′A∂bU
B
= ∂aU
′ALN ′
(
Nµ̄qq
abhAB(U)∂bU
B
)
− LN ′N
(
hABµ̄qq
ab∂bU
′B)− LN ′
(
µ̄qq
ab
)
NhAB∂aU
′A∂bU
B,
likewise
U ′ALN ′
(
∂b
(
Nµ̄qq
abhAB∂aU
B
))
= U ′ALN ′hAB∂a
(
Nµ̄qq
ab∂bU
B
)
+ U ′ALN ′
(
Nµ̄qq
ab∂bU
B∂UChAB∂aU
C
)
.
32 N. Gudapati
Collecting the terms above, while using the background field equations (2.14) and computa-
tions analogous to the ones in Section 3 and
∂a
(
U ′ALN ′
(
Nµ̄qq
abhAB∂aU
A
))
= ∂aU
′ALN ′
(
Nµ̄qq
abhAB∂aU
A
)
+ LN ′
(
∂a
(
Nµ̄qq
abhAB∂aU
A
))
= ∂aU
′ALN ′
(
Nµ̄qq
abhAB∂aU
A
)
+ ∂a
(
LN ′
(
Nµ̄qq
abhAB∂aU
A
))
,
we have
∂te
Reg = ∂b
(
N2µ̄−1
q
(
µ̄q0q
ab
0 p′A∂aU
′A)+ U ′ALN ′
(
µ̄q0q
ab
0 hAB∂bU
B
))
× LN ′(N)
(
2µ̄q0q
ab
0 ∂aν
′ + 2LNν ′µ̄qq
ab∂aN − 2N ′bµ̄qq
bc∂aν
′∂cN
)
−H ′LN ′(N)
for
{(
q′ab,π
′ab), (U ′A, p′A
)}
∈ CH′ this reduces to
= ∂b
(
N2µ̄−1
q
(
µ̄q0q
ab
0 p′A∂aU
′A)+ U ′ALN ′
(
µ̄q0q
ab
0 hAB∂bU
B
))
× LN ′(N)
(
2µ̄q0q
ab
0 ∂aν
′)+ 2LNν ′µ̄qq
ab∂aN − 2N ′bµ̄qq
bc∂aν
′∂cN.
Thus, if we define
(J t)Reg := eReg,
(Jb)Reg := N2e−2ν
(
qab0 p′A∂aU
′A)+ U ′ALN ′
(
Nµ̄q0q
ab
0 hAB∂bU
B
)
× LN ′(N)
(
2µ̄q0q
ab
0 ∂aν
′)+ 2LN ′ν ′µ̄qq
ab∂aN − 2N ′bµ̄qq
bc∂aν
′∂cN,
it follows that JReg is a divergence-free vector field density for{
(q′ab,π
′ab),
(
U ′A, p′A
)}
∈ CH′ ∩ CH′
a
∩ Cτ ′ . ■
5 Boundary behaviour of the dynamics in the orbit space
In any dynamical physical theory, the concept of a conserved energy plays a fundamental role
in the analysis of stability of its solutions. For instance, in the Lyapunov theory of stability, the
notion of energy, its positive-definiteness and its dynamical behaviour act as important basis for
various notions of stability. Likewise, PDE techniques are typically based on a conserved and
positive energy.
In the previous sections, we constructed a positive-definite ‘bulk’ Hamiltonian energy func-
tional for the axially symmetric linear perturbative theory of Kerr black hole spacetimes for the
entire subextremal range (|a| < M); and the associated variational principles.
We used a special ‘Weyl–Papapetrou’ gauge that provides additional structure in Einstein’s
equations for general relativity:
Rµν = 0, µ, ν = 0, 1, 2, 3,
(
M̄, ḡ
)
,
when the 3 + 1 Lorentzian spacetime
(
M̄, ḡ
)
admits a rotational isometry
ḡ = |Φ|−1g + |Φ|(dϕ+Aνdx
ν)2, ν = 0, 1, 2, (Weyl–Papapetrou gauge), (5.1)
where g is the Lorentzian metric on the orbit space M := M̄/SO(2). A, g are independent
of the parameter ϕ corresponding to the rotationally symmetric Killing vector Φ := ∂ϕ, whose
spacetime norm squared is represented as |Φ|.
As we already discussed, the effects of the geometry of the Kerr black hole spacetime man-
ifest themselves in the Kerr black hole stability problem in a fundamental way. In particular,
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 33
as a result of the ergo-region the perturbations (scalar wave, Maxwell and linearized gravity)
do not necessarily have a positive-definite and conserved energy. This significantly limits the
immediate use of the standard PDE techniques in establishing the boundedness and asymptotic
decay of the perturbations.
A standard technique in the literature to construct a positive-definite energy functional is
to consider an energy current from the linear combination of the time-translational ∂t and
rotational ∂ϕ vector fields
∂t + χ∂ϕ.
In view of the fact that the corresponding energy is not necessarily conserved, a separate, intri-
cate Morawetz spacetime integral estimate is needed to control this energy in time. Moreover,
these methods are suitable for small angular-momentum |a| ≪ M .
Thus, for our Hamiltonian energy to be efficacious, it is important to establish that it is
strictly conserved in time. In the process of construction of the positive-definite Hamiltonian
energy in our work, we pick up several boundary terms, at various stages. From both concep-
tual and technical perspectives these boundary terms play an important role in our dynamical
stability problem. These issues do not arise in the corresponding black hole uniqueness theo-
rems of stationary black holes. For the convenience of the reader, let us elaborate on why it
is fundamental to rigorously understand the behaviour of these boundary terms and provide
a motivation for this article.
As we already remarked, there is a possibility that the energy density in the original construc-
tion can be made to be locally negative. The positive-definite energy is constructed using the
transformations that involve boundary terms from a Hamiltonian energy with an indefinite sign,
while preserving the ‘bulk’ Hamiltonian structure of the equations. Therefore, in our argument
it is fundamental to rule out the ‘negativity’ or the ambiguity of sign does not ‘secretly get
hidden’ in a plethora of boundary terms that occur in both the Carter–Robinson type identities
and the linearization stability methods.
In the usual PDE theory, we perform a variation with respect to smooth compactly sup-
ported C∞
0 ‘test functions’ to evaluate the field equations and understand the critical points.
However, in the Einstein equations one cannot ignore the boundary terms, because they can
have a physical and geometric interpretation.
From the perspective of calculus of variations, a positive-definite second variation mass-energy
corresponds to the mass of the Kerr black hole being a minimizer, for fixed angular momentum,
in the space of admissible metrics. This interpretation combines well with the mass-angular
momentum inequalities for axisymmetric spacetimes [21]. However, as we already discussed, the
boundary terms cannot be ignored in this interpretation.
Most PDE techniques in establishing the uniform boundedness and decay, work at their natu-
ral best if there exists a positive-definite and (strictly) conserved energy, that goes together with
the evolution of the PDEs. In case there exists a residual, dynamically non-vanishing surface
integral (especially with an indefinite sign), together with a positive-definite ‘bulk’ Hamilto-
nian (4.2) and (4.1), then this could significantly impede the efficacy of our positive-definite
‘bulk’ Hamiltonian (4.2) and (4.1) in PDE methods that establish uniform-boundedness and
decay.
The Weyl–Papapetrou gauge (5.1), although it plays a fundamental role in the construction
of our bulk Hamiltonian energy (4.2) and (4.1), presents significant regularity issues at the axes
and at the infinity. This is due to the behaviour of the rotationally symmetric Killing vector
field, i.e., |Φ|−1 blows up at the axes Γ and |Φ| blows up at the infinity ι0. These expressions
routinely occur in our formulas due to the form of the Weyl–Papapetrou gauge (5.1). It may be
noted that these regularity issues manifest themselves in the form of boundary terms that occur
precisely at these boundaries Γ, ι0 and H+.
34 N. Gudapati
The Einstein equations in the Weyl–Papapetrou gauge are not purely hyperbolic in nature.
In particular, the Weyl–Papapetrou gauge has coupled elliptic-hyperbolic PDE structure. This
causes gauge-related causality issues. Even if we start with a compactly supported initial data,
away from the boundaries Γ, ι0, H+, thereby ensuring the regularity and the vanishing of
boundary flux integrals initially, it is not necessary that the fields stay compactly supported at
later times. In other words, ‘pure gauge’ perturbations and the associated boundary terms, can
‘kick-in’ in the asymptotic regions, away from the causal future of the support of initial data,
at later times, thus affecting the boundary behaviour of the dynamics.
On account of the boundary related issues and complications mentioned above, the ques-
tion whether the Weyl–Papapetrou gauge is even compatible with the axially symmetric evo-
lution problem of the Einstein equations can legitimately arise. In this work, we shall provide
a favourable answer to this question. We explain how we overcome these complications and
discuss our methods in the following.
We formulate the initial value problem of the linearized Einstein equations in the harmonic
gauge and transform the solutions to the Weyl–Papapetrou gauge. In the harmonic gauge, the
constraints and the gauge condition propagate in time automatically, as long they are satisfied
on the initial data. Indeed, in the harmonic gauge the linearized Einstein equations system is
purely a hyperbolic system of equations, which in turn implies propagation of regularity and
causality, from standard theory of hyperbolic PDE. We take advantage of the global regularity
in harmonic gauge including at the axes and infinity in harmonic gauge. We would like to
point out that, when we make such gauge transformations, we can lose regularity but in our
construction we show in fact that there exists a (C∞-)diffeomorphism. In our work, we also
construct a variational formulation of the linearized Einstein equations in harmonic gauge, which
may be of independent interest and fits well with the theme of our approach.
In view of such conceptual and technical subtleties, the ‘safest’ way to prove that the positive-
energy HReg we constructed is strictly conserved is to use the argument that the time-derivative
of the energy HReg vanishes dynamically in time. In other words, as we have already shown
that the time derivative of the energy density eReg is a pure spatial divergence (a fact that is
associated to (N, 0)T being the kernel of the adjoint of the dimensionally reduced constraint map)(
J t
)Reg
:= eReg,(
Jb
)Reg
:= N2e−2ν
(
qab0 p′A∂aU
′A)+ U ′ALN ′
(
Nµ̄q0q
ab
0 hAB∂bU
B
)
× LN ′(N)
(
2µ̄q0q
ab
0 ∂aν
′)+ 2LN ′ν ′µ̄qq
ab∂aN − 2N ′bµ̄qq
bc∂aν
′∂cN,
where JReg is a divergence-free vector field density, we need to prove that the fluxes of Jb at the
boundaries vanish dynamically in time.
Using Fredholm theory and that transverse-traceless 2-tensors vanish3 for our geometry and
topology, we reduce the elliptic operators into conformal Killing operators. Furthermore, benefit-
ing from conformal invariance of these operators, we reduce the elliptic operators into (tensorial)
Poisson equations. We are then able to obtain the desired decay rates of the fields using both
the fundamental solution and Fourier methods.
In the fundamental solution approach, we construct regularity and decay rate in our orbit
space geometry, using the method of images in such a way that the total ‘charge’ of the source is
zero. In the Fourier methods, the regularity conditions imply that the frequency corresponding
to the logarithmic blow up of the solution does not occur. In both methods, we recover a faster
decay rate than for the usual Poisson equation in two dimensions. This faster decay rate, in
contrast to the translational symmetric dimensional reduction, plays a fundamental role in our
work.
3This is in contrast with the (2+ 1)-dimensional relativistic Teichmüller theory (for, e.g., Σ is compact and of
genus > 1), where transverse-traceless tensors play an important role (see [60]).
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 35
The diffeomorphism invariance of the Einstein equations allows us the gauge-freedom. In the
3 + 1 (vis-á-vis 2+1) picture, this gauge freedom is reflected in terms of the ‘lapse’ and the ‘shift’
vector field. After fixing the gauge, we estimate the behaviour of gauge dependent quantities
in the asymptotic regions where the perturbations are assumed to be pure gauge, starting from
the independent wave map phase space variables and then moving on to dependent phase space
variables.
Following the estimates on the fundamental phase space variables, we estimate the fluxes of
each of the quantities in Jb term by term and we prove that regularity holds and that they
dynamically vanish at all the three boundaries Γ, ι0 and H+, including at the corner Γ ∩H+.
In establishing the boundary behaviour of these terms, we pay special attention to the quan-
tity ν that is related to a conformal factor. We note that ν ′ does not transform like a scalar.
This result may be of independent interest in conformal geometry. Secondly, we also establish
that an integral quantity Y0(H) that vanishes for all times. These two results are crucial in
resolving the regularity issues on the axes and at the corners.
In the current work, in keeping with the spirit of part I, we pay special attention to the
covariance of our analysis with respect to the target metric and not rely on a specific gauge on
the target. Apart from the aesthetics, this can be used as a basis for studying the stability of
higher (n+ 1) dimensional black holes with toroidal symmetry Tn−2.
The problem of stability of Kerr black hole spacetimes is currently an area of very active
research among several groups worldwide. In this connection, there are several important and
remarkable works (see, e.g., [1, 2, 3, 16, 19, 40, 48, 49, 83]). Furthermore, there have been
recent advances in the nonlinear black hole stability problem. In [18], nonlinear stability of
Schwarzschild black holes was announced. In [32], the full proof of slowly rotating black holes
is announced (see also [31, 46]). In [17], Dafermos–Holzegel–Rodnianski have established linear
stability of Schwarzschild black hole spacetimes for gravitational perturbations, where a positive-
definite energy played a fundamental role (see [43]). A positive-definite energy functional for
both even and odd gravitational perturbations of Schwarzschild black holes was constructed
by Moncrief [53] using mode decomposition. We should point out that even in the case of
Schwarzschild black holes, which do not contain the ergo-region, establishing the existence of
a positive-definite energy functional for gravitational is non-trivial.
In the case of axial symmetry, wave map behaviour in Kerr spaces has been studied in [45].
As we already pointed out, a positive-definite energy was first constructed by Dain–de Aus-
tria [22] for extremal Kerr black holes (|a| = M) using Brill mass formula [8, 21].
These works (except [22]) are dedicated to the stability of Kerr black hole spacetimes with
‘small’ or ‘very small’ angular momentum (|a| ≪ M). Relatively less is understood about the
stability of Kerr black hole spacetimes for large, but subextremal angular momentum |a| < M .
This is mainly attributed due to the effects of the ergo-region that always surrounds a Kerr
black hole spacetime with a non-vanishing angular momentum.
In the case of the large and sub-extremal angular-momentum of Kerr black hole spacetimes
(|a| < M), the effects of the ergo-region become even more subtle and counterbalancing its
effects to obtain uniform-boundedness and decay of propagating fields is even more difficult
from a PDE perspective. The decay of a linear wave equation on Kerr black hole spacetimes
with |a| is studied in the remarkable works [20, 24, 25, 27].
