From pp-Waves to Galilean Spacetimes
We exhibit all spatially isotropic homogeneous Galilean spacetimes of dimension (𝑛 + 1) ≥ 4, including the novel torsional ones, as null reductions of homogeneous pp-wave spacetimes. We also show that the pp-waves are sourced by pure radiation fields and analyse their global properties.
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| description | We exhibit all spatially isotropic homogeneous Galilean spacetimes of dimension (𝑛 + 1) ≥ 4, including the novel torsional ones, as null reductions of homogeneous pp-wave spacetimes. We also show that the pp-waves are sourced by pure radiation fields and analyse their global properties.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 035, 16 pages
From pp-Waves to Galilean Spacetimes
José FIGUEROA-O’FARRILL a, Ross GRASSIE b and Stefan PROHAZKA a
a) Maxwell Institute and School of Mathematics, The University of Edinburgh,
James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, Scotland, UK
E-mail: j.m.figueroa@ed.ac.uk, stefan.prohazka@ed.ac.uk
b) Laboratory for Foundations of Computer Science, School of Informatics,
The University of Edinburgh, Informatics Forum, 10 Crichton Street, Edinburgh EH8 9AB,
Scotland, UK
E-mail: rgrassie@ed.ac.uk
Received October 27, 2022, in final form May 22, 2023; Published online June 03, 2023
https://doi.org/10.3842/SIGMA.2023.035
Abstract. We exhibit all spatially isotropic homogeneous Galilean spacetimes of dimension
(n+ 1) ≥ 4, including the novel torsional ones, as null reductions of homogeneous pp-wave
spacetimes. We also show that the pp-waves are sourced by pure radiation fields and analyse
their global properties.
Key words: pp-waves; Galilean spacetimes; null reduction
2020 Mathematics Subject Classification: 22F30; 53C30; 53Z05
1 Motivation and introduction
Maximally symmetric Lorentzian spacetimes provide a natural arena for many areas of physics
ranging from high energy physics (Minkowski space) to cosmology (de Sitter space) and quan-
tum gravity (anti-de Sitter space). Beyond these well-known Lorentzian spacetimes, there is
a plethora of equally interesting Galilean, Carrollian and Aristotelian spacetimes. These space-
times were classified in [14], completing the pioneering work of Bacry and Lévy-Leblond [1].
One of the salient features of the classification in [14] is the emergence of two novel families of
generically torsional1 Galilean spacetimes, containing the (A)dS–Galilei spacetimes as special
(non-torsional) points, see Figure 1.
Unlike many of the spacetimes in the classification, very little is known about these torsional
Galilean spacetimes; however, there are various lines of investigation in which they may play
interesting roles.
Holography. Particles in (A)dS–Galilei spacetimes have recently appeared as instructive toy
models in the holography literature [4, 20, 22]. It is, therefore, natural to ask how these torsional
geometries, which interpolate between AdS–Galilei and dS–Galilei, could appear in these models.
Intrinsically Galilean. The torsional Galilei algebras are a purely Galilean feature with no
Lorentzian or Carrollian counterpart [14]. They are the only one-parameter family of non-
equivalent maximally symmetric spaces, and they do not arise naturally as limits of Lorentzian
spaces. Therefore, it could well be that effects due exclusively to the presence of torsion may be
intrinsically Galilean, in some sense.
1These reductive homogeneous spacetimes are torsional in the sense that their canonical invariant connections
have nonzero torsion. This should not be confused with the intrinsic torsion [9, 19] of the Galilean geometry, which
is given by the exterior derivative of the clock one-form. The intrinsic torsion vanishes in all cases considered, but
that does not mean that every connection has vanishing torsion, just that there is at least one connection which
does.
mailto:j.m.figueroa@ed.ac.uk
mailto:stefan.prohazka@ed.ac.uk
mailto:rgrassie@ed.ac.uk
https://doi.org/10.3842/SIGMA.2023.035
2 J. Figueroa-O’Farrill, R. Grassie and S. Prohazka
This paper and the accompanying work [10] aim to deepen our understanding of these novel
geometries and demystify them.
dS
M
AdS
dSG−1 = dSG dSG1 = AdSG∞
G
AdSG = AdSG0
dSGγ∈[−1,1]
AdSGχ≥0
Figure 1. Spatially isotropic homogeneous Galilean spacetimes in dimension n+1 ≥ 4. The blue arrows
denote limits arising from Lie algebra contractions and show that (A)dS–Galilei arise naturally as limits
of (A)dS, while the families of torsional Galilean spacetimes (green arrows) do not.
It is well known that we can obtain non-relativistic spacetimes by null reductions of Bargmann
spacetimes in one dimension higher [7, 18]; namely, Lorentzian spacetimes admitting a nowhere-
vanishing null vector field. In Duval, Burdet, Künzle and Perrin [7] this null vector field is
assumed to be parallel with respect to the Levi-Civita connection; in other words, the Bargmann
spacetime is a Brinkmann spacetime [5] or, equivalently, a pp-wave [8]. Starting with Minkowski
spacetime, the null reduction along a parallel null vector field gives Galilei spacetime. In [18],
the null vector field need not be parallel, but only Killing, and hence the Lorentzian manifolds
go beyond the class of pp-waves. Later, Gibbons and Patricot [17] extended these ideas to
obtain the non-relativistic limits of (anti) de Sitter spaces, which they called Newton–Hooke
spacetimes, as null reductions of homogeneous pp-waves. These (A)dS–Galilei spacetimes can
be interpreted as non-relativistic spacetimes with a non-vanishing cosmological constant. More
recently Bekaert and Morand [3] studied conditions under which the null reduction by proper
and free, but not necessarily isometric, action of the additive reals on a Lorentzian manifold
results in a Galilean spacetime. We shall see examples of both the Julia–Nicolai null reduction
and the more general null reduction of Bekaert–Morand in this paper.
Before we start, a word of nomenclature. From now on, by the word ‘spacetime’ we shall
always mean a reductive homogeneous spacetime. Any reductive homogeneous space carries
a canonical invariant connection (defined by the vanishing of the Nomizu map, as explained
in the current context, for example, in [11]), as well as a number of homogeneous tensor fields
arising by the holonomy principle from any invariant tensors of the linear isotropy representation
at any chosen “origin” of the spacetime, as will be recalled below. By reduction we shall mean
first and foremost reduction of homogeneous spaces, but in some cases this reduction also results
directly in a reduction of the additional structure: connection and/or homogeneous tensor fields.
