Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations
We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України
2022
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| Цитувати: | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations. Igor Khavkine. SIGMA 18 (2022), 011, 57 pages |
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| author | Khavkine, Igor |
| author_facet | Khavkine, Igor |
| citation_txt | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations. Igor Khavkine. SIGMA 18 (2022), 011, 57 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form complicated systems of coupled linear ODEs. We outline a precise abstract strategy to decouple these systems into a sparse triangular form, where the diagonal blocks consist of spin- scalar Regge-Wheeler equations (for spins = 0,1,2). Building on the example of the vector wave equation, which we have treated previously, we complete a successful implementation of our strategy for the Lichnerowicz wave equation. Our results go a step further than previous, more ad-hoc attempts in the literature by presenting a full and maximally simplified final triangular form. These results have important applications to the quantum field theory of and the classical stability analysis of electromagnetic and gravitational perturbations of the Schwarzschild black hole in harmonic gauges.
|
| first_indexed | 2026-03-19T12:55:46Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 011, 57 pages
Explicit Triangular Decoupling of the Separated
Lichnerowicz Tensor Wave Equation on Schwarzschild
into Scalar Regge–Wheeler Equations
Igor KHAVKINE ab
a) Institute of Mathematics of the Czech Academy of Sciences,
Žitná 25, 115 67 Praha 1, Czech Republic
b) Charles University in Prague, Faculty of Mathematics and Physics,
Sokolovská 83, 186 75 Praha 8, Czech Republic
E-mail: khavkine@math.cas.cz
Received March 16, 2021, in final form January 22, 2022; Published online February 04, 2022
https://doi.org/10.3842/SIGMA.2022.011
Abstract. We consider the vector and the Lichnerowicz wave equations on the Schwarz-
schild spacetime, which correspond to the Maxwell and linearized Einstein equations in
harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete
separation of variables, the radial mode equations form complicated systems of coupled
linear ODEs. We outline a precise abstract strategy to decouple these systems into sparse
triangular form, where the diagonal blocks consist of spin-s scalar Regge–Wheeler equations
(for spins s = 0, 1, 2). Building on the example of the vector wave equation, which we
have treated previously, we complete a successful implementation of our strategy for the
Lichnerowicz wave equation. Our results go a step further than previous more ad-hoc
attempts in the literature by presenting a full and maximally simplified final triangular form.
These results have important applications to the quantum field theory of and the classical
stability analysis of electromagnetic and gravitational perturbations of the Schwarzschild
black hole in harmonic gauges.
Key words: Schwarzschild black hole; linearized gravity; harmonic gauge; Regge–Wheeler
equation; rational ODE; computer algebra; rational solution; decoupling
2020 Mathematics Subject Classification: 35Q75; 34L99; 34L05; 68W30
1 Introduction
It is textbook knowledge that a standard self-adjoint Sturm–Liouville spectral problem (pϕ′)′ −
qϕ+ω2wϕ = 0, with p, q, w > 0, has (i) a real and positive ω2-spectrum (or just real ω-spectrum)
on a natural function space, with (ii) the generalized eigenfunctions providing a resolution of
the identity. In some applications (which we describe below) it is crucial to provably know
that properties (i) and (ii) hold (or do not hold) for some equations that do not quite fit into
this standard class. Unfortunately, in those cases, the standard Sturm–Liouville theory is no
longer applicable and each case has to be studied individually. This work is dedicated to a deep
study of a specific ordinary differential equation (ODE) system that naturally appears in the
context of linearized gravity on the spherically symmetric Schwarzschild black hole, which is
non-standard in the above sense: the coefficients p, q, w are now matrices, either no longer self-
adjoint or no longer positive definite, and the spectral parameter ω also appears linearly in the q
coefficient. Despite these complications and its superficially very unstructured form, this system
turns out to be highly special. Our main result is an explicit transformation of this complicated
ODE system into a much simpler (sparse, upper triangular) form, with standard self-adjoint
Sturm–Liouville operators on the diagonal. From this simplified form, it becomes obvious that
mailto:khavkine@math.cas.cz
https://doi.org/10.3842/SIGMA.2022.011
2 I. Khavkine
properties (i) and (ii) can be proven to hold by standard theory, while as a byproduct a number
of interesting geometric features of the equations of linearized gravity Schwarzschild (under
the harmonic gauge condition) are discovered, including an alternative variational formulation,
non-trivial generalized symmetries and the existence of Debye potentials (all of these results
are concisely collected in Section 2). Our methods are rather special to this ODE system, but
having synthesized previous ad-hoc results from the literature into an abstract strategy based
on ideas from homological algebra, the geometry and algebra of differential equations, and the
theory of rational ODEs, they may also be applicable in other cases. Last but not least, we
know of no other tools that can produce any of the same results for the given ODE system.
Since the above topics do not often appear together, let alone applied to problems motivated
by linearized gravity, we have aimed for the presentation to be self-contained, though as concise
as possible. That balance and the intrinsic complexity of the ODE system under study bear
responsibility for the length of this work.
Physical motivation. The study of linear metric perturbations around a Schwarzschild
black hole was initiated in [40]. Since then, a rich literature has developed on this topic, including
extensions to other linear fields and more general kinds of black holes [10, 14, 23], with important
applications, including the modeling of gravitational wave forms [42], rigorous linear stability of
black holes [18, 46] (geared towards the nonlinear stability problem), and the study of quantum
effects around black holes [23].
Due to diffeomorphism invariance, the linearized Einstein equations require gauge fixing to
get well-posed initial value or inhomogeneous problems. Different gauge choices have different
merits. The mathematical literature on linear metric perturbations of Schwarzschild has been
dominated by the question of classical stability of the classical Cauchy problem. Rather deci-
sive, positive results have already been obtained in a gauge-invariant formalism [21], as well as
directly for the metric in a special double null-foliation gauge [17], neither approach using mode
decompositions. So why bother studying a different gauge, using mode decomposition, as we
do below? The answer is that there is a wider class of interesting questions that we can ask
about black hole perturbations, where such information is useful. For instance, applications in
quantum field theory (QFT) require an explicit construction of the linear Green function, which
is difficult to obtain other than by a mode decomposition. Also, QFT favors specific choices
of gauge, with mode stability in those gauges being a necessary condition for the existence of
a reasonable vacuum state, while stability in one gauge (favored by the classical Cauchy prob-
lem), does not imply stability in a different gauge (favored by QFT). See more precise remarks
at the end of Section 2.
In general theoretical treatments of linear metric perturbations, a common choice is the so-
called harmonic1 gauge (also known as de Donder gauge, wave (map/coordinate) gauge, or by
analogy with electrodynamics as Lorenz gauge) ∇νpµν = 0, with pµν = pµν − 1
2p
λ
λgµν the trace
reversed metric perturbation, preferred for its local regularity properties [6] and in applications
to quantum field theory [8, 12, 22]. In this gauge, the linearized Einstein equations take the
form of a tensor wave equation, the Lichnerowicz wave equation □pµν −2R(µ
λκ
ν)pλκ = 0, which
is both covariant and hyperbolic, with both properties of significant theoretical importance.
However, due to its algebraic complexity, until recently, relatively little work has been done in
this gauge for black hole perturbations, compared to Regge-Wheeler gauge and its variations.
Notable early uses of harmonic gauge include [5, 9, 25]. Following [5], this gauge (there mostly
called Lorenz ) has found important applications in the gravitational self-force literature [7].
More recently, the work [26] has used harmonic gauge in the proof of global non-linear stability
1For any background metric ḡ, the perturbed metric g satisfies the non-linear harmonic (or wave map) condition
when ∇̄µ(ϵgg
µν) = 0, where ϵg is the perturbed volume form, but ∇̄ is the background covariant derivative. This
choice of gauge also reduces the full Einstein equations to wave-like form and linearizes to our choice of gauge.
The coordinate based condition □gx
µ = 0 is only equivalent when the background metric is Minkowski [11].
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 3
of Kerr-de Sitter black holes, while in [29, 30]2 and [27, 28] the vector field method was used,
directly in real-space, to study stability and decay of perturbations of Schwarzschild, with [31]
an addendum that implicitly considers an upper triangular decoupling of the Lichnerowicz wave
equation under the transverse-traceless conditions, ∇νpµν and pλλ = 0, without referring to or
comparing with the fully diagonal decoupling of the same system by [9].
Mathematical problem. To date, despite the strong motivations listed above and the
known complete separability of the Lichnerowicz d’Alembertian, there still does not exist an
explicit mode-level construction of a harmonic gauge Green function for metric perturbations
on Schwarzschild. A necessary step in such a construction would be a determination of the
spectrum of the separated radial mode equation and a proof of completeness of its generalized
eigenfunctions. In this work, we take a significant step in that direction. Namely, we perform
a highly non-trivial simplification that allows this spectral problem to be studied by standard
Sturm–Liouville theory.
The main obstacle is that the separated radial mode equation on Schwarzschild is a rather
complicated system of ODEs. While it can be put into matrix Sturm–Liouville form, it presents
a highly non-standard spectral problem. For one, it is naturally self-adjoint only with respect to
an indefinite functional inner product. So either it becomes a self-adjoint spectral problem on
a Krein space (analogous to a Hilbert space, but with an indefinite inner product), or a non-self-
adjoint problem on a Hilbert space (where we artificially positivise the natural inner product).
Further, it presents a quadratic eigenvalue problem in the frequency ω, rather than a regular
eigenvalue problem with respect to ω2. In either case, the standard spectral theory of self-adjoint
operators on Hilbert space becomes totally inapplicable, leaving us without any obvious way to
prove (or disprove) the reality of the spectrum (absence of modes exponentially unstable in
time) or even to check the completeness of (generalized) eigenfunctions. The indefiniteness of
the functional inner product is ultimately due to the Lorentzian signature of the metric, meaning
that this complication does not occur in the analogous problem in Riemannian geometry [25].
Methodology. In this work, building on the strategy outlined and successfully implemented
in [32, 34] for the simpler example of the vector wave equation □vµ = 0, we prove that the
harmonic gauge radial mode equation can be decoupled into a triangular system, where the
diagonal blocks are the spin-s Regge–Wheeler equations, with s = 0, 1 or 2. These Regge–
Wheeler equations are then of standard scalar Sturm–Liouville form, with very well-understood
spectral properties [14, 20]. Such a decoupling essentially reduces the non-standard spectral
problem of the radial mode equation to the standard and well-understood spectral problem of
the Regge–Wheeler equation. The details of such a spectral analysis are left for to future work.
To our knowledge, prior to [32], which was our warm-up for this work, such a triangular
decoupling has never been explicitly discussed in the literature on linearized gravity. Indirect
hints of it have previously appeared only in [9] (Remark 5.10) and independently in [41] (though
only for the vector wave equation in the latter). Very recently, [27, 28] has used some of the
resulting formulas from [9]. Unfortunately, though pioneering, the original works [9, 41] were
based on rather ad-hoc, extensive, explicit computations and the full details of how the original
radial mode equations transform into the decoupled form and back are not easy to understand
from these references. We have strived to distill their overall strategy into conceptually clear
terms3 and to understand why certain key ad-hoc steps were successful, as recorded and imple-
mented in [32, 34] for the vector wave equation. In the end, our results also go a step further
2Actually, [29, 30] use a generalized harmonic/wave gauge, ∇νpµν = fµ[p]. In the earlier preprint [29] fµ[p]
was chosen to be a local zeroth order expression, while in the later published work [30] fµ[p] was chosen to be
non-local with respect to the orbits of spherical symmetry of Schwarzschild.
3It is by no means a simple task to extract a guiding strategy from [9], as it only becomes apparent as the
common pattern in the detailed and explicit calculations done for a sequence of examples of increasing complexity.
Still, that work should be credited as (to our knowledge) the first to carry out this strategy explicitly for the
Lichnerowicz wave equation, and also incidentally for the vector wave equation on Schwarzschild.
4 I. Khavkine
than [9, 41], by reducing the resulting triangular form as much as possible and proving that no
further reduction is possible (in the context of rational ODEs).
Contents overview. In Section 3, we start by reviewing a precise notion of morphism
between differential equations, as well as of equivalence up to homotopy between morphisms or
between equations (or isomorphism), where we synthesize some elementary notions from homo-
logical algebra [49], D-modules [43, Section 10.5] and the categorical approach to differential
equations [35, Section VII.5]. Essentially, a morphism is a differential operator mapping solu-
tions to solutions, but also equipped with extra data preserving the structure of the equations.4
Then, we review how a hierarchical separation a differential equations into gauge modes, gauge
invariant modes and constraint violating modes can lead to an equivalence with an equation
in block upper triangular form. Our general decoupling strategy is to use this step recursively
until a full triangular form is reached. The second step in the strategy is to simplify this initial
triangular form further by transforming as many as possible off-diagonal components to zero or
to some non-zero canonical form by the methods to be presented in Section 4. It is remarkable
that all these equations and transformations between them involve only differential operators
with rational coefficients. This review is a condensed version of the more detailed discussion
given in [32].
In Section 4, we review some tools from the study of rational solutions of rational ODEs
(ODEs with rational coefficients) and apply them to the spin-s Regge–Wheeler equations.
Namely, we reduce the problem of deciding when a triangularly coupled Regge–Wheeler sys-
tem can be made diagonal to a finite dimensional linear problem that can be easily solved with
computer algebra. Incidentally, for spins s = 0, 1, 2 that are of interest to us, we prove for the
first time that the only non-trivial rational morphisms (in the sense of Section 3) between spin-s
Regge–Wheeler equations are the identity morphisms for equal s. (In other words, there do not
exist rational spin raising and lowering operators [44] at least between radial modes of spins
s = 0, 1, 2, and also likely any other spins.) These methods were first applied in [34]. In this
work, they are further improved and refined by new results on the characterization of the coker-
nel of a rational ODE. Finally, in Section 4.1, we also discuss the Chandrasekhar transformation
as an equivalence between the s = 2 Regge–Wheeler and Zerilli equations, with special attention
to its dependence on the frequency ω.
In Section 5, we first review the complete separation of variables of the scalar, vector and
Lichnerowicz wave equations into spherical and time harmonic modes. Our notation and con-
ventions are compared with the literature in Appendix A. Then we present the resulting radial
mode equations, related differential operators and identities between them that are needed to
implement the decoupling strategy from Section 3. In Section 5.2 we recall from [32] and slightly
improve the explicit full triangular decoupling for the radial vector wave equation. In Section 5.3
we give for the first time a explicit full triangular decoupling for the radial Lichnerowicz wave
equation, which follows from the decoupling strategy of Section 3 and builds on the results of
Section 5.2. Since some of the explicit formulas needed for the even sector of the Lichnerowicz
equation are quite lengthy, they are relegated to Appendix B. In both the vector and Lichnerow-
icz wave equation cases, we improve on the original results of [9] by giving a more complete and
simpler triangular form.
In Section 2 we formulate our main results as a concise theorem. Then we state several im-
mediate qualitative corollaries, which indicate important applications of the explicit knowledge
of our decoupling formulas. Finally, these applications and other potential further developments
are discussed in Section 6.
4Those familiar with the SE = OT identity that is used prominently in Wald’s adjoint method [47] might be
comforted to know that such an identity is in fact an example of a morphism. Another example is Chandrasekhar’s
transformation (Section 4.1). Many examples of similar transformations in black hole perturbation theory are
reviewed in [24].
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 5
2 Results and applications
Here we concisely state our main result (Theorem 2.1) and then discuss several important
applications in the form of corollaries that follow straightforwardly from the main theorem. The
commentary given along the way can be seen as a guide to reading the rest of the paper. There
are of course further potential applications that require less straightforward consequences of
Theorem 2.1. They will be pursued in future works.
We must start by introducing the minimum of the notations and definitions needed to state
our results. We may be informal here, but will give pointers to the formal details in the body
of the paper.
The central objects that enter the hypothesis of our main theorem are specific linear ordi-
nary second order differential operators (equivalently, differential equations). They have matrix
coefficients (of given size) whose entries are rational functions depending on parameters, which
are M (mass), l (angular momentum quantum number) and ω (frequency). They are □0 (1× 1),
□1 o (1 × 1), □1 e (3 × 3), □2 o (3 × 3) and □2 e (7 × 7), where notation reflects that they are re-
lated to the mode separation of the tensor wave operator □ = ∇µ∇ν on the static spherically
symmetric Schwarzschild black hole (full details in Section 5) for different spins (s = 0, 1, 2)
and belong either to the even or odd sectors under antipodal reflection of the spherical orbits of
symmetry. Each operator is defined on the interval r ∈ (2M,∞), with a regular singularity at
r = 2M and an irregular one at r = ∞. The dependence on all the parameters is polynomial,
with ω considered the spectral parameter. Since the explicit form of some of these operators
is rather complicated, we write them only in the sections where they are derived, namely Sec-
tions 5.1, 5.2 and 5.3. However, each of them has the following generic form (see Section 3.1 for
our conventions regarding differential operators):
E = ∂rP (r)∂r +Q(r) + iωA(r) + ω2W (r),
where P (r), Q(r), iA(r) and W (r) are self-adjoint matrices (supposing that M and ℓ have
real values) of appropriate size, meaning that E∗ = E is formally self-adjoint (Section 3.3).
As discussed in the Introduction, the matrix W (r) is of indefinite signature (except for the 1×1
cases), meaning that it cannot be used to define a Hilbert space (only a Krein space) on which E
could define a self-adjoint unbounded operator. Even if that were possible, the dependence of E
on both ω and ω2 implies that we would have needed to consider it as a quadratic eigenvalue
problem, where verifying the reality of the ω-spectrum would have already been non-trivial.
In our main result, each of these complicated differential operators will be transformed to
simplified upper diagonal form, which involves the family of spin-s Regge–Wheeler operators
(Section 4)
Dsϕ := ∂rf∂rϕ− 1
r2
[
Bl +
(
1− s2
)
f1
]
ϕ+
ω2
f
ϕ,
where for convenience
f(r) := 1− 2M
r
, f1(r) := r∂rf(r) =
2M
r
.
The following convenient combinations of the parameters often accompany the Regge–Wheeler
operators:
Bl := l(l + 1), Al := (l − 1)l(l + 1)(l + 2) = Bl(Bl − 2), α := (12Mω)2 +A2
l .
When s = 2, the alternative Zerilli operator is often used
D+
2 ϕ := ∂rf∂rϕ− Al + 3(Bl − 2)f1(1 + 3f1) + 9f31
r2(Bl − 2 + 3f1)2
+
ω2
f
ϕ.
6 I. Khavkine
In Section 4.1 we study in detail the equivalence of D2 and D+
2 , and explain why we prefer not to
use the Zerilli operator. The Regge–Wheeler operators are in fact of standard Sturm–Liouville
type [20].
2.1 Main result
To state Theorem 2.1 below, besides the notations we have just introduced above, one also needs
the notion of equivalence between two differential equations. Informally, for our purposes, two
differential equations are equivalent when there exist differential operators that map solutions to
solutions, in both directions, and are moreover mutually inverse on the solutions. More precisely,
our notion of equivalence involves the existence of a number of auxiliary differential operators
satisfying some compositional identities, which are conveniently summarized in an equivalence
diagram such as (2.1) below. The full details of the definition and the rationale for it are given
in Section 3.1.
To prove that such an equivalence exists, it is of course sufficient to write down all the
differential operators and verify the required identities between them. That is precisely our proof
strategy, with full details presented in Section 5, with Sections 3 and 4 building up preliminary
material for it.
The main content of the theorem is that each of our differential equations of interest is equiva-
lent to a much simpler upper triangular form. To appreciate the magnitude of the simplification,
it is sufficient to compare the upper triangular forms in (2.2) with the original forms of □0 (5.1c),
□1 o (5.3d), □1 e (5.3c), □2 o (5.6d) and □2 e (5.6c).
Theorem 2.1. Let E be one of the □0 , □1 o, □1 e, □2 o or □2 e operators, representing the sys-
tems of radial mode equations for the scalar (Section 5.1), vector (Section 5.2) or Lichnerowicz
(Section 5.3) wave operators on the Schwarzschild black hole of mass M , mode separated as
described in Section 5, in either the odd or even sector. Then the following is true:
(i) With respect to the Schwarzschild radial coordinate r, E is a rational ODE, with singular
points only at r = 0, 2M,∞. E also depends polynomially on the frequency ω and angular
momentum Bl = l(l + 1) spectral parameters, as well as the mass parameter M .
(ii) There exists a rational ODE system Ẽ in upper triangular form and an equivalence diagram
(Section 3)
• •
• •
E
k
k̄
Ẽ
k′
h
k̄′
h̃ , (2.1)
such that the diagonal of Ẽ consists of Regge–Wheeler operators Ds (up to constant, ω-
and Bl-dependent factors), while the equivalence maps are rational differential operators
of order at most 1 and poles only at r = 0, 2M .
(iii) The equivalence
(
k, k′, k̄, k̄′
)
and homotopy
(
h, h̄
)
operators in (2.1) depend rationally on ω
and Bl, with poles only at ω = 0 and Bl = 0, 2, respectively, with one exception. The
exception is the case E = □2 e, where some of these operators have additional poles at
the algebraically special frequencies ω = ±i Al
12M . In that case, using the Chandrasekhar
transformation (Section 4.1) to replace the Regge–Wheeler operator D2 by the Zerilli ope-
rator D+
2 in Ẽ removes the poles at the algebraically special frequencies at the price of
introducing factors of (Bl−2+3f1) in some of the denominators in the equivalence diagram.
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 7
(iv) The equivalence diagram (2.1) has finite limit as M → 0. Ẽ becomes diagonal in that limit.
(v) The non-vanishing off-diagonal elements of Ẽ can all be put into the form ∆0(r)
f +∆1(r)∂r,
with ∆0(r) and ∆1(r) uniformly bounded on r ∈ (2M,∞).