We expect that our results will be useful to fill this gap for Maxwell and gravitational (i.e.,
Einsteinian) perturbations. Even among the methods for large |a| < M , our approach is different
in the sense that the (Hamiltonian) flow of our phase space variables is restricted to positive-
definite energy surfaces. Thus, we have a relation (equality) between the energy at different
time levels without the need for a spacetime ‘bulk’ integral or Morawetz estimate.
The more general class of 3 + 1 Lorentzian spacetimes is the Kerr–Newman family of space-
times which is a solution of coupled 3 + 1 Einstein–Maxwell equations for general relativity.
36 N. Gudapati
In a series of classical works [54, 55, 56], the stability of Reissner–Nordström spacetimes is stud-
ied for the entire sub-extremal range |Q| < M . Indeed, the fact that the Hamiltonian stability
results of Reissner–Nordström spacetimes hold for the full sub-extremal range was an early in-
dication that the current results and the Kerr–Newman results [61] are feasible. In view of the
rigidity of the Reissner–Nordström spacetimes, ‘non-trivial’ perturbations of the RN spacetimes
belong to the Kerr–Newman family of black hole spacetimes.
We expect that the results in [61] shall allow us to venture in this direction. Equivalent
results for the stability of Kerr–Newman–de Sitter spacetimes (|a|, |Q| < M), which has different
asymptotics and gauge issues compared to the Kerr–Newman problem in [61], is a work in
progress (cf. [37, 38] for now).
Hollands and Wald [42] have developed a notion of canonical energy, which was later extended
by Prabhu–Wald [62], where they showed that if the energy is not positive-definite for axisym-
metric perturbations of Kerr black hole spacetimes, it would blow up at later times. This work,
based on the 2+1 Einstein-wave map formalism and dimensional reduction in Weyl–Papapetrou
gauge, confirms their criterion for axisymmetric stability.
6 Global existence and propagation of regularity
Suppose
(
M̄, ḡ
)
is a Lorentzian spacetime, then consider the Einstein–Hilbert action on
(
M̄, ḡ
)
,
SH[ḡ] :=
∫
R̄ḡµ̄ḡ.
Consider a curve γs : [0, 1] → C(ḡ) where C(ḡ) is a space of smooth Lorentzian metrics ḡ. We
shall use the following notation ḡ′ := Dγs
· ḡ and define
Dγs
· SH
[
ḡ′, ḡ
]
:=
∫ (
−Ric(g)µν ḡ′ +
1
2
Rgg
µνg′
µν +
(ḡ)∇µ(ḡ)∇ν ḡ′
µν − (ḡ)∇2tr
(
ḡ′))µ̄ḡ.
Now, consider another analogous curve γλ : [0, 1] → C(ḡ) and let us define ḡ′′ := D2
γsγλ
· ḡ, we
then have for the Kerr background metric. The functional (6.1) can be simplified as follows:
D2
γλγs
· SH
[
ḡ′′, ḡ′, ḡ
]
:=
∫ (
ḡµν (ḡ)∇γ(ḡ)∇δḡ′
γδ − ḡµν (ḡ)∇γ(ḡ)∇γtr
(
ḡ′)− (ḡ)∇γ
(ḡ)∇µḡ′γν
− (ḡ)∇γ
(ḡ)∇ν ḡ′γµ + (ḡ)∇µ(ḡ)∇νtr
(
ḡ′)
+ (ḡ)∇α(ḡ)∇αḡ
′µν)1
2
ḡ′
µν µ̄ḡ
+ (ḡ′)∇µ
(
(ḡ)∇ν ḡ′
µν − ḡγδ(ḡ)∇µḡ
′
γδ
)
µ̄ḡ
+ (ḡ)∇µ
(
(ḡ)∇ν ḡ′
µν − ḡγδ(ḡ)∇µḡ
′
γδ
)
µ̄ḡ
+ (ḡ)∇µ
(
(ḡ)∇ν ḡ′
µν − ḡγδ(ḡ)∇µḡ
′
γδ
)(1
2
µ̄ḡḡ
µν ḡ′
µν
)
. (6.1)
We are interested in a variational principle, so that we get the Euler–Lagrangian field equations
for the linearized Einstein equations around the Kerr black hole spacetimes. The functional (6.1)
can be simplified as follows:
SLEE
[
ḡ′, ḡ
]
=
∫
1
2
(
(ḡ)∇αḡ
′
µν
(ḡ)∇µg′αν + (ḡ)∇αḡ
′
µν
(ḡ)∇ν ḡ′αµ − (ḡ)∇µg′
µν
(ḡ)∇νtr
(
g′) (6.2)
− (ḡ)∇αg
′
µν
(ḡ)∇αg′µν − (ḡ)∇αtr
(
g′)(ḡ)∇βg′
αβ + (ḡ)∇αtr
(
g′)(ḡ)∇αtr
(
g′))µ̄ḡ.
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 37
Suppose D is the local continuous group of diffeomorphisms generated by the vector field Ȳ .
It follows that the 2-tensor(
ḡ′)
αβ
= (LȲ ḡ)αβ = (g)∇αȲβ + (g)∇βȲα, α, β = 0, 1, 2, 3,
is a critical point of the variational principle (6.2), corresponding to the pure-gauge perturbations
of the Kerr metric. It may be noted that the gauge transformations are abelian. In general, the
computation of the Euler–Lagrange field equations corresponding to the variation principle (6.2)
involves the terms listed below.
� The terms (ḡ)∇ḡ′′
µν
(ḡ)∇µḡ′αν + (ḡ)∇αḡ
′
µν
(ḡ)∇µḡ′′αν can be transformed as
(ḡ)∇αḡ
′′
µν
(ḡ)∇µḡ′αν = (ḡ)∇α
(
ḡ′′(ḡ)∇µḡ′αν)− ḡ′′
µν
(ḡ)∇α
(ḡ)∇µḡ′αν ,
(ḡ)∇αḡ
′
µν
(ḡ)∇µḡ′′αν = (ḡ)∇µ
(
ḡ′′αν (ḡ)∇αḡ
′
νν
)
− (ḡ)∇µ(ḡ)∇aḡ
′
µν ,
� likewise, the terms (ḡ)∇αḡ
′′
µν
(ḡ)∇ν ḡ′αµ + (ḡ)∇αḡ
′
µν
8(ḡ)∇ν ḡ′′αµ can be rewritten as
(ḡ)∇αḡ
′′
αν
(ḡ)∇ν ḡ′αν = (ḡ)∇a
(
(ḡ)∇ḡ′
µν
(ḡ)∇ν ḡ′αµ)− ḡ′′
µν
(ḡ)∇α
(ḡ)∇ν ḡ′αµ,
(ḡ)∇aḡ
′
µν
(ḡ)∇ν ḡ′′αν = (ḡ)∇ν
(
ḡ′′αν (ḡ)∇αḡ
′
µν
)
− ḡ′′(ḡ)∇ν (ḡ)∇αḡ
′
µν ,
� the terms −(ḡ)∇µḡ′′
µν
(ḡ)∇νtrḡ′ − (ḡ)∇µḡ′
µν
(ḡ)∇νtrḡ′′, can be transformed conveniently as
− (ḡ)∇µḡ′′
µν
(ḡ)∇νtrḡ′ = −(ḡ)∇µ
(
ḡ′′
µν
(ḡ)∇νtrḡ′)+ (ḡ)∇′′
µν
(ḡ)∇µ
(
(ḡ)∇νtrḡ′),
− (ḡ)∇µḡ′
µν
(ḡ)∇νtrḡ′′ = −(ḡ)∇ν
(
trḡ′′(ḡ)∇µḡ′
µν
)
+ trḡ′′(ḡ)∇ν (ḡ)∇µḡ′
µν .
� Finally, the terms −(ḡ)∇aḡ
′′µν(ḡ)∇αḡ′µν and (ḡ)∇αtrḡ′′(ḡ)∇αtrḡ
′ can be transformed as
− (ḡ)∇αḡ
′′
µν
(ḡ)∇αḡ′µν = −(ḡ)∇a
(
ḡ′′(ḡ)∇αḡ′µν)+ ḡ′′
µν
(ḡ)∇α
(ḡ)∇αḡ′µν
and
(ḡ)∇αtrḡ′′(ḡ)∇αtrḡ
′ = (ḡ)∇α
(
trḡ′′(ḡ)∇αtrḡ
′)− trḡ′′(ḡ)∇α(ḡ)∇αtrḡ
′
respectively.
Assembling the terms from above, we obtain the Euler–Lagrange equations for the variational
principle (6.2) as
(ḡ)∇γ(ḡ)∇µḡ
′
γν +
(ḡ)∇γ(ḡ)∇ν ḡγµ − (ḡ)∇µ
(ḡ)∇νtr
(
ḡ′)− (ḡ)∇γ(ḡ)∇γ ḡ
′
µν
− ḡµν
(ḡ)∇γ(ḡ)∇δḡ′
γδ + ḡµν
(ḡ)∇γ(ḡ)∇γtrḡ
′ = 0,
(
M̄, ḡ
)
,
which are the field equations for the linearized Einstein equations around the Kerr background
spacetime.
Harmonic coordinates
Consider a coordinate system xα
x̄α :
(
M̄, ḡ
)
→ (M, ḡ), α = 0, 1, 2, 3.
It follows that the coordinate functions xα satisfy the equations
□ḡx
α + (ḡ)Γα
βγ ḡ
βγ = 0,
(
M̄, ḡ
)
, α, β, γ = 0, 1, 2, 3. (6.3)
38 N. Gudapati
We would like to point out that this equation (6.3) is reminiscent of the wave map equations.
With the harmonic gauge condition □gx
α = 0, it follows from (6.3) that (ḡ)Γα
βγ ḡ
βγ = 0. We de-
fine an equivalent gauge condition in the perturbative theory using Dγs
· (ḡ)Γα
βγ ḡ
βγ , i.e.,
(ḡ)∇µḡ′
µν =
1
2
∂νtrḡ
′ (6.4)
and if we define g̃′ := ḡ′ − 1
2 ḡ
αβḡ′
αβ, the ‘trace-reversed’ metric perturbation ḡ′, then this con-
dition can be compactly represented as
(ḡ)∇αg̃
αβ = 0.
If we consider the variational principle for the linearized gravity in the Einstein equations for
general relativity and transform using harmonic coordinates, we get
SHG
[
ḡ′, ḡ
]
=
∫
1
2
(
(ḡ)∇αḡ
′
µν
(ḡ)∇µg′αν + (ḡ)∇αḡ
′
µν
(ḡ)∇ν ḡ′αµ − (ḡ)∇αg
′
µν
(ḡ)∇αg′µν)µ̄ḡ.
It may be noted that the structure of the variational functional, is closely related to the
deformed wave map action that we considered previously, e.g., in [35]. The quantities in the
variational principle can be transformed, using the identities
(ḡ)∇γ
(ḡ)∇µḡ
′
αν − (ḡ)∇µ
(ḡ)∇γ ḡ
′
αν = −Riem(ḡ)σαγµḡ
′
σν − Riemσ
νγµḡ
′
ασ,
(ḡ)∇γ
(ḡ)∇ν ḡ
′
αµ − (ḡ)∇ν
(ḡ)∇γ ḡ
′
αµ = −Riem(ḡ)σαγν ḡ
′
σµ − Riemσ
µγν ḡ
′
ασ
then the Euler–Lagrange linearized Einstein equations in harmonic coordinates are
(ḡ)∇γ(ḡ)∇γ ḡ
′
µν +Riem(ḡ)σ α
µ ν ḡ
′
ασ +Riem(ḡ)σ α
ν µḡ
′
ασ = 0,
(
M̄, ḡ
)
. (6.5)
A similar equation is satisfied by the trace-reversed metric g̃,
(ḡ)∇γ(ḡ)∇γ g̃
′
µν +Riem(ḡ)σ α
µ ν g̃
′
ασ +Riem(g̃)σ α
ν µg̃
′
ασ = 0,
(
M̄, ḡ
)
. (6.6)
In the initial value framework, we need to solve the field equations (6.5) or (6.6) together with
the gauge determining equations (6.4). Now consider the divergence of the linearized Einstein
tensor,
1
2
(ḡ)∇µ
(
(ḡ)∇γ(ḡ)∇µḡ
′
γν +
(ḡ)∇γ(ḡ)∇ν ḡ
′
γµ − (ḡ)∇µ
(ḡ)∇νtrḡ
′ − (ḡ)∇γ(ḡ)∇γ ḡ
′
µν
− ḡµν
(ḡ)∇γ(ḡ)∇δḡ′
γδ + ḡµν
(ḡ)∇γ(ḡ)∇γtrḡ
′).
If we define the gauge-fixing quantity, Fν = (ḡ)∇µg̃′
µν , we can construct a propagation equa-
tion for Fν as
(ḡ)∇γ(ḡ)∇γFν ≡ (ḡ)∇µ
(
(ḡ)∇γ(ḡ)∇γ ḡ
′
µν +Riem(ḡ)σ α
µ ν ḡ
′
ασ +Riem(ḡ)σ α
ν µḡ
′
ασ
)
= 0,
(
M̄, ḡ
)
.
If we consider the initial value problem
(ḡ)∇γ(ḡ)∇γFν = 0,
(
M̄, ḡ
)
,
Fν |Σ0
= 0, ∂t⃗Fν |Σ0
= 0,
(
Σ0, q̄0
)
,
it is straightforward to show that Fν and ∂t⃗Fν ≡ 0 for all times in the domain of outer com-
munications of the Kerr black hole spacetime. In other words, the gauge condition (6.4) is
propagated for all times if it holds on the initial data and the linearized Einstein equations
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 39
hold in the harmonic gauge. Analogously, consider the propagation of the constraint equations,
defined as
H ′ := G′(n, n)
= nµnν
(
(ḡ)∇γ(ḡ)∇µḡ
′
γν +
(ḡ)∇γ(ḡ)∇ν ḡ
′
γµ − (ḡ)∇µ
(ḡ)∇νtrḡ
′ − (ḡ)∇γ(ḡ)∇γ ḡ
′
µν
− ḡµν
(ḡ)∇γ(ḡ)∇δḡ′
γδ + ḡµν
(ḡ)∇γ(ḡ)∇γtrḡ
′)
the expression for the propagation of the Hamiltonian constraint H ′ follows from the transfor-
mations analogous to the above
= nµnν
(
(ḡ)∇µ
(
(ḡ)∇γ ḡγν −
1
2
(ḡ)∇νtrḡ
′
)
+ (ḡ)∇ν
(
(ḡ)∇γ ḡγµ − 1
2
(ḡ)∇µtrḡ
′
))
− nµnν
(
(ḡ)∇γ(ḡ)∇γ ḡ
′
µν +Riem(ḡ)σ α
µ ν ḡ
′
ασ +Riem(ḡ)σ α
ν µḡ
′
ασ
)
+ nνnν
(ḡ)∇γ
(
(ḡ)∇γ(ḡ)∇δ − (ḡ)∇γ(ḡ)∇γtrḡ
′)
after imposing the linearized Einstien field equations in the harmonic gauge
=nµnν
(
(ḡ)∇µFν +
(ḡ)∇νFµ
)
+ nνnν
(ḡ)∇γFγ = 0
for all times, due to the Harmonic gauge propagation. Likewise, for the momentum constraint,
after plugging in the relevant formulas
H ′
i := G′(n,X)
= nµXj
(
(ḡ)∇γ(ḡ)∇µḡ
′
γj +
(ḡ)∇γ(ḡ)∇j ḡ
′
γµ − (ḡ)∇µ
(ḡ)∇jtrḡ
′ − (ḡ)∇γ(ḡ)∇γ ḡ
′
µj
− ḡµj
(ḡ)∇γ(ḡ)∇δḡ′
γδ + ḡµj
(ḡ)∇γ(ḡ)∇γtrḡ
′)
= nµXj
(
(ḡ)∇µ
(
(ḡ)∇γ ḡγj −
1
2
(ḡ)∇νtrḡ
′
)
+ (ḡ)∇j
(
(ḡ)∇γ ḡγµ − 1
2
(ḡ)∇µtrḡ
′
))
− nµXj
(
(ḡ)∇γ(ḡ)∇γ ḡ
′
µj +Riem(ḡ)σ α
µ j ḡ
′
ασ +Riem(ḡ)σ α
j µḡ
′
ασ
)
+ nµXj ḡµj
(ḡ)∇γ
(
(ḡ)∇γ(ḡ)∇δ − (ḡ)∇γ(ḡ)∇γtrḡ
′)
= nµXj
(
(ḡ)∇µFj +
(ḡ)∇jFµ
)
+ nνXj ḡνj
(ḡ)∇γFγ = 0, Xi ∈ T (Σ), i = 1, 2, 3.
