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Classical and Quantum Gravity
Class. Quantum Grav. 40 (2023) 184002 (33pp) https://doi.org/10.1088/1361-6382/acecce
Finsler gravitational waves of (α,β)-type
and their observational signature
Sjors Heefer∗and Andrea Fuster
Department of Mathematics and Computer Science, Eindhoven University of
Technology, Eindhoven, The Netherlands
E-mail: s.j.heefer@tue.nl
Received 16 February 2023; revised 24 July 2023
Accepted for publication 2 August 2023
Published 14 August 2023
Abstract
We introduce a new class of (α,β)-type exact solutions in Finsler gravity
closely related to the well-known pp-waves in general relativity. Our class
contains most of the exact solutions currently known in the literature as spe-
cial cases. The linearized versions of these solutions may be interpreted as
Finslerian gravitational waves, and we investigate the physical effect of such
waves. More precisely, we compute the Finslerian correction to the radar dis-
tance along an interferometer arm at the moment a Finslerian gravitational
wave passes a detector. We come to the remarkable conclusion that the effect
of a Finslerian gravitational wave on an interferometer is indistinguishable
from that of standard gravitational wave in general relativity. Along the way
we also physically motivate a modication of the Randers metric and prove
that it has some very interesting properties.
Keywords: Finsler geometry, Finsler gravity, (α,β)-metrics, exact solutions,
gravitational waves, Randers metrics, Berwald spacetimes
(Some gures may appear in colour only in the online journal)
∗Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution
4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the
title of the work, journal citation and DOI.
1361-6382/23/184002+33$33.00 © 2023 The Author(s). Published by IOP Publishing Ltd Printed in the UK 1
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
Contents
1. Introduction 2
1.1. Structure of this article 4
2. Finsler gravity 4
2.1. Finsler spacetime geometry 5
2.2. Berwald spacetimes 6
2.3. A note about causal structure and physical interpretation 7
2.4. The eld equations 7
3. (α, β)-metrics 8
3.1. (α,β)-metrics—basic denitions 8
3.2. Exact (α,β)-metric solutions in Finsler gravity 8
3.3. Plane wave solutions in Brinkman and Rosen coordinates 11
3.4. Linearized gravitational wave solutions 12
3.5. Linearized (α,β)-metrics are Randers metrics 13
4. Modied Randers metrics 14
4.1. Motivation and denition 15
4.2. Causal structure 16
4.3. Regularity and signature 17
5. Radar distance for a Finsler gravitational wave 20
5.1. Finslerian null geodesics 21
5.2. Radar distance 22
6. Discussion 26
Data availability statement 28
Acknowledgments 28
Appendix A. Some properties of the metric aµν +bµbν28
A.1. Proof of Lorentzian signature 28
A.2. Afne structure 28
Appendix B. Determinant of a not necessarily positive denite (α, β)-metric 30
References 31
1. Introduction
Even though in the general theory of relativity (GR) the geometry of spacetime is modelled
by a (pseudo-)Riemannian metric of Lorentzian signature, there is no clear physical principle,
nor experimental evidence, that tells us that this spacetime geometry should necessarily be
(pseudo-)Riemannian. In fact, as suggested already in 1985 by Tavakol and Van den Bergh
[1–3], the axiomatic approach by Ehlers et al [4] is compatible with Finsler geometry, a nat-
ural extension of (pseudo-)Riemannian geometry. This was originally overlooked due to too
restrictive differentiability assumptions, as recently pointed out in [5] and then worked out in
detail in [6]. Other axiomatic approaches also allow for types of geometry more general than
the type used in GR, see e.g. [7]. This indicates that such types of geometries should not a pri-
ori be excluded from our theories and motivates the study of extensions of general relativity
based on more general spacetime geometries.