(vi) The rows and columns of Ẽ can be permuted such that it remains upper triangular and
block diagonal with respect to blocks of equal s of the Regge–Wheeler operators Ds. The
most general rational automorphism of Ẽ takes the form ẼA = AẼ, where A is a constant
matrix that is block diagonal with respect to equal spin blocks and is upper triangular within
each such block, with the single exception of the s = 0 block when E = □2 e.
(vii) There exists a constant matrix Σ, with Σ2 = id and Σ∗ = Σ (Section 3.3), such that ΣẼ
is formally self-adjoint,
(
ΣẼ
)∗
= ΣẼ.
In each of these cases, the upper triangular Regge–Wheeler system has the following form:
(a) when E = □0 , then
Ẽ =
1
ω2
D0, (2.2a)
(b) when E = □1 o, then
Ẽ =
Bl
ω2
D1, (2.2b)
(c) when E = □1 e, then
Ẽ =
1
ω2
D0 0 −f1
r2
(
Bl +
1
2f1
)
0 BlD1 0
0 0 D0
, (2.2c)
(d) when E = □2 o, then
Ẽ = − 2
ω2
BlD1 0 1
3B
2
l
f1
r2
0 AlD2 0
0 0 BlD1
, (2.2d)
(e) when E = □2 e, then
Ẽ = − 2
ω2
D0 0 −f1
r2
(
Bl+
1
2f1
)
0 f1
r2
(
Bl+
1
2f1
)
0
f2
1
8r2
(7Bl+2)
0 BlD1 0 0 0 −5
3B
2
l
f1
r2
0
0 0 D0 0 0 0 f1
r2
(
Bl+
1
2f1
)
0 0 0 αAlD2 0 0 0
0 0 0 0 D0 0 −f1
r2
(
Bl+
1
2f1
)
0 0 0 0 0 BlD1 0
0 0 0 0 0 0 D0
. (2.2e)
Proof. Section 5 gives a detailed presentation of the explicit formulas for the operators in
the equivalence diagram (2.1), thus proving (ii), for each case being considered: (5.2) =⇒ (a),
(5.5) =⇒ (b), (5.2.2) =⇒ (c), (5.10) =⇒ (d), (5.12) =⇒ (e). The rest of the theorem is simply
a summary, collected here in a way convenient for future reference, of properties of these opera-
tors that can be gleaned from their explicit formulas. ■
8 I. Khavkine
2.2 Spectral problem
One might notice that the explicit form (2.2) of the triangular Regge–Wheeler systems in Theo-
rem 2.1 gives much more detailed information about the off-diagonal elements than point (v) of
Theorem 2.1. But the reason we have phrased it like that is to draw attention to the following
almost immediate
Corollary 2.2. Let Ẽ be one of □0 , □1 o, □1 e, □2 o and □2 e from Theorem 2.1. Then, suppo-
sing that Ẽ is an n × n operator matrix, the bounded inverse Ẽ−1 : L2(2M,∞; f dr)⊕n →
L2
(
2M,∞; drf
)⊕n
exists for any value of ω for which the bounded inverse D−1
s : L2(2M,∞; f dr)
→ L2
(
2M,∞; drf
)
also exists.
Proof. The corollary follows from a direct generalization to larger operator matrices of Re-
mark 4.7, about the relative boundedness of off-diagonal matrix elements with respect to the
Regge–Wheeler operators on the diagonal, which here follows from Theorem 2.1(v). ■
As point out in Remark 4.7, the novel results of Section 4 concerning the cokernel of the
differential operator in (4.4) are crucial for guaranteeing the possibility of choosing the off-
diagonal elements to be relatively bounded while simplifying an upper triangular Regge–Wheeler
system. In practice, we have found that at the initial stages of this simplification, which is part of
the general strategy of Section 3, the off-diagonal elements were generally not relatively bounded.
Using only the results from [34] about only the solutions of (4.4), or other off-the-shelf tools
from the literature, it would have been impossible to decide a priori whether those off-diagonal
elements that cannot be chosen to be zero could be chosen to be relatively bounded.
From Corollary 2.2 one can essentially conclude that all the properties of the resolvent Ẽ−1
(like its dependence on ω), and by equivalence hence also of E−1, can be deduced from the
properties of the resolvent D−1
s , which has been very well studied. In particular, we can conclude
that the ω-spectrum of each E in Theorem 2.1 is purely real, as it is for Ds [20], once the
relevant function spaces are chosen to respect the equivalences that we have constructed. This
information is an important starting point for the integral representation of the solutions of the
corresponding Cauchy problem and of the large time asymptotics of such solutions [20]. It is
also crucial for the explicit construction of propagators in quantum field theory [8, 12, 22].
We now make the above reasoning slightly more precise. When E is one of □0 , □1 o, □1 e, □2 o
or □2 e, both it and its decoupled triangular form Ẽ are rational ODEs with a regular singular
point at r = 2M and an irregular singular point at r = ∞. Hence, for each system, and at
each singular point, using the methods of asymptotic analysis of ODEs [48], each solution can
be uniquely identified by its asymptotics. Being bijective on solutions, the equivalence mor-
phisms k, k̄ of Theorem 2.1 must hence be bijective on these asymptotics. Moreover, since these
morphisms are differential operators, they can be applied to the asymptotics directly, without
knowledge of the full solution. This means that if we know the connection coefficients between
the singular points at r = 2M and r = ∞ (essentially the transmission and reflection coefficients
for incoming and outgoing waves) at frequency ω for Ẽ, the equivalence morphisms transfer them
to the corresponding connection coefficients for E. This argument of course works only when
the equivalence morphisms are well-defined, which excludes l = 0, 1 in some cases and ω = 0,
as well as the algebraically special ω = ±i Al
12M in the case E = □2 e. But these excluded cases
can be studied individually. For instance, for the algebraically special frequencies, it is sufficient
to replace the Regge–Wheeler operator D2 with the Zerilli operator D+
2 in □̃2 e, as pointed out
in Theorem 2.1(iii). For ω = 0 and l = 0, 1 one can get more information by studying the
coefficients of the Laurent expansion of the equivalence morphisms. Thus, we arrive at
Corollary 2.3. For ω ̸= 0 and l ≥ 2, the equivalence morphisms k, k̄ from Theorem 2.1 are
bijective on solution asymptotics of each E, Ẽ pair near r = 2M and r = ∞. Hence, after
bijectively identifying their solutions, the connection coefficients for E and Ẽ are equal.
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 9
2.3 Symmetries and potentials
Each of Section 5.1, 5.2 and 5.3 starts with separation of variables on the static spherically sym-
metric Schwarzschild black hole background, which turns spacetime partial differential operators
into ordinary differential operators acting on radial mode functions. The time derivative i∂t is
replaced by the frequency parameter ω, while the spherical Laplacian DAD
A is replaced by the
angular momentum parameter −Bl (shifted by a constant depending on the tensor rank of its
argument), among other substitutions. Not every operator on radial mode functions comes from
such a separation of variables, but it is straightforward to work out the rules for recognizing
those operators that do. Thus, directly from the statement of Theorem 2.1, we can also draw
conclusions about spacetime partial differential operators.
The first conclusion is about symmetries. A symmetry of a differential equation E[u] = 0 is
a pair of differential operators S, S′ such that ES = S′E (Section 3.1). Such symmetries are
extremely important in understanding the structure of a differential equation [39].
The second conclusion is about a variational formulation [39]. A formally self-adjoint differ-
ential operator E = E∗ (Section 3.3) is the Euler–Lagrange equation of a variational principle∫
⟨u,E[u]⟩ dx. Conversely, the Euler–Lagrange equation of a variational principle is always for-
mally self-adjoint. For linear differential equations the two notions are essentially equivalent.
Having a variational formulation leads to the possibility of using Noether’s theorem to convert
symmetries to conservation laws. Even more interesting is when the same equation has more
than one variational principle. This is the case for a self-adjoint equation E[u] = 0 when there
exists another self-adjoint equation Ẽ[v] = 0, which are equivalent in a non-trivial way (E and Ẽ
are not just multiples of each other). Multiple variational formulations can lead to interesting
integrability properties and bi-Hamiltonian structures.
Corollary 2.4.
(a) It is easy to see that, reversing the mode separation performed in Section 5, each the tri-
angular systems □̃1 o, □̃1 e, □̃2 o and □̃2 e constitutes the radial mode equation of a triangularly
coupled system of scalar spacetime differential equations. The diagonal components of this
system consist of the spacetime version of Regge–Wheeler wave equations, possibly multi-
plied by a polynomial in the spherical Laplacian DAD
A and i∂t with constant coefficients.
(b) By similar arguments, the morphisms k, k′ and k̄, k̄′ from (2.1) can be promoted to space-
time differential operators, possibly after being multiplied by a constant coefficient polyno-
mial in Bl and ω of sufficiently high degree. These spacetime differential operators will
effect a morphism between the original harmonic gauge tensor wave equation and the tri-
angular system of Regge–Wheeler wave equations from (a).
(c) The systems of Regge–Wheeler wave equations from (a) are self-adjoint on spacetime after
composition with the Σ operator from Theorem 2.1(vii). Hence they possess a variational
formulation.
Remark 2.5. Note that Corollary 2.4 cannot hold for the Zerilli form of □̃2 e mentioned in
Theorem 2.1(iii), because no spacetime differential operator has a radial mode decomposition
with Bl-dependent factors like (Bl − 2 + 3f1) in the denominator.
A Debye potential is a differential operator that maps solutions of a simple auxiliary PDE
equation (typically a scalar wave equation with a potential) to solutions of another PDE of inter-
est (typically a more complicated tensor wave equation), with the usual additional requirement
that the image solutions are non-trivial (typically not pure gauge, in an appropriate sense).
In the terminology of Section 3, a Debye potential a special kind of morphism between PDEs.
Debye potentials for Maxwell and linearized Einstein equations, on both flat and some curved
10 I. Khavkine
backgrounds (including black hole backgrounds), have been known for some time, but have also
gained more attention recently [3, 4, 45, 47]. In the case of black hole backgrounds, the Debye
potentials for Maxwell and linearized Einstein equations typically give solutions in some version
of radiation gauge [45, 47], which is adapted to the algebraically special nature of these back-
grounds, but may not be optimal for other purposes. The recent work [30] produced a Debye
potential in a (non-local) generalized harmonic gauge. Of course, a solution in one gauge can
be (at least locally) transformed into any other gauge, but it is by no means obvious when such
a transformation can be done by a differential operator. With that in mind, we draw attention
to the next corollary (first implicitly obtained in [9]):
Corollary 2.6. On the exterior spacetime of a Schwarzschild black hole, there exist spacetime
Debye potentials generating solutions of (a) Maxwell equations in harmonic (Lorenz) gauge and
(b) linearized Einstein equations in traceless harmonic (de Donder) gauge, starting from solutions
of Regge–Wheeler wave equations of appropriate spin.
Proof. From Theorem 2.1, it is clear that the spin s = 1 components decouple from the other
components in □̃1 o and □̃1 e, so they satisfy independent Regge–Wheeler equations. Hence, the
corresponding columns of the k̄1o and k̄1e operators, once promoted to spacetime differential
operators as in Corollary 2.4, are Debye potentials for the vector wave equation □1 . However,
the composition properties of k1o, k1e with T1o, T2e (as computed in Sections 5.2.1 and 5.2.2)
imply that the image of the Debye potential is annihilated by the spacetime operator T1. In other
words, the Debye potentials actually generate solutions of Maxwell equations in harmonic gauge.
For linearized Einstein equations, the discussion is completely analogous, starting with the
Debye potentials for the Lichnerowicz wave equation generated by solutions of Regge–Wheeler
equations of spin s = 2. In that case, the composition properties with T2 and tr operators (as
computed in Sections 5.3.1 and 5.3.2) imply that the image of these Debye potentials is both
harmonic and trace free. ■
3 Formal properties of differential equations and operators
3.1 Equations and morphisms
For the purposes of this work, a (partial) differential operator, say E, is always linear, with
smooth coefficients, which we will write as E[u], where u is a possibly vector valued function.
Later on we will even specialize to ordinary differential operators with rational coefficients, but
all abstract statements expressed in terms of operator identities or commutative diagrams will
be valid at the greater level of generality. Compositions of differential operators will be written
E1 ◦ E2 or just E1E2 when no confusion could arise.
An operator E can always be thought of as a matrix of scalar differential operators acting
on the components of u. By a scalar differential operator, we mean an operator that takes
possibly vector valued functions into scalar valued functions (corresponding to a matrix with
a single row). Scalars are real or complex numbers (for the sake of generality we will generally
allow complex scalars). Differential operators could be of any order, including order zero, which
just corresponds to multiplication by some matrix valued function. While the operators can
be thought of as acting on smooth (vector valued) functions, we will be mostly concerned
with composition identities among operators, and so we will not bother specifying precisely
the domain or codomain of each operator. This information can always be deduced from the
context (e.g., the size of the matrix). Also, for the abstract discussion below, it is also not
necessary to fix the number of independent variables, but in the rest of this work we will be
mostly concerned with applications to ordinary differential operators (i.e., acting on functions
of a single independent variable).
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 11
Quick example: the operator E = r−2∂rr
2 should be interpreted as E[u] = r−2
(
∂r
(
r2u
))
=
∂ru+ 2u/r, so alternatively it can be rewritten as E = ∂r + 2/r.
Below, we state some basic definitions and results concerning differential equations and mor-
phisms between them, which essentially correspond to differential operators that map solutions
to solutions. The presentation is logically self-contained and does not extend far beyond what
is needed in the rest of the paper. We generalize and streamline slightly the presentation pre-
viously given in [32, Section 2], which can be consulted for a more pedagogical exposition. For
proper context, these ideas can be seen as simple special cases of concepts coming from the more
general frameworks of D-modules [43, Section 10.5], the category of differential equations [35,
Section VII.5] or homological algebra [49]. We will not delve into the precise connection with
these larger frameworks, but it is useful to mention that all arrows/morphisms need to be re-
versed when our statements are interpreted in the language of D-modules.
Central objects of our attention are differential equations, like E[u] = 0. Sometimes we will
refer just to the operator E as the equation itself and vice versa. This should not lead to any
confusion. Technically, it becomes convenient to consider not just an equation E(0), but also
a set of Noether (or compatibility) identities that comes with it, namely an operator E′ such
that E′ ◦ E(0) = 0. One can equally consider higher stage Noether identities, for example an
operator E′′ such that E′′ ◦ E′ = 0, etc. Such a sequence of operators E(0), E′, E′′, . . . , E(n),
is also called a complex (a term from homological algebra [49]), but we might as well still refer
to it as a differential equation, albeit equipped with the extra data of Noether and higher
stage Noether identities. We get back to a notion of solutions via the idea of cohomology
H(n)(E) := kerE(n)/ imE(n−1) of the complex, which may be non-vanishing in any location
and can be interpreted as the space of solutions of E(n)[u] = 0 modulo a gauge symmetry
generated by u ∼ u+E(n−1)[v]. The complex is said to be exact (in location n) if H(n)(E) = 0.
When E(n−1) = 0 or is absent, by definition imE(n−1) = 0, meaning that H(n)(E) = kerE(n)
coincides with the usual notion of solutions. Note however that the precise spaces of solutions
and cohomologies depend on the choice of the function space on which our differential operators
act. The various transformations and constructions involving complexes that we introduce below
are designed to preserve cohomologies, hence also solutions, once the function spaces have been
chosen, which is the main reason that we abstract from solutions themselves and work only at
the level of complexes of differential operators.
The next most important concept is that of a morphism between two differential equations.
Referring again to terminology from homological algebra, a morphism for us will be a cochain
map between the complexes corresponding to the two equations. In more detail, given two com-
plexes E(0), E′, . . . and F (0), F ′, . . ., a cochain map consists of differential operators k(0), k′, . . .
with domains and codomains prescribed by the diagram in Figure 1(a), which also needs to
be commutative (any sequence of operators starting and ending at the same point must com-
pose to the same thing). We have denoted by bullets the appropriate function spaces, which
are less important to us than the structure and properties of the differential operators acting
between them. When needed, e.g., for the purposes of defining the cohomology or for defining
a cochain map between complexes of different lengths, any complex can be freely extended by
zero maps in both directions. This explains the extra zero maps appended and prepended to
the complexes in Figure 1(a). But since this can be done implicitly, we will not show such
extensions and rather use more compact diagrams like the rest of Figure 1. When a complex
consists of a single operator E (E′ = E′′ = · · · = 0, and we drop the now unnecessary su-
perscript (0)), the cohomology has the obvious identification H(E) = kerE/ im 0 = kerE and
H ′(E) = ker 0/ imE = cokerE. Once again, this highlights the interpretation of a complex as
a generalization of a linear differential equation and of its cohomology as generalizing the space
of solutions. It is not hard to check that the defining properties of a morphism (cochain map)
k, k′, . . . imply that it descends to well-defined maps in cohomology k(n) : H(n)(E) → H(n)(F ).
12 I. Khavkine
0 0
• •
• •
• •
• •
• •
0 0
E(0)
k(0)
F (0)
E′
k′
F ′
...
k′′
...
E(n)
k(n)
F (n)
k(n+1)
• •
• •
• •
• •
• •
E(0)
k(0)
F (0)
E′
k′
h(0)
F ′
...
k′′
h′
...
E(n)
k(n)
F (n)
k(n+1)
h(n)
• •
• •
• •
• •
• •
E(0)
k(0)
k̄(0)
Ẽ(0)
k′
h(0)
E′
k̄′
h̃(0)
Ẽ′
k′′
h′
...
k̄′′
h̃′
...
k(n)
E(n)
k̄(n)
Ẽ(n)
k(n+1)
h(n)
k̄(n+1)
h̃(n)
(a) (b) (c)
Figure 1. Diagrams of morphisms, homotopies and equivalences (see text).
In the simplest example of a single operator, this means that k : kerE → kerF maps solutions
of E[u] = 0 to solutions of F [v] = 0 (also k′ : cokerE → cokerF in this case). An automorphism
is a morphism from a given equation to itself, in which case the operator k mapping solutions
of the equation to solutions is also known as a symmetry operator.
A morphism is induced by a homotopy if there exists a homotopy, that is, a sequence of
operators h(0), h′, . . . (again, extended by zero operators in either direction, as necessary) fitting
into the morphism diagram in Figure 1(b), such that each horizontal arrow satisfies the formula
k(n) = h(n) ◦ E(n) + F (n−1) ◦ h(n−1),
which is indeed a morphism. It is a straightforward exercise to check that a morphism induced
by a homotopy always descends to the zero map in cohomology. Two morphisms k
(0)
1 , k′1, . . .
and k
(0)
2 , k′2, . . . are equivalent up to homotopy when the difference
(
k
(0)
2 − k
(0)
1
)
, (k′2 − k′1), . . . is
induced by a homotopy. Two equations are equivalent up to homotopy if there exists morphisms
between them that are mutually inverse up to homotopy (their compositions are equivalent to
the identity morphisms up to homotopy). All the relevant operators can be exhibited in an
equivalence diagram, like in Figure 1(c), where the solid arrows commute and the homotopy
corrections enter the identities
k̄(n) ◦ k(n) = id− h(n) ◦ E(n) − E(n−1) ◦ h(n−1),
k(n) ◦ k̄(n) = id− h̃(n) ◦ Ẽ(n) − Ẽ(n−1) ◦ h̃(n−1). (3.1)
Equivalence up to homotopy, both for operators and equations, of course defines an equivalence
relation.
Remark 3.1. Unwinding the definition of a morphism k between two single operator complexes
corresponding to the equations E[v] = 0 and F [w] = 0, we find that when E[v] = 0 and w =
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 13
k(0)[v], then F [w] = F
[
k(0)[v]
]
= k(1)[E[v]] = 0. Hence, as promised, in this case, a morphism
induces a map from solutions to solutions, which is a familiar notion in PDE theory. Similarly,
a symmetry (or automorphism) in our sense induces a map from solutions to solutions of a
given equation. Extending our earlier example by an inverse morphism k̄ up to homotopy
implies a bijection between the solutions of E[v] = 0 and F [w] = 0, since then k̄(0)
[
k(0)[v]
]
=
v − h(0)[E[v]] = v and k(0)
[
k̄(0)[w]
]
= w − h̃(0)[F [w]] = w. But even more than that, an
equivalence allows us to transfer the solutions of inhomogeneous problems between equations,
namely E[v] = u is solved by v = k̄(0)[w] + h(0)[u] if w solves F [w] = k(1)[u], which can be
checked by direct calculation.
Essentially, our notion of morphism is the same as the usual PDE notion of a map between
solution spaces induced by a differential operator, but equipped with extra structure. For some
equations that are not in some sense pathological, the extra operators k(n) may be reconstructed
(perhaps up to homotopy equivalence) from the knowledge of a single operator k(0). Then, the
above two notions actually coincide. See [32] for a bit more discussion on this point. Considering
subtle cases when differences between the two notions actually arise is beyond the scope of this
work. We will restrict ourselves to using the more structured notion of morphism, which is
happens to be most convenient for our work.
3.2 Triangular decoupling strategy
In this section, we outline our general decoupling strategy. We basically summarize the more
pedagogical discussion from [32]. There are two small differences. We use slightly different nota-
tion, more adapted to the later needs of Section 5. Also, we take advantage of having defined, in
the preceding section, morphisms for complexes of possibly more than one differential operators.
Part of the discussion in [32] was unnecessarily awkward because we insisted on using morphisms
only with single differential operators (namely, the homotopy corrections on the right-hand side
of (3.1) did not have the same uniform structure because of the truncation).