It follows that, in the harmonic gauge, the constraints are automatically propagated as long
as they are satisfied on the initial data, analogous to the propagation of the harmonic gauge
condition. This is an analogous statement for the nonlinear theory developed in the classic work
of Choquet-Bruhat.
Let us now discuss the degrees of freedom of linearized gravity. The number of independent
degrees of freedom, modulo the gauge degrees of freedom, is 6. We have shown that the constraint
propagation in time follows from the harmonic gauge condition. As a consequence, it follows
that these remaining degrees of freedom are all independent and unconstrained.
It may be noted that the system of equations for the linearized Einstein equations is purely
a hyperbolic partial differential equation system, from which it follows that the evolution of the
data is causal.
Proposition 6.1. Suppose
(
Σ′, q̄′
)
is the linearized initial data for the linearized Einstein equa-
tions, satisfying the constraint equations, then
(1) The linearized Einstein equations in harmonic gauge is purely a hyperbolic differential
equation system, with the independent degrees of freedom being dynamically unconstrained.
40 N. Gudapati
(2) The future (and past) development of the linearized initial data in harmonic gauge is
globally regular and globally hyperbolic.
Let us make a couple of comments about the aforementioned global existence theorem. As we
already discussed, the Weyl–Papapetrou gauge offers significant benefits in terms of geometry
and topology, that allows us to construct a positive-definite energy in the first place, but presents
(gauge-related) causality and regularity issues at the boundaries.
On the other hand, as we already discussed above, the global development of the linearized
Einstein equations in the harmonic gauge is untroubled by the (gauge-related) causality and
regularity issues at the axes (Γ), infinity
(
ῑ0
)
and at the corners Γ ∩ H+. In other words, the
global development of the linearized Einstein equations from regular initial data is regular (C∞)
for all times, including at the pathologies like axes, infinity and at the corner of axes and horizon.
For example, we have
∂n⃗A = 0,
for a scalar, and
A⊥ = 0, ∂nA
∥ = 0
for a vector A, near the axes, globally in time. In our work, our approach is to combine these
two gauges and take advantage of benefits offered in each.
Recall that the metric q in the orbit space can be expressed in harmonic gauge as q = e2νq0,
where q0 is the flat metric.
The preservation of the flatness condition is
D ·R′
q0
= µ̄q0
(
(q0)∇a(q0)∇b
(
q′0
)
ab
− (q0)∇a(q0)∇aq
cd
0
(
q′0
)
cd
)
= 0, on each Σ. (6.7)
The tensor q′0 can be decomposed as (q′0)ab =
(
q′TT
0
)
ab
+ (q0)∇aYb +
(q0)∇aYa +
1
2(q0)tr q
′
0. In
particular, it may be noted that the pure gauge perturbations (q′0)ab =
(q0)∇aYb +
(q0)∇bYa sat-
isfy the condition (6.7). The perturbed metric (q′0) has the following regularity conditions on
the axes Γ, expressed in (R, θ) coordinates [64]
∂θq
′
0(∂R, ∂R) = 0, ∂θq
′
0(∂θ, ∂θ) = 0, q′0(∂θ, ∂R) = 0, at the axes Γ.
Our construction can also be used to study stability problem of Kerr black hole spacetimes
in harmonic gauge. However, in principle, we can use our energy to study the stability problem
in any preferred gauge. If we are given a solution (M ′, ḡ′) in any gauge, we can transform the
solution to a harmonic gauge by solving the linearized wave map equations xα : M̄ → M̄ , for
which it can be proven that they admit smooth solutions for all times. Subsequently, we can
transform our the solution to the Weyl–Papapetrou gauge using a (C∞-)diffeomorphism, by
making use of the abelian nature of gauge-transformations.
7 Canonical phase space variables and Lagrange multipliers
in the Weyl–Papapetrou gauge
Suppose, the group SO(2) acts on the 3+1 Lorentzian spacetime
(
M̄, ḡ
)
such that the orbits of
the group SO(2) are closed and the group action has a nonempty fixed point set, denoted by Γ.
These conditions are satisfied by the Kerr metric
(
M̄, ḡ
)
ḡ := −Σ−1
(
∆− a2 sin2 θ
)
dt2 − 4aΣ−1 sin2 θmrdtdϕ
+Σ−1
((
r2 + a2
)2 −∆a2 sin2 θ
)
sin2 θdϕ2 +∆−1Σdr2 +Σdθ2,
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 41
where
Σ := r2 + a2 cos2 θ and ∆ = r2 − 2Mr + a2,
which can be represented as
ḡ = |Φ|
(
∆sin−2 θdt2 +R−2 sin2 θ
((
r2 + a2
)2 − a2∆sin2 θ
)(
dρ2 + dz2
))
+ |Φ|
(
dϕ2 − 2mar
((
r2 + a2
)2 − a2∆sin2 θ
)−1)2
,
where
ρ = R sin θ, z = R cos θ, R := 2
(
r −m+
√
∆
)
, θ ∈ [0, π],
|Φ| = sin2 θ
r2 + a2 cos2 θ
((
r2 + a2
)2 − a2∆sin2 θ
)
,
qab = sin2 θ
(
r2 + a2
)2 − a2∆sin2 θ
R2
.
Now consider the conjugate harmonic functions (ρ̄, z̄) such that
ρ̄ := ρ
(
1−
(
m2 − a2
)
4
(
ρ2 + z2
)), z̄ := z
(
1 +
m2 − a2
4
(
ρ2 + z2
)),
so that the Jacobian is
J :=
1 + m2−a2
4(ρ2+z2)
(
2ρ2
ρ2+z2
− 1
)
ρ(1 + (m2−a2)z
2(ρ2+z2)2
z
(
1− (m2−a2)ρ
2(ρ2+z2)2
)
1 + m2−a2
4(ρ2+z2)
(
1− 2z2
ρ2+z2
) .
In these coordinates, the Kerr black hole horizon H+ corresponds to a ‘cut’ on the {ρ̄ = 0}
curve and its complement on the {ρ̄ = 0} curve corresponds to the union of two axes, Γ. This
coordinate system and the (ρ, z) coordinate system in the extremal case are the ones originally
used by Carter [10].
Let us briefly recall our construction. In the Weyl–Papapetrou gauge for the Einstein equa-
tions, we can reduce the Einstein–Hilbert action into the reduced Einstein-wave map system:∫ (
Rg − hAB(U)gαβ∂αU
A∂βU
B
)
µ̄g.
It is straightforward to verify that the Kerr metric is a critical point of the variational func-
tional. In the Hamiltonian version of the dimensional reduction, we also encounter the interme-
diate phase space XMax:
XMax :=
{
Ai, E i
}
so that the Hamiltonian and momentum constraints for the combined phase space{
(q,π), (A, E),
(
|Φ|1/2, p
)}
are
H := µ̄−1
q
(
∥π∥2q − tr(π)2
)
+
1
8
p2 +
1
2
|Φ|−2EaEa
+ µ̄q
(
−Rq +
1
2
qab∂a log |Φ|∂b log |Φ|+
1
4
qabqbd∂[bAa]∂[dAc]
)
,
42 N. Gudapati
Ha = −2(q)∇bπ
b
a +
1
2
p∂a log |Φ|+ Eb
(
∂[aAb]
)
with the Lagrange multipliers
{N,Na, A0}.
Subsequently, the phase space X was introduced
X := {(q,π), (|Φ|, p), (ω, r)},
which resulted in the Hamiltonian and momentum constraint equations:
H = µ̄−1
q
(
|π|2q − tr(π)2 +
1
2
pAp
A
)
+ µ̄q
(
−Rq +
1
2
hABq
ab∂aU
A∂bU
B
)
,
Ha = −2(q)∇bπ
b
a + pA∂aU
A,
where the Lagrange multipliers are now
{N,Na}. (7.1)
The fact that the Lagrange multiplier set (7.1) is now simplified is due to the special topological
structure of the orbit space M of the Kerr metric. This plays a convenient role in the properties
of the adjoint of the dimensionally reduced constraint map. The main result of Section 4 was
to obtain a positive-definite energy functional for the linear perturbative theory of Kerr black
hole spacetimes within the assumption of axial symmetry, which allows the aforementioned
dimensional reduction. The regularized Hamiltonian energy functional is
HReg :=
∫
Σ
eRegd2x,
where
eReg := Nµ̄−1
q0 e
−2ν
(
∥ϱ′∥2q0 +
1
2
p′Ap
′A
)
− 1
2
Ne2ν µ̄q0τ
′2
+
1
2
Nµ̄q0q
ab
0 hAB(U)(h)∇aU
′A(h)∇bU
′B
− 1
2
Nµ̄q0q
ab
0 hAEU
′A(h)RE
BCD∂aU
B∂bU
CU ′D.
Let us now formally define the Weyl–Papapatrou gauge.
Definition 7.1. Suppose
(
M̄, ḡ
)
is a Lorentzian spacetime such that M̄ admits the ADM
decomposition M̄ = Σ × R and the group SO(2) acts on Σ through isometries such that the
fixed point set is nonempty and the orbits of its action on Σ are closed. Then we define ḡ to be
in Weyl–Papapetrou form if
1. ḡ admits the decomposition
ḡ = |Φ|−1g + |Φ|A2, (7.2)
where the 1-form A in M̄ is defined as
A = dϕ+Aνdx
ν , and
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 43
∂ϕ is the (Killing) vector field corresponding to the SO(2) symmetry of M̄ the scalar
|Φ| = ḡαβ(∂ϕ)
α(∂ϕ)
β is the spacetime norm of the Killing vector ∂ϕ. g, A, |Φ| are indepen-
dent of ϕ, i.e., L∂ϕg = ∂ϕA = 0. It may be noted that g is a metric of Lorentzian signature
in the orbit space M := M̄/SO(2) such that it further admits the ADM decomposition
M = Σ× R, where now Σ = Σ/SO(2):
g = −N2dt2 + qab
(
dxa +Nadt
)
⊗
(
dxb +N bdt
)
and q is the induced metric of Σ.
2. There exists a coordinate basis e1 and e2 in (Σ, q) such that
q
(
e1, e1
)
− q
(
e2, e2
)
= 0, and q
(
e1, e2
)
= 0, on each Σ
and
q = e2νq0, where q0 is the flat 2-metric.
The metric ḡ represented in the above coordinate conditions is referred to as in ‘Weyl–
Papapetrou’ form. We would like to remark that the representation of the metric ḡ in terms of
a Weyl–Papapetrou form is not unique. We would also like to emphasize that on the fixed point
set of the SO(2) action on M̄ we have |Φ| → 0
(
|Φ|−1 → ∞
)
and at the outer asymptotic end
of M̄ we have |Φ| → ∞
(
and |Φ|−1 → 0
)
. Counterbalancing these effects to obtain well-defined,
convergent, gauge-independent quantities, in the context of the initial value problem, is one of
the main aspects in our work. We shall apply this construction for the perturbative theory.
We shall pay particular attention to the implications of the perturbed Weyl–Papapetrou
gauge on Σ and the basis {e1, e2}. Now then, we have the following conditions on the gauge-
transformed perturbed metric ḡ′ as follows:
LY ḡ(e1, e1)−
1
2
q0(e1, e1)q
ab
0 (LY ḡ)ab = −
(
ḡ′(e1, e1)−
1
2
q0(e1, e1)q
ab
0 ḡ′ab
)
,
LY ḡ(e1, e2) = −ḡ′(e1, e2),
LY ḡ(e2, e2)−
1
2
q0(e1, e2)q
ab
0 (LY ḡ)ab = −
(
ḡ′(e1, e2)−
1
2
q0(e1, e1)q
ab
0 ḡ′ab
)
,
which can further be expressed as, using the form of (7.2),
LY q0(e1, e1)−
1
2
q0(e1, e1)q
ab
0 (LY q0)ab = −e−2ν |Φ|
(
ḡ′(e1, e1)−
1
2
q0(e1, e1)q
ab
0 ḡ′ab
)
,
LY q0(e1, e2) = −e−2ν |Φ|ḡ′(e1, e2),
LY q0(e2, e2)−
1
2
q0(e1, e2)q
ab
0 (LY q0)ab = −e−2ν |Φ|
(
ḡ′(e1, e2)−
1
2
q0(e1, e1)q
ab
0 ḡ′ab
)
.
The above system can be expressed compactly in a covariant form as follows
(LY q0)ab −
1
2
(q0)ab (q0)
cd(LY q0)cd = −e−2ν |Φ|
(
ḡab −
1
2
(q0)ab(q0)
cdḡ′cd
)
.
We have established the following.
Lemma 7.2. Suppose the perturbations of Einstein’s equations in axial symmetry are compactly
supported in the harmonic gauge then
44 N. Gudapati
(1)
LY (µ̄qq)
ab = LY (µ̄q0q0)
ab = µ̄qq
acqbd|Φ|
(
ḡ′cd −
1
2
qcdq
ef ḡ′ef
)
(7.3)
the gauge transformation vector field Y ∈ T (Σ) is the projection of the spacetime gauge
transformation vector field Ȳ .