In this regard Finsler geometry is the natural candidate as it provides the most general
geometric framework that is still compatible with the clock postulate in the usual sense, namely
that the proper time interval measured by an observer between two events can be dened as the
length of its worldline connecting these events, in this case the Finslerian length rather than
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
the (pseudo-)Riemannian length. We remark that Weyl geometry, another generalization of
Lorentzian geometry, is also compatible with the clock postulate, but in that case the denition
of proper time has to be revised [8].
Further motivation for the study of Finsler spacetime geometry comes from quantum grav-
ity phenomenology [9]. Inspired by various approaches to quantum gravity, a generic fea-
ture of phenomenological or effective quantum gravity models is the presence of Planck-scale
modied dispersion relations (MDRs), related to departure from (local) Lorentz symmetry [9–
11], which may manifest either in the sense of Lorentz invariance violation or in the sense of
deformed Lorentz symmetry. It turns out that such MDRs generically induce a Finsler geo-
metry on spacetime [12]. The mathematical details of this were investigated in [13,14]; see
e.g. [15–17] for applications to specic quantum gravity phenomenology models.
Here we consider the (action-based) approach to Finsler gravity outlined in [18,19].
Structurally the theory is completely analogous to general relativity, but Einstein’s eld
equation is replaced by Pfeifer and Wohlfarth’s eld equation. For (pseudo-)Riemannian
spacetimes the latter reduces to the former. Although any solution to the eld equations of
GR is a solution in Finsler gravity, not many exact, properly Finslerian solutions are known
as of yet. To the best of our knowledge the only ones currently known in the literature are
the (m-Kropina type) Finsler pp-waves [20] and their generalization as very general relativity
(VGR) spacetimes [21], and the Randers pp-waves [22].
Here we introduce a large class of exact vacuum solutions that contains most of the
aforementioned solutions as special cases, the only exception being those solutions in [21]
that are not of pp-wave type. Namely, we prove that any Finsler metric constructed from a
(pseudo-)Riemannian metric αand a 1-form βthat is covariantly constant with respect to α,
is an exact vacuum solution in Finsler gravity if αis a vacuum solution in general relativity.
We classify all such solutions, leading to two possibilities: either αis at Minkowski space,
or αis a pp-wave. Our solutions are (α,β)-metrics of Berwald type.
The natural question that arises is whether and how such spacetimes can be physically dis-
tinguished from their general relativistic counterparts. To answer this question we consider the
linearized versions of our exact solutions, which may be interpreted as Finslerian gravitational
waves, and we study their physical effect. More precisely, we ask the question what would
be observed in an interferometer experiment when such a Finslerian gravitational wave would
pass the earth, and what would be the difference with a classical general relativistic gravit-
ational wave. The relevant observable measured in interferometer experiments is essentially
the radar distance, so we compute this radar distance for our Finslerian gravitational waves,
reproducing in the appropriate limit the radar distance formula for a standard gravitational
wave in GR [23]. Although at rst sight the expression for the Finsler radar length looks dif-
ferent from the corresponding expression GR, we show that this is nothing but a coordinate
artifact. Remarkably, when the two expressions are interpreted correctly in terms of observable
quantities, it becomes clear that there is in fact no observational difference between the Finsler
and GR case, at least as far as radar distance measurements are concerned. We discuss the sig-
nicance of this. To the best of our knowledge this is the rst time an explicit expression for the
Finslerian Radar length has been obtained in the case of nite spacetime separations, and as
such our work may be seen as a proof of concept. In contrast, the radar length for innitesimal
separations has been studied in [24,25].
We do point out that our results rely on the assumption that the amplitude of the gravitational
wave, as well as the parameter λthat characterized the departure from (pseudo)-Riemannian
geometry, are sufciently small, so that a certain perturbative expansion is valid. This never-
theless seems physically justied. We argue in a heuristic manner that up to rst order in λ, any
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
physically viable (α,β)-metric can be equivalently described by a slightly modied version
of a standard Randers metric.