The symmetric Galilean spacetimes – namely, Galilei and (anti) de Sitter–Galilei – are ho-
mogeneous spaces of the Galilei and Newton–Hooke groups. They are known to arise as null
reductions of certain homogeneous pp-wave spacetimes and hence it may be expected that the
torsional Galilean spacetimes also admit such a description. In this paper, we show that this ex-
From pp-Waves to Galilean Spacetimes 3
pectation is correct and exhibit explicit homogeneous pp-wave spacetimes whose null reductions
agree with all the spatially isotropic homogeneous Galilean spacetimes in [14], including the
torsional cases. We also show that these pp-waves are solutions of the Einstein field equations
sourced by pure radiation fields. For convenience, we collect Galilean spacetimes in Table 1,
whose notation we now briefly explain.
Table 1. Spatially isotropic homogeneous Galilean spacetimes (n > 2).
Spacetime Additional nonzero Lie brackets Name
G Galilei
dSG [H,P ] = −B de Sitter–Galilei (also dSGγ=−1)
dSGγ [H,P ] = γB + (1 + γ)P torsional de Sitter–Galilei (γ ∈ (−1, 1))
dSG1 [H,P ] = B + 2P torsional de Sitter–Galilei (γ = 1)
AdSG [H,P ] = B anti de Sitter–Galilei (also AdSGχ=0)
AdSGχ [H,P ] =
(
1 + χ2
)
B + 2χP torsional anti de Sitter–Galilei (χ > 0)
In this table we provide the nonzero Lie brackets in addition to [J ,J ] = J , [J ,B] = B,
[J ,P ] = P and [B, H] = P , in a basis where the stabiliser subalgebra h is spanned by ⟨J ,B⟩.
A kinematical Lie algebra (in spatial dimension n) is a real Lie algebra with so(n) rotations
accompanied by two vectors (boosts and spatial translations) and one scalar (temporal transla-
tion). More concretely, g is spanned by Jij = −Jji, Bi, Pi and H, where i, j = 1, . . . , n and the
Lie brackets include the following
[Jij , Jkℓ] = δjkJiℓ − δikJjℓ − δjℓJik + δiℓJjk,
[Jij , Bk] = δjkBi − δikBj ,
[Jij , Pk] = δjkPi − δikPj ,
[Jij , H] = 0, (1.1)
and any other brackets consistent with the Jacobi identities. It is convenient to use an abbrevi-
ated notation in which we avoid the indices and write the above brackets as
[J ,J ] = J , [J ,B] = B, [J ,P ] = P , [J , H] = 0.
In this work we focus on homogeneous kinematical spacetimes; that is, homogeneous spacetimes
of a kinematical Lie group G. Every such spacetime is described infinitesimally by a Klein pair2
(g, h), where g, the Lie algebra of G, is the transitive Lie algebra, and h, the isotropy subalgebra,
is the Lie algebra of the subgroup H of G which fixes a chosen origin for the homogeneous
spacetime. It is this choice of homogeneous spacetime that provides a physical interpretation to
the abstract Lie group G and gives meaning to what we call a boost (whose generator belongs
to h along with the rotations) and what we call a translation (which is in the complement). For
example, the Poincaré group gives rise to a plethora of interesting inequivalent homogeneous
spacetimes with different physical interpretations, see, e.g., [13].
In Table 1, we have chosen a basis for g in such a way that the Lie subalgebra h is spanned
by Jij and Bi, the rotations and (Galilean) boosts. Every row in the table lists the additional
(i.e., not included in equation (1.1) or [B, H] = P , which is common to Galilean transitive Lie
algebras) nonzero Lie brackets in such a basis in the above abbreviated notation. As shown
in [14], the Klein pairs in the table are in bijective correspondence with (isomorphism classes of)
simply-connected spatially isotropic homogeneous Galilean spacetimes. These spacetimes form
2Originally, and unwisely, called a Lie pair in [14].
4 J. Figueroa-O’Farrill, R. Grassie and S. Prohazka
two continua dSGγ , for −1 ≤ γ < 1, and AdSGχ, for χ ≥ 0. They both have a common limit
limγ→1 dSGγ = limχ→∞ AdSGχ, which is no contraction and is represented by the red dot in
Figure 1. All torsional Galilean spacetimes share a common geometric limit (i.e., a contraction
at the level of the kinematical Lie algebras) to the Galilei spacetime G, as illustrated by the
blue arrows that leave the green arrows in Figure 1. That figure also contains the maximally
symmetric Lorentzian spacetimes and their Galilean (non-relativistic) and zero-curvature limits.
The spacetime denoted dSG1 is more properly thought of as a limit limγ→1 dSGγ and it
coincides with the limit limχ→∞ AdSGχ. All Galilean spacetimes are reductive and G, dSG and
AdSG are symmetric. They are the ones which can be obtained as non-relativistic limits of
Minkowski, de Sitter and anti de Sitter spacetimes, respectively.
This note is organised as follows. In Section 2, we introduce some Klein pairs correspond-
ing to homogeneous pp-wave spacetimes; that is, homogeneous Lorentzian manifolds admitting
a parallel (relative to the Levi-Civita connection) null vector field. In Section 3, we describe
the invariant Lorentzian metrics relative to (modified) exponential coordinates. In Section 4, we
calculate the curvature of the pp-wave metrics and show that they are solutions of the Einstein
field equations sourced by pure radiation fields. In Section 5, we exhibit explicit expressions
for the Killing vector fields. In Section 6, we discuss the null reductions along the parallel null
vector and show that they give the homogeneous Galilean spacetimes described in the Introduc-
tion. The Galilean structure induced by the reduction only partially agrees with the invariant
Galilean structure: the spatial cometric is the invariant one, but the clock one-form is only
homothetic to the invariant one; that is conformal to it but with a constant conformal factor.