The input is a differential equation E[u] = 0, together with the following morphisms
• •
• •
DD
D
E
D′
,
• •
• •
E
T
DT
T ′
,
• •
• •
E
T
Φ
DΦ
[
Φ′ −∆ΦT
]
(3.2)
from/to “simpler” equations DD, DT , DΦ. The end result is an equation in block upper tri-
angular form with DD, DΦ and DT on the diagonal, where both the diagonal blocks and the
off-diagonal ones have also been simplified as much as possible, keeping upper triangular struc-
ture, as shown in (3.4). For our purposes, we may refer to the degrees of freedom captured by
the D operator as pure gauge modes, those by the Φ operator as gauge invariant modes, and
those by the T operator as constraint violating modes. This terminology will be justified in
how the strategy will be applied in Section 5 to Maxwell and linearized Einstein equations in
harmonic gauge (see also the detailed discussion in [32]). At the abstract level, we have simply
introduced convenient names to the degrees of freedom corresponding to the 3 sectors of the
eventual block upper triangular decoupling Ē in (3.4).
The first non-trivial step is to find an equivalence diagram, illustrated in (3.3a), between DD
(the residual gauge equation) and the joint system E[u] = 0, Φ[u] = 0 (vanishing of gauge
invariant modes), T [u] = 0 (gauge fixing condition). That is, we must complement the input
14 I. Khavkine
differential operators already introduced in (3.2) by a bunch of new operators that are defined
by their relation to the following diagram being an equivalence up to homotopy:
• •
• •
• •
E
Φ
T
D̄
DD
D
−Φ′ DΦ ∆ΦT
−T ′ 0 DT
[
D̄′ −∆DΦ −∆DT
]
[
hE Φ̄ T̄
]
0
D′
HΦ
HT
hD
0
−Φ̄′ −T̄ ′
hΦΦ hΦT
hTΦ hTT
0
(3.3a)
supplemented by two more operators H̄Φ, H̄T satisfying the relation
D̄
[
hE Φ̄ T̄
]
− hD
[
D̄′ −∆DΦ −∆DT
]
=
[
H̄Φ H̄T
] [−Φ′ DΦ ∆ΦT
−T ′ 0 DT
]
,
[
D̄′ −∆DΦ −∆DT
] −Φ̄′ −T̄ ′
hΦΦ hΦT
hTΦ hTT
= DD
[
H̄Φ H̄T
]
.
(3.3b)
If such a diagram does not exist, than our strategy cannot proceed directly. In this work, we
will only consider cases where this step of the strategy succeeds.
Remark 3.2. There are a few potential points of failure, which should all be checked.
First, the gauge-invariant and constraint violating modes should be compatible and inde-
pendent from the pure gauge modes, namely
[
Φ
T
]
◦ D ∼ 0 up to homotopy. This condition is
captured by the following subset of identities from (3.3a):[
Φ
T
]
D =
[
HΦ
HT
]
DD,
[
Φ′
T ′
]
D′ =
[
DΦ ∆ΦT
0 DT
] [
HΦ
HT
]
DD.
Strictly speaking, to justify the adjective gauge-invariant, we should have Φ ◦D = 0 (HT = 0),
but here we are allowing Φ to mix with E and T , that is, terms that vanish on-shell or by the
gauge-fixing condition. In practive, within this work we will always be able to make an initial
choice of Φ such that HT = 0, and then update it to fit our needs while maintaining the above
relations.
Note that we have chosen to represent the E-Φ-T equation in diagram (3.3a) by a complex
of length 2, with the second operator representing Noether identities for the first one. If we
do not choose this complex correctly, then the construction of the equivalence diagram may be
obstructed, for instance when some Noether identity for the first operator is not a consequence of
the rows of the second operator. When such obstructions are absent, the existence of operators
H̄Φ, H̄T in (3.3b) actually follows from the relations already exhibited by (3.3a). In this work,
we will only encounter cases where both sides of the equivalence diagram (3.3a) are correctly
chosen, but to save space we simply postulate the factorizations (3.3b) separately.
Finally, another point of failure could be that the T [u] = 0 and Φ[u] = 0 conditions are
simply not strong enough to reduce the number of remaining degrees of freedom to equal that
of the DD equation. Again, in this work we will only encounter cases where this step succeeds.
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 15
• • •
• • •
• • •
E
id
Φ
T
E 0 0
Φ −id 0
T 0 −id
0 DΦ ∆ΦT
0 0 DT
[
id 0 0
]
D̄ 0 0
0 id 0
0 0 id
DD ∆DΦ ∆DT
0 DΦ ∆ΦT
0 0 DT
D Φ̄ T̄
0 id 0
0 0 id
0
id
0
0
Φ′
T ′
0
[
id 0 0 0 0
]
−Φ′ DΦ ∆ΦT id 0
−T ′ 0 DT 0 id
D̄′ −∆DΦ −∆DT 0 0
0 0 0 id 0
0 0 0 0 id
0 0 0 0 0
0 −id 0 0 0
0 0 −id 0 0
hE Φ̄ T̄ 0 0
0 0 0 0 0
0 0 0 0 0
0
D′ Φ̄′ T̄ ′
HΦ −hΦΦ −hΦT
HT −hTΦ −hTT
0 id 0
0 0 id
hD −H̄Φ −H̄T
0 0 0
0 0 0
0 0
0
0 0
0 0
0 0
id 0
0 id
−Φ̄′ −T̄ ′
hΦΦ hΦT
hTΦ hTT
0 0
0 0
0
Figure 2. Concatenation of diagrams of morphisms showing the equivalence between E[u] = 0 and
a system of equations in block upper triangular form.
Next, using only operators that have already been introduced above, together with their
defining properties, we can construct two more equivalence diagrams that are concatenated and
explicitly shown in Figure 2. Composing the morphisms in Figure 2, in both directions, we obtain
the much more compact equivalence with an upper triangular form, which we denote by Ẽ,
• •
• •
E
D̄
Φ
T
Ẽ=
DD ∆DΦ ∆DT
0 DΦ ∆ΦT
0 0 DT
[
D Φ̄ T̄
]
D̄′
Φ′
T ′
hE
[
D′ Φ̄′ T̄ ′]
hD −H̄Φ −H̄T
−HΦ hΦΦ hΦT
−HT hTΦ hTT
. (3.4)
16 I. Khavkine
Conversely, knowing the final equivalence diagram (3.4) of E with the decoupled form Ẽ, simply
by placing individual operators in the appropriate places, we can reproduce the equivalence (3.3a)
of the E-Φ-T system with DD and the relations (3.3b)
The triangular system Ẽ may still be rather complicated. So the strategy proceeds with two
kinds of simplifications: simplifying the diagonal blocks, and simplifying the off-diagonal blocks
(cf. [32, Sections 2.2–3]). The first kind is summarized in the following
Lemma 3.3. If there exists an equivalence diagram between equations E2 and Ē2, then it implies
the following equivalence diagram for an upper triangular system, where E2 is the middle block:
• •
• •
E2
k
k̄
Ē2
k′
h
k̄′
h̄
=⇒
• •
• •
E1 ∆12 ∆13
0 E2 ∆23
0 0 E3
id 0 0
0 k 0
0 0 id
id 0 0
0 k̄ −h∆23
0 0 id
E1 ∆̄12 ∆̄13
0 Ē2 ∆̄23
0 0 E3
id −∆12h 0
0 k′ 0
0 0 id
0 0 0
0 h 0
0 0 0
id 0 0
0 k̄′ 0
0 0 id
0 0 0
0 h̄ 0
0 0 0
,
where ∆̄12 = ∆12k̄, ∆̄23 = k′∆23 and ∆̄13 = ∆13−∆12h∆23, provided the extra identity kh = h̄k′
also holds (though it is only needed to verify the form of the barred homotopy operator on the
right).
The proof is by direct calculation. Though, we should note that when E2 has no non-
trivial Noether identities (for example, in the ODE case, when it can be solved for all highest
derivatives), then the extra identity on the homotopy operators is automatic. Indeed, then(
kh− h̄k′
)
E2 =
(
k − hĒ2k
)
− (k − khE2) =
(
kk̄
)
k − k
(
k̄k
)
= 0,
and since E2 has no non-trivial Noether identities, we must have kh− h̄k′ = 0.
Note that Lemma 3.3 remains valid when either of the untransformed diagonal blocks E1
or E3 is zero dimensional (a 0× 0 matrix, or just absent, in other words).
The second kind of simplification can be achieved by applying the next
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 17
Lemma 3.4. For any operators δ and δ′ the following equivalence diagram holds:
• •
• •
E1 ∆
0 E2
id δ
0 id
id −δ
0 id
E1 ∆̄
0 E2
id δ′
0 id
0 0
0 0
id −δ′
0 id
0 0
0 0
,
where ∆̄ = ∆− E1δ + δ′E2.
The proof is again by direct calculation. This Lemma implies that by a clever choice of δ
and δ′ it might be possible to set ∆̄ = 0, thus decoupling the two diagonal blocks, or at least set
it some preferred canonical form. For instance, in the case when E1 and E2 are ODEs that can
be solved for their higher derivatives, say respectively of orders n1 and n2, then by an obvious
choice of δ and δ′ the off-diagonal term ∆̄ can be reduced to order min(n1, n2).
Our decoupling strategy proceeds by applying the last two lemmas recursively, repartitioning
our triangular equation into blocks as necessary, until no further simplification is possible or
practical. In the successful applications of this strategy in Section 5 we stop when the system Ẽ
has been reduced to fully upper triangular form, where each operator on the diagonal is a spin-s
Regge–Wheeler operator (Section 4), while off-diagonal terms are mostly zero, with those that
are non-vanishing expressed in terms of a small number of canonical representatives. We dedicate
the following section (Section 4) to giving an effective method of deciding when Lemma 3.4 can
be used to cancel (or at least “maximally simplify”) specific off-diagonal terms in a triangular
system with Regge–Wheeler operators on the diagonal.
3.3 Formal adjoints
Given a local sesquilinear scalar paring ⟨−,−⟩ =
∫
dx ⟨−,−⟩x, with ⟨−,−⟩x the corresponding
pointwise non-degenerate pairing, the formal adjoint E∗ of a differential operator E is defined by
the usual relation ⟨E∗[v], u⟩ = ⟨v,E[u]⟩ for smooth compactly supported test functions u and v.
If an operator satisfies E∗ = E, then it is (formally) self-adjoint. As defined, formal adjoints
always exist and are unique. In local coordinates they may be computed by the standard rules
(EF )∗ = F ∗E∗, ∂∗
xi = −∂xi and f(x)∗ for a multiplicative matrix operator coinciding with its
usual pointwise adjoint with respect to ⟨−,−⟩x.
On a manifold with a metric, for tensor fields v and u, we define the paring ⟨v, u⟩ to be the
integral with respect to the metric volume form of the metric contraction of v∗ (the complex
conjugate of v) and u. We use this convention in Section 5, both for the spacetime metric 4gµν
on M = R2 × S2 and for the metric gab on the radio-temporal R2-factor.
In Sections 4 and 5, we will also be dealing with ordinary differential operators with respect
to the Schwarzschild radial variable r. In that context, we take the pairing ⟨v, u⟩ =
∫∞
2M v∗u dr,
where u is a column vector valued function and v∗ is the conjugate transpose of a column vector
valued function.
When the operators depend on spectral parameters, like ω and l, they are considered to be
real, so that ω∗ = ω and l∗ = l. However, that does not prevent us from considering complex
values of these parameters once all the adjoints have been computed.
18 I. Khavkine
4 Morphisms and extensions of Regge–Wheeler equations
In this section, we present the theory needed for simplifying the off-diagonal elements of rational
ODE systems in upper triangular form. We first concisely summarize some necessary notions
and results from [34] (where a more detailed and pedagogical treatment can be found) and then
extend them. In this section, we will work exclusively with rational differential operators.
The following functions will appear in the Schwarzschild metric and its derivatives:
f(r) = 1− 2M
r
, f1(r) = r∂r
2M
r
.
Let also l, s and ω be real (or more generally complex) constants, with Bl = l(l+1) a convenient
notation. Throughout this work, we use only ∂r to denote differentiation, while primes are used
only as decorations (to distinguish two objects from each other, like to operators k and k′).
Define the (generalized) spin-s Regge–Wheeler operator with mass parameter M , angular
momentum quantum number l and frequency ω by (following the notation conventions of Sec-
tion 3.1)
Dsϕ = ∂rf∂rϕ− 1
r2
[
Bl +
(
1− s2
)
f1
]
ϕ+
ω2
f
ϕ. (4.1)
We will usually assume that l = 0, 1, 2, . . ., that s is a non-negative integer (in fact, only s = 0, 1, 2
will concern us), and that ω ̸= 0. Of course, Ds has rational coefficients in r and all of these
parameters.
Consider the following equivalence of upper triangular Regge–Wheeler systems:
• •
• •
Ds0 ∆
0 Ds1
id δ
0 id
Ds0 ∆̄
0 Ds1
id ε
0 id
,
• •
• •
Ds0 ∆̄
0 Ds1
id −δ
0 id
Ds0 ∆
0 Ds1
id −ε
0 id
.
As discussed in Section 3, this system is reducible to diagonal form by the above equivalence
(Lemma 3.4) if and only if the following equation is satisfied by some δ and ε with rational
coefficients:
Ds0 ◦ δ = (∆− ∆̄) + ε ◦ Ds1 . (4.2)
For simplicity of notation, where it does not introduce any ambiguities, we set ∆̄ = 0 below.
By [34, Theorem 3.1], without loss of generality, we can consider this problem restricted to first
order δ, ϵ and ∆, with δ uniquely determining ε and vice versa. For instance, if ∆ is of order
higher than one, we can always reduce the order by writing ∆ = ∆1+∆′ ◦Ds1 with ∆1 of order
at most one, and absorb ∆′ into ε (or analogously with δ).
Let us parametrize these operators as follows:
∆ =
iωr
r2
(−∆− + {rf∆+, ∂r}), (4.3a)
δ = δ+ − 2∂r(rfδ−) + f1δ− + {rfδ−, ∂r}, (4.3b)
ε = δ+ + 2∂r(rfδ−)− f1δ− + {rfδ−, ∂r}, (4.3c)
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 19
where δ±, ∆± are arbitrary rational functions and we have used the anti-commutator notation
{g, h} = g◦h+h◦g for differential operators, namely {g0, ∂r} = 2g0∂r+(∂rg0) for any function g0.
Plugging this parametrization into (4.2) and comparing coefficients of powers of ∂r, we find the
equivalent ODE system
1
iω
∂r
(s20 − s21)f1
iωr
−(s20−s21)f1
iωr
∂rrf∂rrf∂r−
{
f(2Bl−(s20+s
2
1+1)f1+
1
2f−
2
f ω
2r2), ∂r
}
iω
︸ ︷︷ ︸
D(s0,s1)
[
δ+
δ−
]
=
[
∆+
∆−
]
. (4.4)
The parametrization we have used for ∆ and δ is different from the one we used previously
in [34], but it is more convenient for some simple technical reasons. If ∆ is parametrized
by (∆+,∆−), then ∆∗ is parametrized by (∆+,−∆−). Moreover, given a solution δ of (4.4)
parametrized by (δ+, δ−), if we simultaneously replace ∆ by ∆∗ and exchange s0 with s1, then
the solution to the new problem is parametrized in the same way but by (δ∗+,−δ∗−). The last
property is due to the formal self-adjointness of D∗
s = Ds. With our new parametrization, it is
also easy to check that the operator D(s0, s1) in (4.4) is itself formally self-adjoint, and setting
s0 = s1 puts it into diagonal form. The version of the decoupling equation used in [34] did not
manifestly exhibit these properties.
Furthermore, the ODE coefficients in (4.4) are not just rational functions, but even more
conveniently they are Laurent polynomials in r. The rationality of the coefficients means that
we can use the results from [34] to systematically solve (4.4), or check that it has no solution.
Actually, we will need to extend the results of [34] in an important way. When ∆ is such that
no solution δ exists, we would like to find some canonical choice of ∆̄ such that replacing the
right-hand side by
(
∆− ∆̄
)
does make the equation solvable. That is, we would like to be able
to pick some convenient representatives of the equivalence classes in the cokernel of D(s0, s1)
in (4.4). In fact, exploiting the fact that its coefficients are actually Laurent polynomials, we
will show in Theorem 4.5 that fixing the locations of the poles of ∆, the cokernel of the operator
in (4.4) is finite dimensional and a complete set of representatives can be explicitly selected.
First, let us recall some useful terminology and notions from [34], where further details and
more references to the literature on this topic can be found. We will be working with5 complex
valued rational functions in r, C(r), polynomials in r and r−1, C[r] and C
[
r−1
]
, formal power
series in r and r−1, C[[r]] and C
[[
r−1
]]
, as well as formal Laurent series in r. Let us distinguish
between unbounded Laurent series C
[[
r, r−1
]]
, bounded (from below) Laurent series C
[
r−1
]
[[r]],
bounded from above Laurent series C[r]
[[
r−1
]]
and Laurent polynomials C[r, r−1]. Of course,
we could also consider Laurent series centered at some other r = ρ ̸= 0, but for convenience of
notation whenever possible we will stick with ρ = 0.
By convention, matrix ordinary differential operators E[ϕ] act on column vectors ϕ with
components either in C(r), or belonging to one of these classes of Laurent series. When E[ϕ]
has bounded Laurent series coefficients, its action on bounded Laurent series produces bounded
Laurent series (from above or below, respectively). Rational coefficients become bounded Lau-
rent series upon appropriate expansion. For the action of E[ϕ] to be well-defined on unbounded
Laurent series, its coefficients must be Laurent polynomials. In each case, let us call such an E
a compatible differential operator.
We call a Laurent polynomial matrix S = S(r) Laurent unimodular if its inverse S(r)−1 is
also Laurent polynomial (equivalently, the determinant of S is proportional to a single power of r
and non-vanishing). We say that Laurent unimodular matrices S and T are respectively source
5What follows is standard notation in commutative algebra, where in general C can be replaced by any
commutative ring R and the variable r could be replaced by any set of independent symbolic variables.
20 I. Khavkine
and target leading multipliers of E when, after expanding all rational coefficients as bounded
Laurent series, we have
E
[
S(r)ϕnr
n
]
= T (r)
(
Enϕnr
n + rnO(r)
)
,
with the components of O(r) all in rC[[r]] and En an r-independent matrix that is invertible
for almost all n (i.e., all but finitely many). We call En the leading characteristic matrix of E
with respect to the given multipliers. Similarly, we say that S and T are respectively source and
target trailing multipliers of E when, after expanding all rational coefficients as bounded from
above Laurent series, we have
E
[
S(r)ϕnr
n
]
= T (r)
(
Enϕnr
n + rnO
(
r−1
))
,
with the components of O
(
r−1
)
all in r−1C
[[
r−1
]]
and En an r-independent matrix that is
invertible for almost all n. We call En the trailing characteristic matrix of E with respect to
the given multipliers.
Those integers n ∈ Z such that detEn = 0, which is a polynomial equation in n, are called
(respectively leading or trailing) (integer) characteristic roots or exponents of E with respect to
given multipliers S, T . We denote the set of such leading characteristic exponents by σ̌(E) and
the set of such trailing characteristic exponents by σ̂(E), with implicit dependence on the S, T
multipliers, of course.
If the leading (trailing) multipliers S, T of E are known, the formal adjoint operator E∗
obviously has SE∗ = T−∗ and TE∗ = S−∗ as corresponding multipliers, where A−∗ = (A∗)−1 is
the inverse of the conjugate transpose matrix. Dependence on n in En is due to the action of
differential operators r∂rr
n = nrn. Due to the identity (r∂r)
∗ = −(r∂r + 1), using the above
multipliers for E∗ allows us identify the characteristic exponents of E∗ as σ̂(E∗) = −σ̂(E) − 1
and σ̌(E∗) = −σ̌(E)− 1.
We will not dwell on when leading or trailing multipliers exist or on their uniqueness, but will
just assume that they are given for any particular problem. Our focus, instead, is to use these
data as a certificate that a concrete, finite dimensional linear algebra computation can prove
some statement about a rational ODE. In practice, often S and T may be taken to be diagonal,
with appropriately chosen powers of r on the diagonal. Otherwise, they could be determined by
a recursive procedure similar to that used in the analysis of regular and irregular singularities
for ODEs with meromorphic coefficients [48]. Related algorithms can be found in the recent
papers [1, 2].
For the inhomogeneous problem E[ϕ] = β, the solution ϕ may only have poles at a subset of
the union of the poles of β and of the (regular or irregular) singular points of E[ϕ]. When the
number of such singular points is finite, the knowledge of the leading and trailing multipliers at
the singular points (including r = ∞) and at the poles of β is sufficient to reduce the problem of
finding all rational solutions ϕ for rational β to a finite dimensional linear algebra problem [34,
Theorem 2.4]. For our purposes, we now need to extend this result to characterize both the
kernel and cokernel of E[ϕ] when acting on the different classes of Laurent series. There exist
results in the literature [15, 38] that are, on an abstract level, equivalent or even more general
than what we present below. But, being written in the abstract language of D-modules, they
would require substantial interpretation to be applicable to our very concrete situation.
Let us fix the following notation. We denote the kernel of E by kerE in the C
[
r, r−1
]
class,
by ker+E in the C
[
r−1
]
[[r]] class, by ker−E in the C[r]
[[
r−1
]]
class, and by ker−+E in the
C
[[
r, r−1
]]
class. For the cokernel of E, we write cokerE, coker+E, coker−E and coker−+E,
using the same conventions. In each case, the operator E[ϕ] must have a well-defined action
on the Laurent series of the appropriate class. If we have in mind Laurent series centered at
r = ρ instead of r = 0, then we write kerρ and cokerρ with appropriate subscripts. Finally,
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 21
we write kerρ1,ρ2,...rat and cokerρ1,ρ2,...rat when we restrict our attention to rational functions with
possible poles only at r = ρ1, ρ2, . . ..