Indeed, the causal structure of the standard Randers metric does not in general have a
straightforward physical interpretation. We therefore propose to modify the Randers metric
slightly, only changing some relative signs in different subsets of the tangent bundle. We then
prove that these modied Randers metrics have the nice property that their causal structure is
completely equivalent to the causal structure of some auxiliary (pseudo-)Riemannian metric.
This analysis is done in full generality, i.e. not just for our exact solutions. In the special case,
however, that the dening 1-form of the Randers metric is covariantly constant (as is the case
for our solutions) we prove that not only the causal structure, but also the afne structure of
the Finsler metric coincides with that of the auxilliary (pseudo)-Riemannian metric, i.e. the
timelike, spacelike and null geodesics of the Finsler metric can be understood, respectively, as
the timelike, spacelike and null geodesics of the auxiliary (pseudo)-Riemannian metric. This
leads to the particularly nice property that the existence of radar neighborhoods is guaranteed
[26], i.e. that given an observer and any event in spacetime, there is (at least locally) exactly
one future pointing light ray and one past pointing light ray that connect the event to the world-
line of the observer. This is of essential importance in our work, because without this property
the notion of radar distance would not even make sense.
1.1. Structure of this article
The paper is organized as follows. We begin in section 2with a discussion of Finsler geometry
and the core ideas behind Finsler gravity. Then in section 3we introduce (α,β)-metrics, and
in particular Randers metrics and discuss their relevance to Finsler gravity. We then introduce
our new solutions to the eld equations and show that after linearization these solutions may be
interpreted as Finslerian gravitational waves. Next, in section 4we propose our modication
of the standard Randers metric and prove that it has very satisfactory properties with respect
to its causal structure, afne structure, Lorentzian signature, etc. section 5is devoted to the
calculation of the radar distance at the moment a Finsler gravitational wave passes, say, the
Earth. We clearly point out the differences with the general relativity case. We conclude in
section 6.
2. Finsler gravity
In this section we recall the basic denitions in Finsler geometry and the basic ideas that under-
lie Finsler gravity. We will be brief and to the point; for a slightly more detailed introduction we
refer the reader to our previous article [22]. We start with some notational remarks. Throughout
this article will use induced local coordinates on the tangent bundle of a smooth manifold that
can be introduced in the following way. Given a chart ϕ:U⊂M→Rnon a smooth manifold
M, we identify any p∈Uwith its image ϕ(p)=(x0,...,xn−1)∈ϕ(U)⊂Rnunder ϕ. For p∈U
and Yp∈TpM, where TpMis the tangent space to Mat p, we can express Yp=yµ∂µ|pin terms
of the holonomic basis ∂µ≡∂/∂xµof TpM. This decomposition induces local coordinates
(x,y)≡(x0,...,xn−1,y0,...,yn−1)∈ϕ(U)×Rnon the tangent bundle TM (and any open sub-
manifold thereof). We will thus generally represent any point (p,Yp)∈TM by the tuple (x,y).
The holonomic basis vectors, corresponding to these coordinates, of the tangent space T(x,y)TM
to TM at (x,y) will be denoted by ∂µ=∂/∂xµand ¯
∂µ=∂/∂yµ. Throughout the article we will
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
also assume that the spacetime dimension is 1 +3, and we will use the signature convention
(−,+,+,+).
2.1. Finsler spacetime geometry
A Finsler spacetime is a triple (M,A,F), where Mis a smooth manifold, Ais a conic
subbundle1of TM \0 with non-empty bers, and F, the so-called Finsler metric, is a map
F:A → Rthat satises the following axioms:
•Fis (positively) homogeneous of degree one with respect to y:
F(x,λy) = λF(x,y),∀λ > 0;(1)
•The fundamental tensor, with components gµν =¯
∂µ¯
∂ν1
2F2, has Lorentzian signature
on A.