We then introduce a different null reduction resulting in the same homogeneous Galilean space-
time, but inducing the invariant Galilean structure on the nose. Two of the pp-wave metrics
in Section 3 are flat and hence locally isometric, but their reductions result in non-isomorphic
homogeneous Galilean spacetimes. This is discussed in Section 7. Finally, in Section 8, we offer
some conclusions. In Appendix A, we provide a complementary set of coordinates and study
global properties of the homogeneous pp-waves.
2 Lorentzian Klein pairs of the pp-waves
In this section, we introduce a number of Klein pairs corresponding to homogeneous Lorentzian
manifolds admitting a parallel null vector field; that is, homogeneous pp-wave spacetimes. We
will later show that their associated null reductions give all the homogeneous Galilean spacetimes
in Table 1.
Before describing them, it is perhaps useful to say something about where they come from.
They arose initially in a forthcoming follow-up paper to [12] in which we discuss geometries
associated to Lie algebras obtained from Lifshitz Lie algebras by the addition of boosts. The
homogeneous pp-wave spacetimes in question are geometric realisations of effective Klein pairs
(g, h) where g is a deformation of the centrally extended static kinematical Lie algebra and h
is spanned by what could be interpreted as spatial rotations and boosts. The Lie algebras g
had already appeared in [16, Table 2] for spatial dimension n = 3 and in [15, Table 18] for
n > 3. These Lie algebras also arise naturally in the description of particle dynamics [10] on the
homogeneous Galilean spacetimes in Table 1.
Let us now describe the Lorentzian Klein pairs (g, h) in question. The Lie algebra g is
spanned by Jij = −Jji, Bi, Pi, H and Z; although despite the notation Z is not necessarily
central. The Lie brackets are written in the abbreviated form and we list only the nonzero Lie
brackets in g in addition to [J ,J ] = J , [J ,B] = B, [J ,P ] = P , [B,P ] = Z and [B, H] = P , in
a basis where the stabiliser subalgebra h is spanned by Jij and Bi. They are listed in Table 2,
where the spacetimes have been labelled as PX with X a Galilean spacetime, foreshadowing their
interpretation as a principal (right) R-bundle over X.
From pp-Waves to Galilean Spacetimes 5
Table 2. Homogeneous Lorentzian spacetimes.
Spacetime Additional nonzero Lie brackets Comments
PG
PdSG [H,P ] = −B
PdSGγ [H,P ] = γB + (1 + γ)P [H,Z] = (1 + γ)Z γ ∈ (−1, 1)
PdSG1 [H,P ] = B + 2P [H,Z] = 2Z
PAdSG [H,P ] = B
PAdSGχ [H,P ] =
(
1 + χ2
)
B + 2χP [H,Z] = 2χZ χ > 0
In this table we provide the nonzero Lie brackets in addition to [J ,J ] = J , [J ,B] = B,
[J ,P ] = P , [B,P ] = Z and [B, H] = P , in a basis where the stabiliser subalgebra h is spanned
by ⟨J ,B⟩.
For each pair (g, h) above we have a reductive decomposition g = m⊕ h with m = ⟨P , H, Z⟩.
Let πa, η, ζ be the canonically dual basis for m∗, i.e., the nonzero relations are ⟨πa, Pb⟩ = δab ,
⟨η,H⟩ = 1 and ⟨Z, ζ⟩ = 1. Then for all Klein pairs in Table 2, there is an H-invariant Lorentzian
inner product on m given by π2 − 2ηζ and an H-invariant vector given by Z. The holonomy
principle associates to the inner product a G-invariant Lorentzian metric g on the homogeneous
space with Klein pair (g, h) and to the invariant vector a nowhere-vanishing null vector field ζ.
Both the metric and the null vector field are parallel with respect to the canonical invariant
connection on the homogeneous space, whose holonomy representation is the linear isotropy
representation of H. It is only in the case of a symmetric space ([m,m] ⊂ h) that the canonical
invariant connection agrees with the Levi-Civita connection of g. In that case, the null vector
field ζ is parallel with respect to the Levi-Civita connection and indeed agrees with the parallel
null vector of the pp-wave. In this case, and this case alone, is ζ also, in particular, Killing.
The above reductive split lets us introduce the coset parametrisation3 σ : m → G
σ(u, v,x) = exp(vZ) exp(x · P ) exp(uH). (2.1)
This parametrisation is of course not unique and we provide a complementary set of coordinates
in Appendix A. The pull-back σ∗ϑ = σ−1dσ of the Maurer–Cartan one-form on G decomposes
into σ∗ϑ = θ+ω, where θ is a local coframe, in terms of which the G-invariant Lorentzian metric
is given by g =
(
π2 − 2ηζ
)
(θ, θ). The one-form ω is the canonical invariant connection and it
only agrees with the Levi-Civita connection of the invariant metric in the symmetric case; that
is, when [m,m] ⊂ h. This is only the case for PG, PdSG and PAdSG. Not coincidentally, these are
precisely the cases where Z is central and hence g is a central extension of a Galilean kinematical
Lie algebra: Bargmann in the case of PG and (anti) de Sitter–Bargmann in the cases of PdSG
and PAdSG which are sometimes also called Bargmann–Newton–Hooke.
3 The pp-wave metrics and the invariant vector fields
We now construct the explicit pp-wave metrics g and the invariant vector fields ζ in the modified
exponential coordinates
(
u, v, xi
)
in equation (2.1). We calculate the coframe θ and then we
evaluate the metric g =
(
π2 − 2ηζ
)
(θ, θ). The value at p ∈ M of the invariant vector field ζ is
the inverse image of Z under the isomorphism TpM → m defined by the coframe.
For PG we find that
σ∗ϑ = duH + dvZ + dx · P = θ,
3We refer to (u, v,x) as modified exponential coordinates, to distinguish them from the truly exponential
coordinates which would be defined by σ(u, v,x) = exp(vZ + uH + x · P ).
6 J. Figueroa-O’Farrill, R. Grassie and S. Prohazka
so that ω = 0. We recognise the metric
g = dx · dx− 2dudv (3.1)
as Minkowski spacetime in light-cone coordinates. This metric is of course invariant under the
full Poincaré group, but here we are reducing the structure to the Bargmann subgroup singled
out by the distinguished null vector ζ = ∂v corresponding to Z.