Proposition 4.1. If a compatible equation E[ϕ] = 0 (with given multiplier matrices) has finitely
many singular points, then kerE, ker+E, ker−E, ker−+E or kerρ1,ρ2,...,ρNrat E, as appropriate, is
finite dimensional.
Proof. For rational functions with prescribed poles, this is the statement of [34, Corollary 2.5].
For the other cases, the proof can be attacked with similar methods, so we merely sketch it.
For ker+E, we can split ϕ = ϕ′ + Sϕ+, using the leading multiplier S, where the powers of r
in S−1ϕ′ are bounded from below by ň and from above by n̂ satisfying ň≪ σ̌(E) ≪ n̂, while the
powers of r in ϕ+ are only bounded from below by n̂. Then there are no obstructions to solving
E[Sϕ+] = −E[ϕ′] for ϕ+ order by order, since no characteristic exponents will appear by our
conditions on n̂. Then finding all ϕ′ such that E[ϕ′ + Sϕ+] = 0 reduces to a finite dimensional
linear algebra problem. The ker−E case is completely analogous. For ker−+E, one needs to
start with the splitting ϕ = Šϕ− + ϕ′ + Sϕ+, where now Š is the trailing multiplier and again
proceed analogously. ■
To deal with cokernels, it is helpful to introduce the sesquilinear residue pairing
⟨α, ϕ⟩ρ = Resρ(α
∗ϕ), (4.5)
where we take the residue at r = ρ (defined as the coefficient of (r − ρ)−1) of the product of
a column vector ϕ and the conjugate transposed column vector α (r∗ = r and its powers are
treated as real under conjugation). We may drop the subscript when ρ = 0, ⟨−,−⟩ = ⟨−,−⟩0.
The residue pairing is of course well-defined for rational functions, but it is also well-defined
for the following pairs of Laurent series classes: (bounded from above and below, unbounded),
(bounded from below, bounded from below), and (bounded from above, bounded from above).
This pairing has several nice properties. It is easy to see that it is non-degenerate in either
argument, for each possible combination of pairs Laurent series classes. If S is a Laurent
unimodular matrix, then the pairing is obviously invariant under ⟨S−∗α, Sϕ⟩ = ⟨α, ϕ⟩. The
following lemma is slightly less obvious and will be particularly helpful.
Lemma 4.2. For a compatible differential operator E[ϕ], the formal adjoint E∗[α] is also the
adjoint with respect to the residue pairing, ⟨α,E[ϕ]⟩ = ⟨E∗[α], ϕ⟩. The same is true with the
arguments of the pairing reversed.
Proof. The last statement holds simply because ⟨α, ϕ⟩∗ = ⟨ϕ, α⟩. To prove the adjoint relation,
we need to make use of the defining formula (4.5). First, note that it is obviously true that
⟨α,Cϕ⟩ = ⟨C∗α, ϕ⟩, when C is a constant matrix, and that ⟨α, rnϕ⟩ = ⟨rnα, ϕ⟩, for any power
of r. Then, due to the Leibniz identity α∗∂rϕ − (−∂rα)∗ϕ = ∂r(α
∗ϕ) and the observation that
Res0(∂rw) = 0 for any Laurent series w, it is also true that ⟨α, ∂rϕ⟩ = ⟨−∂rα, ϕ⟩. Hence, the
adjoitness property follows for all differential operators, since any operator can be written as
a sum of compositions of these three kinds of operators. ■
We can now state and prove the following lemmas characterizing cokernels in the bounded
Laurent series and Laurent polynomial cases. In each case, since E∗∗ = E, we are free to
exchange the roles of E and E∗.
Lemma 4.3. Let E[ϕ] = 0 be an ODE with coefficients that are Laurent series bounded from
below (above), with given multiplier matrices. Then the induced sesquilinear pairing ⟨[α], ϕ⟩ =
Res0(α
∗ϕ) is non-degenerate between coker+E
∗ and ker+E (between coker−E
∗ and ker−E).
22 I. Khavkine
Proof. Obviously, E[ϕ] is a compatible differential operator. We will consider Laurent series
bounded from below (the bounded from above case is completely analogous).
Let L and R denote the spaces of left and right arguments of the residue pairing. Denote by
Ln ⊂ L and Rn ⊂ R the subspaces consisting of Laurent series containing powers of r only equal
to n or greater. Note that L⊥
n = R−n and vice versa. By the invariance of the residue pairing,
without loss of generality, we can assume that E[ϕ] has trivial source and target multipliers (they
are just identity matrices). Then E[Rn] ⊂ Rn for any n. Fix some exponents ň≪ σ̌(E) ≪ n̂.
By invoking Lemma 4.2 and using standard arguments, we can conclude that the residue
pairing descends to the non-degenerate pairing ⟨[α], [ϕ]⟩ = ⟨α, ϕ⟩ on the finite-dimensional sub-
quotients L−n̂/L−ň ×Rň/Rn̂, with the induced operators [E][ϕ] = [E[ϕ]] and [E∗][α] = [E∗[α]]
still satisfying the adjoint relation ⟨[α], [E][ϕ]]⟩ = ⟨[E∗][α], [ϕ]⟩. Hence, by standard finite di-
mensional linear algebra, this induced pairing descends also to a non-degenerate pairing between
ker[E] and coker[E∗], for the operators induced on these sub-quotients. But by the same argu-
ments as in the proof of Proposition 4.1, the sub-quotients provide isomorphisms ker+E = ker[E]
and coker+E
∗ = ker[E∗]. Hence, since these isomorphisms respect the hierarchy of induced pair-
ings, the residue pairing induces a non-degenerate pairing between coker+E
∗ and ker+E. ■
Lemma 4.4. Let E[ϕ] = 0 be an ODE with Laurent polynomial coefficients (with given multiplier
matrices). Then the induced sesquilinear pairing ⟨α, [ϕ]⟩ = Res0(α
∗ϕ) is non-degenerate between
ker−+E
∗ and cokerE.
Proof. Clearly, E is a compatible differential operator, the induced pairing is well-defined and
the non-degeneracy in the first argument follows directly from the non-degeneracy of the residue
pairing. It remains to show non-degeneracy in the second argument.
Suppose ψ is a Laurent polynomial and ⟨β, [ψ]⟩ = 0, for any β ∈ ker−+E
∗ ⊃ ker±E
∗. Then,
by Lemma 4.3, there exist bounded (respectively from above or below) Laurent series ϕ± such
that ψ = E[ϕ±]. Pick any Laurent polynomial α ∈ (ker+E)⊥ ∩ (ker−E)⊥. Then, again by
Lemma 4.3, there exist bounded (respectively from above or below) Laurent series β± such that
α = E∗[β±], and hence β+ − β− ∈ ker−+E
∗. By the hypotheses on ψ,
0 = ⟨β+ − β−, [ψ]⟩ = ⟨β+, E[ϕ+]⟩ − ⟨β−, E[ϕ−]⟩ = ⟨E∗[β+], ϕ+⟩ − ⟨E∗[β−], ϕ−⟩
= ⟨α, ϕ+ − ϕ−⟩.
Since the residue paring is non-degenerate and α was arbitrary up to being annihilated by
ker±E (finite dimensional subspaces, by Proposition 4.1), there must be ϕ′± ∈ ker±E such that
ϕ+ −ϕ− = ϕ′+ −ϕ′−. But then ϕ+ −ϕ′+ = ϕ− −ϕ′− and hence, by their boundedness properties,
both sides are equal to a Laurent polynomial ϕ such that E[ϕ] = ψ, implying as desired that
[ψ] = 0 ∈ cokerE. ■
Finally, we can package the above results in the following, for our purposes, conveniently
phrased
Theorem 4.5. Consider an ordinary differential operator E[ϕ] with Laurent polynomial coef-
ficients with a finite number of singular points (with given multiplier matrices). Suppose that
the leading multipliers and integer characteristic exponents are known at each point ρ ∈ C. The
formal adjoint E∗[α] will share the same properties.
(a) Given ρ ∈ C, there is a finite basis
(
αρ
j
)
spanning a subspace of kerρ+E
∗ and as many
polynomials
(
ψρ
k
)
in (r − ρ)−1 such that the residue pairing Rρ
jk = Resρ
((
αρ
j
)∗
ψρ
k
)
is full
rank and such that, for any rational ψ, there exist constants ckρ and a Laurent polynomial ϕρ
such that
ψ̄ρ = ψ −
∑
k
ckρψ
ρ
k − E[ϕρ],
has no poles at r = ρ.
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 23
(b) There is a finite basis (αj) for ker−+E
∗ and as many Laurent polynomials (ψk) such that
Rjk = Resρ(α
∗
jψk) is full rank, for any Laurent polynomial ψ, there exist constants ck and
a Laurent polynomial ϕ such that
E[ϕ] = ψ −
∑
k
ckψk.
Proof. Part (a) follows from Proposition 4.1 and Lemma 4.3. The implication is that both
kerρ+E
∗ and cokerρ+E are finite dimensional. Hence, the dimension of the subspace of cokerρ+E
represented by polynomials in (r − ρ)−1 must also be finite. Let
(
ψρ
k
)
be a set of polynomials
in (r − ρ)−1 that represent a basis of this subspace. Then, by non-degeneracy of the residue
pairing, we can pick a finite subset
(
αk
ρ
)
satisfying the desired non-degeneracy condition. So by
subtracting the appropriate linear combination
∑
k c
k
ρψ
ρ
k from ψ we can find a representative ψ̄ρ
from the same equivalence class in cokerρ+E that has no poles at r = ρ. This argument gives
us ϕρ that is merely bounded from below as a Laurent series, but once we have it, we can simply
truncate it after some sufficiently high finite power or r.
Part (b) follows from Proposition 4.1 and Lemma 4.4 similarly. Once we know that ker−+E
∗
and cokerE are finite dimensional, it remains only to pick a basis (αj) for ker−+E
∗ and a finite
set (ψk) whose representatives form a basis of cokerE. ■
Remark 4.6. Theorem 4.5 also implies that cokerρ1,ρ2,...rat E, the cokernel of ϕ 7→ ψ = E[ϕ]
for ψ and ϕ rational with potential pole locations fixed at ρ1, ρ2, . . ., is finite dimensional. The
functions
(
ψk, ψ
ρ1
k , ψ
ρ2
k , . . .
)
then constitute a complete set of independent representatives of
the cokernel equivalence classes exactly when they satisfy the conditions in (a) and (b) of the
theorem.
We conclude this section by applying Theorem 4.5 to the operator D(s0, s1) from the Regge–
Wheeler decoupling equation (4.4). Let us denote the multiplier and characteristic matrices as
in the following identity:
D(s0, s1)Sρδn(r − ρ)n = Tρ
(
Eρ,nδn +O(r − ρ)
)
(r − ρ)n.
The only singular points of D(s0, s1) are r = 0, 2M , ∞. They are accompanied by the
following data:
� ρ = 0: σ̌0(D(s0, s1)) = {±s0 ± s1}, with
S0 =
[
1 0
0 r
2M
]
, T0 =
[2M
r 0
0 4M2
r2
]
, E0,n =
1
2iMω
[
n s20 − s21
s21 − s20 n(n2 − 2s20 − 2s21)
]
,
detE0,n = −(n+ s0 + s1)(n+ s0 − s1)(n− s0 + s1)(n− s0 − s1)
4M2ω2
.
� ρ = 2M : σ̌2M (D(s0, s1)) = {0}, with
S2M =
[
1 0
0 1
]
, T2M =
[
1
f 0
0 1
f
]
, E2M,n =
1
2iMω
[
n 0
0 n(n2 + 16M2ω2)
]
,
detE2M,n = −n
2(n2 + 16M2ω2)
4M2ω2
.
� ρ = ∞: σ̂∞(D(s0, s1)) = {0}, with
S∞ =
[
1 0
0 1
ωr
]
, T∞ =
[
1
ωr 0
0 1
]
, E∞,n =
1
i
[
n 0
0 4n
]
, detE∞,n = −4n2.
24 I. Khavkine
� ρ ̸∈ {0, 2M,∞}: σ̂ρ(D(s0, s1)) = {0, 1, 2}, with
Sρ =
[
1 0
0 1
]
, Tρ =
[
1
(r−ρ) 0
0 1
(r−ρ)3
]
, Eρ,n =
1
iω
[
n 0
0 ρ2f(ρ)2n(n− 1)(n− 2)
]
,
detEρ,n = −ρ
2f(ρ)2n2(n− 1)(n− 2)
ω2
.
Notice that, besides r = 0 and r = ∞, the only singular point of D(s0, s1) is r = 2M . In prac-
tice (in Section 5), we will only deal with rational functions that have poles at these three points,
so we will restrict our attention only to them. Given the above data, it is possible to explicitly
calculate (using computer algebra) the dimensions of the solution spaces ker2M+ D(s0, s1)
∗ and
ker−+D(s0, s1)
∗, as well as their bases. Since these solutions are presented by infinite series, it is
impractical to write them here explicitly. Instead, we will simply report their dimensions, while
explicitly writing a choice corresponding cokernel representatives
(
∆k,∆
2M
k
)
. In practice, the
self-adjointness D(s0, s1)
∗ = D(s0, s1) turns out to be extremely useful and saves a significant
amount of effort since the same computer algebra code can be used for both operators.
For ∆ involving poles at points other than {0,∞}, we can report the following.
� ρ ̸∈ {0, 2M,∞}: dimkerρ+D(s0, s1)
∗ = 2, while
(∆ρ
k) =
([ 1
r−ρ
0
]
,
[
0
1
r−ρ
])
.
Although we will not need this case later (in Section 5), we report it here for completeness.
� ρ = 2M : dimker2M+ D(s0, s1)
∗ = 2, while
(∆ρ
k) =
([
1
r−2M
0
]
,
[
0
1
r−2M
])
=
([ 1
rf
0
]
,
[
0
1
rf
])
.
For ∆ a Laurent polynomial, we can report the following. This data depends on the values
of s0 and s1, so we concentrate only on the cases that will be needed later (in Section 5).
� (s0, s1) = (0, 0): dimker−+D(s0, s1)
∗ = 5, while
(∆k) =
([
1
r
0
]
,
[
0
1
]
,
[
0
1
r
]
,
[
0
1
r2
]
,
[
0
1
r3
])
.
� (s0, s1) = (1, 1): dimker−+D(s0, s1)
∗ = 5, while
(∆k) =
([
1
r
0
]
,
[
0
1
]
,
[
0
1
r
]
,
[
0
1
r2
]
,
[
0
1
r4
])
.
� (s0, s1) = (2, 2): dimker−+D(s0, s1)
∗ = 5, while
(∆k) =
([
1
r
0
]
,
[
0
1
]
,
[
0
1
r
]
,
[
0
1
r2
]
,
[
0
1
r6
])
.
� (s0, s1) = (0, 1) or (1, 0): dimker−+D(s0, s1)
∗ = 4, while
(∆k) =
([
1
r
0
]
,
[
0
1
]
,
[
0
1
r
]
,
[
0
1
r3
])
.
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 25
� (s0, s1) = (0, 2) or (2, 0): dimker−+D(s0, s1)
∗ = 4, while
(∆k) =
([
1
r
0
]
,
[
0
1
]
,
[
0
1
r
]
,
[
0
1
r4
])
.
� (s0, s1) = (1, 2) or (2, 1): dimker−+D(s0, s1)
∗ = 4, while
(∆k) =
([
1
r
0
]
,
[
0
1
]
,
[
0
1
r
]
,
[
0
1
r5
])
.
Remark 4.7. Obviously, the choice of cokernel representatives given above is not unique. Our
choice was mainly dictated by simplicity and practicality. However, it satisfies another im-
portant requirement. When each cokernel representative is rewritten as a differential operator
∆ = ∆0+∆1f∂r
f via the parametrization (4.3a), the resulting coefficients ∆0 and ∆1 are bounded
functions of r on the interval (2M,∞). This requirement is sufficient to ensure that the ∆ is rela-
tively bounded byDs as a (possibly unbounded) operator L2
(
2M,∞; f−1 dr
)
→ L2(2M,∞; f dr),
which means that ∥∆ϕ∥ ≤ C2∥fDsϕ∥ + C0∥ϕ∥ for each ϕ ∈ D(Ds), with some constants
C0, C2 > 0. In turn, relative boundedness ensures that the operator matrix inverse[
Ds0 ∆
0 Ds1
]−1
=
[
D−1
s0 −D−1
s0 ∆D−1
s1
0 D−1
s1
]
is itself bounded whenever the inverses D−1
s0 and D−1
s1 are bounded.
To see the desired relative boundedness, it is obviously equivalent to establish the relative
boundedness of f∆ with respect to fDs. If we introduce the Regge–Wheeler tortoise coordinate
r∗ = r∗(r), defined by dr∗ =
dr
f , then L2
(
2M,∞; f−1dr
) ∼= L2(−∞,∞; dr∗) =: H and fDs is an
unbounded operator on H (in fact [19], it is also essentially self-adjoint starting from the core of
compactly supported test functions, with its ω-resolvent set equal to C\R). By the boundedness
conditions on ∆0 and ∆1, f∆ can be relatively bounded onH by a constant coefficient differential
operator in ∂r∗ = f∂r of order one or higher. But since the ∂r∗-coefficients of fDs are bounded
and the second order coefficients even tend to positive constants as |r∗| → ∞, fDs can relatively
bound any constant coefficient operator of order two or lower, which hence includes f∆.
Finally, we would like to record here another result that was actually already implicit in
the examples presented in [34, Section 4.1], but not made explicit there. If we consider the
Regge–Wheeler decoupling equation (4.4) with zero right-hand side, then solutions constitute
morphisms from Ds1ϕ1 = 0 to Ds0ϕ0 = 0. Such a morphism, if it existed between different s0
and s1 it could be called a spin raising/lowering operator for radial modes. Such spin raising
and lowering operators are known to exist for angular modes [44] and their existence for radial
modes would have important applications. Unfortunately, explicit computation shows that the
result is negative, at least when we restrict ourselves to operators with rational coefficients.
Theorem 4.8. For ω ̸= 0 and for s0, s1 ∈ {0, 1, 2} there exist no non-vanishing morphisms
between the equations Ds0ϕ0 = 0 and Ds1ϕ1 = 0, with the exception of the identity morphism
ϕ0 7→ ϕ1 = ϕ0 when s0 = s1.
The proof follows from direct computation, by using the methods of [34, Theorem 2.4] to
find all rational solutions of the Regge–Wheeler decoupling equation (4.4). We have only done
the computation for the above values of s0 and s1 because the location of the characteristic
exponents of the equation depends on these spins in such a way that the dimension of the
corresponding linear algebra problem grows as O(|s0| + |s1|). But it is likely that the above
result actually holds for all s0 and s1.
26 I. Khavkine
4.1 Equivalence of Regge–Wheeler and Zerilli equations
We have defined the spin-s Regge–Wheeler operator Ds in (4.1). When s = 2, D2 is closely
related to the so-called Zerilli operator
D+
2 ϕ = ∂rf∂rϕ− Al + 3(Bl − 2)f1(1 + 3f1) + 9f31
r2(Bl − 2 + 3f1)2
+
ω2
f
ϕ.
In fact, the equations D2ϕ = 0 and D+
2 ϕ = 0 are equivalent (in the sense of Section 3) for all
values of ω except the algebraically special frequencies [16, 36] satisfying
α := (6ωrf1)
2 +A2
l = (12Mω)2 +A2
l = 0. (4.6)
This can be seen from the following equivalence diagram:
• •
• •
D+
2
(12M)2
α
fC−
fC+
α
(12M)2
D2
C−f
(12M)2
α
f
(12M)2
α
C+f
(12M)4
α2 f, (4.7)
where
C± = ±∂r +
Al
12Mf
+
3f1
r(Bl − 2 + 3f1)
are the Chandrasekhar transformation operators [14, Section 4.26], [21, 24]. The validity of the
above equivalence diagrams may be checked by direct calculation, or can be seen more abstractly
from the identities
D2 = −C−fC+ +
α
(12M)2f
, D+
2 = −C+fC− +
α
(12M)2f
.
The necessity of the failure of the equivalence identities in (4.7) when α = 0 (due to division
by α) can be explained as follows. When α = 0, the two operators simplify to D2 = −C−fC+
and D+
2 = −C+fC−. Reducing each of them to a first order system gives the equivalences
D2 ∼
[
C+ 1
f
0 C−
]
,
[
C− 1
f
0 C+
]
∼ D+
2 .
Then, using the same machinery that we applied earlier to upper triangular Regge–Wheeler
systems (Theorem 4.8 and other results), one can conclude that (a) there are no non-vanishing
morphisms between the equations C+ϕ+ = 0 and C−ϕ− = 0 and (b) neither of the upper
triangular systems above can be decoupled to diagonal form. On the other hand, given (a), it
is not hard to see that the existence of an equivalence D2 ∼ D+
2 would imply that both of the
above upper triangular systems decouple to the diagonal form
[
C+ 0
0 C−
]
, which contradicts (b).
Thus, no manipulation will be able to get rid of the poles at α = 0 from the equivalence
in (4.7).
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 27
Remark 4.9. Since both D2 and D+
2 are formally self-adjoint, while α and M are real, taking
formal adjoints of all the operators in the equivalence diagram (4.7) gives another equivalence
diagram. The way that we have distributed various factors of α and the adjoint identity C+ = C∗
−
makes the new equivalence diagram identical to the original one, and hence the whole diagram
formally self-adjoint. While we could have chosen to distribute various constant factors among
the operators in (4.7) in different ways, the choice of no prefactor for D+
2 and the self-adjointness
of the whole diagram dictates the coefficient α
(12M)2
in front of D2.
5 Tensor wave equations on Schwarzschild
Consider the exterior Schwarzschild spacetime (M, 4g) of mass M > 0, where M ∼= R2 × S2.