The denition of a Finsler spacetime given above is a very weak one in the sense that most other
denitions appearing in the literature are more restrictive (see e.g. [18,27–32]). Accordingly,
our denition allows for a lot of instances, many of which will not be physically viable. This
is, in our opinion, a feature rather than a bug, as most of the results in this article can be proven
without further restrictions. It should be understood, however, that in order to guarantee that a
viable physical interpretation is possible, the geometry should be subjected to more stringent
requirements.
Given a Finsler metric F, the length of a curve γ:λ7→ γ(λ)on Mcan be dened as
L(γ) = ˆF( ˙γ)dλ=ˆF(x,˙
x)dλ, ˙γ=dγ
dλ,(2)
which, due to homogeneity, is invariant under (orientation-preserving) reparameterization. It
follows by Euler’s theorem for homogeneous functions that
gµν (x,y)yµyν=F(x,y)2,(3)
and hence the length of curves is formally identical to the length in Riemannian geometry, the
difference being that now the ‘metric tensor’ may depend not only on position xbut on the
direction yas well. In fact if gµν =gµν (x), or, equivalently, if F2is quadratic in y, then Finsler
spacetime geometry reduces to classical Lorentzian geometry.
The fundamental theorem of Riemannian geometry generalizes to what’s sometimes called
the fundamental lemma of Finsler geometry. It states that any Finsler metric admits a unique
homogeneous (nonlinear) connection on the subbundle A ⊂ TM \0, characterized by its con-
nection coefcients Nρ
µ, that is torsion-free, ¯
∂νNρ
µ=¯
∂µNρ
ν, and metric-compatible, δµF2=0,
where δµ≡∂µ−Nρ
µ¯
∂ρis the horizontal derivative induced by the connection. This connection
is usually referred to as the Cartan nonlinear connection or the canonical nonlinear connection
and its connection coefcients are given by
Nρ
µ(x,y) = 1
4¯
∂µgρσyν∂ν¯
∂σF2−∂σF2(4)
1By a conic subbundle with non-empty bers we mean an open subset A ⊂ TM \0 such that (x,λy)∈ A for any
(x,y)∈ A and any λ > 0, and such that π(A) = M, where π:TM →Mis the canonical projection of the tangent
bundle.
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
where gρσ is the matrix inverse of the fundamental tensor gµν. Parallel transport of a vector V
along a curve γis then characterized by the parallel transport equation2
˙
Vµ+Nµ
ν(γ, V) ˙γµ=0,(5)
and consequently, autoparallels are those curves that satisfy
¨γµ+Nµ
ν(γ, ˙γ) ˙γν=0.(6)
The curvature tensor, Finsler Ricci scalar and the Finsler Ricci tensor of (M,F) are dened,
respectively, as
Rρµν (x,y) = δµNρ
ν(x,y)−δνNρ
µ(x,y),Ric(x,y) = Rρρµ(x,y)yµ,
Rµν (x,y) = 1
2¯
∂µ¯
∂νRic.(7)
2.2. Berwald spacetimes
A Berwald spacetime is a Finsler spacetime for which the Cartan nonlinear connection reduces
to a linear connection3, which is the case if and only if the connection coefcients are of the
form
Nρ
µ(x,y)=Γρ
µν (x)yν(8)
for a set of functions Γρ
µν :M→R. If so, the functions Γρ
µν can be identied as the Christoffel
symbols of a torsion-free afne connection on M. We will refer to this afne connection as the
associated afne connection, or simply the afne connection on the Berwald spacetime. Since
any (pseudo)-Riemannian spacetime is of Berwald type (with Γρ
µν given by the Levi-Civita
connection), we have the following inclusions:
(pseudo-)Riemannian ⊂Berwald ⊂Finsler.