For PdSGγ , we have
σ∗ϑ = duH + exp(−u adH)(dvZ + dx · P )
= duH + e−u(1+γ)dvZ + (exp(−u adH)P ) · dx.
From
−u adH
(
B P
)
=
(
B P
)(0 −γu
u −(1 + γ)u
)
we find that for γ ∈ [−1, 1),
exp(−u adH)
(
B P
)
=
(
B P
) 1
1− γ
(
e−uγ − γe−u γ
(
e−u − e−uγ
)
e−uγ − e−u e−u − γe−uγ
)
and for γ = 1,
exp(−u adH)
(
B P
)
=
(
B P
)(e−u(1 + u) −ue−u
ue−u e−u(1− u)
)
,
from where we read off
(exp(−u adH)P ) =
γ
1− γ
(
e−u − e−uγ
)
B +
1
1− γ
(
e−u − γe−uγ
)
P , γ ∈ [−1, 1),
−ue−uB + (1− u)e−uP , γ = 1.
The local coframe is then given by
θ = duH +
e−u(1+γ)dvZ +
1
1− γ
(
e−u − γe−uγ
)
P · dx, γ ∈ [−1, 1),
e−2udvZ + (1− u)e−uP · dx, γ = 1
(3.2)
and the connection one form is given by
ω =
γ
1− γ
(
e−u − e−uγ
)
B · dx, γ ∈ [−1, 1),
−ue−uB · dx γ = 1.
For γ ≤ 0, the coframe is invertible for any u ∈ R whereas for γ ∈ (0, 1) we have the restriction
u < ln γ
γ−1 . From equation (3.2) we can determine the invariant vector field ζ as well as the
resulting metric g. With γ ∈ (−1, 1) they are given by (we have singled out the two extreme
points):
(PdSG) g = −2dudv + (coshu)2dx · dx, ζ = ∂v,
(PdSGγ) g = −2e−u(1+γ)dudv +
1
(1− γ)2
(
e−u − γe−uγ
)2
dx · dx, ζ = e(1+γ)u∂v,
(PdSG1) g = −2e−2ududv + e−2u(1− u)2dx · dx, ζ = e2u∂v.
From pp-Waves to Galilean Spacetimes 7
Lastly, for PAdSGχ we have
σ∗ϑ = duH + exp(−u adH)(dvZ + dx · P )
= duH + e−2uχdvZ + (exp(−u adH)P ) · dx.
From
−u adH
(
B P
)
=
(
B P
)(0 −u
(
1 + χ2
)
u −2uχ
)
we find that
exp(−u adH)
(
B P
)
=
(
B P
)
e−uχ
(
cosu+ χ sinu −
(
1 + χ2
)
sinu
sinu cosu− χ sinu,
)
from where we read off
(exp(−u adH)P ) = e−uχ(cosu− χ sinu)P − e−uχ
(
1 + χ2
)
sinuB.
The local coframe is given by
θ = duH + e−2uχdvZ + e−uχ(cosu− χ sinu)P · dx
and the connection one-form is given by
ω = −e−uχ
(
1 + χ2
)
sinuB · dx.
The invariant metrics and vector fields, where now χ > 0 and we have singled out the case
χ = 0, are given by
(PAdSG) g = −2dudv + (cosu)2dx · dx, ζ = ∂v
(PAdSGχ) g = −2e−2uχdudv + e−2uχ(cosu− χ sinu)2dx · dx, ζ = e2χu∂v.
For χ > 0 the coframe is invertible in the range −π
2 < u < arctan
(
1
χ
)
and for χ = 0 in
−π/2 < u < π/2.
In Appendix A, using a different set of (global) coordinates, we show that the simply-
connected Lorentzian spacetimes are actually diffeomorphic to Rn+2.
We summarise the results of this section in Table 3.
Table 3. Invariant metrics and vector fields.
Spacetime Metric g Vector field ζ Comments
PG −2dudv + dx · dx ∂v
PdSG −2dudv + (coshu)2dx · dx ∂v PdSGγ=−1
PdSGγ −2e−(1+γ)ududv +
( e−u−γe−γu
1−γ
)2
dx · dx e(1+γ)u∂v γ ∈ (−1, 1)
PdSG1 −2e−2ududv + e−2u(1− u)2dx · dx e2u∂v
PAdSG −2dudv + (cosu)2dx · dx ∂v PAdSGχ=0
PAdSGχ −2e−2χududv + e−2χu(cosu− χ sinu)2dx · dx e2χu∂v χ > 0
8 J. Figueroa-O’Farrill, R. Grassie and S. Prohazka
4 Curvature tensors
Every metric in Table 3 is of the following form
g = −2a(u)dudv + b(u)dx · dx,
for some functions a(u) and b(u), which can easily be read off. Our first observation is that such
a metric is conformally flat. Indeed, away from the set of points where b(u) = 0, we may factor
out b(u) and write
g = b(u)
(
−2a(u)
b(u)dudv + dx · dx
)
and then simply define a new coordinate t by dt = −a(u)/b(u)du so that the metric becomes
g = b(u)(2dtdv + dx · dx),
which is manifestly conformally flat. The Weyl tensor vanishes outside the set of points where
b(u) = 0, which for the metrics in Table 3 is either the whole space (when b never vanishes, as
in PG and PdSG) or a dense open subset (in the remaining metrics). In either case, if the Weyl
tensor vanishes in an open dense subset, it vanishes everywhere.
The nonzero components of the Riemann and Ricci tensor of the Levi-Civita connection of g
are given by
Ruxiuxi = 1
4
(
2a′b′
a + b′2
b − 2b′′
)
,
Ruu = n
bRuxiuxi
and any other components related to these by the symmetries of the Riemann tensor, where
i = 1, . . . , n and where we do not sum over the i indices.
For the case at hand this reduces to
(PdSG) Ruxiuxi = −(coshu)2, Ruu = −n,
(PdSGγ) Ruxiuxi =
γ
(1− γ)2
(
e−u − γe−uγ
)2
, Ruu = nγ,
(PdSG1) Ruxiuxi = e−2u(1− u)2, Ruu = n
and
(PAdSG) Ruxiuxi = (cosu)2, Ruu = n,
(PAdSGχ) Ruxiuxi =
(
1 + χ2
)
e−2uχ(cosu− χ sinu)2, Ruu = n
(
1 + χ2
)
.