When
(
S2,Ω
)
is the unit round sphere and (t, r), with −∞ < t < ∞ and 2M < r < ∞, are
the Schwarzschild coordinates on the (R2, g) factor, the exterior Schwarzschild metric has the
following (2 + 2)-warped product form:
4g := g + r2Ω, g := −f(r) dt2 + dr2
f(r)
, f := 1− 2M
r
.
For convenience, we also define (consistently with Section 4)
f1 := r∂rf =
2M
r
.
This is a vacuum spacetime
(
the Einstein tensor 4Gµν = 4Rµν − 1
2
rR 4gµν = 0 vanishes
)
descri-
bing the asymptotically flat region outside the horizon of a static spherically symmetric black
hole of mass M .
The goal of this section is to write down the vector and Lichnerowicz wave equations
□1 vµ := 4□vµ = 0 and □2 pµν := 4□pµν − 2 4R(µ
λκ
ν)pλκ = 0,
on this spacetime and to reduced them to their radial mode equations, with respect to spher-
ical and time harmonic modes. Eventually we will also show their equivalence to a simplified
triangular system of Regge–Wheeler equations (4.1).
For background, let us recall that the vector wave operator □1 can be obtained from the
Maxwell equation by adding a term factoring through the harmonic (Lorenz) gauge fixing con-
dition 4∇νvν = 0, namely
□1 vµ = 4∇ν
(
4∇νvµ − 4∇µvν
)
+ 4∇µ
4∇νvν .
Similarly, the Lichnerowicz wave operator □2 can be obtained from the linearized Einstein equa-
tion by adding a term factoring through the harmonic (de Donder) gauge fixing condition
4∇λpνλ = 0, namely
□2 pµν = −2 4Ġµν [p] + 2 4∇(µ
4∇λpν)λ,
where pµν = pµν − 1
2p
λ
λ
4gµν is the trace reversal operation and 4Ġµν [p] is the linearized Einstein
tensor, defined by 4Gµν
[
4g + p
]
= 4Ġµν [p] + O
(
p2
)
, keeping in mind that 4Gµν
[
4g
]
= 0 for the
background Schwarzschild metric.
We will proceed by first using the covariant formalism of [37] for (2+2)-warped products and
spherical harmonics, then by using Schwarzschild coordinates, then by presenting the ingredients
needed to implement the general decoupling strategy from Section 3.2, and finally by giving the
explicit equivalence. The intermediate calculations in the decoupling strategy are rather long,
28 I. Khavkine
unenlightening and best performed by computer algebra. Thus, we present only the final result,
but along the way giving some guiding comments on how it could be reproduced. The reader
is referred to [32], where the intermediate steps were worked out in more detail for the vector
wave operator □1 .
To start, let us briefly introduce the covariant 2+2 formalism from [37]. We use Greek indices,
(µ, ν, . . .), for spacetime or M-tensors, lower or upper case Latin indices, (a, b, . . .) or (A,B, . . .),
respectively, for tensors on the R2 or S2 factors. The tangent and cotangent subspaces of R2
and S2 are naturally identified with the appropriate subspaces of TM and T ∗M, allowing us to
map tensors from its factors to M. Thus, any spacetime tensor can be decomposed into sectors
with possibly mixed lower-upper case Latin indices, e.g.,
vµ →
[
va
rvA
]
, wν →
[
wb r−1vB
]
.
Note the extra factors of r, which make it convenient to raise and lower Greek indices with 4gµν ,
while their sector components have their indices raised and lowered respectively by gab and ΩAB.
This convention extends to higher rank tensors by respecting tensor products. We define the
following spacetime tensors by their decomposition into sectors:
gµν →
[
gab 0
0 0
]
, Ωµν →
[
0 0
0 ΩAB
]
.
We denote respectively by ∇a and DA the Levi-Civita connections on
(
R2, g
)
and
(
S2,Ω
)
.
In terms of them and the decomposition into sectors, the spacetime Levi-Civita connection is
determined by
4∇µvν →
[
∇avb r∇avB
rDAvb r2DA
vB
r
]
−
[
0 0
r
rb
r
vA −r2ΩAB
rcvc
r
]
, where ra = ∇ar = (dr)a.
Now, working in Schwarzschild (t, r) coordinates, we define ta = (∂t)
a, which is a timelike
Killing vector. We also introduce the notation ta = gabt
b = −f(dt)a and εab = 2(dt)[a(dr)b] =
−2f−1t[arb]. Then rar
a = f , tat
a = −f , and ta = −εabrb. The Levi-Civita connection ∇a is
determined by
∇atb =
f1
2r
εab and ∇arb =
f1
2r
gab.
The respective Riemann tensors of the
(
R2, g
)
and
(
S2,Ω
)
factors are
Rabcd =
f1
r2
(gacgbd − gadgbc) and RABCD = ΩACΩBD − ΩADΩBC .
Finally, the spacetime Riemann tensor (computed in [33, equation (23)], for instance) is conve-
niently expressed as
4Rµνλκ =
f1
r2
(gµλgνκ − gµκgνλ) +
f1
r2
(ΩµλΩνκ − ΩµκΩνλ)
− f1
2r2
(gµλΩνκ − gνλΩµκ − gµκΩνλ + gνκΩµλ).
Scalar spherical harmonics Y lm, where l = 0, 1, 2, . . . and −l ≤ m ≤ l are respectively
the angular momentum and magnetic quantum numbers, are normalized eigenfunctions of the
Laplacian on the unit sphere
(
S2,Ω
)
, defined by the conditions
DAD
AY lm = −l(l + 1)Y lm, (Y lm)∗ = Y l(−m), ⟨(Y l′m′
)∗ Y lm⟩ = δl′lδm′m,
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 29
where we have used the short-hand ⟨−⟩ =
∫
S2 ϵ (−), with ϵAB the volume form on
(
S2,Ω
)
.
Where no confusion would arise, we shall regularly omit the l and m indices from Y lm and their
coefficients. We use the convenient notations (already mentioned at the top of Section 2)
Bl := l(l + 1) and Al := (l − 1)l(l + 1)(l + 2) = Bl(Bl − 2).
By taking covariant derivatives of the Y scalar harmonics and combining them with ΩAB
and ϵAB, we can define also vector and tensor spherical harmonics:
YA := DAY, YAB := DAYB +
Bl
2
ΩABY,
XA := ϵBAD
BY, XAB := DAXB +
Bl
2
ϵABY.
The above definitions are chosen so that the following convenient orthogonality and normaliza-
tion properties hold:〈(
Y A
l′m′
)∗
Y lm
A
〉
= Blδl′lδm′m,〈(
XA
l′m′
)∗
X lm
A
〉
= Blδl′lδm′m,
〈(
XA
l′m′
)∗
Y lm
A
〉
= 0,〈(
Y AB
l′m′
)∗
Y lm
AB
〉
=
Al
2
δl′lδm′m,
〈(
Y l′m′
ΩAB
)∗
Y lm
AB
〉
= 0,〈(
XAB
l′m′
)∗
X lm
AB
〉
=
Al
2
δl′lδm′m,
〈(
XAB
l′m′
)∗
Y lm
AB
〉
= 0.
Note the consequent vanishing of Y 0m
A = X0m
A = Y 0m
AB = X0m
AB = 0 and Y 1m
AB = X1m
AB = 0. The
tensorial spherical harmonics also satisfy the following eigenvalue and divergence identities:
DCD
CYA = −(Bl − 1)YA, DAYA = −BlY,
DCD
CXA = −(Bl − 1)XA, DAXA = 0,
DCD
CYAB = −(Bl − 4)YAB, DAYAB = − Al
2Bl
YB,
DCD
CXAB = −(Bl − 4)XAB, DAXAB = − Al
2Bl
XB.
Under the parity inversion of the sphere, all the Y -harmonics transform as Y 7→ (−)lY and so
are called even, while all the X-harmonics transform as X 7→ (−)l+1X and so are called odd [40].
Our conventions for spherical harmonics follow [37], whose Appendix A may be consulted for
comparison with other conventions in the literature.
When needed, we will also decompose sectorial tensors on the
(
R2, g
)
factor into coordinate
components with respect to the Schwarzschild coordinates (t, r) with the conventions
vb = vt(dt)b + vr(dr)b, hcd = htt(dt)c(dt)d + 2htr(dt)(c(dr)d) + hrr(dr)c(dr)d,
va = vt(∂t)
a + vr(∂r)
a, hab = htt(∂t)
a(∂t)
b + 2htr(∂t)
(a(∂r)
b) + hrr(∂r)
a(∂r)
b.
If we raise and lower indices with gab, this convention implies the relations vt = −fvt, vr = f−1vr,
htt = (−f)2htt, etc.
Remark 5.1. In the sequel, while working at the mode level, we allow ourselves the freedom
of dividing by the expressions ω, Bl or Al in the mode parameters. The expressions for various
radial mode operators will be valid for any value of the mode parameters where they are well-
defined, which might exclude ω = 0, l = 0 or l = 1, when dividing by ω or Bl. These singular
cases would need to be investigated separately. However, since all expressions will be rational,
it would be sufficient to consider Laurent expansions in the mode parameters and just compare
corresponding coefficients.
30 I. Khavkine
Harmonic dependence on time with frequency ω will refer to being proportional to e−iωt with
time independent coefficients.
Remark 5.2. We will use the convention that r and ω−1 will carry the dimensions of length, r−1,
ω and ∂r will carry the dimensions of inverse length, while f , f1, Bl and Al will be dimensionless.
Clearly, by this convention, length dimension is additive under multiplication and composition
of differential operators. For practical convenience, upon radial mode decomposition, we will try
to make use of only dimensionless differential operators, by for instance inserting extra factors
of ω, where needed.
Remark 5.3. Whenever we need to take a formal adjoint of a differential operator, we will
follow the conventions from Section 3.3.
Before proceeding to mode decomposing concrete equations, let us make some general re-
marks about the notation. If sµν... is a tensor field, then schematically we will denote its mode
coefficients (omitting indices) as
s = seven · Y + sodd ·X =
(
seven · Y + sodd ·X
)
e−iωt.
The s coefficients represent the even and odd multiplets obtained after the decomposition of
a spacetime tensor into sectors, absorbing all angular dependence and spherical tensor indices
into the spherical harmonics Y and X. The s coefficients represent the even and odd multiplets
after decomposing the remaining sectorial tensors in v into coordinate components with respect
to the Schwarzschild coordinates (t, r) on (R2, g), absorbing all time dependence into the har-
monic factor e−iωt. When convenient, the s and s coefficients may be defined with extra r-, l-
and ω-dependent normalization factors.
Given a spacetime differential operator s′ = O[s] (in general s and s′ need not be of the same
tensor rank), which is even under inversion of the
(
S2,Ω
)
factor, we will schematically denote
its action on the mode coefficients as
s′even = Oe[s
even], s′even = Oe[s
even],
s′odd = Oo[s
odd], s′odd = Oo[s
odd].
The Oe and Oo operators acting on the s coefficients are precisely the radial mode operator
versions of the original spacetime operator O that we will eventually be working with.
In principle, the intermediate stage of the decomposition involving s and s′ can be omitted
from the presentation. But we believe that it is worth keeping them, because these expres-
sions would be useful for obtaining the explicit radial mode equations for other coordinate
systems on the
(
R2, g
)
factor, for instance the Eddington–Finkelstein coordinates instead of the
Schwarzschild one. In addition, these intermediate expressions would be useful checkpoints for
anyone trying to reproduce our calculations.
5.1 Scalar wave equation
Before proceeding to tensor wave equations, for illustration purposes, let us first handle the
simplest case of the scalar wave equation
z′ = □0 z := 4□z = 0. (5.1a)
Following earlier conventions, we use the notation z = zY = zY e−iωt and z′ = z′
r2
Y = z′
r2
Y e−iωt.
First, we separate out S2 dependence via spherical harmonics:
1
r2
z′ Y = □0
(
z(t, r)Y
)
=
(
□0 z(t, r)
)
Y, with z′ = □0 z(t, r) :=
(
∇ar2∇a − Bl
)
z. (5.1b)
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 31
Then, we separate out the t coordinate via harmonic time dependence:
z′e−iωt = □0
(
z(r)e−iωt
)
=
(
□0 z(r)
)
e−iωt, z′ = □0 z(r) :=
(
∂rfr
2∂r +
ω2r2
f
− Bl
)
z. (5.1c)
The last formula defines the radial mode ODE for the scalar wave equation (5.1a). Note that,
to economize notation, we have reused the symbol □0 , though it changes meaning when acting
on different function spaces, which should be clear from context.
Note that the final expression for □0 z(r) is formally self-adjoint, □0 ∗ = □0 . That is no accident,
since we have chosen our definitions such that the sesquilinear forms (with compactly supported
y(r) and z(r))∫
R2
ε r2
∫
S2
ϵ y(r)∗(Y l′m′
)∗eiω
′t□0 z(r)Y lme−iωt = δl′lδm′m
∫
R2
ε y(r)∗eiω
′t□0 z(r)e−iωt
= 2πδ(ω′ − ω)δl′lδm′m
∫ ∞
2M
dr y(r)∗□0 z(r)
all agree and are symmetric up to complex conjugation, because the original □0 operator is
formally self-adjoint on spacetime. We will follow the same convention in the rest of the section.
This means that some of the components of the radial mode equations in the tensor case might
have spurious looking factors of Bl or Al. Such factors arise from absorbing the norms of the
corresponding tensor harmonics and are needed to maintain manifest formal self-adjointness of
the radial mode ODEs.
The transformation
ϕ0 = −iωr z, z = − 1
iωr
ϕ0,
puts □0 z = 0 in direct equivalence with the s = 0 Regge–Wheeler equation D0ϕ0 = 0:
□0
(
− 1
iωr
)
ϕ0 = (iωr)
1
ω2
D0ϕ0,
1
ω2
D0(−iωr)z =
(
1
iωr
)
□0 z.
In the sense of Section 3 we have an equivalence of □0 with D0, exhibited by the diagram
• •
• •
□0
k0=−iωr
k̄0=− 1
iωr
□̃0
k′0=
1
iωr
h0=0
k̄′0=iωr
h̃0=0, where □̃0 :=
1
ω2
D0 . (5.2)
Notice that the original operator □0 is dimensionless. We have chosen to include various
factors of ω to keep all the other operators in the diagram dimensionless as well. We will follow
the same convention throughout all further calculations.
Remark 5.4. Note also that the above equivalence diagram is self-adjoint in the following sense:
□0 ∗ = □0 , □̃0
∗
= □̃0 , h∗0 = h0 and h̃∗0 = h̃0, while k̄0 = (k′0)
∗ and k̄′0 = (k0)
∗. That is, taking formal
adjoints of all operators in the diagram, we obtain the same diagram. The various factors of i
and ω were chosen also to exhibit this property. In particular, the powers of ω in k0, k
′
0 and □̃0
are not all independent. There is an obvious economy to self-adjoint equivalence diagrams: the
morphism need only be specified in one direction, with the inverse morphism recovered by taking
adjoints. In subsequent cases, we have succeeded in identifying a self-adjoint equivalences, but
with slightly modified notion of self-adjointness for the decoupled triangular form. It is not
obvious to us at the moment why we were successful in that sense and whether that is due to
a fortuitous coincidence or a general pattern.
32 I. Khavkine
5.2 Vector wave equation
To present the radial mode equation for □1 vµ = 0, we follow the same pattern as in the case of
the scalar wave equation (Section 5.1).
The tensor spherical harmonic decomposition of a covariant vector field is
vµ = vevenµ + voddµ , vevenµ →
∑
lm
[
vlm
a Y lm
ulm rY lm
A
]
, voddµ →
∑
lm
[
0
wlm rX lm
A
]
.
For conciseness, as before, we will omit the lm indices. Following the notation conventions given
at the top of Section 5, we first separate out the spherical harmonics (bold coefficients) and
then harmonic time dependence (fraktur coefficients). We will need to use two conventions to
normalize the coefficients, which we distinguish by primes (recall that primes are decorations
and not derivatives):
vν →
[
vbY
u rYB
]
+
[
0
w rXB
]
=
([
vbY
u rYB
]
+
[
0
w rXB
])
e−iωt,
v′µ → 1
r2
([
v′aY
u′
Bl
r−1Y A
]
+
[
0
w′
Bl
r−1XA
])
=
1
r2
([
v′aY
u′
Bl
r−1Y A
]
+
[
0
w′
Bl
r−1XA
])
e−iωt.
With these conventions, again reusing the symbol □1 at each stage of mode separation of v′µ =
□1 vµ, we have[
v′a
u′
]
= □1 e
[
vb
u
]
:=
[
∇mr
2∇mva
∇mBlr
2∇mu
]
− Bl
[
va
Blu
]
+
[
−2rarb 2Blr
a
2Blr
b 0
] [
vb
u
]
+ (1− f)
[
0
Blu
]
= 0, (5.3a)
w′ = □1 ow := ∇mBlr
2∇mw − Bl(Blw) + Bl(1− f)w = 0, (5.3b)
after separating out the S2 dependence, andv′tv′r
u′
= □1 e
vtvr
u
:=
−∂rf−1r2f∂r vt
∂rfr
2f∂r vr
∂rBlr
2f∂r u
+
(
ω2
f
− Bl
r2
)−f−1r2 vt
fr2 vr
Blr
2 u
+ iωr
f1
f
0 1 0
−1 0 0
0 0 0
vtvr
u
+
0 0 0
0 −2f2 2Blf
0 2Blf Blf1
vtvr
u
= 0, (5.3c)
w′ = □1 ow := ∂rBlr
2f∂rw+
(
ω2
f
− Bl
r2
)
Blr
2w+ Bl(1− f)w = 0, (5.3d)
after separating out the time dependence. Our normalization conventions for the mode coeffi-
cients were chosen precisely to maintain manifest formal self-adjointness, □1 ∗
e = □1 e and □1 ∗
o = □1 o,
at each stage of the mode separation.
Remark 5.5. Note also that these expressions are valid for all values of ω and l. When l = 0,
Y l=0
A = X l=0
A = 0, meaning that their coefficients are not well-defined. By consistency, the
corresponding components of □1 e and □1 o simply vanish.
Next, we will introduce the operators needed to decouple radial modes into pure gauge and
constraint violating modes, to set up the decoupling strategy (cf. Section 3.2). The key identities,
analogous to E0T = T ′E1 and E1D = D′E0, hold already at the spacetime level:
□0 4∇µvµ = 4∇µ□1 vµ and □1 4∇µz =
4∇µ□0 z .
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 33
In addition, analogous to TD = HTE0, we have the composition identity
4∇µ4∇µz = □0 z.
Now we define the operators T1, T
′
1, D1, D
′
1 and their successive mode decompositions:
for z = 4∇µvµ =: T1[vν ],
z = T1e
[
vb
u
]
:=
[
1
r2
∇br2 −Bl
r
] [
vb
u
]
,
z
iω
= T1e
vtvr
u
:= [ 1
f
1
iωr2
∂rfr
2 − Bl
iωr
]vtvr
u
,
T1o := 0;
for z′ = 4∇µv
′µ =: T ′
1[v
′],
z′ = T ′
1e
[
v′b
u′
]
:=
1
r
[
r∇b −1
][v′b
u′
]
,
z′
iω
= T ′
1e
v′tv′r
u′
:=
[
−1
1
iω
∂r − 1
iωr
]v′tv′r
u′
,
T ′
1o := 0;
for vµ = 4∇µz =: D1[z],[
va
u
]
= D1e[z] :=
1
r
[
r∇a
1
]
z,
1
iω
vtvr
u
= D1e[z] :=
−1
1
iω∂r
1
iωr
z, D1o := 0;
for v′µ = 4∇µz
′ =: D′
1[z
′],
[
v′a
u′
]
= D′
1e[z
′] :=
[
r2∇a 1
r2
Bl
r
]
z′,
1
iω
v′tv′r
u′
= D′
1e[z
′] :=
1
f
fr2∂r
1
iωr2
Bl
iωr
z′, D′
1o := 0.
The extra iω factors in the normalizations have been chosen so that all operators acting on
radial modes are dimensionless and the first two of the following commutative diagrams remain
valid at each stage of mode separation:
• •
• •
DD1
=□0
D1
□1
D′
1
,
• •
• •
□1
T1
DT1
=□0
T ′
1
,
• •
• •
□1
T1
Φ1
DΦ1
[
Φ′
1 −∆Φ1T1
]
. (5.4)
The last diagram holds at the radial mode level and shows the decoupling of gauge invariant
modes (cf. Section 3.2). It decomposes into independent even and odd sectors as follows:
Φ1e :=
[
0− ff∂rr
]
, Φ1o := −iωr,
Φ′
1e :=
1
r2
[
0− Blr
2∂r
f
r
]
, Φ′
1o :=
1
iωr
,
DΦ1e :=
Bl
ω2
D1, DΦ1o :=
Bl
ω2
D1,
∆Φ1eT1e := −Blf1
iωr
, ∆Φ1oT1o := 0.
We are now ready to give the full triangular decoupling of the even and odd sectors of the
vector wave equation □1 vµ = 0.
34 I. Khavkine
5.2.1 Odd sector
This sector is structurally similar to the scalar wave case (Section 5.1). It is particularly simple;
since it does not contain any gauge modes, it is not constrained by the harmonic gauge condition
and the operators Φ1o, Φ
′
1o are directly invertible. Thus, the transformation
ϕ1 = −iωrw, w = − 1
iωr
ϕ1,
puts □1 ow = 0 in direct equivalence with the s = 1 Regge–Wheeler equation D1ϕ1 = 0:
□1 o
(
− 1
iωr
)
ϕ1 = (iωr)
Bl
ω2
D1ϕ1,
Bl
ω2
D1(−iωr)w =
(
1
iωr
)
□1 ow.