The parallel transport (5) and autoparallel equation (6) on a Berwald space reduce to the famil-
iar equations
˙
Vi+ Γi
jk(γ) ˙γjVk=0,¨γi+ Γi
jk(γ) ˙γj˙γk=0 (9)
in terms of the Christoffel symbols. The curvature tensors (7) of a Berwald space can be
written as
Rjkl =¯
Rijkl(x)yi,Ric =¯
Rij(x)yiyj,Rij =1
2(¯
Rij(x) + ¯
Rji(x)) ,(10)
in terms of the Riemann tensor ¯
Rlijk =2∂[jΓi
k]l+2Γi
m[jΓm
k]land Ricci tensor ¯
Rlk =¯
Rliik of the
associated afne connection, where we have used the notation T[ij]=1
2(Tij −Tji)and T(ij)=
1
2(Tij +Tji)for (anti-)symmetrization. In fact, for positive denite Finsler spaces, it follows by
Szabo’s metrization theorem that Rij =1
2(¯
Rij +¯
Rji) = ¯
Rij, but this does not extend to Finsler
spacetimes in general [37].
2Note that the parallel transport map is in general nonlinear. Some authors (e.g. [33]) choose to dene parallel trans-
port differently, namely by requiring a priori that parallel transport should be linear, which leads to the alternative
parallel transport equation ˙
Vi+Ni
j(γ, ˙γ)Vi=0. This approach, however, seems unnatural to us. Here we follow e.g.
[34], where parallel transport of a vector is dened via its unique horizontal lift along a given curve. In this case
parallel transport is linear if and only if the connection is linear.
3See [35] for an overview of the various equivalent characterizations of Berwald spaces and [36] for a more recent
equivalent characterization.
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
2.3. A note about causal structure and physical interpretation
Given a Finsler spacetime geometry, it is natural to postulate, in analogy with GR, that matter
travels along timelike geodesics and light travels on null geodesics. The generalization of the
notion of null direction is mathematically straightforward. A vector yuat a point xµis said
to be null (or lightlike) if F(x,y)2=gµν (x,y)yµyν=0. However, the structure of the light
cone, composed of such null vectors, may be non-trivial. In GR it is always the case that the
light cone separates the tangent space at each point into three connected components, that we
may interpret as forward-pointing timelike vectors, backward-pointing timelike vectors, and
spacelike vectors, respectively. It is then a consequence that a timelike vector is one that has
positive (or negative, depending on the convention) Riemannian norm. For a generic Finsler
spacetime geometry these properties of the lightcone structure are by no means guaranteed and
as such it is not obvious in general how to even dene what one means a by timelike vector.
It certainly does not sufce to dene them as positive length vectors. We do not discuss this
issue any further in its full generality here. Only in the specic case of the Randers metric,
in section 4, will we dive into the details. We argue that the causal structure of the standard
Randers metric does not have a straightforward physical interpretation, but we prove that,
by modifying the denition only slightly, the causal structure of such a modied Randers
metric has exactly the desirable properties mentioned above in the case of GR, allowing for a
straightforward physical interpretation. This will be exploited in section 5, where we compute
the radar distance for a Finslerian gravitational wave of (modied) Randers type passing an
interferometer.
It is worth mentioning that in the ideal case the (forward and backward) timelike cones
should be contained in the subbundle A. This statement is essentially the condition that geo-
metry is well-dened for all timelike innitesimal spacetime separations. This property is sat-
ised by our modied Randers metrics (up to a set of measure zero). It can be argued that it is
not strictly necessary for spacelike vectors to be contained in A, as it would not be possible,
not even in principle, to perform any physical experiment that probes such directions. Whether
the lightcone should be contained in Ais a more delicate question, which we will not further
explore here.
2.4. The eld equations
In the context of Finsler gravity, arguably the simplest and cleanest proposal for a vacuum
eld equation was the one by Rutz [38]. The Rutz equation, Ric =0, can be derived from the
geodesic deviation equation in complete analogy to the way Einstein’s vacuum eld equation,
Rµν =0 (to which it reduces in the classical (pseudo-)Riemannian setting), can be derived by
considering geodesic deviation.