Notice that for γ = 0, the Riemann tensor of the metric in PdSGγ vanishes and thus PdSGγ=0
and PG are locally isometric. We shall contrast these two cases in more detail in Section 7.
By inspection we also see that the Ricci scalar vanishes in all cases. This means that the
only non-vanishing components of the Einstein tensor are the null Ricci tensor components Ruu.
This suggests that these metrics can be understood as solutions of the Einstein field equations
which are sourced by pure radiation fields (null dust) for which the energy momentum tensor
satisfies (see, e.g., [21, Section 5.2])
Tµν = ϕ2kµkν , kµk
µ = 0.
For our metrics this means that k = du and ϕ2 = nγ or ϕ2 = n
(
1 + χ2
)
, which would seem to
require γ ≥ 0. The physical interpretation of the metrics PdSGγ<0 is not so clear.
From pp-Waves to Galilean Spacetimes 9
5 The Killing vectors
We will now exhibit explicit expressions for the Killing vector fields of the metrics in Table 3.
For each generator X ∈ g, we will exhibit vector fields ξX such that [ξX , ξY ] = −ξ[X,Y ] for all
X,Y ∈ g.
In all cases ξJij = −xi∂j + xj∂i, ξPi = ∂i and ξZ = ∂v. It then remains to give expressions
for ξBi and ξH . We find that
ξBi = xi∂v + f(u)∂i,
ξH = ∂u + h(u)xi∂i +
(
λv + 1
2µx
2
)
∂v, (5.1)
for functions f , h of u and constants λ, µ. The value of λ is determined from the [ξH , ξZ ]
bracket and that of µ by [ξH , ξPi ], which also gives an algebraic relation allowing us to solve
for h in terms of f . Finally, the bracket [ξBi , ξH ] gives a first-order ODE for f , which we can
solve in each case. The results of these calculations are summarised in Table 4. As a check on
the calculations, one can show that the invariant vector fields in Table 3 commute with all the
Killing vectors.
Table 4. Data for ξBi
and ξH in equation (5.1).
Spacetime λ µ f(u) h(u)
PG 0 0 u 0
PdSG 0 −1 tanhu − tanhu
PdSGγ∈(−1,1) 1 + γ γ − e−u−e−γu
e−u−γe−γu
e−u−γ2e−γu
e−u−γe−γu
PdSG1 2 1 u
1−u
2−u
1−u
PAdSG 0 1 tanu tanu
PAdSGχ>0 2χ 1 + χ2 sinu
cosu−χ sinu 2χ+
(
1 + χ2
)
sinu
cosu−χ sinu
6 The null reductions
We now show that the homogeneous pp-waves null reduce to the torsional Galilean spaces
as homogeneous spaces (spacetimes). We do this first via Killing reduction, which is always
guaranteed to result in a Galilean structure in the quotient. Doing so, however, we find that the
reduced invariant structure matches the clock one-form of the torsional spaces only up to scale.
This is then remedied by performing a reduction by the invariant vector field ζ. This results
in the same homogeneous quotient (since ζ is invariant) but now with the invariant Galilean
structure.
6.1 Killing reduction of spacetime
Each of the Lorentzian metrics in Table 3 possesses a nowhere vanishing null Killing vector
field ξ = ∂v in the modified exponential coordinates employed above. This Killing vector field
generates a one-parameter subgroup Γ of the isometry group of (M, g): the one generated by
Z ∈ g. The space of orbits M/Γ can be given the structure of a smooth manifold in such a way
that the canonical projection π : M → M/Γ taking a point to its orbit under Γ is a smooth
map. This allows us to pull back functions and differential forms from M/Γ to M and sets up
an isomorphism of C∞(M/Γ)-modules between the forms on M/Γ and the basic forms on M ,
where we remind the reader that basic forms are those forms α which are horizontal, so that
ıξα = 0, and invariant, so that Lξα = 0. Equivalently, α is basic if both ıξα = 0 and ıξdα = 0.
10 J. Figueroa-O’Farrill, R. Grassie and S. Prohazka
Let ξ♭ = g(ξ,−) be the one-form metrically dual to ξ. Then ıξξ
♭ = g(ξ, ξ) = 0 because ξ
is null and Lξξ
♭ = (Lξξ)
♭ = 0, where we have used that ξ is a Killing vector field, so that
it commutes with the musical isomorphisms. This shows that ξ♭ = π∗τ , for some one-form
τ ∈ Ω1(M/Γ).
If α ∈ Ω1(M/Γ), we may construct a vector field (π∗α)♯ on M by pulling α back to M and
then applying the musical isomorphism. Given α, β ∈ Ω1(M/Γ) we get a function
g
(
(π∗α)♯, (π∗β)♯
)
= (π∗α)
(
(π∗β)♯
)
= π∗α
(
π∗(π
∗β)♯
)
.
This defines ψ ∈ Γ
(
⊙2T (M/Γ)
)
by
ψ(α, β) = α
(
π∗(π
∗β)♯
)
.
Clearly, if β = τ , then (π∗β)♯ = ξ and since π∗ξ = 0, we see that ψ(α, τ) = 0 for all α ∈ Ω1(M).
The pair (τ, ψ) defines a Galilean structure on M . Galilean structures can be classified into
several types according to their intrinsic torsion dτ [2, 6, 9, 19]. In all examples in this section,
the null killing vector ξ = ∂v is actually parallel, so that dξ♭ = 0 and hence dτ = 0. So that the
intrinsic torsion of the Galilean spacetimes vanishes in all cases.
Below we will calculate local coordinate expressions for τ and ψ, so let us unpack the previous
discussion and re-express everything in a local chart. It is often convenient to work in local
coordinates which are adapted to the reduction. This means choosing local coordinates xµ =
(xa, v) forM such that xa are local coordinates for the baseM/Γ. Since functions on the base lift
to Γ-invariant functions on M , it follows that ξxa = 0, so that ξ ∝ ∂v. It is convenient to choose
the coordinate v to be adapted to ξ, so that ξ = ∂v. Nevertheless we will write ξ = ξµ∂µ, with the
tacit understanding that ξa = 0 and ξv = 1. The metric has a local expression g = gλρdx
λdxρ
and gvv = 0 since ξ is null. The dual one-form ξ♭ has a local expression ξ♭ = ξµdx
µ, where
ξµ = gµρξ
ρ. Since ξ is null, it follows that ξv = 0 and the only nonzero components are ξa =: τa,
the clock one-form. It is locally a one-form on M/Γ since ξ is also a Killing vector. To obtain
a local coordinate expression for ψ, let α, β be two one-forms on the base, whose local expressions
are α = αadx
a and similarly for β. Then ψ(α, β) = ψabαaβb = gλρg
λagρbαaβb, so that ψab = gab.