In diagrammatic form, we have
• •
• •
□1 o
k1o=−iωr
k̄1o=− 1
iωr
□̃1 o
k′1o=
1
iωr
0
k̄′1o=iωr
0, where □̃1 o :=
Bl
ω2
D1. (5.5)
This equivalence diagram is self-adjoint in the sense explained in Section 5.1 for the scalar wave
equation.
5.2.2 Even sector
This sector is more complicated and is the first prototype for the abstract approach to triangular
decoupling that we outlined earlier in Section 3. All that we need to feed into it are the even
sectors of the diagrams (5.4) and the additional composition identities
T1eD1e = − 1
ω2r2︸ ︷︷ ︸
HT1e
□0 and Φ1eD1e = 0︸︷︷︸
HΦ1e
□0 .
Since the implementation of our approach in this particular case was explained in detail in [32]
(in particular, see the step-by-step discussion there in Section 3.2), we only state the final result,
which has only been slightly adjusted for the purposes of this work (the end of the section shows
how). The final decoupled triangular form is
□1 e
vtvr
u
= 0 ⇐⇒ 1
ω2
D0 0 −f1
r2
(
Bl +
1
2f1
)
0 BlD1 0
0 0 D0
︸ ︷︷ ︸
□̃1 e
ϕ0ϕ1
ϕ′0
= 0.
While □̃1 e is obviously not formally self-adjoint, its equivalence with the formally self-adjoint
□1 e survives in the existence of an operator Σ1e effecting the equivalence between □̃1 e and □̃1
∗
e ,
namely □̃1 eΣ1e = Σ1e□̃1
∗
e , where
Σ1e =
0 0 1
0 1 0
1 0 0
.
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 35
The precise equivalence identities take the form
• •
• •
□1 e
k1e
k̄1e
□̃1 e
k′1e
h1e
k̄′1e
h̃1e
,
where
k̄1e =
−iωr iωrBl
1
2 iωr(Bl + f)
∂rr −Bl
r ∂rr
2 −1
2
(
(Bl − f) 1
r2
∂rr
3 + 2f + f1
)
1 −f
r ∂rr
2 − Bl −1
2(2f∂rr + Bl + f)
1
ω2r2
,
k̄′1e =
iωr
f − iωr
f − iωr(Bl+f)
2f
fr2∂r
1
r −fr∂r −1
2f((Bl − f)∂rr + 2f − f1)
Bl −r∂rf − Bl −1
2Bl(2r∂rf + Bl − f)
,
h1e =
1
ω2r2
0 0 0
0 1 0
0 0 − f
Bl
, h̃1e =
−
1
2(Bl − f) 1
2(Bl + f) 1
4(Bl + f)2
−1 Bl+f
Bl
1
2(Bl + f)
[1ex]1 −1 −1
2(Bl − f)
,
while the remaining operators can be recovered from the identities
k1e = Σ1e(k̄
′
1e)
∗, k′1e = Σ1e(k̄1e)
∗,
where also
h∗1e = h1e, h̃∗1e = Σ1eh̃1eΣ1e.
Remark 5.6. These last identities embody the modified sense in which the equivalence dia-
gram (5.2.2) is self-adjoint, in the sense discussed earlier in Section 5.1, but with a twist provided
by the matrix Σ1e. In fact, if we replace □̃1 e by Σ1e□̃1 e, it is no longer upper triangular, but it is
formally self-adjoint(
Σ1e□̃1 e
)∗
= Σ1e□̃1 e,
and the twist by Σ1e is no longer necessary. Since we place importance on the upper triangular
form, we will not take advantage of this replacement.
The above upper triangular form of □̃1 e allows us to classify all of its symmetries (or auto-
morphisms in the sense of Section 3). The key result is the absence of non-vanishing morphisms
between the Regge–Wheeler equations Ds0 and Ds1 , except the identity morphism when s0 = s1
(Theorem 4.8). This prevents the coupling of the D0 and D1 blocks by an automorphism. Also,
the single non-vanishing (and non-removable) off-diagonal element in □̃1 e prevents the exchange
of the order of the D0 blocks. Hence, any automorphism of □̃1 e must also be upper triangular,
with every non-vanishing matrix element proportional to the identity.
With the above logic in mind, the most general automorphism takes the form □̃1 eA = A□̃1 e,
with
A =
a1 0 b1
0 a2 0
0 0 a1
,
parametrized by the 3 constants a1, b1, a2. It is invertible when a1, a2 ̸= 0.
36 I. Khavkine
With the above choice of k1e and k′1e, up to homotopy, we have the following equivalences of
operators:
k1e ◦D1e ◦ k̄0 ∼
10
0
, k0 ◦ T1e ◦ k̄1e ∼
[
0 0 1
]
.
Finally, for the record, denoting by
(
□̃1 e
)old
,
(
k̄1e
)old
and
(
k̄′1e
)old
by corresponding operators
from [32], let us note the explicit relations
□̃1 e =
1 0 0
0 Bl 0
0 0 1
(□̃1 e)
old, k̄1e = (k̄1e)
old
1 0 1
2Bl
0 1 0
0 0 1
, k̄′1e = (k̄′1e)
old
1 0 1
2Bl
0 1
Bl
0
0 0 1
.
5.3 Lichnerowicz wave equation
To present the radial mode equation for □2 pµν = 0, we follow the same pattern as in the case of
the scalar and vector wave equations (Sections 5.1 and 5.2).
The tensor spherical harmonic decomposition of a symmetric covariant 2-tensor is
pµν = pevenµν + poddµν ,
pevenµν →
∑
lm
[
hlm
ab Y
lm r jlma Y lm
B
r jlmb Y lm
A r2 (K lmΩABY +GlmY lm
AB)
]
,
poddµν →
∑
lm
[
0 rhlm
a X lm
B
rhlm
b X lm
A r2 hlm
2 X lm
AB
]
.
For conciseness, as before, we will omit the lm indices. Following the notation conventions given
at the top of Section 5, we first separate out the spherical harmonics (bold coefficients) and
then harmonic time dependence (fraktur coefficients). We will need to use two conventions to
normalize the coefficients, which we distinguish by primes (recall that primes are decorations
and not derivatives):
pλκ →
[
hcd Y jc rYD
jd rYC r2(K ΩCDY +GYCD)
]
+
[
0 hc rXD
hd rXC h2 r
2XCD
]
=
([
hcd Y jc rYD
jd rYC r2(KΩCDY +GYCD)
]
+
[
0 hc rXD
hd rXC h2 r
2XCD
])
e−iωt,
p′µν → 1
r2
h′ab Y j′a
Bl
1
rY
B
j′b
Bl
1
rY
A 1
r2
(K ′ΩABY + 2G′
Al
Y AB)
+
0 h′a
Bl
1
rX
B
h′b
Bl
1
rX
A 2
h′
2
Al
1
r2
XAB
=
1
r2
[ h′ab Y j′a
Bl
1
rY
B
j′b
Bl
1
rY
A 1
r2
(K′ΩABY + 2G′
Al
Y AB)
]
+
0 h′a
Bl
1
rX
B
h′b
Bl
1
rX
A 2
h′2
Al
1
r2
XAB
e−iωt.
With these conventions, again reusing the symbol □2 at each stage of mode separation of
p′µν = □2 pµν , we have
h′ab
j′a
K ′
G′
= □2 e
hcd
jc
K
G
:=
∇mr
2∇mhab
∇m2Blr
2∇mja
∇m2r2∇mK
∇m
Al
2 r
2∇mG
− Bl
hab
2Blj
a
2K
Al
2 G
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 37
+
−4r(agb)(crd) 4Blr
(agb)c 4rarb 0
4Blg
a(crd) −8Blr
arc −4Blr
a 2Alr
a
4rcrd −4Blr
c −4 0
0 2Alr
c 0 Al
hcd
jc
K
G
+ f1
0
2Blj
a
4K
AlG
− 2f1
gc(agb)d − gabgcd 0 gab 0
0 −Blg
ac 0 0
gcd 0 −2 0
0 0 0 Al
2
hcd
jc
K
G
, (5.6a)
[
h′a
h′
2
]
= □2 o
[
hc
h2
]
:=
[
∇m2Blr
2∇mha
∇m
Al
2
r2∇mh2
]
− Bl
[
2Blh
a
Al
2 h2
]
+
[
−8Blr
arc 2Alr
a
2Alr
c Al
] [
hc
h2
]
+ f1
[
2Blh
a
Alh2
]
− 2f1
[
−Blg
ac 0
0 Al
2
] [
hc
h2
]
, (5.6b)
after separating out the S2 dependence, and after separating out the time dependence the final
result is
h′tr
j′t
h′tt
h′rr
K′
j′r
G′
= □2 e
htr
jt
htt
hrr
K
jr
G
:=
−∂r2r2f∂r htr
−∂r2Bl
r2
f f∂r jt
∂r
r2
f2 f∂r htt
∂rf
2r2f∂r hrr
∂r2r
2f∂r K
∂r2Blfr
2f∂r jr
∂r
Al
2 r
2f∂r G
+
(
ω2
f
− Bl
r2
)
−2r2 htr
−2Bl
r2
f jt
r2
f2 htt
f2r2 hrr
2r2 K
2Blfr
2 jr
Al
2 r
2G
− 2iωr
f1
f
0 0 − 1
f −f 0 0 0
0 0 0 0 0 −Bl 0
1
f 0 0 0 0 0 0
f 0 0 0 0 0 0
0 0 0 0 0 0 0
0 Bl 0 0 0 0 0
0 0 0 0 0 0 0
htr
jt
htt
hrr
K
jr
G
(5.6c)
+
2
f (f
2+1) −4Bl 0 0 0 0 0
−4Bl −4Blf1
f 0 0 0 0 0
0 0
f2
1
2f3 − f1
2f (f1+4f) 2f1
f 0 0
0 0 − f1
2f (f1+4f) 1
2f(f
2
1−8f2) −2f(3f1−2) 4Blf
2 0
0 0 2f1
f −2f(3f1−2) 4(2f1−1) −4Blf 0
0 0 0 4Blf
2 −4Blf 4Blf(3f1−2) 2Alf
[1ex]0 0 0 0 0 2Alf Al
htr
jt
htt
hrr
K
jr
G
,
h′th′r
h′2
= □2 o
hthr
h2
:=
−∂r2Bl
r2
f f∂r ht
∂r2Blfr
2f∂r hr
∂r
Al
2 r
2f∂r h2
+
(
ω2
f
− Bl
r2
)−2Bl
r2
f ht
2Blfr
2 hr
Al
2 r
2 h2
− 2iωr
f1
f
0 −Bl 0
Bl 0 0
0 0 0
hthr
h2
+
−4Bl
f1
f 0 0
0 4Blf(3f1 − 2) 2Alf
0 2Alf Al
hthr
h2
. (5.6d)
38 I. Khavkine
The operators □2 e and □2 o defined in (5.6c) and (5.6d) are our radial mode equations.
Remark 5.7. Note that the final radial mode equations are manifestly self-adjoint, □2 ∗
e = □2 e
and □2 ∗
o = □2 o. These expressions are also valid for all values of ω and l. When l = 0, Y l=0
A =
X l=0
A = Y l=0
AB = X l=0
AB = 0, and when l = 1, Y l=1
AB = X l=1
AB = 0, meaning that their coefficients are
not well-defined. By consistency, the corresponding components of □2 e and □2 o simply vanish.
Next, we will introduce the operators needed to decouple radial modes into pure gauge and
constraint violating modes, to set up the decoupling strategy (cf. Section 3.2). The key identities,
analogous to E1T = T ′E2, E2D = D′E1 and so on, hold already at the spacetime level:
4∇ν□2 pµν = □1 4∇νpµν , □2 4∇(µvν) =
4∇(µ□1 vν),
4gµν□2 pµν = □0 4gµνpµν .
In addition, analogous to TD = HTE1 and so on, we have the composition identities
4∇ν4∇(µvν) =
1
2
□1 vµ,
4gµν4∇(µvν) =
4∇µvµ.
Now we define the operators T2, T
′
2, D2, D
′
2, tr, tr
′ and their successive mode decompositions
(following the same notational conventions as in Section 5.2 for mode decomposing vectors):
for vµ = 4∇νpµν =: T2[p],
[
va
u
]
= T2e
hcd
jc
K
G
:=
1
r
[1
r
∇br2
(
hab − 1
2gabg
cdhcd + gabK
)
1
r2
∇cr3jc
]
+
1
r
[
rag
cd −Blδ
c
a 0 0
−1
2g
cd 0 0 − Al
2Bl
]
hcd
jc
K
G
,
w = T2o
[
hc
h2
]
:=
1
r
[
1
r2
∇cr3 − Al
2Bl
] [
hc
h2
]
,
1
iω
vtvr
u
= T2e
htr
jt
htt
hrr
K
jr
G
:=
1
iωr
1
r∂rfr
2 −Bl
iωr
2
1
f
iωr
2 f
iωr
f 0 r
2
1
f ∂r
1
2r3f
∂rr
4f2
0 iωr
f
1
2
1
f −1
2f
htr
jt
htt
hrr
+
1
iωr
iωr 0 0
−1
r∂rr
2 −Bl 0
0 1
r2
∂rfr
3 − Al
2Bl
Kjr
G
,
1
iω
w = T2o
hthr
h2
:=
1
iωr
[
iωr
f
1
r2
∂rfr
3 − Al
2Bl
]hthr
h2
;
for v′µ = 4∇νp
′µν =: T ′
2[p
′],
[
v′a
u′
]
= T ′
2e
h′cd
j′c
K ′
G′
:=
1
2r
[
2rδa(c∇d) −δac 2ra 0
0 ∇cr Bl −2
]
h′cd
j′c
K ′
G′
,
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 39
w′ = T ′
2o
[
h′c
h′
2
]
:=
1
2r
[
∇cr −2
] [h′c
h′
2
]
,
1
iω
v′tv′r
u′
= T ′
2e
h′tr
j′t
h′tt
h′rr
K′
j′r
G′
:=
1
2iωr
r
f ∂rf −1 −2iωr 0 0 0 0
−iωr 0 f1f 2r∂r − f1
f −2f −1 0
0 −iωr 0 0 Bl ∂rr −2
h′tr
j′t
h′tt
h′rr
K′
j′r
G′
,
1
iω
w = T ′
2o
h′th′r
h′2
:=
1
2iωr
[
−iωr ∂rr −2
] h′th′r
h′2
;
for pµν = −(4∇µvν +
4∇νvµ) =: D2[v],
hab
ja
K
G
= D2e
[
vc
u
]
:= −1
r
r(∇avb +∇bva)
r2∇a
1
ru
0
0
− 1
r
0 0
δca 0
2rc −Bl
0 2
[vcu
]
,
[
ha
h2
]
= D2o[w] := −1
r
[
r2∇a
1
r
2
]
w,
1
iω
htr
jt
htt
hrr
K
jr
G
= D2e
vtvr
u
:= − 1
iωr
fr∂r
1
f −iωr 0
1 0 −iωr
−2iωr −f1f 0
0 2r∂r +
f1
f 0
0 2f −Bl
0 1 r2∂r
1
r
0 0 2
vtvr
u
,
1
iω
hthr
h2
= D2o[w] := − 1
iωr
−iωr
r2∂r
1
r
2
w;
for p′µν = −(4∇µv′ν + 4∇νv′µ) =: D′
2[v
′],
h′ab
j′a
K ′
G′
= D′
2e
[
v′c
u′
]
:= −2
r
r3∇(a 1
r2
v′b) − 1
2g
abr3∇c
1
r2
v′c
r4∇a 1
r3
u′
r3∇c
1
r2
v′c
0
− 2
r
−rcgab 1
2g
ab
Blδ
a
c 0
0 0
0 Al
2Bl
[v′c
u′
]
,
[
h′a
h′
2
]
= D′
2o[w
′] := −2
r
[
r4∇a 1
r3
Al
2Bl
]
w′,
1
iω
h′tr
j′t
h′tt
h′rr
K′
j′r
G′
= D′
2e
v′tv′r
u′
:= − 2
iωr
fr3∂r
1
r2
iωr
f 0
Bl 0 iωr
f
iωr
2f
1
2r∂r
1
f − 1
2f
iωrf
2
1
2f
2r5∂r
1
r4f
f
2
iωr −r3∂r 1
r2
0
0 Bl fr4∂r
1
r3
0 0 Al
2Bl
v′tv′r
u′
,
40 I. Khavkine
1
iω
h′th′r
h′2
= D′
2o[w
′] := − 2
iωr
iωr
f
fr4∂r
1
r3
Al
2Bl
w′.
Some of the above radial mode operators are related by taking adjoints, namely
T ′
2e = −1
2
D∗
2e, D′
2e = −2T ∗
2e,
T ′
2o = −1
2
D∗
2o, D′
2o = −2T ∗
2o.
For the trace operators again we return to the convention for scalars from Section 5.1: using
zY e−iωt = z Y = 4gλκpλκ =: tr[p] we get
z = tre
hcd
jc
K
G
:=
[
gab 0 2 0
]
hcd
jc
K
G
,
z = tre
htr
jt
htt
hrr
K
jr
G
:=
[
0 0 − 1
f
f 2 0 0
]
htr
jt
htt
hrr
K
jr
G
, tro := 0;
and using 1
r2
z′ Y e−iωt = 1
r2
z′ Y = 4gλκp
′λκ =: tr′[p′], we get
z′ = tr′e
h′cd
j′c
K ′
G′
:=
[
gab 0 1 0
]
h′cd
j′c
K ′
G′
,
z = tr′e
h′tr
j′t
h′tt
h′rr
K′
j′r
G′
:=
[
0 0 −f 1
f
1 0 0
]
h′tr
j′t
h′tt
h′rr
K′
j′r
G′
, tr′o := 0.
The extra iω factors in the normalizations have been chosen so that all operators acting on
radial modes are dimensionless and the first three of the following commutative diagrams remain
valid at each stage of mode separation:
• •
• •
DD2
=□1
D2
□2
D′
2
,
• •
• •
□2
T2
DT2
=□1
T ′
2
,
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 41
• •
• •
□2
tr
Dtr=□0
tr′
,
• •
• •
□2
T2
Φ2
DΦ2
[
Φ′
2 −∆Φ2T2
]
. (5.7)
The last diagram holds at the radial mode level and shows the decoupling of gauge invariant
modes (cf. Section 3.2). It decomposes into independent even and odd sectors as follows. The
odd sector is described by the following radial mode operators:
Φ2o =
[
0 2f −fr∂r
]
, Φ′
2o =
Al
ω2r2
[
0
1
Bl
−2r3
Al
∂r
f
r2
]
,
DΦ2o =
Al
ω2
D2, ∆Φ2oT2o = 2f1
Al
iωr
.
Remark 5.8. The operator Φ2o is indeed gauge invariant, Φ2o ◦D2o = 0. It is essentially the
well-known odd sector Regge–Wheeler scalar [9, equation (2.15)], whose decoupling was first
identified in [40].
In the even radial mode sector, we could choose DΦ2e to be proportional to D2 as well, but the
corresponding Φ2e and Φ′
2e would be rather large and, at the same time, their complexity would
obscure some important properties that we would like to highlight. Instead, we choose DΦ2e to
be a 2 × 2 first order system, with corresponding Φ2e and Φ′
2e that are simple6 and satisfy the
gauge invariance condition Φ2e ◦D2e = 0. The corresponding formulas are
Φ2e =
1 −f∂r r
f 0 0 iωr
f −iωr iωr(Bl−2+3f1)
2f
0 4iωr 2 0 f1 0 −4ω2r2−Blf1
2
,
Φ′
2e =
1
2iωr
[
1
iωr 0 f 1
f −1 0 0
1
2f
1
Bl
0 0 0 0 0
]
+
Bl
4Al
[
4ω2r2−Blf1
iωrf
−Bl−2+3f1
f2
] [
− f1
2iωr
0 −f − 1
f
1 − 2
Bl
0
]
,
DΦ2e =
[
0 f
iω∂r
1
f
1
iωf ∂rf
]
− Bl
4Alω2r2f
×
(4ω2r2−Bl(2f+f1))
2+8f(4ω2r2−Bl)
iωr(4ω2r2(Bl−2+3f1)−3Blf
2
1−Al(2f+f1))
f
− iωr(4ω2r2(Bl−2+3f1)−3Blf
2
1−Al(2f+f1))
f
ω2r2(Bl−2+3f1)2
f2
,
∆Φ2eT2e =
Bl
Al
×
(4ω2r2−Blf1)f1−4(Bl−2)f
4iωrf2
(
f2r∂r
1
f
−ω2r2
)
Blf1−2(2ω2r2−(Bl−2)f)
2
(
r∂r+
f1
2f
)
(Blf1−4ω2r2)
(
r2∂r
1
r
+
Bl
2f
)
− (Bl−2+3f1)f1+2(Bl−2)f
4f2
(
f
r
∂r
r2
f
)
iωr(Bl−2+3f1)
2f
(
r∂r+
f1
2f
)
iωr(Bl−2+3f1)
f
(
r2∂r
1
r
+
Bl
2f
)
+
Bl
Al
[
iωr(4ω2r2−Blf1)(2f+f1)
4f2
(Bl−2)(f1−2f)
2 Al
(ω2r2+f1f)(Bl−2+3f1)
2f3
iωr(Bl−2)
2f
iωr(Bl−2)
f
]
.
We will now see that this choice of DΦ2e is ultimately equivalent, with minor caveats, to
both the Regge–Wheeler operator D2 as well as the corresponding Zerilli operator D+
2 [9, equa-
tions (3.30) and (3.31)], which is commonly used to decouple gauge invariant perturbations in
6Such a choice is also practically helpful for reducing the size and complexity of certain expressions in the
computer algebra implementation of intermediate steps of our decoupling strategy.