However, it turns out that Rutz’s equation is not variational, i.e. it cannot be obtained by
extremizing an action functional. In fact, its variational completion (i.e. the variational equation
that is as close as possible to it, in a well-dened sense [39]) turns out to be the eld equation
that was proposed by Pfeifer and Wohlfarth in [18] using a Finsler extension of the Einstein–
Hilbert action [19]. This is again in complete analogy to the situation in GR, where the vacuum
Einstein equation in the form Rµν −1
2gµν R=0 is also precisely the variational completion
of the equation Rµν =0 [39]. While in the GR case the completed equation happens to be
equivalent to the former, this is not true any longer in the Finsler setting.
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
Although several other proposals have been made as well [40–48], we consider the Pfeifer-
Wohlfarth equation4[18] to be by far the most promising, and from here onwards we will refer
to it simply as the vacuum eld equation in Finsler gravity. We do not show the eld equation
in full generality here, as its general form is not required for our present purposes. In the case
of Berwald spacetimes it can be expressed relatively simply as [21]
F2gµν −3yµyνRµν =0,(11)
where Rµν is the Finsler Ricci tensor and since we are in a Berwald setting, Rµν =Rµν (x)
only depends on x. Clearly the vanishing of the Finsler Ricci tensor is a sufcient condition
for a Berwald spacetime to be a solution to equation (11). In many cases of interest (but not
always) it is a necessary condition as well. For instance, for Randers metrics of Berwald type
it is known that the eld equation (11) is equivalent to Rµν =0 [22], and similar results have
been obtained for Finsler metrics satisfying strict smoothness requirements [50]. There exist
Finsler metrics, however, that have a non-vanishing Finsler Ricci tensor, yet for which (11)
holds. An explicit example illustrating this will be provided in forthcoming work.
3. (α, β)-metrics
3.1. (α, β)-metrics—basic denitions
An important class of Finsler geometries is given by the so-called (α,β)-metrics. Here α=
p|aµν ˙
xµ˙
xν|and β=bµ˙
xνare scalar variables dened in terms of a (pseudo-)Riemannian
metric aµν on Mand a 1-form bµon M, and an (α,β)-metric is simply a Finsler metric that is
constructed only from αand β, i.e. F=F(α,β). Due to homogeneity it follows that any such
Fcan be written in the standard form F=αϕ(β/α)for some function ϕ, at least whenever
α6=0. Well-known examples of (α,β)-metrics are:
•Pseudo-Riemannian Finsler metrics F=α;
•Randers metrics F=α+β;
•Kropina metrics F=α2
β;
•m-Kropina metrics F=α1+mβ−mwith msome real number. Also referred to as generalized
Kropina metrics, Bogoslovsky metrics or Bogoslovsky–Kropina metrics.
For each of these types of (α,β)-metrics certain conditions need to be fullled in order to
satisfy the denition of a Finsler spacetime [51].
3.2. Exact (α,β)-metric solutions in Finsler gravity
From the physical viewpoint, (α,β)-metrics allow us to deform a GR spacetime αinto a Finsler
spacetime by the 1-form β. And it turns out, as we will prove below, that these types of metrics
can be used to generalize some of the vacuum solutions to Einstein’s eld equations to properly
Finslerian vacuum solutions in Finsler gravity. This procedure is possible whenever such a
solution admits a covariantly constant vector eld, or equivalently, 1-form. Namely: if the
Lorentzian metric αsolves the classical Einstein equations and the 1-form βis covariantly
constant with respect to αthen any (α,β)-metric constructed from the given αand βis a
4In the positive denite setting a similar eld equation has been obtained by Chen and Shen [49].
8
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
solution to the Finslerian eld equations. To see why this is true, we rst recall the following
well-known result (see e.g. section 6.3.2. in [52]):
Proposition 1. Let F be an (α,β)-metric. If βis covariantly constant with respect to αthen
F is of Berwald type and the afne connection of F coincides with the Levi-Civita connection
of α.