Below we will actually choose local coordinates xa =
(
u, xi
)
and we will see that the only nonzero
components of ψ are ψij and that τ is proportional to du.
Let (g, h) be one of the pp-wave Klein pairs in Table 2. It is clear that in all cases, the
generator Z ∈ g spans an ideal ⟨Z⟩ of g. Quotienting by the action of the one-parameter subgroup
generated by Z gives rise to a homogeneous space with Klein pair
(
g, h
)
, where g = g/⟨Z⟩ and
h = h/h ∩ ⟨Z⟩ ∼= h. It is clear by inspection of Table 2 that for each such Klein pair (g, h), the
quotient Klein pair
(
g, h
)
is the corresponding one in Table 1. In other words, the homogeneous
pp-wave spacetimes in Table 2 can be realised as the total space of principal (right) R-bundles
over the homogeneous Galilean spacetimes in Table 1. In summary, this exhibits the Galilean
spatially isotropic homogeneous spacetimes in Table 1 as null reductions of homogeneous pp-
wave spacetimes with metrics given by Table 3.
6.2 Reduction of invariant structure
A natural question is whether we recover them not just as homogeneous spaces, but whether the
induced Galilean structure is the invariant one. The clock one-form τ pulls back to the one-form
metrically dual to ξZ . It follows that if X ∈ g does not commute with Z, the fundamental vector
field ξX in the quotient does not preserve τ . Indeed, we have that
π∗LξX
τ = LξXπ
∗τ = LξX ξ
♭
Z = [ξX , ξZ ]
♭ = ξ♭[Z,X].
It is clear from the Lie brackets in Table 2, that H is the only generator of g which may fail
to commute with Z and, when this happens, it simply rescales Z by a constant. Suppose that
From pp-Waves to Galilean Spacetimes 11
[Z,H] = wZ for some weight w. Then it follows from the above calculation that LξH
τ = wτ ,
so that it acts homothetically on the clock one-form.
It is a simple matter to read off the Galilean structure (τ, ψ) relative to the coordinates(
t = −u, xi
)
for each of the above spacetimes. These are listed in Table 5. That table also lists
the homogeneous Galilean spacetimes corresponding to the null reductions, using the notation
of [14]. They cannot be immediately compared with the ones in [11] since we are using a different
coordinate system, but they can be recognised by their Klein pair as explained above.
Table 5. Galilean structures of null reductions.
Spacetime τ ψ
G dt δij∂i ⊗ ∂j
dSG dt (cosh t)−2δij∂i ⊗ ∂j
dSGγ e(1+γ)tdt
( et−γeγt
1−γ
)−2
δij∂i ⊗ ∂j
dSG1 e2tdt e−2t(1 + t)−2δij∂i ⊗ ∂j
AdSG dt (cos t)−2δij∂i ⊗ ∂j
AdSGχ e2χtdt e−2χt(cos t+ χ sin t)−2δij∂i ⊗ ∂j
The fundamental vector fields ξX in the quotient are easy to determine from the expressions
of the Killing vector fields: all we need to do is drop any component along ξZ = ∂v. Doing
so, we see that ξJij and ξPi
are given formally by the same expression as the Killing vector
fields, whereas ξBi
= f(u)∂i and ξH = ∂u + h(u)xi∂i, where f(u) and h(u) can be read off from
Table 4. It is easy to check that these fundamental vector fields define an anti-representation of
the Galilean kinematical Lie algebra g. It follows from an explicit calculation that the Galilean
structure (τ, ψ) in the quotient is invariant under the first derived ideal [g, g], which is spanned
by Jij , Bi, Pi. On the other hand, the generator H leaves invariant ψ, but acts homothetically
on τ , as explained above.
Indeed, it is a simple calculation to check that for G, dSG and AdSG, the Galilean structure
induced from the null reduction is the invariant one, whereas for dSGγ , one finds LξH
τ = (1+γ)τ
and for AdSGχ, one finds LξH
τ = 2χτ . It follows that the invariant clock one-form is conformal
to the one induced by the null-reduction. Indeed, for dSGγ , it is e−(1+γ)tτ which is invariant,
whereas for AdSGχ, the invariant clock one-form is e−2χtτ . In [10, Appendix D], it is shown that
there exists a conformal rescaling of the Lorentzian metric such that the vector field ξZ remains
Killing and the corresponding null reduction does give the invariant clock-one form on the nose;
although one now pays the price that the induced spatial cometric is only homothetic to the
invariant one.
6.3 Reduction by invariant vector field
We may solve the above problems by reducing via the action of the reals generated by the
invariant vector field ζ. This vector field is parallel with respect to the canonical invariant
connection, which is compatible with the invariant metric g, but has typically nonzero torsion.
It is therefore not Killing in the torsional cases, but since it commutes with Killing vector fields,
its flow commutes with the G-action. This means that the quotient M/Γζ , with Γζ the one-
dimensional subgroup of diffeomorphisms of M generated by ζ, admits an action of G. However
the normal subgroup Γ ⊂ G generated by Z acts trivially onM/Γζ . Indeed, the unparametrized
integral curves of ζ and ξZ agree. To see this, notice from Table 3, that ζ = eβu∂v in all cases,
for some β ∈ R. The integral curves of ζ are given by
c(s) =
(
u0, v0 + eβu0s,x0
)
,
12 J. Figueroa-O’Farrill, R. Grassie and S. Prohazka
whereas the integral curves of ξZ = ∂v are given by
c(s) = (u0, v0 + s,x0).