42 I. Khavkine
the even sector. More precisely, the equivalence is established by concatenating the following
equivalence diagrams, where the middle vertical arrow corresponds to a self-adjoint first order
reduction of D2:
• • •
• • •
DΦ2e
q
q̄
α
ω2r2
f
−∂rr
r∂r
1
f
− Bl − 2
ω2r2
[
0 1
]
f
ω2r2
∂rr
1
αD2
q′
q̄′
[
−r∂r
f
ω2r2
1
]
0
1
(5.8)
where the operators in the first square are given by
q := − Bl
αAl
[
Al(4ω
2r2−2f(Bl−2−f1)−Blf1)
2ω2r2
−Al(Bl−2+3f1)−6(Bl−2)ff1
2iωrf
2Alf − 3f1(4ω
2r2 − Blf1) −3iωrf1(Bl−2+3f1)
f
]
,
q′ :=
Bliωr
Al
[
−3iωrf1(Bl−2+3f1)
f 2Alf − 3f1(4ω
2r2 − Blf1)
Al(Bl−2+3f1)−6(Bl−2)ff1
2iωrf −Al(4ω
2r2−2f(Bl−2−f1)−Blf1)
2ω2r2
]
= α iωr
[
0 −1
1 0
]
q
[
0 1
1 0
]
,
q̄ := −Al
Bl
(q′)∗, q̄′ := −Al
Bl
q∗,
and α (equation (4.6)) is the frequency dependent constant
α := (12Mω)2 +A2
l ,
which vanishes at so-called algebraically special frequencies [16]. Notice that α appears in the
denominator of q and q̄′, thus creating poles at the algebraically special frequencies, the only poles
in frequency other than ω = 0 that enter the equivalence morphisms with eventual triangular
decoupling of □2 e (Section 5.3.2). The same constant α and the poles created by it appears in
the Chandrasekhar transformations that relate the Regge–Wheeler D2 and Zerilli D+
2 operators.
We give a complete discussion of this relation in Section 4.1 and the necessity of these poles.
Remark 5.9. Using the Zerilli operator D+
2 (and its first order reduction) instead of the Regge–
Wheeler operator D2 in (5.8) results in equivalence maps that are free of poles at α = 0, the
algebraically special frequencies (in fact, free of all poles in frequency except ω = 0). We have
checked that this change also eliminates the α = 0 from the equivalence morphisms with the
triangular decoupled form in Section 5.3.2. However, a different undesirable property appears.
As is well-known and can be explicitly seen from our discussion in Section 4.1, both the Zerilli
operator D+
2 and the corresponding decoupling morphisms from □2 e contain r-dependent poles
in the angular momentum quantum number l. This means that the corresponding radial mode
operators cannot be easily interpreted as the mode separated form of differential operators on
spacetime. On the contrary poles at α = 0 (since it is a constant independent of r and depending
polynomially on ω and Bl) do not prevent us from finding such an interpretation in terms of
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 43
differential operators on spacetime (Corollary 2.4). Also, besides being simpler, we already have
a good understanding of D2 with respect to Section 4, while the analogous treatment of D+
2
would have to be done separately. For these reasons, we refrain from using the Zerilli operator
in this work.
We are now ready to give the full triangular decoupling of the even and odd sectors of the
Lichnerowicz wave equation □2 pµν = 0.
5.3.1 Odd sector
This sector is structurally very similar to the even sector of the vector wave equation (Sec-
tion 5.2.2). What we need to feed into the decoupling strategy of Section 3.2 are the odd sectors
of the diagrams (5.7) and the additional composition identities
T2oD2o =
1
ω2r2
1
Bl︸ ︷︷ ︸
HT2o
□1 o and Φ2oD2o = 0︸︷︷︸
HΦ2o
□1 o .
The key equivalence diagram (3.3a) for the E2o-Φ2o-T2o system is then obtained by following
exactly the same steps as for the vector even sector (see [32, Section 3.2]). To save space, we
do not give these intermediate details here. As explained in Section 3.2, a version of these
intermediate steps can be recovered from the full triangular simplification diagram (5.10). The
final decoupled triangular form is
□2 o
hthr
h2
= 0 ⇐⇒ − 2
ω2
BlD1 0 1
3B
2
l
f1
r2
0 AlD2 0
0 0 BlD1
︸ ︷︷ ︸
□̃2 o
ψ1
ψ2
ψ′
1
= 0. (5.9)
While □̃2 o is obviously not formally self-adjoint, its equivalence with the formally self-adjoint
□2 o survives in the existence of an operator Σ2o effecting the equivalence between □̃2 o and □̃2
∗
o,
namely □̃2 oΣ2o = Σ2o□̃2
∗
o, where
Σ2o =
0 0 1
0 −1 0
1 0 0
.
The precise equivalence identities take the form
• •
• •
□2 o
k2o
k̄2o
□̃2 o
k′2o
h2o
k̄′2o
h̃2o ,
(5.10)
where the operators k̄2o, k̄
′
2o, h2o, h̃2o are
k̄2o =
iωr iωrBl
3
iωr(3
1
r2
∂rf1fr3+Al+3(3Blf−f2
1 ))
18Bl
−r∂r − (Bl−2)
3
1
r2
∂rr
3 3f1ω2r2−r∂rf(Al−3f(3Bl+2f1))−6Alf−3ff1(7Bl−2(3f−f1))
18Blf
−2 −2 f
r2
∂rr
3− 2(Bl−2)
3 −3∂rf(2Bl+f1)r+Al+3(Bl(f−2f1)+f1(2f−f1))
9Bl
1
ω2r2
,
44 I. Khavkine
k̄′2o =
iωr
f
iωr
3f
iωr(3∂rff1+9Blf+Al)
18Blf
r3f∂r
1
r2
f∂rr
3 −3f1r2w2+
1
r ∂rf(3f(3Bl+2f1)−Al)r
2−Al(2f−f1)−6f(2Bl(3f+f1)+3ff1)
18Bl
Bl−2
2
3∂rfr+(Bl−2)
6
Al(3f+Bl−2)+3r2∂rf(2Al+(Bl−2)f1)
1
r
36Bl
,
h2o =
1
12B2
l ω
2r2
0 iωrf1 0
−iωrf1 2(3Bl + 2f1) −2f1
0 −2f1
24B2
l f
Al
,
h̃2o =
1
36 − 1
108 −2(2−9f)−Bl
648
0 − f
2Al
1
108
0 0 1
36
+
1
Bl
−
2+9f
36
2−9f
108
49−102f1−57f2
1
648
1
23 − 1
18 −2−9f
108
1
2 − 1
23 −2+9f
36
+
f1f
B2
l
−
1
6 − 1
36 −33f1−20
216
0 0 1
36
0 0 −1
6
+
f21
B3
l
0 0 −ω2r2−2+2f1(3f+f1)
72
0 0 0
0 0 0
,
while the remaining operators can be recovered from the identities
k2o = Σ2o(k̄
′
2o)
∗, k′2o = Σ2o(k̄2o)
∗,
where also
h∗2o = h2o, h̃∗2o = Σ2oh̃2oΣ2o.
When it comes to the properties of diagram (5.10) with respect to taking formal adjoints,
Remark 5.6 applies here equally well, with appropriate transposition of notation.
The above upper triangular form of □̃2 o allows us to classify all of its symmetries (or auto-
morphisms in the sense of Section 3). The key result is the absence of non-vanishing morphisms
between the Regge–Wheeler equations Ds0 and Ds1 , except the identity morphism when s0 = s1
(Theorem 4.8). This prevents the coupling of the D1 and D2 blocks by an automorphism. Also,
the single non-vanishing (and non-removable) off-diagonal element in □̃2 o prevents the exchange
of the order of the D1 blocks. Hence, any automorphism of □̃2 o must also be upper triangular,
with every non-vanishing matrix element proportional to the identity.
With the above logic in mind, the most general automorphism takes the form □̃2 oA = A□̃2 o,
with
A =
a1 0 b1
0 a2 0
0 0 a1
,
parametrized by the 3 constants a1, b1, a2. It is invertible when a1, a2 ̸= 0.
With the above choice of k2o and k′2o, up to homotopy, we have the following equivalences of
operators:
k2o ◦D2o ◦ k̄1o ∼
10
0
, k1o ◦ T2o ◦ k̄2o ∼
[
0 0 1
]
.
5.3.2 Even sector
The even sector of the Lichnerowicz wave equation is by far the most complicated case of
those we presented in this work. It requires the full arsenal of tools that we have outlined in
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 45
Sections 3 and 4. If one blindly attacks it, even with the systematic strategy of Section 3.2, the
intermediate formulas can easily become very large, significantly slowing down even computer
algebra calculations. So, before giving the final result below, we first make some comments
about how we implemented the decoupling strategy. In particular, points 2–6 below describe
how to obtain the key equivalence diagram (3.3a) for the E2e-Φ2e-(tre, T2e) system.
1. We start by taking full advantage of the simultaneous decoupling of both T2e and tr from
□2 e and the relation tre ◦ D2e[v] ∝ T1e[v]. Namely, we extend the gauge fixing condition
T2[p] = 0 by tr[p] = 0. So the choice of pure gauge, constraint violating and gauge invariant
modes that we feed into the strategy of Section 3.2 corresponds to the diagrams
• •
• •
□1 e
T1e
D2e
□2 e
[
D′
2e 0
]
,
• •
• •
□2 e
tre
T2e
□0 0
0 □1 e
tr′e
T ′
2e
,
• •
• •
□2 e
tre
T2e
Φ2e
DΦ2e
[
Φ′
2e 0 −∆Φ2eT2e
]
,
and the composition identities
T2eD2e =
1
ω2r2
−f 0 0 0
0 1
f 0 0
0 0 1
Bl
0
︸ ︷︷ ︸
HT2e
[
□1 e
T1e
]
,
treD2e =
[
0 −2
]︸ ︷︷ ︸
Htre
[
□1 e
T1e
]
,
Φ2eD2e =
[
0 0
]︸ ︷︷ ︸
HΦ2e
[
□1 e
T1e
]
.
2. Note that by the results of Section 5.2.2,
[
□1 e
T1e
]
is clearly equivalent to the diagonal operator[
D0 0
0 BlD1
]
. We use this equivalence to simplify the first diagram above before proceeding.
3. Next, we need to simplify the joint system □2 ep = 0, Φ2e[p] = 0, tre[p] = 0 and T2e[p] = 0.
It is of mixed second, first and zeroth orders. We reduce it to mixed first and zeroth orders
only, as follows. We use □2 e to eliminate second order derivatives from ∂rT2e, giving us six
independent first order equations. Adding to the list the first component of Φ2e, we get
seven independent first order equations, from which we can solve for the first derivatives of
the seven metric components htr, jt, htt, hrr, K, jr and G. Using these first order equations
to eliminate all derivatives from ∂rtr, we get a zeroth order equation. Including also tr
itself and the second component of Φ2e, we get three independent zeroth order equations.
4. Next, we use these three zeroth order equations to eliminate the htr, htt and hrr components
completely. The choice of these three components is dictated by the simplicity of the
46 I. Khavkine
denominators needed for the inversion. What remains is the following first order system
for the remaining four metric components:
∂rr
Bl
iωr −ω2r2+Blf
iωr −2ω2r2−Bl(Bl−2f)
2iωr
2iωr
f
1
r2
∂rr
3 − Bl
2f Bl −4ω2r2+Bl(Bl−2f)
4f
iωr
f2
1
f
1
fr2
∂rr
3f −Bl−2
2f
0 1
f −2 r∂r +
Bl
2f
jt
K
jr
G
= 0.
5. The above 4× 4 first order system is then equivalent to the 2× 2 second order
[
D0 0
0 BlD1
]
.
In one direction, the equivalence is given by the appropriately modified D2e. Obviously,
the 2 × 2 second order system can also be reduced to a 4 × 4 first order system, while
homotopy equivalence reduces the morphism induced by D2e to a zeroth order operator
(a 4 × 4 matrix, in fact) between these two first order systems. This 4 × 4 matrix is
invertible and its inverse, after undoing the order reductions, gives the equivalence in the
other direction.
6. In the above steps, the various eliminations of components introduce divisions only by ω, r
and f , except the final 4× 4 matrix inversion, which introduces division by Bl. Whenever
encountering choices, e.g., which components to eliminate, we are guided by the desire
not to introduce other factors into the denominators, in order to keep the intermediate
expressions in the calculations more manageable.
7. From this point on, it is enough to continue blindly following the strategy from Section 3.2
to end up with a block upper triangular form of □2 e. At this stage, the diagonal blocks
are not all proportional to Regge–Wheeler operators Ds, but include also □0 , □1 e and DΦ2e .
Now we can use the results of Sections 5.1, 5.2.2 and the equivalence (5.8) to transform
each of these blocks themselves to upper triangular form (using the transformation rules
from Lemma 3.3), now with only Regge–Wheeler operators on the diagonals.
8. It remains now only to apply the methods of Section 4 to systematically simplify the
off-diagonal components of the above upper triangular system. Practically speaking, it
turns out to be convenient to first decouple the D2 equations from the rest, then the
D1 equations from the remaining D0 equations. Within each still coupled block, it is
convenient to complete the simplification by going through successive super-diagonals.
The final decoupled triangular form (recalling the definition of α in (4.6)) of □2 ep
even = 0 is
− 2
ω2
D0 0 −f1
r2
(
Bl +
1
2f1
)
0
f1
r2
(
Bl +
1
2f1
)
0
f21
8r2
(7Bl + 2)
0 BlD1 0 0 0 −5
3B
2
l
f1
r2
0
0 0 D0 0 0 0
f1
r2
(
Bl +
1
2f1
)
0 0 0 αAlD2 0 0 0
0 0 0 0 D0 0 −f1
r2
(
Bl +
1
2f1
)
0 0 0 0 0 BlD1 0
0 0 0 0 0 0 D0
︸ ︷︷ ︸
□̃2 e
ϕ0
ϕ1
ϕ′0
ϕ2
χ0
χ1
χ′
0
= 0. (5.11)
Remark 5.10. At this point, it is worth noting that equation (5.11) is the first appearance of the
full triangular decoupled form □̃2 e in the literature. Previously, the original investigations in [9]
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 47
had only produced the (ϕ0, ϕ1, ϕ2) subsystem, without studying how it couples to the remaining
degrees of freedom, or considering the maximal simplification of the off-diagonal couplings as
we have done here. A similar remark can be made about □̃2 o from equation (5.9).
While □̃2 e is obviously not formally self-adjoint, its equivalence with the formally self-adjo-
int □2 e survives in the existence of an operator Σ2e effecting the equivalence between □̃2 e and □̃2
∗
e ,
namely □̃2 eΣ2e = Σ2e□̃2
∗
e , where
Σ2e =
0 0 0 0 0 0 1
0 0 0 0 0 1 0
0 0 −1 0 0 0 0
0 0 0 −1 0 0 0
0 0 0 0 −1 0 0
0 1 0 0 0 0 0
1 0 0 0 0 0 0
.
The precise equivalence identities take the form
• •
• •
□2 e
k2e
k̄2e
□̃2 e
k′2e
h2e
k̄′2e
h̃2e,
(5.12)
where the (rather lengthy) explicit formulas for k̄2e, k̄
′
2e and h2e, h̃2e can be found in Ap-
pendix B,7 while the remaining operators can be recovered from the identities
k2e = Σ2e
(
k̄′2e
)∗
, k′2e = Σ2e
(
k̄2e
)∗
,
where also
h∗2e = h2e, h̃∗2e = Σ2eh̃2eΣ2e. (5.13)
When it comes to the properties of diagram (5.12) with respect to taking formal adjoints,
Remark 5.6 applies here equally well, with appropriate transposition of notation.
Just in the case of □̃2 o (Section 5.3.1), the above upper triangular form of □̃2 e allows us to
classify all of its symmetries (or automorphisms in the sense of Section 3). Following the same
logic, the most general automorphism takes the form □̃2 eA = A□̃2 e, with
A =
a0 0 c0 + d0 0 c0 − e0 0 g0
0 a1 0 0 0 b1 0
0 0 1
2(b0 + a0) 0 1
2(b0 − a0) 0 c0 − d0
0 0 0 a2 0 0 0
0 0 1
2(b0 − a0) 0 1
2(b0 + a0) 0 c0 + e0
0 0 0 0 0 a1 0
0 0 0 0 0 0 a0
, (5.14)
parametrized by the 9 constants a0, b0, c0, d0, e0, g0, a1, b1, a2. It is invertible when a0, b0, a1, a2
̸= 0. Note that it is allowed to deviate from upper triangular form by mixing the two inner D0
blocks.
7While undeniably lengthy, the formulas in Appendix B are relatively compact. Analogous formulas in [9,
Appendix A] take up 10 pages and correspond only to the ϕ0, ϕ1 and ϕ2 columns of our k̄2e operator.
48 I. Khavkine
With this choice of k2e and k′2e, up to homotopy, we have the following equivalences of
operators:
iωr
2
tr ◦ k̄2e ∼
[
0 0 1 0 1 0 0
]
,
k2e ◦D2e ◦ k̄1e ∼
1 0 1
16(7Bl + 2)
0 1 0
0 0 1
0 0 0
0 0 0
0 0 0
0 0 0
,
k1e ◦ T2e ◦ k̄2e ∼
0 0 0 0 1 0 1
16(7Bl + 2)
0 0 0 0 0 1 0
0 0 0 0 0 0 1
.
Note that the adjointness properties of the equivalence diagram (5.12) do not uniquely fix the
operators k2e and k
′
2e. It turns out that a subfamily of (5.14), the symmetries of □̃2 e, respects the
adjointness properties of the equivalence when A−1 = Σ2eA
∗Σ2e. This constraint is satisfied for
instance by the family a0 = a1 = b0 = 1, d0 = e0 = 0, and g0 = c20. The remaining c0 parameter
may be uniquely fixed by requiring that the operator equivalent to iωr
2 tr involves only the two
inner spin-0 components ϕ′0 and χ0, and the operator equivalent to D2e has the right sign. No
other ambiguity remains after that.
6 Discussion
Inspired by previous more ad-hoc work [9, 41] on decoupling the radial mode equations of the
vector wave and Lichnerowicz wave equations on the Schwarzschild spacetime, in [32, 34] we
have initiated a research program of finding a systematic approach to the problem, as well
as simplifying and extending the results. This work completes part of our program, which
consists of successfully applying the developed methods to both the odd and even sectors of
the Lichnerowicz wave equation on Schwarzschild. The results include explicit formulas for
reducing the complicated mode separated radial equations to a triangular system of sparsely
coupled spin-s Regge–Wheeler equations, by separating the degrees of freedom into pure gauge,
gauge invariant, and constraint violating modes. Our systematically derived results extend those
of [9, 41] because these earlier works did not attempt to uncover the full triangular structure of
the decoupled equations or their maximal simplification (in the context of rational ODEs).
Our main results are summarized in Theorem 2.1 in Section 2, where we also list a number
of quick corollaries that indicate important applications of our results to the spectral analy-
sis of electromagnetic and gravitational perturbations of the Schwarzschild black hole in har-
monic gauges. One application that we intend to pursue in future work, by leveraging the
well-understood spectral theory of Regge–Wheeler operators [20], is an explicit mode-level con-
struction of the retarded/advanced Green functions for these perturbations. Such a construction
is a first step toward classical stability analysis and the construction of quantum field theoretic
propagators. The absence decoupling results, such as ours, have so far constituted a great
obstacle in these applications.
An open question is whether our decoupling strategy could be applied to Kerr black holes,
where so far the harmonic gauges have received very little attention. The main obstacle to
a direct generalization seems to be the lack of a complete separation of the vector wave and
Lichnerowicz wave equations on the Kerr background. Other than that, the basic ingredients
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 49
that would allow the separation of the modes into pure gauge, gauge invariant, and constraint
violating already exists. After all, metric perturbations are related to pure gauge and constraint
violating modes by the same covariant operators on any spacetime, while the role of the Regge–
Wheeler equations satisfied by gauge invariant field combinations can be played by the Teukolsky
equations [14]. This question should be addressed in future work.
A Tensor spherical harmonics
Here we compare our conventions and notation for mode decomposition into spherical harmonics,
summarized in Section 5, to other closely related work in the literature. For convenience of
notation, we omit the angular momentum spectral parameters lm, where it causes no confusion.
Any spherical harmonic decomposition of vectors and symmetric 2-tensors takes the form
vµ =
∑
lm
4∑
i=1
v(i)lm(r, t)Y (i)lm
µ and pµν =
∑
lm
10∑
i=1
p(i)lm(r, t)Y (i)lm
µν ,
where the precise choice of the spherical harmonic bases Y
(i)
µ and Y
(i)
µν depend on the convention.
In this work our notation in the vector case corresponds to
v(1) = vt, Y (1)
µ →
[
(dt)aY
0
]
,
v(2) = vr, Y (2)
µ →
[
(dr)aY
0
]
,
v(3) = u, Y (3)
µ →
[
0
rYA
]
,
v(4) = w, Y (4)
µ →
[
0
rXA
]
,
and in the tensor case to
p(1) = htr, Y (1)
µν →
[
2(dt)(a(dr)b)Y 0
0 0
]
,
p(2) = jt, Y (2)
µν →
[
0 (dt)arYB
(dt)brYA 0
]
,
p(3) = htt, Y (3)
µν →
[
(dt)a(dt)bY 0
0 0
]
,
p(4) = hrr, Y (4)
µν →
[
(dr)a(dr)bY 0
0 0
]
,
p(5) = K, Y (5)
µν →
[
0 0
0 r2ΩABY
]
,
p(6) = jr, Y (6)
µν →
[
0 (dr)arYB
(dr)brYA 0
]
,
p(7) = G, Y (7)
µν →
[
0 0
0 r2YAB
]
,
p(8) = ht, Y (8)
µν →
[
0 (dt)arXB
(dt)brXA 0
]
,
p(9) = hr, Y (9)
µν →
[
0 (dr)arXB
(dr)brXA 0
]
,
50 I. Khavkine
p(10) = h2, Y (10)
µν →
[
0 0
0 r2XAB
]
.