If the afne connection of Fis the same as the connection of α, the associated curvature
tensors and (afne) Ricci tensors are also the same. So if αhappens be a vacuum solution to
Einstein gravity, i.e. its Ricci tensor vanishes, then it follows that the afne Ricci tensor of F
vanishes as well, which implies, by equation (11), that Fis a vacuum solution to Pfeifer and
Wohlfarth’s eld equation in Finsler gravity. We may summarize this result in the following
theorem.
Theorem 2. Let F be any (α, β)-metric such that αsolves the classical vacuum Einstein
equations and βis covariantly constant with respect to α. Then F is a vacuum solution to
the eld equation in Finsler gravity.
In this way (α,β)-metrics provide a mechanism to Finslerize any vacuum solution to
Einstein’s eld equations, as long as the solution admits a covariantly 1-form, or equivalently
a covariantly constant vector eld. The theorem generalizes some of the results obtained in
[22] for Randers metrics and in [20,21] for m-Kropina metrics (i.e. VGR spacetimes) to arbit-
rary Finsler spacetimes with (α,β)-metric. In particular, all pp-wave type solutions in Finsler
gravity currently known in the literature are of this type.
Let’s investigate this type of solution in some more detail. It turns out that if a vacuum
solution αto Einstein’s eld equations admits a covariantly constant 1-form β, then either α
is at, or βis necessarily null [53] (see also [54,55]). We remark that this result assumes that
the spacetime dimension is 1 +3 and generally is not true in higher dimensions. This leads to
two classes of solutions.
First class of solutions
The rst of these possibilities, where αis at, leads to a class of solutions that can always
be written in suitable coordinates in the following way.
(α, β)-metric solutions (Class 1). Let the metric Aand 1-form βbe given by
A=−(dx0)2+ (dx1)2+ (dx2)2+ (dx3)2, β =bµdxµ,(12)
where bµ=const. Then any (α,β)-metric constructed from α=p|A|and βis a vacuum
solution to the eld equations in Finsler gravity. The resulting geometry is of Berwald
type with all afne connection coefcients vanishing identically in these coordinates.
Right below equation (12) we have used the notation α=p|A|=p|aijdxidxj|. This should
be understood pointwise, i.e.
α=α(y) = q|aijdxidxj|(y) = q|aij dxi(y)dxj(y)|=q|aijyiyj|.(13)
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
In other words, we sometimes write αfor the function p|aijdxidxj|:y7→ p|aij yiyj|, and at
other times we write αfor its value p|aijyiyj|at y. It should always be clear from context
what is meant.
Second class of solutions
The second possibility, that βis null, leads to a class of solutions that seems to be more inter-
esting. In this case αis covariantly constant null vector (CCNV) spacetime metric, meaning
that it admits a CCNV, namely in this case β, or rather its vector equivalent via the isomorph-
ism induced by α. CCNV metrics are also known as pp-waves (plane-fronted gravitational
waves with parallel rays) and have been studied in detail in [53,56] (see section 24.5 in [57]
for a summary).
It is an elementary result that by choosing suitable coordinates (u,v,x1,x2), such αand β
can always be expressed in the form
A=−2du(dv+H(u,x)du+Wa(u,x)dxa) + hab(u,x)dxadxb,(14)
β=du,(15)
where xa=x1,x2and hab is a two-dimensional Riemannian metric. This holds irrespective of
whether αis a solution to Einstein’s eld equations or not. If αis additionally assumed to be a
vacuum solution, as in theorem 2, it turns out that the expression (14) for Acan be simplied
even more without changing the form (15)of β. To see this, we rst consider only the metric A.