As a consequence, M :=M/Γζ is a homogeneous space of G = G/Γ and provides a geometric
realisation of the Klein pair
(
g, h
)
, where g = g/⟨Z⟩ and h = h/h ∩ ⟨Z⟩ ∼= h, as above.
One can check that the induced Galilean structure agrees with the invariant one, but there
is no need to do this explicitly, since the fact that the action of Γζ on M commutes with the G-
action, any projectable G-invariant tensor field onM (i.e., one descending toM) is automatically
G-invariant.
For example, it follows from the explicit expressions of ζ in Table 3, that the metrically dual
one-form ζβ is −du in all cases, which agrees with the invariant clock one-forms on the Galilean
spacetimes in the coordinates
(
t = −u, xi
)
, as can be checked from the formulae in Section 3,
but ignoring the Z-component of the soldering forms. The expressions for the induced spatial
cometric ψ are precisely those in Table 5.
7 Is G = dSG0?
As we saw above, the invariant Lorentzian metric in PdSGγ=0 is flat. This means that the
spacetime PdSGγ=0 is (perhaps only a portion of) Minkowski spacetime. The coordinates u, v, x
are unconstrained, but they are not flat coordinates. Indeed, relative to these coordinates, the
metric takes the form
g = −2e−ududv + e−2udx · dx.
This suggests first of all defining a new coordinate t = e−u, which takes only positive real values.
Relative to
(
t, v, xi
)
the metric becomes
g = 2dtdv + t2dx · dx.
We wish to find flat coordinates for this metric. Flat coordinates are characterised by the fact
that the coordinate vector fields are parallel and the metrically dual one-forms are exact. Hence
to find them, we must first determine the parallel vectors. A vector field ξ = ξµ∂µ is parallel if
∂µξ
ρ + Γµν
ρξν = 0.
We may find the Christoffel symbols by comparing the Euler–Lagrange equations of the la-
grangian
L = ṫv̇ + 1
2 t
2ẋ · ẋ
with the geodesic equation
ẍρ + Γµν
ρẋµẋν = 0.
Doing so, we find the following nonzero Christoffel symbols
Γij
v = −tδij and Γti
j = Γit
j = 1
t δ
j
i .
The vector field ξ = ξµ∂µ is parallel if and only if the components ξµ satisfy the following partial
differential relations:
∂µξ
t = 0, ∂tξ
i + 1
t ξ
i = 0, ∂tξ
v = 0,
∂vξ
µ = 0, ∂iξ
j + 1
t δ
j
i ξ
t = 0, ∂iξ
v − tδijξ
j = 0.
From pp-Waves to Galilean Spacetimes 13
Solving these relations, we find the following parallel vector fields
∂v, xi∂v +
1
t ∂a, ∂t − 1
tx
i∂i − 1
2x
2∂v.
The metrically dual one-forms are, respectively,
dt, d
(
txi
)
and d
(
v − 1
2 tx
2
)
,
from where we can read off the flat coordinates
T = t, Xi = txi and V = v − 1
2 tx
2,
which we may invert to write
t = T, xi = Xi
T and v = V + 1
2
X2
T .
Expressing the metric relative to these coordinates, we find
g = 2dTdV + dX · dX,
which is indeed manifestly flat. Notice that since t > 0, it is also the case that T > 0 so the
original metric covers half of Minkowski spacetime and the new coordinates allow us to extend
the metric to the whole of Minkowski spacetime.
The null vector ξZ = ∂v becomes
∂v =
∂T
∂v
∂T +
∂V
∂v
∂V +
∂Xi
∂v
∂Xi = ∂V
in the new coordinates. The null reduction of this metric along ∂V is clearly the same as the
null reduction of the metric in equation (3.1) along ∂v: simply change coordinates to t = −u
and notice that the metrics agree under
(
t, v, xi
)
7→
(
T, V,Xi
)
.
This does not mean, however, that the homogeneous Galilean spacetimes dSG0 and G are
the same. One way to see this is to notice that under the improved reduction by the invariant
vector field ζ, which results in the invariant Galilean structure, the reductions are not the
same. For G, it is still the case that ζ = ∂v, but for dSG0, ζ = eu∂v or, after the change of
coordinates, ζ = 1
T ∂V , which does not extend smoothly to the full Minkowski spacetime in
(T, V,X) coordinates.
8 Discussion and conclusions
In this note we have exhibited the spatially isotropic homogeneous Galilean spacetimes (of spatial
dimension n > 2) classified in [14] as null reductions of homogeneous pp-wave spacetimes. We
have performed two null reductions with identical quotients as homogeneous spacetimes: one by
a Killing vector field and one by an invariant vector field. These two reductions agree for the
non-torsional (i.e., symmetric) homogeneous spacetimes, but are different for the torsional ones.
In the former reduction, we find that although the null reduction gives the homogeneous Galilean
spacetimes as homogeneous spaces, the Galilean structure induced from the null reduction is
not the invariant one in the torsional cases (dSGγ for −1 < γ ≤ 1 and AdSGχ for χ > 0), but
only homothetic to it. This is a reflection of the fact that the null Killing vector is not central
in the torsional cases. In contrast, the null reduction by the invariant vector field results in the
invariant Galilean structure by design.
This work was focused on Galilean spacetimes and their intricate properties. As final note
let us advertise their Carrollian curved counterparts (A)dS–Carroll, which arise as a ultra-
relativistic limit of (A)dS. They are complementary equally interesting candidates for the study
of holography in a possibly more tractable setup (besides the intriguing relation of AdS-Carroll
to time-like infinity [13]). They can be seen as null hypersurfaces of (A)dS [14] and AdS–Carroll
shares the box-like (its spatial metric is hyperbolic) and dS–Carroll the cosmological character
(its spatial metric is the sphere) of their Lorentzian parents [11].
14 J. Figueroa-O’Farrill, R. Grassie and S. Prohazka
A Global properties of the Lorentzian spacetimes
In this appendix we show that every simply-connected homogeneous Lorentzian spacetime M
whose Klein pair is listed in Table 2 is diffeomorphic to Rn+2.