(2) Suppose ḡ′ is such that it is compactly supported away from the horizon H+ and the spatial
infinity ι0, then Y is a conformal Killing vector field in (Σ, q), i.e., CK(Y, q) = 0 in the
asymptotic regions (i.e., in the complement of the support of ḡ′, Σ \ Supp(ḡ)|Σ).
Now, for estimates, let us choose a gauge for the dimensionally reduced Cauchy hypersur-
face (Σ, q). If we choose the polar coordinates (R, θ), it follows from the condition (7.3) that:
∂θY
θ = R∂R
Y R
R
+
1
R
(
µ̄qq
aRqbR|Φ|
(
ḡ′ab −
1
2
qabq
cdḡ′cd
))
, (7.4a)
∂RY
θ = − 1
R
∂θ
Y R
R
− 1
R
(
µ̄qq
aRqbθ|Φ|
(
ḡ′ab −
1
2
qabq
cdḡ′cd
))
. (7.4b)
It may be noted that above system of differential equations is an overdetermined system.
It follows from the Picard theorem and the Frobenius theorem that the necessary and sufficient
conditions for the existence of the solutions is the compatibility condition:
1
R
∂R
(
R∂R
Y R
R
)
+
1
R2
∂2
θ
Y R
R
= − 1
R
∂R
(
1
R
µ̄qq
aRqbR|Φ|
(
ḡ′ab −
1
2
qabq
cdḡ′cd
))
− 1
R2
∂θ
(
µ̄qq
aRqbθ|Φ|
(
ḡ′ab −
1
2
qabq
cdḡ′cd
))
.
In formal terms, this corresponds to vanishing of the commutator of the vector fields corre-
sponding to the differential equations (7.4). It may be noted that the compatibility condition
fortuitously turns out to be a Poisson equation for Y R
R , for the Laplacian ∆0
∆0 =
1
R
∂
∂R
(
R
∂
∂R
)
+
1
R2
∂2
∂θ2
.
in the R, θ gauge. Likewise, equations (7.4), which are equivalent to equation (7.3) in the
Lemma 7.2, can be transformed into an overdetermined system of equations for Y R
R
∂θ
Y R
R
= −R∂RY
θ −
(
µ̄qq
aRqbθ|Φ|
(
ḡ′ab −
1
2
qabq
cdḡ′cd
))
,
∂R
Y R
R
=
1
R
∂θY
θ − 1
R2
(
µ̄qq
aRqbR|Φ|
(
ḡ′ab −
1
2
qabq
cdḡ′cd
))
for which the compatibility condition is
1
R
∂R
(
R∂RY
θ
)
+
1
R2
∂2
θY
θ =
1
R3
∂θ
(
1
R
µ̄qq
aRqbR|Φ|
(
ḡ′ab −
1
2
qabq
cdḡ′cd
))
− 1
R
∂R
(
1
R
µ̄qq
aRqbθ|Φ|
(
ḡ′ab −
1
2
qabq
cdḡ′cd
))
,
which is again a Poisson equation for Y θ. For the reasons of regularity on the axes, we impose
the conditions [64]
Y θ = 0, ∂θY
R = 0, on the axes Γ.
In particular, we assume that the behaviour of Y θ ∼ sin θ and ∂θY
R ∼ sin θ close to the axes Γ.
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 45
Now consider the boundary value problem:
∆0
Y R
R
= − 1
R
∂R
(
1
R
µ̄qq
aRqbR|Φ|
(
ḡ′ab −
1
2
qabq
cdḡ′cd
))
− 1
R2
∂θ
(
µ̄qq
aRqbθ|Φ|
(
ḡ′ab −
1
2
qabq
ef ḡ′ef
))
on (Σ)
Y R
R
= Y R
H+ on
(
H+
)
(D− BVP)Y R
We solve the Dirichlet problem (D− BVP)Y R above with the method of images. Let us first
consider the Poisson equation
∆0u = F, in any Lipschitz domain (Σ).
From elliptic theory, it follows that if F ∈ C∞(Σ) for regular Dirichlet or Neumann boundary
data, then u ∈ C∞(Σ). Suppose Ku is the fundamental solution such that
∆0Ku = δ
(
x− x′
)
where δ is a Dirac-delta function with a Euclidean metric on Σ. Then consider the quantity,
∂a(u∂
aKu −Ku∂
au) = (∂au∂
aKu + u∆Ku)− (∂aKu∂
au+Ku∆u)
= u∆Ku −Ku∆u
and upon integration over the domain Σ, we get∫
Σ
∂a(u∂
aKu −Ku∂
au) =
∫
∂Σ
n · (u∂aKu −Ku∂
au)
=
∫
∂Σ
u∂nKu −Ku∂nu =
∫
Σ
uδ
(
x− x′
)
−KuF = u−
∫
Σ
KuF.
Therefore, the general representation formula for u is
u =
∫
Σ
KuF +
∫
∂Σ
u∂nKu −Ku∂nu. (7.5)
The formula (7.5) will be useful for us throughout our work, in different contexts. We can tailor
this general formula for both Dirichlet and Neumann boundary value problems. In the following,
we shall discuss two configurations that would be particularly relevant for us.
The orbit space Σ geometry of Kerr black hole spacetime resembles that of the complement
of a half-disk (with the boundary representing the horizon) in a half plane. We solve the
Dirichlet problem using the method of images. The regularity and compatibility conditions
for our problem imply that the ‘image’ is reflection antisymmetric. This applies for the image
charge as well as the Dirichlet data. Thus, with this picture, we have the complement of a full
disk in a full plane, with reflection (with respect to the axes) antisymmetric data at the disk. It
follows that the asymptotic decay rate for this problem is O
(
1
R
)
for this Dirichlet problem. This
decay rate can be independently verified using the separation of variables. It may be noted that
this decay rate is faster than that of the Poisson equation in a plane
(
i.e., 1
2π logR asymptotic
behaviour
)
. This faster decay rate plays a fundamental role in our problem.
Likewise, consider the Poisson equation with Neumann boundary conditions in the orbit
space Σ. The regularity at the axes implies that the Neumann data in the extended picture is
reflection anti-symmetric. We thus recover theO
(
1
R
)
decay rate for the solution with appropriate
decay conditions for the source function f , e.g.,
− 1
R2
∂θ
(
µ̄qq
aRqbθ|Φ|
(
ḡ′ab −
1
2
qabq
ef ḡ′ef
))
,
46 N. Gudapati
− 1
R
∂R
(
1
R
µ̄qq
acqbd|Φ|
(
ḡ′ − 1
2
qabq
ef ḡ′ef
))
is compactly supported (or with appropriate decay rate consistent with asymptotically flat
conditions) for our problem. It follows that the solution of the Dirichlet problem (D− BVP)Y R
decays as Y R
R ∼ 1
R asymptotically, for large R. It follows from analogous arguments that the
solutions for the boundary value problem
∆0Y
θ = − 1
R
∂R
(
1
R
µ̄qq
Rcqθd|Φ|
(
ḡ′ − 1
2
qabq
ef ḡ′ef
))
+
1
R3
∂θ
(
1
R
µ̄qq
RcqRd|Φ|
(
ḡ′ − 1
2
qabq
ef ḡ′ef
))
on (Σ)
Y θ = Y θ
H+ on
(
H+
)
(D− BVP)Y θ
are unique, regular (well-posed) and decay Y θ ∼ 1
R asymptotically for large R. On the other
hand, due to the regularity conditions Y admits the expansion:
Y θ =
∞∑
n=1
Y θ
n sin(nθ), Y R =
∞∑
n=0
Y R
n cos(nθ)
for the solutions of the conformal Killing vector Y . Likewise, for regularity reasons, the behaviour
of inhomogeneities in the boundary value problems mentioned above is restricted on the axes.
In particular, written in explicit form, they behave as follows
µ̄qq
RRqθθ|Φ|
(
ḡ′
(
∂R, ∂θ
))
∼ sin θ,
∂R
(
µ̄qq
RRqRR|Φ|
(
ḡ′(∂R, ∂R)−
1
2
qRRtrq
(
ḡ′
)))
∼ sin θ
close to the axes Γ. Now that we clarified the structure of the source terms, let us introduce the
notation, for (R, θ) coordinates
MRθ := µ̄qq
RRqθθ|Φ|ḡ′(∂R, ∂θ),
MRR := µ̄qq
RRqRR|Φ|
(
ḡ′(∂R, ∂R)−
1
2
qRRtrqḡ
′
)
.
As a consequence of the above arguments, they must admit a Fourier decomposition of
the form:
− 1
R
MRR =
∞∑
n=0
In(R, t) cosnθ,
−RMRθ =
∞∑
n=1
Jn(R, t) sinnθ.
Now, plugging in these decompositions in the first order equations (7.4), we get
∂RY
R
0 (R, t)− 1
R
Y R
0 (R, t) = I0(R, t),
which admits the solution,
Y R
0 (R, t) =
R
R+
Y R
0 (R+, t) +
∫ R
R+
I0(R
′, t)
R′ dR′
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 47
for the lowest frequency quantity Y R
0 . Now for higher frequencies,
∂RY
R
n (R, t)− 1
R
Y R
0 (R, t)− nY θ
n (R, t) = In(R, t),
R2∂RY
θ
n (R, t)− nY R
n (R, t) = Jn(R, t),
which again follow from the first-order equations respectively. These equations can be decou-
pled as
1
R
∂R
(
R∂RY
θ
n (R, t)
)
− n2Y θ
n (R, t)
R2
=
nIn(R, t)
R2
+
1
R
∂R
Jn(R, t)
R
. (7.6)
The characteristic equation admits two real roots and it may be noted that the fundamental
set of solutions is given by
fundamental solutions set for Y θ in (7.6) = {Rn, R−n}.
The corresponding Wronskian is 2n
R ̸= 0 ∀R ∈ (R+,∞), → 0 as R → ∞ and → 2n
R+
for R → R+.
It follows that Y θ = A(t)R−n +B(t)R−n.
We get the following asymptotic behaviour of the Wronskian near the horizon H+:
WH+
(
Y θ
n
)
=
2n
R+
,
WH+
(
Y R
n
R
)
=
2n
R+
as R → R+,
and near the outer asymptotic region,
Wῑ0
(
Y θ
n
)
=
2n
R
,
Wῑ0
(
Y R
n
R
)
=
2n
R
→ 0 as R → ∞,
so that, for the conformal Killing vector Y , the asymptotic behaviour is
Y θ
n (R, t) = Y θ
n (t)
(
Rn −R2n
+ R−n
)
, R near R+, n ≥ 1,
→ 0 as R → R+, (7.7a)
Y θ
n (R, t) = Y θ
n (t)R
−n for large R, n ≥ 1,
→ 0 as R → ∞, (7.7b)
and
Y R
n (R, t) = Y R
n (t)
(
Rn+1 −R2n
+ R−n+1
)
, R near R+, n ≥ 1,
→ 0 as R → R+, (7.8a)
Y R
n (R, t) = Y R
n (t)R−n+1 for large R, n ≥ 1,
Y R
n
R
→ 0 as R → ∞. (7.8b)
For the estimates in this work, the quantities Y θ(t), Y R(t) in the right-hand sides of (7.7)
and (7.8) are treated as constants (in each Σt) and thus there is a slight abuse of notation. The
behaviour of Y R
0 (R, t) is a bit subtle and it is directly related to the regularity issues of our
problem. This will be studied separately later.
48 N. Gudapati
Wave map phase space X
The general gauge transforms of the quantities look like
|Φ|′ = |Φ|′ + LȲ|Φ|
subsequently the wave map canonical pairs
U ′A = U ′A + LȲU
A,
p′A = p′
A + LȲpA ∀A
likewise, the (spacetime) gauge transform of the metric on the target
h′AB(U) = h′
AB(U) + LȲhAB(U) ∀A,B,
which is analogous to the transformation of a scalar. In the case of the axially symmetric and
stationary Kerr black hole spacetime the operator LȲ = LȲ |Σ. As a consequence, the formulas
above reduce to
U ′A = U ′A + LȲU
A,
p′A = p′
A + LȲpA.
In the asymptotic regions we have the above formulas reduce to
U ′A = LȲ|ΣUA,
p′A = LȲ|ΣpA.
After noting that in the (R, θ) coordinates for (Σ, q)
∂R|Φ| ∼ O(R), ∂θ|Φ| ∼ O
(
R2
)
for large R,
∂R|Φ| ∼ O(1), ∂θ|Φ| ∼ O(1) for R close to R+.
As a consequence, we have
|Φ|′ ∼ O
(
1
R
)
for large R,
|Φ|′ ∼ O(1) for R close to R+.
Next, the other component of the wave map U : (M, g) → (N,h) is constituted by the twist
potential. It follows from background Kerr geometry that
∂Rω ∼ O
(
1
R3
)
, ∂θω ∼ O(1) for large R,
∂Rω ∼ O(1), ∂θω ∼ O(1) for R close to R+
and
|ω|′ ∼ O
(
1
R
)
for large R,
|ω|′ ∼ O(1) for R close to R+.
Now, then let us turn to the conjugate momenta, we have from the Hamiltonian equation,
N
µ̄q
pA = hAB(U)∂tU
′B − hAB(U)LN ′UB
in the asymptotic regions
= hAB(U)∂tLY U
B − hAB(U)LN ′UB.
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 49
Lagrange multipliers
Now let us turn to the remanding quantities that occur in the ADM formalism, Lagrange
multipliers {N ′, N ′a} in our Weyl–Papapetrou gauge, as constructed using a gauge transform
from harmonic coordinates. It may noted that, for our background Kerr metric,
ḡ′(∂ϕ, ∂ϕ) = |Φ|′, N = |Φ|
1
2 (−ḡ(dt,dt))−
1
2
noting that
|Φ|′ = ḡ′(∂ϕ, ∂ϕ) + Y α∂α|Φ|
and
ḡ′(∂t, ∂t) = ḡ(dt,dxα)ḡ
(
dt,dxβ
)(
ḡ′(∂a, ∂β) + LY ḡ(∂α, ∂β)
)
we get
N ′ =
1
2
|Φ|−1N(ḡ(∂ϕ, ∂ϕ) + Y α∂α|Φ|)
− 1
2
|Φ|
1
2N3
(
ḡ(dt,dxα)ḡ
(
dt,dxβ
)(
ḡ′(∂a, ∂β) + LY ḡ(∂α, ∂β)
))
. (7.9)
In the asymptotic regions, we have
N ′ = Y a∂aN +N∂tY
t.
Likewise, for the shift vector, we have(
|Φ|−1qab + |Φ|AaAb
)
N̄ b = |Φ|qabN b + |Φ|AtAa
consequently
|Φ|−1N̄ ′b = |Φ|−1N ′b + qab|Φ|AtA′
a.
Recall
N̄ b = q̄abḡ0a = |Φ|qabḡ0a,
then
N ′b = N̄ ′b + qab|Φ|2AtA′
a
= |Φ|qabḡ′ta + qab|Φ|Atḡ
′
aϕ
= |Φ|qab
(
ḡ′
ta + (LY ḡ)ta
)
+ qab|Φ|At
(
ḡ′
aϕ + (LY ḡ)aϕ
)
.