Our conventions on the definition of the spherical harmonics Y , YA, XA, YAB and XAB
are taken directly from [37] (MP), where they are compared to spin-weighted and pure spin
harmonics in [37, Appendix A]. However, our normalizations for the coefficients differ by some
factors of r [37, equations (4.1)–(4.5), (5.1)–(5.4)]:
ξMP
t = vt, ξMP
r = vr, ξMP
even = ru, ξMP
odd = rw,
hMP
tt = htt, hMP
rr = hrr, hMP
tr = htr,
jMP
t = rjt, jMP
r = rjr, KMP = K, GMP = G,
hMP
t = rht, hMP
r = rhr, hMP
2 = r2h2.
The conventions for the spherical harmonics used in [5, Appendix A] (BL) are different. In their
notation [5, equation (8)], the trace-reversed metric perturbation p̄µν = pµν − 1
2p
λ
λ
4gµν is
expanded as
p̄µν =
∑
lm
10∑
i=1
(Bl − 2)
r
p̄(i)lm(r, t) a(i)l BLY (i)lm
µν ,
where they factored out some normalizing constants a(i)l [5, equations (9,10)] from their basis.
The relation of their basis to ours is the following, which can be read off by comparing the
corresponding coordinate formulas in [37, Appendix A] and [5, equations (9,10),Appendix A]:
a(1) BLY (1)
µν =
1
2
[Y (3)
µν + f−2Y (4)
µν ], a(2) BLY (2)
µν =
f−2
2
Y (1)
µν ,
a(3) BLY (3)
µν =
1
2
[Y (3)
µν − f−2Y (4)
µν ], a(4) BLY (4)
µν =
1
2Bl
Y (2)
µν ,
a(5) BLY (5)
µν =
f−1
2Bl
Y (6)
µν , a(6) BLY (6)
µν =
1
2
Y (5)
µν , a(7) BLY (7)
µν =
1
2Al
2Y (7)
µν ,
a(8) BLY (9)
µν = − 1
2Bl
Y (8)
µν , a(9) BLY (9)
µν = −f
−1
2Bl
Y (9)
µν , a(10) BLY (10)
µν = − 1
2Al
2Y (10)
µν .
B Operators for the Lichnerowicz even sector
Below, we give explicit formulas for the morphism operators k̄2e, k̄
′
2e and homotopy operators h2e,
h̃2e that appear in the equivalence diagram (5.11) in Section 5.3.2. The morphism operators are
first order. Hence, they can be written as
k̄2e = k̄
(1)
2e
1
iω
∂r + k̄
(0)
2e and k̄′2e = k̄
′(1)
2e
1
iω
∂r + k̄
′(0)
2e .
Let us denote by C and C̃ the coefficient matrices of ∂2r in □2 e and □̃2 e, respectively. They
are clearly diagonal and invertible. By comparing the highest order coefficients in the identity
□2 ek̄2e = k̄′2e□̃2 e, we find the relation
Ck̄(1)2e = k̄
′(1)
2e C̃,
which means that there exists a matrix K such that
k̄
′(1)
2e = K and k̄
(1)
2e = C−1KC̃.
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 51
More explicitly
C = r2
−2f 0 0 0 0 0 0
0 −2Bl 0 0 0 0 0
0 0 f−1 0 0 0 0
0 0 0 f3 0 0 0
0 0 0 0 2f 0 0
0 0 0 0 0 2Blf
2 0
0 0 0 0 0 0 Al
2 f
, C̃ = −2f
ω2
1 0 0 0 0 0 0
0 Bl 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 αAl 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 Bl 0
0 0 0 0 0 0 1
.
The expression for K is somewhat large, so we display it split into groups of columns:
K
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 α
0 0 0 0
0 0 0 0
0 0 0 0
=
2irω −2irω − i(15Bl+2)rω
8
i(3f−Bl−1)(3f+Bl−3)rω
6
0 −irω −iBlrω
iBl(3f−Bl−1)rω
3
− f1
2f
f1
2f − (8f−15Bl−2)f1
32f −−12f1r2ω2−Bl(6f+Bl−2)f1
24f
−f(3f+1)
2
f(f+1)
2 −
(
f(24f2 − 13Blf
+2f − 15Bl − 2)
)
32 −
(
−12ff1r
2ω2 − Blf(6f
2
−5Blf + 4f + Bl − 2)
)
24
2f −f f(8f−7Bl−2)
8 −Blf(3f−Bl−1)
6
2Blf 2f(f − Bl)
Blf(24f−15Bl−2)
8 −
(
−12ff1r
2ω2 − Blf(6f
2
−2Blf − 2f − B2
l + 2Bl)
)
6
0 − (Bl−2)f
2 − (Bl−2)Blf
2 − (Bl−2)Blf(3f+2Bl−1)−36ff1r2ω2
12
,
K
0 0
0 0
0 0
0 0
1 0
0 1
0 0
=
i(15Bl+2)rω
8
i(3f−98B2
l −20Bl−3)rω
18Bl
iBlrω − i(15f+19Bl−8)rω
18
(8f−15Bl−2)f1
32f
−6f1r2ω2−Bl(15f−49Bl−10)f1
36Blf
f(24f2−13Blf+2f−15Bl−2)
32
−6ff1r2ω2−Blf(45f
2−109Blf−25f−49Bl−10)
36Bl
−f(8f−7Bl−2)
8
f(30f−79Bl−25)
18
−Blf(24f−15Bl−2)
8
2Blf(34Blf−5f−49B2
l −10Bl)−3ff1r2ω2
18Bl
(Bl−2)Blf
2 − (Bl−2)(19Bl−5)f
36
,
K
0
0
0
0
0
0
1
=
i(16f2−471B2
l −84Bl+20)rω
128
− iBl(2f+15Bl+4)rω
8
−−64f1r2ω2−(32f2+16Blf+32f+471B2
l +84Bl−4)f1
512f
−−64ff1r2ω2−f(96f3+240Blf
2+128f2+453B2
l f+76Blf+20f+471B2
l +84Bl−4)
512
−f(32f2+64Blf+32f+231B2
l +36Bl−4)
128
−Blf(128f
2−464Blf−64f+471B2
l +84Bl−4)
128
3(Bl−2)Blf(f−5Bl−1)
16
.
Note that the 4th column of K is proportional to 1/α, which is reflected in the notation in
a special way.
It is easy to see (comparing the coefficients of highest order terms of the relevant identities)
that the homotopy corrections h2e and h̃2e depend only on k
(1)
2e and k̄
(1)
2e coefficients. More
explicitly, plugging in their dependence on K, we obtain
h2e = C−1KC̃Σ2eK∗C−1, and h̃2e = Σ2eK∗C−1K.
Since C̃Σ2e = Σ2eC̃, while C∗ = C, C̃∗ = C̃, Σ∗
2e = Σ2e and Σ2
2e = id, the operators h2e and h̃2e
manifestly satisfy the expected self-adjointness properties (5.13).
52 I. Khavkine
The lower order terms k̄
(0)
2e and k̄
′(0)
2e are also quite lengthy, so we present them column by
column:8
k̄
(0)
2e
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
0 0 0
0 0 0
=
1
(iωr)3
i(f+1)rω
f − iBlrω
f − i(8f2−Blf−6f+15Bl+2)rω
16f
−2irω 2iBlrω
i(15Bl+2)rω
8
ff1 − 2r2ω2 2Blr
2ω2 2(8f+15Bl+2)r2ω2+(Bl+6)ff1
16
−
(
2r2ω2
−f(3f + 2Bl + 1)
)
f2
2Blr
2ω2−2B2
l f
f2 −
(
2(8f − 15Bl − 2)r2ω2
−f(19Blf+18f−30B2
l −3Bl+6)
)
16f2
−2f − Bl B2
l −10Blf+12f−15B2
l −2Bl
16
−4 r2ω2+2Blf
f
4r2ω2−f(4f−7Bl+2)
4f
2 −2Bl
8f−15Bl−2
8
,
k̄
′(0)
2e
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
0 0 0
0 0 0
=
1
iωr
i(f−3)rω
f − i(2f−3)rω
f
i(8f2−31Blf−10f+45Bl+6)rω
16f
2iBlrω
f −2iBlrω
f − iBl(15Bl+2)rω
8f
−2r2ω2−(4f+1)f1
2f2
2r2ω2−(3f+1)f1
2f2
2(8f + 15Bl + 2)r2ω2
+(40f2 − 44Blf − 8f
−15Bl − 2)f1
32f2
−
(
2r2ω2 − 12f2
−2Blf + f − 1
)
2
2r2ω2−3f2−2Blf−1
2 −
2(8f − 15Bl − 2)r2ω2 − 120f3
+20Blf
2 + 48f2 + 30B2
l f
+5Blf − 2f + 15Bl + 2
32
−8f − Bl + 2 3f + Bl − 1 −80f2−32Blf−48f−15B2
l +12Bl+4
16
−8Blf r2ω2−2f(4f−3Bl−2) 2Blr
2ω2−Blf(26f−11Bl−12)
2
(Bl−2)Bl
2
(Bl−2)(3f−Bl−1)
2
(Bl−2)Bl(56f−15Bl−18)
32
,
k̄
(0)
2e
0
0
0
1
0
0
0
=
1
(iωr)3
2iB2
l rω(−9f3+24f2−9f−2)+36iBlff
2
1 rω−2iB3
l (3f
2+3f−2)rω+iB4
l (3f−1)rω
12f
−12iBlff1rω−2iB2
l (3f
2−3f−2)rω+iB4
l rω−4iB3
l rω
6
12B2
l ff
2
1+6B3
l ff1(f−2)+3B4
l ff1
12
12B2
l ff1(1+f)−2B3
l (−3f3−3f2+10f)+B4
l (−3f2+11f)−2B5
l f
12f2
−12B2
l f1f−2B3
l (3f
2−3f−2)+B5
l −4B4
l
12
6Blf
2f1−B2
l (−3f3+3f2+4f)+4fB3
l −B4
l f
3f
2B2
l (−3f2+9f−2)−12Blff1−2B3
l (3f−2)−B4
l
6
− 1
iωr
2iB2
l rω(6f−6)+24iBlf1rω
12f
24if1rω−12iBlf1rω
6
−2B2
l (3f
2−6f−1)−8B3
l +2B4
l +12Blf
2
1
12
−2B2
l (15f
2−18f−1)−8B3
l +2B4
l −12Bl(5f−1)f1
12f2
−12B2
l f1+24Blf1
12
−2Bl(3f
2−6f+1)−B2
l (3f+1)+B3
l −12ff1
3f
12B2
l −24Blf+24(3f−2)f1
6
,
8These formulas are copied from the output of computer algebra.
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 53
k̄
′(0)
2e
0
0
0
1
0
0
0
=
1
α
1
iωr
− i(36f3−6Blf
2−42f2+5B2
l f−4Blf−3B2
l +6Bl+6)rω−12if1r3ω3
12f
− iBl(6f
2−6f+B2
l −2Bl)rω−12if1r3ω3
6f
−2(9f2−18f−B2
l +2Bl+9)r2ω2+Bl(24f
2−6f+Bl−2)f1
24f2
−2(21f2−30f−B2
l +2Bl+9)r2ω2+Bl(24f
3−12Blf
2−6f2+2B2
l f−Blf+Bl−2)
24
Bl(24f
2−6Blf−18f+B2
l +2)−12f1r2ω2
12
− (6f2+3Blf−6f−B2
l −Bl)r
2ω2+Blf(15f
2−4Blf−13f+2Bl+2)
3(
12(3f2 − 2Blf − 8f + B2
l + 5)r2ω2 + 24B2
l f
2 − 48Blf
2
+6B3
l f − 30B2
l f + 36Blf − 4B3
l + 10B2
l − 4Bl − α
)
24
,
k̄
(0)
2e
0
0
0
0
1
0
0
= 1
(iωr)3
i(8f2−Blf−6f+15Bl+2)rω
16f
− i(15Bl+2)rω
8
−2(15Bl+2)r2ω2+(Bl+6)ff1
16
−2(15Bl+2)r2ω2+f(19Blf+18f−30B2
l −3Bl+6)
16f2
−16r2ω2−10Blf−12f+15B2
l +2Bl
16
−4r2ω2−f(4f−7Bl+2)
4f
−8f−15Bl−2
8
,
k̄
′(0)
2e
0
0
0
0
1
0
0
=
1
iωr
− i(8f2−31Blf−10f+45Bl+6)rω
16f
iBl(15Bl+2)rω
8f
−2(15Bl+2)r2ω2+(40f2−44Blf−8f−15Bl−2)f1
32f2
−2(15Bl+2)r2ω2+120f3−20Blf
2−48f2−30B2
l f−5Blf+2f−15Bl−2
32
−16r2ω2−80f2+32Blf+48f+15B2
l −12Bl−4
16
−2Blr
2ω2−Blf(26f−11Bl−12)
2
− (Bl−2)Bl(56f−15Bl−18)
32
,
k̄
(0)
2e
0
0
0
0
0
1
0
=
1
(iωr)3
− iBl(30Blf+3f+49Bl+10)rω−6if1r3ω3
18f
−−3if1r3ω3−2iB2
l (49Bl+10)rω
18Bl
(3f2+18Blf−6f+98B2
l +20Bl+3)r2ω2−30B2
l ff1
18
− (3f2+18Blf−98B2
l −20Bl−3)r2ω2+2B2
l f(30f+49Bl+25)
18f2
3(f+6Bl−1)r2ω2+B2
l (45f+49Bl+10)
18
− (6f2+3Blf−6f−19B2
l +8Bl)r
2ω2+2B2
l f(15f−79Bl−10)
18Blf
3(f+2Bl−1)r2ω2+B2
l (15f−49Bl−10)
9Bl
,
54 I. Khavkine
k̄
′(0)
2e
0
0
0
0
0
1
0
=
1
iωr
−6if1r3ω3−i(3f2+68B2
l f+17Blf−147B2
l −30Bl−3)rω
18Blf
2iBl(15f
2−15f−49B2
l −10Bl)rω−3if1r3ω3
18Blf
(9f2+18Blf−18f+98B2
l +20Bl+9)r2ω2+Bl(75f
2−177Blf−45f−49Bl−10)f1
36Blf2(
(3f2 − 18Blf − 12f + 98B2
l + 20Bl + 9)r2ω2
+Bl(225f
3 − 387Blf
2 − 210f2 − 98B2
l f + 10Blf + 15f − 49Bl − 10)
)
36Bl
[0.9ex]
3(f+6Bl−1)r2ω2−Bl(150f
2−282Blf−165f−49B2
l +69Bl+25)
18Bl
− (6f2+3Blf−6f−19B2
l +8Bl)r
2ω2+2Blf(151Blf−20f−177B2
l −98Bl+10)
18Bl(
3(588f2 + Blf − 1178f + 2B2
l − 5Bl + 590)r2ω2 + 72B3
l f
−159B2
l f + 30Blf − 127B3
l + 259B2
l − 10Bl − 49α
)
36Bl
,
k̄
(0)
2e
0
0
0
0
0
0
1
=
1
(iωr)3
i(80Blf
2+64f2+9B2
l f+28Blf+4f−471B2
l −84Bl+4)rω−64if1r3ω3
256f
− i(64f2−32f−471B2
l −84Bl+4)rω
128
2(16f2+8Blf+16f+471B2
l +84Bl−20)r2ω2+f(32f2−96Blf−64f+9B2
l −4Bl−28)f1
256(
2(80f2 − 8Blf − 16f + 471B2
l + 84Bl − 20)r2ω2 + f(96f3 − 160Blf
2
−160f2 − 5B2
l f − 364Blf − 148f − 942B3
l − 159B2
l + 4Bl − 28)
)
256f2
−16(4f−Bl−2)r2ω2+64f3−128Blf
2−128f2+2B2
l f−136Blf−56f−471B3
l −84B2
l +4Bl
256
−8(6f−15Bl−4)r2ω2+f(48f2−56Blf−96f−231B2
l −44Bl−12)
64f
64r2ω2+64f2−16Blf−128f−471B2
l −84Bl+4
128
,
k̄
′(0)
2e
0
0
0
0
0
0
1
=
1
iωr
− i(96f3+80Blf
2−32f2+951B2
l f+196Blf−36f−1413B2
l −252Bl+44)rω
256f
iBl(128f
2−96f−471B2
l −84Bl+4)rω
128f
− (192f3+176Blf
2+128f2+1404B2
l f+240Blf−16f+471B2
l +84Bl−4)f1
512f2(
−576f4 − 1360Blf
3 − 320f3 − 1364B2
l f
2 + 128Blf
2 + 176f2
−942B3
l f − 177B2
l f − 4Blf − 4f − 471B2
l − 84Bl + 4
)
512
−−384f3−768Blf
2−128f2−1384B2
l f+32Blf+160f−471B3
l +378B2
l +76Bl−8
256
−−Blf(168f
2−436Blf−144f+351B2
l +296Bl+36)
32
−416B2
l f
2+832Blf
2+1424B3
l f−2400B2
l f−896Blf−1506B3
l +2920B2
l +184Bl−471α
512
− iωr
64if1rω
256f
0
−2(16f2−8Blf−80f−471B2
l −84Bl+52)
512f2
2(48f2−8Blf+48f+471B2
l +84Bl−52)
512
−16(4f−Bl−2)
256
−4Bl(6f−15Bl−4)
32
4(4239f2−8478f+16B2
l −32Bl+4239)
512
.
Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation 55
Acknowledgments
Research of the author was partially supported by the Praemium Academiae of M. Markl, GAČR
project GA18-07776S and RVO: 67985840. Early stages of this work were completed while the
author was affiliated with the University of Rome II (Tor Vergata) and with the University of
Milan (Statale). Thanks to Claudio Dappiaggi, Felix Finster, Christian Gérard, Dietrich Häfner,
Peter Hintz, Dmitry Jakobson, Niky Kamran, Vesselin Petkov, Iosif Polterovich, Ko Sanders,
Werner Seiler and Artur Sergyeyev for interesting discussions. Thanks to Stefanos Aretakis and
Mihalis Dafermos for their hospitality during a visit to Princeton University, where an early
version of this work was first presented in the Spring of 2016. The author also thanks Francesco
Bussola for checking the mode decompositions (5.6c) and (5.6d) as part of his MSc thesis [13].
And final thanks go to Roman O. Popovych and to anonymous referees, whose comments greatly
improved the presentation of the manuscript.
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1 Introduction
2 Results and applications
2.1 Main result
2.2 Spectral problem
2.3 Symmetries and potentials
3 Formal properties of differential equations and operators
3.1 Equations and morphisms
3.2 Triangular decoupling strategy
3.3 Formal adjoints
4 Morphisms and extensions of Regge–Wheeler equations
4.1 Equivalence of Regge–Wheeler and Zerilli equations
5 Tensor wave equations on Schwarzschild
5.1 Scalar wave equation
5.2 Vector wave equation
5.2.1 Odd sector
5.2.2 Even sector
5.3 Lichnerowicz wave equation
5.3.1 Odd sector
5.3.2 Even sector
6 Discussion
A Tensor spherical harmonics
B Operators for the Lichnerowicz even sector
References
|
| id | nasplib_isofts_kiev_ua-123456789-211534 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T12:55:46Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Khavkine, Igor 2026-01-05T12:28:37Z 2022 Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations. Igor Khavkine. SIGMA 18 (2022), 011, 57 pages 1815-0659 2020 Mathematics Subject Classification: 35Q75; 34L99; 34L05; 68W30 arXiv:2004.09651 https://nasplib.isofts.kiev.ua/handle/123456789/211534 https://doi.org/10.3842/SIGMA.2022.011 We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form complicated systems of coupled linear ODEs. We outline a precise abstract strategy to decouple these systems into a sparse triangular form, where the diagonal blocks consist of spin- scalar Regge-Wheeler equations (for spins = 0,1,2). Building on the example of the vector wave equation, which we have treated previously, we complete a successful implementation of our strategy for the Lichnerowicz wave equation. Our results go a step further than previous, more ad-hoc attempts in the literature by presenting a full and maximally simplified final triangular form. These results have important applications to the quantum field theory of and the classical stability analysis of electromagnetic and gravitational perturbations of the Schwarzschild black hole in harmonic gauges. Research of the author's research was partially supported by the Praemium Academiae of M. Markl, GACR project GA18-07776S, and RVO: 67985840. Early stages of this work were completed while the author was affiliated with the University of Rome II (Tor Vergata) and with the University of Milan (Statale). Thanks to Claudio Dappiaggi, Felix Finster, Christian Gerard, Dietrich Hafner, Peter Hintz, Dmitry Jakobson, Niky Kamran, Vesselin Petkov, Iosif Polterovich, Ko Sanders, Werner Seiler, and Artur Sergyeyev for interesting discussions. Thanks to Stefanos Aretakis and Mihalis Dafermos for their hospitality during a visit to Princeton University, where an early version of this work was first presented in the Spring of 2016. The author also thanks Francesco Bussola for checking the mode decompositions (5.6c) and (5.6d) as part of his MSc thesis [13]. And final thanks go to Roman O. Popovych and to anonymous referees, whose comments greatly improved the presentation of the manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations Article published earlier |
| spellingShingle | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations Khavkine, Igor |
| title | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations |
| title_full | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations |
| title_fullStr | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations |
| title_full_unstemmed | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations |
| title_short | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations |
| title_sort | explicit triangular decoupling of the separated lichnerowicz tensor wave equation on schwarzschild into scalar regge-wheeler equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211534 |
| work_keys_str_mv | AT khavkineigor explicittriangulardecouplingoftheseparatedlichnerowicztensorwaveequationonschwarzschildintoscalarreggewheelerequations |