To do this we introduce another set of coordinates (u, v, xa) and define the parametrisation
σ(u, v,x) = exp(vZ) exp(uH) exp(x · P ),
which is closer in spirit to the one used in [11, Appendix A.2] for the Galilean spacetimes. With
[H,P ] = αB + βP and [H,Z] = βZ and using
e−x·PHex·P = H + αx ·B + βx · P + 1
2αx
2Z,
we obtain
σ∗ϑ = duH +
(
e−βudv + 1
2αx
2du
)
Z + (dx+ βdux) · P + αdux ·B. (A.1)
The soldering form
θ = duH +
(
e−βudv + 1
2αx
2du
)
Z + (dx+ βdux) · P
is everywhere invertible.
Let us define a map j : Rn+2 → M by j(u, v,x) = σ(u, v,x) · o, where o ∈ M is a choice of
origin with stabiliser the subgroup H generated by spatial rotations and boosts. The first thing
to remark is that we may ignore the rotations and think of M simply as a homogeneous space
of the solvable Lie group K generated by B, P , H, Z. The stabiliser of the origin is then the
subgroup B spanned by B. We define an action of K on Rn+2 by requiring j to be equivariant.
Introducing the shorthands
ϖ := 1
2
√
4α− β2, s :=
sin(ϖu)
2ϖ
and f± = cos(ϖu)± βs,
a short calculation reveals that
eaZ · (u, v,x) = (u, v + a,x),
ebH · (u, v,x) =
(
u+ b, ebβv,x
)
,
ew·B · (u, v,x) =
(
u, v + sf+w
2 + euβ/2f+w · x,x+ 2e−uβ/2sw
)
,
ey·P · (u, v,x) =
(
u, v − αsf−y
2 − 2αuuβ/2sy · x,x+ e−uβ/2f−y
)
,
from where we may read the fundamental vector fields:
ξZ = ∂v,
ξH = ∂u + βv∂v,
ξBa = euβ/2f+x
a∂v + 2e−uβ/2s∂a,
ξPa = −2αeuβ/2sxa∂v + e−uβ/2f−∂a.
One can check that these form an anti-representation of the Lie algebra k of K.
Our first observation is that the action of K on Rn+2 is transitive. Indeed, taking as origin
the point with coordinates (0, 0,0), we see that we can reach any other point by the action of
σ(u, v,x) ∈ K:
(u, v,x) = σ(u, v,x) · (0, 0,0),
From pp-Waves to Galilean Spacetimes 15
which follows essentially by definition. The smooth map j : Rn+2 → M is a local diffeomor-
phism, since the soldering form is everywhere invertible, and it is K-equivariant. Together with
transitivity, this says that j is a branched covering map, but the branched locus has to be empty,
otherwise it would not be homogeneous. Therefore j is a covering which, since Rn+2 is simply
connected, is universal. ButM was assumed to be the simply-connected homogeneous spacetime
realising the given Klein pair, hence j is an isomorphism of K-homogeneous spacetimes and, in
particular, M is diffeomorphic to Rn+2.
These coordinates have other advantages. For example, the connection one-form is uniformly
given by ω = αdux ·B, see the last term in (A.1), and the invariant Lorentzian metric g and
vector field ζ can be brought to a more uniform form. To see this, let us write the invariant
Lorentzian metric g =
(
π2 − 2ηζ
)
(θ, θ) as
g = −2
(
e−βudv + α
2x
2du
)
du+ (dx+ βdux)2,
and the invariant vector field ζ as
ζ = eβu∂v.
Acknowledgments
We are grateful to Dieter Van den Bleeken, who spotted an erroneous claim in an earlier version
of this note. JMF would like to acknowledge many useful discussions with Can Görmez and
Dieter Van den Bleeken on related topics leading up to [10]. In addition, we are grateful to an
anonymous referee for a critical reading of the manuscript and a number of insightful comments
and suggestions.
SP was supported by the Leverhulme Trust Research Project Grant (RPG-2019-218) “What
is Non-Relativistic Quantum Gravity and is it Holographic?”.
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1 Motivation and introduction
2 Lorentzian Klein pairs of the pp-waves
3 The pp-wave metrics and the invariant vector fields
4 Curvature tensors
5 The Killing vectors
6 The null reductions
6.1 Killing reduction of spacetime
6.2 Reduction of invariant structure
6.3 Reduction by invariant vector field
7 Is G = dSG_0?
8 Discussion and conclusions
A Global properties of the Lorentzian spacetimes
References
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| id | nasplib_isofts_kiev_ua-123456789-211908 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T05:07:48Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Figueroa-O'Farrill, José Grassie, Ross Prohazka, Stefan 2026-01-16T10:57:02Z 2023 From pp-Waves to Galilean Spacetimes. José Figueroa-O'Farrill, Ross Grassie and Stefan Prohazka. SIGMA 19 (2023), 035, 16 pages 1815-0659 2020 Mathematics Subject Classification: 22F30; 53C30; 53Z05 arXiv:2208.07627 https://nasplib.isofts.kiev.ua/handle/123456789/211908 https://doi.org/10.3842/SIGMA.2023.035 We exhibit all spatially isotropic homogeneous Galilean spacetimes of dimension (𝑛 + 1) ≥ 4, including the novel torsional ones, as null reductions of homogeneous pp-wave spacetimes. We also show that the pp-waves are sourced by pure radiation fields and analyse their global properties. We are grateful to Dieter Van den Bleeken, who spotted an erroneous claim in an earlier version of this note. JMF would like to acknowledge many useful discussions with Can Görmez and Dieter Van den Bleeken on related topics leading up to [10]. In addition, we are grateful to an anonymous referee for a critical reading of the manuscript and a number of insightful comments and suggestions. SP was supported by the Leverhulme Trust Research Project Grant (RPG-2019-218) “What is Non-Relativistic Quantum Gravity and is it Holographic?”. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications From pp-Waves to Galilean Spacetimes Article published earlier |
| spellingShingle | From pp-Waves to Galilean Spacetimes Figueroa-O'Farrill, José Grassie, Ross Prohazka, Stefan |
| title | From pp-Waves to Galilean Spacetimes |
| title_full | From pp-Waves to Galilean Spacetimes |
| title_fullStr | From pp-Waves to Galilean Spacetimes |
| title_full_unstemmed | From pp-Waves to Galilean Spacetimes |
| title_short | From pp-Waves to Galilean Spacetimes |
| title_sort | from pp-waves to galilean spacetimes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211908 |
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