Now then using
(LY ḡ)0a = ∂0Y
bḡba + ∂aY
t
(
−N2 +NϕN
ϕ
)
+ ∂aY
ϕNϕ
= ∂0Y
bḡba + ∂aY
t
(
−N2 + |Φ|A2
0
)
+ |Φ|∂aY ϕA0,
(LY ḡ)aϕ = |Φ|∂aY ϕ + ∂aY
tN̄ϕ
= |Φ|
(
∂aY
ϕ + ∂aY
tA0
)
in the asymptotic regions, we have
N ′b = ∂tY
b −N2qab∂aY
t. (7.10)
The results obtained above are summarized in the following lemma.
50 N. Gudapati
Lemma 7.3 (Lagrange multipliers {N ′, N ′a}). Suppose Ȳ is a gauge transform from the har-
monic coordinates
(
M̄ ′, ḡ′) to the Weyl Papapetrou gauge
(
M̄ ′, ḡ′
)
.
� In the Weyl–Papapetrou gauge, the Lagrange multipliers {N,N ′a} in the asymptotic regions
are given by (7.9) and (7.10) respectively
� The vector field N ′a so constructed is regular at the axes and behaves as
N ′R = O(1), N ′θ = O
(
1
R
)
near the spatial infinity,
and
N ′R = O(1), N ′θ = 0 at the horizon.
Proof. In our gauge, we have
∂tY
t = − 1
N
Y a∂aN, ∀ t
in the asymptotic regions. Thus,
∂bY
t = −∂b
(
1
N
∫ t
0
Y a∂aNdt′
)
.
Now define a quantity Da occurring in (7.10) as
Da := N2qab∂aY
t,
then
Da = −N2qab∂b
(
1
N
∫ t
0
Y a∂aN dt′
)
.
Recall that
e2ν = sin2 θ
(
r2 + a2
)2 − a2∆sin2 θ
R2
, (7.11a)
∂RN
N
=
1 +
R2
+
R2
R
(
1− R2
+
R2
) ∼
(
1− R+
R
)−1
as R → R+
→ 0 as O
(
1
R
)
as R → ∞, (7.11b)
∂θN
N
= cot θ, (7.11c)
and denote
Ya =
∫ t
0
Y a dt. (7.11d)
Let us now compute the behaviour of D at various boundaries. We have the following expressions
for the components of D,
DR = N2e−2ν(q0)
RR∂RY
t
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 51
=
(
1−
R2
+
R2
)2
· R4((
r2 + a2
)2 − a2∆sin2 θ
)∂R(Y t(t = 0)− YR 1 +
R2
+
R2
1− R2
+
R2
− Yθ cot θ
)
→ 0 as R → ∞
at the rate of O
(
1
R2
)
; and
DR = O(1) as R → R+.
Likewise, we have
Dθ = N2e−2ν(q0)
θθ∂θY
t
=
(
1−
R2
+
R2
)2
· R4((
r2 + a2
)2 − a2∆sin2 θ
)∂θ
(
Y t(t = 0)− YR 1 +
R2
+
R2
1− R2
+
R2
− Yθ cot θ
)
→ 0 as R → ∞
at the rate of O
(
1
R3
)
Dθ = O
(
1− R+
R
)
→ 0 as R → R+. ■
It now follows that the conjugate momenta, are given by
N
p′A
µ̄q
= hAB(U)∂tLY U
′A − hAB(U)LN ′∂aU
b
in the asymptotic regions
= hAB(U)N2qab∂bY
t∂aU
B
= hAB(U)R2 sin2 θ
(
1−
R2
+
R2
)2
· R2
sin2 θ
((
r2 + a2
)2 − a2∆sin2 θ
)
×
(
−∂RU
B∂R
(
Ya∂aN
N
)
− 1
R2
∂θU
B∂θ
(
Ya∂aN
N
))
.
As a consequence, we can estimate the values of the conjugate momenta at various boundaries
(cf. equations (7.11b) and (7.11c)).
N
p′A
µ̄q
= O
(
1
R4
)
for large R,
N
p′A
µ̄q
= O
(
1−
R2
+
R2
)
for R close to R+, A = 1, 2.
It can be verified that the decay rates and the boundary behaviour of the shift vector field
agree from separate analysis using the momentum constraint. From the momentum constraint,
we have
−(q0)∇bϱ
b
a +
1
2
p′A∂aU
A = 0
after taking into account that the transverse-traceless tensors vanish for our geometry, where
ϱac = µ̄q0
(
(q0)∇cY
a + (q0)∇aYc − δac
(q0)∇bY
b
)
,
52 N. Gudapati
which can be reduced to an elliptic equation for the shift vector. Now, let us turn our attention
to the Hamiltonian constraint
H = µ̄−1
q0
(
e−2ν∥ϱ∥2q0 −
1
2
τ 2e2ν µ̄2
q0 +
1
2
pAp
A
)
+ µ̄q0
(
2∆0ν +
1
2
hABq
ab
0 ∂aU
A∂bU
B
)
, (Σ, q0),
where
∆0ν :=
1
µ̄q0
∂b
(
qab0 µ̄q0∂bν
)
.
There are a few delicate aspects of the Hamiltonian constraint in our dynamical axisymmetric
problem. For the special case of the Kerr metric,
H = µ̄q0
(
2∆0ν +
1
2
hABq
ab
0 ∂aU
A∂bU
B
)
, (Σ, q0).
If we consider the quantity
∫
H = 0, it may be noted that the inner boundary term involves
the ∂Rν at the horizon H+. This quantity vanishes for the Kerr black hole metric and we recover
the positive mass theorem of Schoen–Yau [66, 67]. This is consistent with the (inner boundary)
horizon being the minimal surface, which is the case with the Kerr black hole spacetime. Let us
now analyze the linearized Hamiltonian constraint
H ′ = µ̄q0
(
2∆0ν
′)+ 1
2
µ̄q0∂UChABq
ab
0 ∂aU
A∂bU
BU ′C + µ̄q0q
ab
0 hAB∂aU
′A∂bU
B, (Σ),
which is an elliptic PDE. We would like to construct ν ′ such that the following proposition holds.
Proposition 7.4. A boundary value problem for ν ′ admits a unique, regular and bounded solu-
tion that decays at the rate of O
(
1
R
)
for large R.
In order to set up a well-posed boundary value problem for this elliptic PDE, we need to
specify appropriate boundary conditions.
Suppose, Σ is a 2-surface that embeds into the Cauchy hypersurface of the Kerr metric
(Σ ↪→)Σ, the Gauss curvature of Σ is given by
K(Σ) = − 1
e2ν
∆0ν
and the mean curvature H of Σ ↪→ Σ
HΣ = 0
on account of the fact that the inner boundary of Σ is a minimal surface. In the perturbative
theory, we need to find an expression of the mean curvature.
H⃗Σ = HΣn, mean curvature vector, Σ ↪→ Σ.
For the conformal transformation q = Ω2q0, we have
HΣ =
1
Ω
(
H0 +
2
Ω
∂Ω
∂n
)
.
It then follows that
HΣ = |Φ|1/2e−ν
(
1
R
+ ∂Rν
)
.
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 53
It may be noted that the preservation of the minimal surface condition at the inner boundary
horizon H+ implies a Neumann boundary condition for ν ′ at the horizon H+. Let us now turn
to the issue of regularity of ν ′ at the axes Γ. Firstly note that, from the form of the Kerr metric,
the quantity ν ′ has the form
ν = 2γ + c− 2 log ρ
in the Weyl–Papapetrou gauge, near the axes. For the reasons of regularity of the Kerr metric
at the axes, we have c ≡ 0. Thus,
ν = 2γ − 2 log ρ.
Consequently, we have the condition
ν ′ = 2γ′ (7.12)
for the linear perturbation theory, thus suggesting the Dirichlet boundary conditions for ν ′ at
the axes
∆0ν
′ =
1
4
∂UChAB(U)qab0 ∂aU
AU ′C + qabhAB∂aU
a∂bU
B on (Σ)
ν ′ = 2γ′ on (Γ)
∂nν
′ = 0 on
(
H+
)
(M− BVP)ν′ .
It must be pointed out that for the regularity of ν′ itself, at the axes, we need to impose
the condition ∂nν
′ = 0 at the axes. A priori, the apparent over-determined boundary data
– both Dirichlet and Neumann – at the axes is a significant issue for the well-posedness of the
boundary value problem for ν ′. It may be recalled that an elliptic problem with both Dirichlet
and Neumann data, specified at the same boundary, is typically ill posed. Importantly, we can
prove that the boundary conditions ν ′ = 2γ′ and ∂nν
′ = 0 are equivalent. In other words,
imposition of one condition automatically satisfies the other and vice versa. This saves us from
the aforementioned issue.
However, we should ask a more fundamental question: Can the set-up of our problem result
in a well-posed boundary value problem for ν ′ (as proposed in Proposition 7.4) in the first
place? and what about the regularity of ν ′ at the corners Γ ∩ H+? Suppose, ν ′ = 2γ′, then
a computation shows that
∂nν
′ = ∂Rν
′ = 2∂Rγ
′ = 0
at the corners, in a limiting sense. On the other hand, suppose ν ′ were a scalar, then ν ′ =
(ν ′)HG + Y a∂aν = Y a∂aν in the asymptotic regions. Then a computation shows that
∂Rν
′ = ∂RY
a∂aν ̸= 0.
In the above, there is an inconsistency for two reasons. Firstly, the quantity ν ′ is not regular at
the corners. Secondly, the Neumann boundary condition at the horizon is not satisfied at the
end points, the corners. The following lemma is crucial and comes to our rescue. In this lemma,
we show that ν ′ does not transform like a scalar.
Lemma 7.5. Suppose the quantity ν ′ is compactly supported in the harmonic gauge, then
(1) ν ′ has the following structure
ν ′ = Y a∂aν +
1
2
(q0)∇aY
a (7.13)
in the asymptotic regions.
(2) Furthermore, ν ′ is regular at the corners Γ ∩H+.
54 N. Gudapati
Proof. Consider D · µ̄q in a conformally flat form, we have
D · µ̄q = e2νD · µ̄q0 + 2e2νν ′µ̄q0 .
Now, then for our choice of the gauge condition, where we hold the flat metric q0 fixed,
2e2νν ′µ̄q0 = (ν ′)HG + LY µ̄q,
in the asymptotic regions
= 2e2νY a∂aνµ̄q0 + e2ν∂aµ̄q0 + e2ν∂aY
aµ̄q0 ,
the right-hand side can be expressed equivalently as
= 2e2νY a∂aνµ̄q0 + e2νLY µ̄q0
= 2e2νY a∂aνµ̄q0 + e2ν∂a(Y
aµ̄q0)
= 2e2νY a∂aνµ̄q0 + e2ν µ̄q0
(q0)∇aY
a.
The result (7.13) follows. Now let us show that the ‘correction term’ is harmonic in the asymp-
totic regions. We have
(q0)∇a
(q0)∇bYc − (q0)∇b
(q0)∇aYc =
(q0)R d
abc Yd
on account of the fact that Y is a conformal Killing vector field in the asymptotic regions,
CK(Y, q0) = 0, it follows that
(q0)∇a(q0)∇a
(
(q0)∇cY
c
)
= 0.
With the correction term in (7.13), the assertion (2) can be also be verified explicitly. It can
also be verified that the Neumann boundary condition for ν ′ at the horizon H+ is satisfied. ■
Firstly, we note that the mixed boundary value problem, with regular boundary data, is well-
posed. We would like to remark that ‘uniqueness up to a constant’ which is usually the case for
a Neumann boundary value problem is not sufficient for our problem because we need the decay
of ν ′ (consistent with the regularity on the axes) to establish the needed boundary behaviour of
our fields. In view of the uniqueness, we claim that the solution to the mixed boundary value
problem above can be decomposed into the following two parts: solution for a inhomogeneous
Dirichlet boundary value problem ν′
D and a homogeneous Neumann problem ν ′
N
ν ′ = ν ′
D + ν ′
N
represented in the same domain. In the following, we shall elaborate on our construction that
ensures that the needed boundary conditions for ν ′ are satisfied. We specify the (regular) data
for Dirichlet problem for ν ′
D in a half plane such that it is consistent with the data for ν ′.
This problem is well-posed on account of the regularity conditions of the source (involves U ′,
h(U), h′(U)) and admits a regular solution. We shall use this to obtain a suitable choice of
data for the Neumann problem for ν ′
N at the horizon, so that the total sum ν ′
D + ν ′
N satisfies
the needed Neumann boundary condition for ν ′. The well-posedness and regularity for ν ′ at
the axes implies that it should vanish on the axes, thus implying that ν ′ satisfies the needed
Dirichlet condition on the axes. This construction also ensures that the regularity of ν ′ at the
corners is also satisfied. Consider
∆0ν
′
D =
1
4
∂UChAB(U)qab0 ∂aU
AU ′C
+ qabhAB∂aU
a∂bU
B on (Σ)
ν ′ = 2γ′ on (Γ)
ν ′ = 2γ′ on ({ρ = 0} \ Γ)
(D− BVP)ν′ . (7.14)
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 55
It may be noted that, for our construction, the data at the ‘cut’ between the axes {ρ = 0} \ Γ can
be specified arbitrarily so that its smooth but for convenience we choose ν ′
D = 2γ′. It follows
that well posed Dirichlet boundary value problem (7.14) is well posed and that there exists
a regular solution. We shall use this to construct Neumann data for ν ′
N . Note that we can
compute ∂nν
′
D at the horizon (boundary) using the ‘Dirichlet to Neumann map’
Λ: ν′ → ∂nν
′
and the conformal inversion map discussed previously. We then formulate the homogeneous
Neumann boundary value problem
∆0ν
′
N = 0 on (Σ)
∂nν
′
N = 0 on (Γ)
∂nν
′
N = ∂nν
′
D
∣∣
H+ on
(
H+
)
(N-BVP)ν′ .
It follows from the use of the representation formulas (7.5) for both Dirichlet and Neumann
problems, that ν ′ decays at the rate of O
(
1
R
)
for large R.
Proposition 7.6. The integral invariant quantity Y0(H+) at the horizon is finite for all times
if and only if it vanishes.
Proof. Recall the expression for Y R
0 (R+)
Y R
0 (R+) =
R+
2π
∫ ∞
R+
1
R′2
∫ 2π
0
R′
2
e2γ−2ν
(
ḡ′(∂R, ∂R)−
1
R′2 ḡ
′(∂θ, ∂θ)
)
dθdR′,
which can be reexpresed as
=
R+
2π
∫ ∞
R+
1
R′2
∫ 2π
0
R′
2
(
q′0(∂R, ∂R)−
1
R′2q
′
0(∂θ, ∂θ)
)
dθdR′, (7.15)
where q0 is the metric perturbation in harmonic gauge.
The quantities, q′0(∂R, ∂R), q
′
0(∂R, ∂θ) and q′0(∂θ, ∂θ) admit the decomposition
q′0(∂R, ∂R) =
∞∑
n=0
{
q′0(∂R, ∂R)
}
n
cosnθ,
q′0(∂R, ∂θ) =
∞∑
n=1
{
q′0(∂R, ∂θ)
}
n
sinnθ,
q′0(∂θ, ∂θ) =
∞∑
n=0
{
q′0(∂θ, ∂θ)
}
n
cosnθ.
Now then (7.15) can be simplified as
Y R
0 (R+) =
R+
2
∫ ∞
R+
1
R′
({
q′0(∂R, ∂R)
}
0
(R′)− 1
R′2
{
q′0(∂θ, ∂θ)
}
0
(R′)
)
dR′
=
R+
2
∫ ∞
R+
1
R′
({
q′0(∂R, ∂R)
}
0
(R′)− ∂R′
(
1
R′
{
q′0(∂θ, ∂θ)
}
0
(R′)
))
dR′
+
R+
2
∫ ∞
R+
∂R′
(
1
R′2
{
q′0(∂θ, ∂θ)
}
0
(R′)
)
dR′. (7.16)
56 N. Gudapati
Now consider (6.7) and note that the operator on the left-hand side is a linear differential
operator. In polar coordinates
− 1
R
∂2
θq0(∂R, ∂R) + ∂Rq0(∂R, ∂R) +
2
R
∂2Rθq0(∂R, ∂θ)−
2
R3
q0(∂θ, ∂θ)
+
2
R2
∂Rq
′
0(∂θ, ∂θ)−
1
R
∂2
RRq0(∂θ, ∂θ) = 0.
It follows that
∂R
{
q′0(∂R, ∂R)
}
0
− 2
R3
{
q′0(∂θ, ∂θ)
}
0
+
2
R2
∂R
{
q′0(∂θ, ∂θ)
}
0
− 1
R
∂2
R
{
q′0(∂θ, ∂θ)
}
0
= 0,
which is a pure divergence,
∂R
({
q′0(∂R, ∂R)
}
0
− ∂R
(
1
R
{
q′0(∂θ, ∂θ)
}
0
))
= 0,
uniformly in Σ for all times. We get the condition that the R derivative of the quantity in the
first integral of (7.16) vanishes. As a result this quantity can make a finite contribution only if
it vanishes. The result follows. ■
8 Strict conservation of the regularized Hamiltonian HReg
In this section, we shall establish that the regularized Hamiltonian energy HReg is strictly
conserved in time. In particular, we shall establish the following.
Theorem 8.1. Suppose we have the initial value problem of Einstein’s equations for general
relativity. Then
(1) There exists a (C∞-)diffeomorphism from harmonic coordinates to the Weyl–Papapetrou
gauge of the maximal development (M ′, g′) of the perturbative theory of Kerr black hole
spacetimes, under axial symmetry.
(2) The positive-definite Hamiltonian HReg is strictly-conserved forwards and backwards in
time.
Consider the vector field density
(Jb)Reg = N2e−2νqab0 p′A∂aU
′A + U ′ALN ′
(
Nµ̄q0q
ab
0 hAB∂bU
B
)
+ LN ′(N)
(
2µ̄q0q
ab
0 ∂aν
′)+ 2N ′a∂aν
′µ̄q0q
ab
0 ∂aN − 2N ′bµ̄q0q
bc
0 ∂aν
′∂cN.
We shall classify the terms in
(
Jb
)Reg
into kinematic, dynamical and conformal terms:
Jb
1 := N2e−2νqab0 p′A∂aU
′A, (dynamical terms)
Jb
2 := U ′ALN ′
(
Nµ̄q0q
ab
0 hAB∂bU
B
)
, (kinematic terms)
Jb
3 := LN ′(N)
(
2µ̄q0q
ab
0 ∂aν
′),
Jb
4 := 2N ′a∂aν
′µ̄q0q
ab
0 ∂aN,
Jb
5 := −2N ′bµ̄q0q
bc
0 ∂aν
′∂cN. (conformal terms)
The asymptotic and decay rates of the fluxes and the associated integrands can be explicitly
evaluated with a specific choice of gauge on the target. In the following, we choose, h =
4dγ2 + e−4γdω2. Analogous computations can be performed for other gauges on the target
manifold. An important aspect of these flux estimates is that most of the integrands of these
flux terms vanish pointwise at the boundaries of the orbit space.
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 57
‘Dynamical’ boundary terms
Consider the dynamical flux term, Flux(J1,Γ), where
(J1)
b := N2e−2νqab0 p′A∂aU
′A.
Let us start by considering the terms
Flux
(
N2e−2ν(q0)
ab
(
4p∂aγ
′),Γ)
and
Flux
(
N2e−2ν(q0)
ab
(
e−4γr∂aγ
′),Γ).
We have
Flux
(
N2e−2ν(q0)
ab
(
4p∂aγ
′),Γ)
=
∫ t
0
∫
(−∞,−R+)∪ (R+,∞)
lim
θ→0,π
(
4N2e−2ν(q0)
abp∂aγ
′)θdRdt
=
∫ t
0
∫
(−∞,−R+)∪ (R+,∞)
lim
θ→0,π
(
4N
(
Np′
µ̄q
)
µ̄q0(q0)
ab∂aγ
′
)θ
dRdt.
If we consider the integrand, within the domain of integration, we have
lim
θ→0,π
4
N
R2
· Np′
µ̄q
µ̄q0∂θγ
′ = 4
∆
1
2 sin θ
R2
Np′
µ̄q
µ̄q0∂θγ
′ → 0 as θ → 0, π,
and
Flux
(
N2e−2ν(q0)
ab
(
e−4γr∂aω
′),Γ)
=
∫ t
0
∫
(−∞,−R+)∪ (R+,∞)
lim
θ→0,π
(
e−4γN2e−2ν(q0)
abr∂aω
′)θdRdt
=
∫ t
0
∫
(−∞,−R+)∪ (R+,∞)
lim
θ→0,π
(
N2(q0)
ab e
−4γr
µ̄q
µ̄q0∂aω
′
)θ
dRdt.
The integrand
lim
θ→0,π
N2
R2
e−4γ r
′
µ̄q
µ̄q0∂θω
′
= lim
θ→0,π
∆
R
sin2 θ
Σ2
sin4 θ
((
r2 + a2
)2 − a2∆sin2 θ
) r′
µ̄q
∂θω
′ → 0 as θ → 0, π,
due to the rapid decay of ∂θω
′ as θ → 0 and π. Next consider, Flux
(
J1,H+
)
.
We have the term, Flux
(
N2e−2νqab0 p∂aγ
′,H+
)
, which can be estimated near the future hori-
zon H+ as
Flux
(
N2e−2νqab0 p∂aγ
′,H+
)
=
∫ t
0
∫ π
0
lim
R→R+
(
N2e−2νqab0 p′∂aγ
′)Rdθdt
=
∫ t
0
∫ π
0
lim
R→R+
(
N · qab0
Np′
µ̄q
µ̄q0∂aγ
′
)R
dθdt
58 N. Gudapati
=
∫ t
0
∫ π
0
lim
R→R+
(
R sin θ
(
1−
R2
+
R2
)
Np′
µ̄q
µ̄q0∂Rγ
′
)
dθdt.
It may be recalled that the behaviour of the canonical pair (γ′,p′) near the future horizon is
4∂Rγ
′ =
(
1−
R2
+
R2
)
and
Np′
µ̄q
= O
(
1−
R2
+
R2
)
,
Flux
(
N2e−2νqab0 p∂aγ
′,H+
)
→ 0 as R → R+.
Likewise,
Flux
(
N2e−2νqab0 r′∂aω
′,H+
)
=
∫ t
0
∫ π
0
lim
R→R+
(
N2e−2νqab0 r′∂aω
′)Rdθdt
=
∫ t
0
∫ π
0
lim
R→R+
(
Ne−4γ · qab0
Ne4γr′
µ̄q
µ̄q0∂aω
′
)R
dθdt.
Again recall that the behaviour of the canonical pair (ω′, r′) near the future horizon is
∂Rω
′ =
(
1−
R2
+
R2
)
and
Ne4γr′
µ̄q
= O
(
1−
R2
+
R2
)
,
the integrand in Flux
(
N2e−2νqab0 p∂aγ
′,H+
)
is
= lim
R→R+
R2 sin θ
(
1−
R2
+
R2
)
Σ2
sin4 θ
((
r2 + a2
)2 − a2∆sin2 θ
)2 c(1− R2
+
R2
)
·
(
1−
R2
+
R2
)
→ 0 as R → R+.
Flux of J1 at the outer boundary, Flux
(
J1, ῑ
0
)
Consider the flux of the term
Flux
(
N2
µ̄q
p′µ̄qq
ab∂aγ
′, ῑ0
)
=
∫ t
0
∫ π
0
(
lim
R→∞
N2
µ̄q
p′µ̄q0∂Rγ
′
)
dθdt,
we have
Np
µ̄q
′
= O
(
1
R3
)
, γ′ =
1
R
, ∂Rγ
′ = O
(
1
R2
)
,
and recall that
N =
(
r(R)2 − 2Mr(R) + a2
)1/2
sin2 θ = O(R) for large R.
Therefore, in the region under consideration, we have for the integrand of
Flux
(
N2
µ̄q
p′µ̄qq
ab∂aγ
′, ῑ0
)
∼ O(R) · O
(
1
R3
)
· O
(
1
R2
)
·R → 0 as R → ∞.
Likewise, consider the following flux terms at the outer boundary:
Flux
(
−4N2
µ̄q
r′γ′µ̄q0q
ab
0 ∂aω, ῑ
0
)
, Flux
(
N2
µ̄q
r′µ̄q0q
ab
0 ∂aω
′, ῑ0
)
,
Flux
(
N2
µ̄q
r′µ̄q0q
ab∂aω
′, ῑ0
)
=
∫ t
0
∫ π
0
(
lim
R→∞
N2
µ̄q
r′µ̄q0∂Rω
′
)
dθdt.
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 59
It may be recalled that
Ne4γr′
µ̄q
= O
(
1
R3
)
,
and
e−2γ =
(
r2 + a2 cos2 θ
sin2 θ
)
1((
r2 + a2
)2 − a2∆sin2 θ
) ∼ O
(
1
R2
)
for large R.
Therefore, for the integrand within, we have
Flux
(
N2
µ̄q
r′µ̄q0q
ab∂aω
′, ῑ0
)
= O(R) · O
(
1
R3
)
· O
(
1
R4
)
R2 · 1
R2
→ 0 as R → ∞.
Next, consider
Flux
(
−4N2
µ̄q
r′γ′µ̄q0q
ab
0 ∂aω, ῑ
0
)
=
∫ t
0
∫ π
0
(
lim
R→∞
−4N2
µ̄q
r′γ′µ̄q0∂Rω
)
dθdt
=
∫ t
0
∫ π
0
(
lim
R→∞
−4N
(
Ne4γr′
µ̄q
)
e−4γγ′µ̄q0∂Rω
)
dθdt.
Using the above estimates again and noting that ∂Rω = O
(
1
R3
)
,
4N2
µ̄q
r′γ′µ̄q0∂Rω = O(R) · O
(
1
R3
)
· O
(
1
R4
)
· O
(
1
R
)
RO
(
1
R3
)
= 0 as R → ∞,
from which it follows that Flux
(
−4N2
µ̄q
r′γ′µ̄q0q
ab
0 ∂aω, ῑ
0
)
= 0.
‘Kinematic’ boundary terms
Consider
(J2)
b := U ′ALN ′
(
Nµ̄qq
abhAB(U)∂bU
B
)
.
In a special gauge, we have
= γ′LN ′
(
4Nµ̄qq
ab∂aγ
)
+ ω′LN ′
(
Nµ̄qq
abe−4γ∂aω
)
.
Let us start with Flux(J2,Γ). We have
Flux(J2,Γ) =
∫ t
0
∫
(−∞,R+)∪(R+,∞)
lim
θ→0,π
(
U ′ALN ′
(
Nµ̄q0q
ab
0 hAB(U)∂bU
B
))θ
dRdt.
We will expand the expression U ′ALN ′
(
Nµ̄q0q
ab
0 hAB(U)∂aU
B
)
as follows:
U ′ALN ′
(
Nqab0 µ̄q0hAB(U)∂θU
B
)
= U ′A(N∂cN
′cµ̄q0q
ab
0 hAB(U)∂aU
B
+N ′c∂c
(
Nµ̄q0q
ab
0 hAB(U)∂aU
B
)
− ∂cN
′bNµ̄q0q
ab
0 hAB∂aU
B
)
. (8.1)
We would like to point out that ω′ vanishes on the axes whereas γ′ does not. Likewise, ∂θω decays
rapidly on the axes whereas ∂θγ does not. This causes a few subtleties for terms involving γ′
60 N. Gudapati
but as will see later, we will have few fortuitous cancellations involving this quantity. Each of
the terms in (8.1) can in turn be decomposed as
U ′A(N∂cN
′cµ̄q0q
ab
0 hAB(U)∂aU
B
)
= γ′
(
4N∂cN
′cµ̄q0q
ab
0 ∂aγ
)
+ ω′(N∂cN
′cµ̄q0q
ab
0 e−4γ∂aω
)
.
As a consequence, we have
Flux
(
γ′
(
4N∂cN
′cµ̄q0q
ab
0 ∂aγ
)
,Γ
)
=
∫ t
0
∫
(−∞,−R+)∪(R+,∞)
lim
θ→0,π
(
γ′
(
4N∂cN
′cµ̄q0q
ab
0 ∂aγ
))θ
dRdt
=
∫ t
0
∫
(−∞,R+)∪(R+,∞)
(
lim
θ→0,π
4
N
R
γ′∂θγ
(
∂θN
′θ + ∂RN
′R))dRdt. (8.2)
Let us note that limθ→0,π ∂θγ
′ = cot θ. Likewise,
Flux
(
ω′(∂cN ′cNµ̄q0q
ab
0 e−4γ∂aω
)
,Γ
)
=
∫ t
0
∫
(−∞,R+)∪(R+,∞)
lim
θ→0,π
(
ω′(∂cN ′cNµ̄q0q
ab
0 e−4γ∂aω∂b
))θ
dRdt
=
∫ t
0
∫
(−∞,R+)∪(R+,∞)
(
lim
θ→0,π
N
R
e−4γω′∂θω∂c
(
N ′c))dRdt = 0,
where we took into account the fact that the ∂θω term has rapid decay as θ → 0, π. Next,
U ′AN ′c∂c
(
Nµ̄q0q
ab
0 hAB(U)∂aU
A
)
= 4γ′N ′c∂c
(
Nµ̄q0q
ab
0 ∂aγ
)
+ ω′N ′c∂c
(
Nµ̄q0e
−4γ∂aω
)
,
Flux
(
4γ′N ′c∂c
(
Nµ̄q0q
ab
0 ∂aγ
)
,Γ
)
=
∫ t
0
∫
(−∞,R+)∪(R+,∞)
lim
θ→0,π
(
4γ′N ′c∂c
(
Nµ̄q0q
ab
0 ∂aγ
)
,
1
R
∂θ
)θ
dRdt
=
∫ t
0
∫
(−∞,R+)∪(R+,∞)
(
lim
θ→0,π
4γ′N ′c∂c
(
N
µ̄q0
R2
∂θγ
))
dRdt.
Consider the integrand
Rγ′N ′c∂c
(
N
R
∂θγ
)
= 4γ′
(
N ′θ∂θ
(
N
R
∂θγ
)
+N ′R∂R
(
N
R
∂θγ
))
. (8.3)
The N ′R term remains while the N ′θ term vanishes at the axes (as θ → 0 or π). For the sake of
brevity, let us combine the N ′R terms in (8.3) and (8.2). We shall revisit this later. Now then
we have∫ t
0
∫
(−∞,−R+)∪(∞,R+)
(
4γ′∂R
(
N ′RN
R
∂θγ
))
dRdt. (8.4)
Similarly,
Flux
(
ω′N ′c∂c
(
Nµ̄q0q
ab
0 e−4γ∂aω
)
,Γ
)
=
∫ t
0
∫
(−∞,R+)∪(R+,∞)
lim
θ→0,π
(
ω′N ′c∂c
(
Nµ̄q0q
ab
0 e−4γ∂aω
)
∂b
)θ
dRdt
=
∫ t
0
∫
(−∞,R+)∪(R+,∞)
(
lim
θ→0,π
ω′N ′c∂c
(
Nµ̄q0
R2
e−4γ∂θω
))
dRdt.
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 61
The integrand
ω′N ′c∂c
(
µ̄q0
R2
e−4γ∂θω
)
= ω′
(
N ′θ∂θ
(
N
µ̄q0
R2
e−4γ∂θω
)
+N ′R∂R
(
N
µ̄q0
R2
e−4γ∂θω
))
vanishes at the axes due to ω′ = 0 on Γ and the rapid decay of ∂θω at the axes. Finally,
−U ′A∂cN
′bµ̄q0q
ac
0 hAB∂aU
B = −4γ′∂cN
′bµ̄q0 q̄0
ac∂aγ − ω′∂cN
′bµ̄q0 q̄0
ace−4γ∂aω,
so that the fluxes
Flux
(
−4γ′∂cN
′bq̄0
ac∂aγ,Γ
)
=
∫ t
0
∫
(−∞,R+)∪(R+,∞)
lim
θ→0,π
(
−4Nγ′∂cN
′bµ̄q0 q̄0
ac∂aγ
)θ
dRdt
=
∫ t
0
∫
(−∞,R+)∪(R+,∞)
(
lim
θ→0,π
(
−4γ′N
(
R∂RN
′θ∂Rγ +
1
R
∂θN
′θ∂θγ
)))
dRdt.
If we consider the integrand
lim
θ→0,π
(
4γ′N
(
N∂RN
′θ∂Rγ +
1
R
∂θN
′θ∂θγ
))
,
we note that the term involving ∂RN
′R∂Rγ vanishes because ∂RN
′θ vanishes on the axes
(
N ′θ is
a constant along the axes
)
and in the second term involving ∂θN
′θ∂θγ cancels with a flux term
in (8.2),
Flux
(
−ω′∂cN
′bq̄0
ace−4γ∂aω,Γ
)
=
∫ t
0
∫
(−∞,R+)∪(R+,∞)
lim
θ→0,π
(
−Nω′∂cN
′bq̄0
ace−4γ∂aω
)θ
dRdt
=
∫ t
0
∫
(−∞,R+)∪(R+,∞)
(
lim
θ→0,π
(
−Nω′(∂RN
′θe−4γ∂Rω + ∂θN
′Re−4γ∂θω
))
dRdt = 0
on account of rapid decay of ∂θω at the axes, vanishing of N ∼ sin θ and ω′ at the axes.
Now then for
Flux
(
J2,H+
)
,
we have
Flux
(
γ′
(
4∂cN
′cµ̄q0q
ab
0 ∂aγ,H+
))
=
∫ t
0
∫ π
0
(
lim
R→R+
4RNγ′∂Rγ∂cN
′c
)
dθdt.
Using
lim
R→R+
∂Rγ ≤ c, lim
R→R+
∂cN
′c ≤ c,
we estimate the integrand
lim
R→R+
4RNγ′∂Rγ∂cN
′c ≤ c
(
1−
R2
+
R2
)2
.
So we have flux
Flux
(
γ′
(
4∂cN
′cµ̄q0q
ab
0 ∂aγ,H+
))
= 0,
62 N. Gudapati
Flux
(
ω′(∂cN ′cµ̄q0q
ab
0 e−4γ∂aω
)
,H+
)
=
∫ t
0
∫ π
0
(
lim
R→R+
ω′∂cN
′cNµ̄q0e
−4γ∂Rω
)
dθdt
=
∫ t
0
∫ π
0
(
lim
R→R+
RNe−4γω′∂Rω∂cN
′c
)
Rdθdt.
Noting that
lim
R→R+
∂Rω ≤ c, lim
R→R+
e−4γ ≤ c
the integrand
RNe−4γω′∂Rω∂cN
′c ≤ c
(
1− R+
R2
)2
.
Thus, again
Flux
(
ω′(∂cN ′cµ̄q0q
ab
0 e−4γ∂aω
))
= 0.
Next,
Flux
(
4γ′N ′c∂c
(
Nµ̄q0q
ab
0 ∂aγ
)
,H+
)
=
∫ t
0
∫ π
0
(
lim
R→R+
4γ′N ′c(∂cNR∂Rγ +N∂c(R∂Rγ))
)
dRdt
the integrand can be estimated close to the horizon as
4γ′N ′c(∂cNR∂Rγ +N∂c(R∂Rγ)) ≤ c
(
1−
R2
+
R2
)
+ c
(
1−
R2
+
R2
)2
,
so we have
Flux
(
4γ′N ′c∂c
(
Nµ̄q0q
ab
0 ∂aγ
)
,H+
)
= 0,
Flux
(
ω′N ′c∂c
(
µ̄q0e
−4γ∂aω
)
,H+
)
=
∫ t
0
∫ π
0
(
lim
R→R+
ω′N ′c∂c
(
NRe−4γ∂Rω
))
dθdt
=
∫ t
0
∫ π
0
(
lim
R→R+
ω′N ′c(∂cNRe−4γ∂Rω +N∂c
(
e−4γR∂Rω
)))
dθdt
the integrand can be estimated as
ω′N ′c(∂cNRe−4γ∂Rω +N∂c
(
e−4γR∂Rω
))
≤ c
(
1−
R2
+
R2
)
+ c
(
1−
R2
+
R2
)2
,
so we have
Flux
(
ω′N ′c∂c
(
µ̄q0e
−4γ∂aω
)
,H+
)
= 0.
Next,
Flux
(
−4Nγ′∂cN
′bq̄0
ac∂aγ,H+
)
=
∫ t
0
∫ π
0
(
lim
R→R+
−4Nγ′
(
∂RN
′R∂Rγ +
1
R2
∂θN
′θ∂θγ
))
dθdt = 0
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 63
in view of the behaviour of the integrand
4Nγ′
(
∂RN
′R∂Rγ +
1
R2
∂θN
′θ∂θγ
)
behaves as c
(
1−
R2
+
R2
)2
as R → R+.
Finally,
Flux
(
−ω′∂cN
′bq̄0
acNe−4γ∂aω,H+
)
=
∫ t
0
∫ π
0
(
lim
R→R+
−Ne−4γω′
(
∂RN
′R∂Rω +
1
R2
∂θN
′θ∂θω
))
dθdt = 0,
where again the integrand can be estimated such that
Ne−4γω′
(
∂RN
′R∂Rω +
1
R2
∂θN
′θ∂θω
)
behaves as
c
(
1−
R2
+
R2
)2
for R close to R+.
Next, for the kinematic fluxes at the spatial infinity
Flux
(
J2, ι
0
)
let us start with estimating the divergence term ∂cN
′c near the spatial infinity. We have
∂cN
′c = ∂θN
′θ + ∂RN
′R
= ∂R
(
∂tY
R −N2e−2ν∂R
(
Y t(t = 0)− Yθ cot θ − YR 1 +
R2
+
R2
R
(
1− R2
+
R2
)
))
+ ∂θ
(
∂tY
θ − N2e−2ν′
R2
(
Y t(t = 0)− Yθ cot θ − YR 1 +
R2
+
R2
R
(
1− R2
+
R2
)
))
.
We have the following behaviour for large R:
∂R
(
N2e−2ν
)
∼ O
(
1
R
)
, ∂θ
N2e−2ν′
R2
∼ O(1)
and
∂R
1 +
R2
+
R2
R
(
1− R2
+
R2
) ∼ O(
1
R2
), ∂2
R
1 +
R2
+
R2
R
(
1− R2
+
R2
) ∼ O(
1
R3
)
from somewhat lengthy but standard computations. It now follows that the coordinate diver-
gence of N ′ is then
∂cN
′c = O
(
1
R
)
,
Flux
(
γ′
(
4∂cN
′cµ̄q0q
ab
0 ∂aγ, ι
0
))
=
∫ t
0
∫ π
0
(
lim
R→∞
4Rγ′∂Rγ∂cN
′cµ̄q0
)
dθdt,
4Rγ′∂Rγ∂cN
′cµ̄q0 = 4R · O
(
1
R
)
· O
(
1
R
)
· O
(
1
R
)
→ 0 as R → ∞.
64 N. Gudapati
Thus,
Flux
(
γ′
(
4∂cN
′cµ̄q0q
ab
0 ∂aγ, ι
0
))
= 0,
Flux
(
ω′(∂cN ′cµ̄q0q
ab
0 e−4γ∂aω
)
, ι0
)
=
∫ t ∫ π
0
(
lim
R→∞
Re−4γω′∂Rω∂c
(
∂tY
c −N2e−2νqac0 ∂aY
t
))
dθdt,
Re−4γω′∂Rω∂c
(
∂tY
c −N2e−2νqac0 ∂aY
t
)
= R · O
(
1
R4
)
· O
(
1
R
)
· O
(
1
R3
)
· O
(
1
R
)
→ 0 as R → ∞,
Flux
(
ω′(∂cN ′cµ̄q0q
ab
0 e−4γ∂aω
)
, ι0
))
= 0.
Analogously, noting that γ′ = O
(
1
R
)
and ω′ = O
(
1
R
)
for large R we have the following:
Flux
(
4γ′N ′c∂c
(
µ̄q0q
ab
0 ∂aγ
)
, ι0
)
=
∫ t
0
∫ π
0
(
lim
R→∞
4γ′N ′c∂c(R∂Rγ)
)
dθdt
=
∫ t
0
∫ π
0
(
4γ′
(
∂tY
c −N2e−2νqac∂aY
t
)
∂c(R∂Rγ)
)
dθdt = 0,
Flux
(
ω′N ′c∂c
(
µ̄q0e
−4γ∂aω
)
, ι0
)
=
∫ t
0
∫ π
0
(
lim
R→∞
ω′(∂tY c −N2e−2νqac∂aY
t
)
∂c
(
Re−4γ∂Rω
))
dθdt = 0,
Flux
(
−4γ′∂cN
′bq̄0
ac∂aγ, ι
0
)
=
∫ t
0
∫ π
0
(
lim
R→∞
−4γ′
(
∂RN
′R∂Rγ +
1
R2
∂θN
′R∂θγ
))
dθdt = 0,
Flux
(
−ω′∂cN
′bq̄0
ace−4γ∂aω, ι
0
)
=
∫ t
0
∫ π
0
(
lim
R→∞
ω′e−4γ
(
∂RN
′R∂Rω +
1
R2
∂θN
′R∂θω
))
dθdt = 0.
‘Conformal’ boundary terms (J3)
b := LN ′(N)
(
2µ̄qq
ab∂aν
′).
Let us start with the flux of this conformal term at the axes:
Flux(J3,Γ)
=
∫ t
0
∫
(−∞,−R+)∪(R+,∞)
lim
θ→0,π
(
LN ′(N)
(
2µ̄q0q
ab
0 ∂aν
))θ
dRdt
=
∫ t
0
∫
(−∞,−R+)∪(R+,∞)
(
2∂θν
′(N ′R∂RN +N ′θ∂θN
))
dRdt
=
∫ t
0
∫
(−∞,−R+)∪(R+,∞)
(
lim
θ→0,π
2∂θν
′
(
N ′R sin θR2
+
R2
+N ′θR cos θ
(
1−
R2
+
R2
)))
dRdt.
Expanding out the integrand within Flux(J3,Γ), near the axes Γ, we get
lim
θ→0,π
2∂θν
′
{(
∂tY
R −
(
1−
R2
+
R2
)2
· R4((
r2 + a2
)2 − a2∆sin2 θ
)
× ∂R
(
Y t(t = 0)− YR 1 +
R2
+
R2
1− R2
+
R2
− Yθ cot θ
))
sin θR2
+
R2
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 65
+
(
∂tY
θ −
(
1−
R2
+
R2
)2
· R4((
r2 + a2
)2 − a2∆sin2 θ
)
× ∂θ
(
Y t(t = 0)− YR 1 +
R2
+
R2
1− R2
+
R2
− Yθ cot θ
))
R cos θ
(
1−
R2
+
R2
)}
from which it follows that
Flux(J3,Γ) = 0
due to the decay rate of the terms in the parenthesis.
Now then, consider the flux of J3 at the horizon H+:
Flux
(
J3,H+
)
=
∫ t
0
∫ π
0
(
lim
R→R+
LN ′(N)
(
2µ̄qq
ab∂aν
′))dθdt
×
∫ t
0
∫ π
0
(
lim
R→R+
2RLN ′N∂Rν
′
)
dθdt
=
∫ t
0
∫ π
0
(
lim
R→R+
2R∂Rν
′
((
∂tY
R −N2e−2ν∂RY
t
)
sin θ
R2
+
R2
+
(
∂tY
θ − N2
R2
e−2ν∂θY
t
)
R cos θ
(
1−
R2
+
R2
)))
dθdt
expanding out the integrand within Flux
(
J3,H+
)
close to the horizon H+, we have
lim
R→R+
2R∂Rν
′
{(
∂tY
R −
(
1−
R2
+
R2
)2
· R4((
r2 + a2
)2 − a2∆sin2 θ
)
× ∂R
(
Y t(t = 0)− YR 1 +
R2
+
R2
1− R2
+
R2
− Yθ cot θ
))
sin θ
R2
+
R2
+
(
∂tY
θ −
(
1−
R2
+
R2
)2
· R4((
r2 + a2
)2 − a2∆sin2 θ
)
∂θ
(
Y t(t = 0)− YR 1 +
R2
+
R2
1− R2
+
R2
− Yθ cot θ
))}
→ 0 as R → R+
after plugging in the expression of Y and Y for R near R+. Likewise, at spatial infinity, we have
Flux(J3, ῑ
0) =
∫ t
0
∫ π
0
(
lim
R→∞
2R∂Rν
′
((
∂tY
R −N2e−2ν∂RY
t
)
sin θ
R2
+
R2
+
(
∂tY
θ − N2
R2
e−2ν∂θY
t
)
R cos θ
(
1−
R2
+
R2
)))
dθdt.
Again the integrand occurring above can be estimated, after plugging in the relevant quantities:
= lim
R→∞
2R∂Rν
′
{(
∂t
∞∑
n=1
Y R
n R−n+1 cosnθ −
(
1−
R2
+
R2
)2
· R4((
r2 + a2
)2 − a2∆sin2 θ
)
× ∂R
(
Y t(t = 0) +
∞∑
n=1
YR
n R−n+1 1 +
R2
+
R2
1− R2
+
R2
−
∞∑
n=1
Yθ
nR
−n+1 sinnθ cot θ
))
sin θ
R2
+
R2
66 N. Gudapati
+
(
∂t
∞∑
n=1
Y θ
nR
−n sinnθ −
(
1−
R2
+
R2
)2
· R4((
r2 + a2
)2 − a2∆sin2 θ
)
× ∂θ
(
Y t(t = 0) +
∞∑
n=1
YR
n R−n+1 cosnθ
1 +
R2
+
R2
1− R2
+
R2
−
∞∑
n=1
Yθ
nR
−n sinnθ cot θ
))}
→ 0 as R → ∞.
We would like to point out that the choice of Y t(t = 0) is at our discretion. Next, let us
consider Flux(J4,Γ).
We have the flux expression at the axes:
Flux(J4,Γ) =
∫ t
0
∫
(−∞,−R+)∪(R+,∞)
lim
θ→0,π
2R
(
LN ′ν ′ 1
R2
∂θN
)
dRdt
=
∫ t
0
∫
(−∞,−R+)∪(R+,∞)
(
lim
θ→0,π
2 cos θ
(
1−
R2
+
R2
)(
∂tY
c −N2e−2νqac∂aY
t
)
∂cν
′
)
dRdt,
expanding out the integrand above, we have
= lim
θ→0,π
2 cos θ
(
1−
R2
+
R2
){
∂tY
R∂Rν
′ −
(
1−
R2
+
R2
)2
· R4∂Rν
′((
r2 + a2
)2 − a2∆sin2 θ
)
× ∂R
(
Y t(t = 0)− YR 1 +
R2
+
R2
1− R2
+
R2
− Yθ cot θ
)
+ ∂tY
θ∂θν
′ −
(
1−
R2
+
R2
)2
· R4∂θν
′((
r2 + a2
)2 − a2∆sin2 θ
)
× ∂θ
(
Y t(t = 0)− YR 1 +
R2
+
R2
1− R2
+
R2
− Yθ cot θ
)}
.
It may be noted that N ′θ now vanishes using the expansion of sinnθ. A term related to YR and
thus N ′R remains. Let us re-compress this term and represent the integrand as
2N ′R∂Rν
′ cos θ
(
1−
R2
+
R2
)
,
which can be re-expressed as
∂R
(
2N ′R∂Rν
′ cos θ
(
1−
R2
+
R2
))
− 2ν′∂R
(
N ′R cos θ
(
1−
R2
+
R2
))
.
Now, the flux corresponding to the second term above fortuitously combines with the remaining
flux term in (8.4) to yield 0 total flux, in view of the regularity condition (7.12). Further, the
first term is a total divergence term and converges to 0 at the boundaries of the axes (R → ∞),
in view of the asymptotic behaviour of N ′R and ν′.
Subsequently,
Flux(J4,H+) =
∫ t
0
∫ π
0
(
lim
R→R+
2LN ′ν ′µ̄qq
ab∂aN
)
dθdt
=
∫ t
0
∫ π
0
(
lim
R→R+
2LN ′ν ′µ̄q0∂RN
)
Rdθdt
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 67
=
∫ t
0
∫ π
0
(
2 sin θ
(
1 +
R2
+
R2
)(
∂tY
c −N2e−2νqac∂aY
t
)
∂cν
′
)
dθdt,
the integrand is
= 2 sin θ
(
1 +
R2
+
R2
){
∂tY
R∂Rν
′ −
(
1−
R2
+
R2
)2
· R4∂Rν
′((
r2 + a2
)2 − a2∆sin2 θ
)
× ∂R
(
Y t(t = 0)− YR 1 +
R2
+
R2
1− R2
+
R2
− Yθ cot θ
)
+ ∂tY
θ∂θν
′ −
(
1−
R2
+
R2
)2
· R4∂θν
′((
r2 + a2
)2 − a2∆sin2 θ
)
× ∂θ
(
Y t(t = 0)− YR 1 +
R2
+
R2
1− R2
+
R2
− Yθ cot θ
)}
.
The second term in the curly brackets rapidly vanishes due to the decay rate of Y θ close
to H+. The non vanishing factor in the second line above is accounted for by the vanishing
boundary condition of ∂Rν
′ at the horizon,
Flux
(
J4, ῑ
0) =
∫ t
0
∫ π
0
(
lim
R→∞
2LN ′ν ′µ̄q0∂RN
)
dθdt
=
∫ t
0
∫ π
0
(
lim
R→∞
2 sin θ
R2
+
R
(
∂tY
c −N2e−2νqac∂aY
t
)
∂cν
′
)
dθdt,
so that the integrand involves
= lim
R→∞
2 sin θ
(
1 +
R2
+
R2
){
∂tY
R∂Rν
′ −
(
1−
R2
+
R2
)2
· R4∂Rν
′((
r2 + a2
)2 − a2∆sin2 θ
)
× ∂R
(
Y t(t = 0)− YR 1 +
R2
+
R2
1− R2
+
R2
− Yθ cot θ
)
+ ∂tY
θ∂θν
′ −
(
1−
R2
+
R2
)2
· R4∂θν
′((
r2 + a2
)2 − a2∆sin2 θ
)
× ∂R
(
Y t(t = 0)− YR 1 +
R2
+
R2
1− R2
+
R2
− Yθ cot θ
)}
.
In the second term in the curly brackets decay only after using the decay rate of ∂θν
′ (because
∂θY
R does not decay near ι0
)
(J5)
b := −2N ′bµ̄qq
ac∂aν
′∂cN.
Now let us look at flux Flux(J5,Γ). We have the flux of J5 at the axes given by
Flux(J5,Γ) =
∫ t
0
∫
(−∞,−R+)∪(R+,∞)
(
lim
θ→0,π
−2R2N ′θqac0 ∂aν
′∂cN
)
dRdt
=
∫ t
0
∫
(−∞,−R+)∪(R+,∞)
(
lim
θ→0,π
−2R2
(
∂tY
θ − N2
R2
e−2ν′
∂θY
t
)
qac0 ∂aν
′∂cN
)
dRdt.
68 N. Gudapati
Now consider the integrand
= lim
θ→0,π
−2R2
(
∂tY
θ − N2
R2
e−2ν′
)
·
×
(
∂Rν
′ sin θ
(
1 +
R2
+
R2
)
+
1
R2
∂θν
′R cos θ
(
1−
R2
+
R2
))
→ 0 as θ → 0, π,
and
Flux
(
J5,H+
)
=
∫ t
0
∫ π
0
(
lim
R→R+
q0
(
−2N ′bµ̄qq
ac∂aν
′∂cN, ∂R
))
dθdt
=
∫ t
0
∫ π
0
(
lim
R→R+
−2N ′Rµ̄qq
ac∂aν
′∂cN
)
dθdt
=
∫ t
0
∫ π
0
(
lim
R→R+
−2
(
∂tY
R −N2e−2ν∂RY
t
)
µ̄qq
ac∂aν
′∂cN
)
dθdt,
the integrand contains the terms
= −2
{
∂t
( ∞∑
n→1
Y R
n RR+
n
(
Rn
Rn
+
−
Rn
+
Rn
cos θ
)
−R2
(
1−
R2
+
R2
)2 1(
r2 + a2
)2 − a2∆sin θ
)
× ∂R
(
Y t(t = 0)−
1 +
R2
+
R2
R
(
1− R2
+
R2
) ∞∑
n=1
YR
n RR+
n
(
Rn
Rn
+
Rn
+
Rn
cos θ
))
− cot θ
∞∑
n=1
Yθ
nR
n
+
(
Rn
Rn
+
+
Rn
+
Rn
)
cosnθ
}
×
(
∂Rν
′ sin θ
(
1 +
R2
+
R2
)
+
1
R2
∂θν
′R cos θ
(
1−
R2
+
R2
))
→ 0 as R → R+
due to vanishing term in the final bracket.
We are now left with the final flux term
Flux(J5, ῑ
0) =
∫ t
0
∫ π
0
(
lim
R→∞
−2
(
∂tY
R −N2e−2ν∂RY
t
)
µ̄qq
ac∂aν
′∂cN
)
dθdt
=
∫ t
0
∫ π
0
(
lim
R→∞
−2
(
∂tY
R −N2e−2ν∂R
(
Y t(t = 0)− YR∂RN/N − Yθ∂θN/N
))
×
(
R∂Rν
′ sin θ
(
1 +
R2
+
R2
)
+
1
R
∂θν
′ cos θ
(
1−
R2
+
R2
)))
dθdt.
Again the integrand can be estimated as
= lim
R→∞
−2
{
−∂t
( ∞∑
n=1
Y R
n R−n+1 cosnθ
)
−R4
(
1−
R2
+
R2
)2 1(
r2 + a2
)2 − a2∆sin θ
×
(
Y t(t = 0)−
1 +
R2
+
R2
R
(
1− R2
+
R2
) ∞∑
n=1
YR
n R−n+1 cosnθ − cot θ
∞∑
n=1
Yθ
nR
−n sinnθ
)}
×
{
R · O
(
1
R
)
O(1) +
1
R
· O
(
1
R
)
· O(1)
}
→ 0 as R → ∞,
and behaves like O
(
1
R
)
for large R. Thus it follows that
Flux
(
J5, ι
0
)
= 0.
On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes 69
Acknowledgements
I acknowledge the gracious hospitality of Institut des Hautes Études Scientifiques (IHES) at
Bures-sur-Yvette in Fall 2016. In a previous work [61], the coauthor Vincent Moncrief had
fittingly paid tribute to A. Taub, J. Marsden and S. Dain for their influence on him and for
making fundamental contributions in this direction. I take this opportunity to pay tribute to his
own outstanding contributions to general relativity and Hamiltonian methods. On an individual
note, for his mentorship, encouragement, and enjoyable interactions, I am indebted to him.
This project has spanned several years, and my gratitude is also due to all the colleagues
who have continually supported me and to the institutions that hosted me. Special thanks are
due to my host Hermann Nicolai at the Albert Einstein Institute at Golm (DFG grant no. GU
1513/2-1), where significant aspects of this work were completed. A part of this work were done
while I was a Postdoc at the Department of Mathematics, Yale University, where I benefited
from conducive working conditions. Finally, I thank the referees for the feedback.
This work was supported by the German Research Foundation (Grant numbers GU 1513/1-1
and GU 1513/2-1).
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1 Geometric mass-energy and perturbations of black holes
2 A Hamiltonian formalism for axially symmetric spacetimes
3 A Hamiltonian formalism for axially symmetric metric perturbations
4 A positive-definite Hamiltonian energy from negative curvature of the target and the Hamiltonian dynamics
5 Boundary behaviour of the dynamics in the orbit space
6 Global existence and propagation of regularity
7 Canonical phase space variables and Lagrange multipliers in the Weyl–Papapetrou gauge
8 Strict conservation of the regularized Hamiltonian H^{Reg}
References
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| id | nasplib_isofts_kiev_ua-123456789-213522 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T22:05:08Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gudapati, Nishanth 2026-02-18T11:24:08Z 2025 On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes. Nishanth Gudapati. SIGMA 21 (2025), 054, 72 pages 1815-0659 2020 Mathematics Subject Classification: 83C57; 58E20; 58E30; 35C15; 35Q75 arXiv:2507.08326 https://nasplib.isofts.kiev.ua/handle/123456789/213522 https://doi.org/10.3842/SIGMA.2025.054 The lack of a positive-definite and conserved energy is a serious obstacle in the black hole stability problem. In this work, we will show that there exists a positive-definite and conserved Hamiltonian energy for axially symmetric linear perturbations of the exterior of Kerr black hole spacetimes. In the first part, based on the Hamiltonian dimensional reduction of 3 + 1 axially symmetric, Ricci-flat Lorentzian spacetimes to a 2 + 1 Einstein-wave map system with the negatively curved hyperbolic 2-plane target, we construct a positive-definite, spacetime gauge-invariant energy functional for linear axially symmetric perturbations in the exterior of Kerr black holes, in a manner that is also gauge-independent on the target manifold. In the construction of the positive-definite energy, various dynamical terms at the boundary of the orbit space occur critically. In the second part, after setting up the initial value problem in harmonic coordinates, we prove that the positive energy for the axially symmetric linear perturbative theory of Kerr black holes is strictly conserved in time by establishing that all the boundary terms dynamically vanish for all times. This result implies a form of dynamical linear stability of the exterior of Kerr black hole spacetimes. I acknowledge the gracious hospitality of Institut des Hautes Études Scientifiques (IHES) at Bures-sur-Yvette in Fall 2016. In a previous work [61], the coauthor Vincent Moncrief had fittingly paid tribute to A. Taub, J. Marsden, and S. Dain for their influence on him and for making fundamental contributions in this direction. I take this opportunity to pay tribute to his own outstanding contributions to general relativity and Hamiltonian methods. On an individual note, for his mentorship, encouragement, and enjoyable interactions, I am indebted to him. This project has spanned several years, and my gratitude is also due to all the colleagues who have continually supported me and to the institutions that hosted me. Special thanks are due to my host Hermann Nicolai at the Albert Einstein Institute at Golm (DFG grant no. GU 1513/2-1), where significant aspects of this work were completed. A part of this work was done while I was a Postdoc at the Department of Mathematics, Yale University, where I benefited from conducive working conditions. Finally, I thank the referees for the feedback. This work was supported by the German Research Foundation (Grant numbers GU 1513/1-1 and GU 1513/2-1). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes Article published earlier |
| spellingShingle | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes Gudapati, Nishanth |
| title | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes |
| title_full | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes |
| title_fullStr | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes |
| title_full_unstemmed | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes |
| title_short | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes |
| title_sort | on axially symmetric perturbations of kerr black hole spacetimes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213522 |
| work_keys_str_mv | AT gudapatinishanth onaxiallysymmetricperturbationsofkerrblackholespacetimes |