Since Ais a vacuum solution to Einstein’s eld equations, it follows that the functions Wacan
be eliminated and hab may be chosen as δab, by a suitable coordinate transformation (section
24.5 in [57]). The metric then takes the form
A=−2du(dv+H(u,x)du) + δabdxadxb.(16)
We are, however, not only interested in the transformation behavior of Aalone, but also in that
of β, because an (α,β)-metric is composed of both. To see why we may assume without loss
of generality that the form of β=duremains invariant we use the fact that any coordinate
transformation
(u,v,x1,x2)7→ (¯
u,¯
v,¯
x1,¯
x2)(17)
that leaves the generic form of the metric (14) invariant, but in general changing the expressions
for the metric functions H,Wa,hab 7→ ¯
H,¯
Wa,¯
hab, has the specic property that u=ϕ(¯
u)for
some function ϕdepending on ¯
ualone (see section 31.2 in [57]). This applies in particular to
the transformation that relates (14) and (16). We can therefore express the 1-form as β=du=
ϕ′(¯
u)d¯
u, or equivalently ¯
bµ=ϕ′(¯
u)δu
µ. However, since βis covariantly constant with respect
to A, we must have ¯
∇µ¯
bν=0. All Christoffel symbols ¯
Γu
µν of the metric (16) with upper index
uvanish identically, however. Hence
¯
∇¯
u¯
b¯
u=∂¯
ub¯
u−¯
Γu
uuϕ′(¯
u) = ϕ′′(¯
u)!
=0.(18)
It follows that ϕ′(¯
u) = C=constant, i.e. β=Cd¯
u. In this case it is easily seen that scaling ¯
u
by Cand scaling ¯
vby 1/Cleaves the metric (16) invariant and brings the 1-form back into its
original form, proving that we may assume without loss of generality that the 1-form remains
invariant under the coordinate transformation.
Finally, the metric (16) is a vacuum solution to Einstein’s eld equations if and only if
(∂2
x1+∂2
x2)H=0. We may therefore characterize the second class of solutions in the following
way.
10
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
(α, β)-metric solutions (Class 2). Let α=p|A|and βbe given by
A=−2du(dv+H(u,x)du) + δabdxadxb,(19)
β=du,(20)
such that δab∂a∂bH=0. Then any (α, β)-metric constructed from the pair (α, β) is a
vacuum solution to the eld equations in Finsler gravity. The resulting geometry is of
Berwald type with afne connection identical to the Levi-Civita connection of α.
Note that when H=0 the geometries in Class 2 are also contained in Class 1. It is not the
case, however, that Class 1 is a subset of Class 2 because in Class 1 the 1-form βneed not be
null, necessarily. The preceding line of argument shows that these two classes of solutions in
fact exhaust all possibilities, which we encapsulate in the following theorem.
Theorem 3. Any vacuum solution of the type of theorem 2must belong to one of the two classes
introduced above.
Before we move on to (α,β)-type solutions of plane-wave type, we end this section by
noting that for specic types of (α,β)-metrics, stronger results have been obtained than the
ones derived above:
•For Randers metrics of Berwald type any vacuum solution to (11) must be of the type
described in theorem 2, that is, αis necessarily a vacuum solution in Einstein gravity and β
is necessarily covariantly constant [22]. Any such solution is therefore either of Class 1 or
Class 2 in the terminology introduced above.
•For m-Kropina metrics some vacuum solutions of a more general type than the one in the-
orem 2have been obtained in the context of VGR [21].
•Any pseudo-Riemannian Finsler metric F=αis trivially a vacuum solution in Finsler grav-
ity if and only if it is a vacuum solution in Einstein gravity.
To the best of our knowledge this list comprises all exact solutions in Finsler gravity currently
known in the literature.
3.3. Plane wave solutions in Brinkman and Rosen coordinates
Equation (19) expresses the pp-wave metric in Brinkmann form [58]. For the description of the
physical effects of (plane) gravitational waves in general relativity, it is sometimes more con-
venient to use a different coordinate system, known as Rosen coordinates [59]. This remains
true in the Finsler case. When we compute the effect on the radar distance of a passing Randers
gravitational wave in section 5, our starting point will be the expression for the gravitational
wave in Rosen coordinates. Therefore we briey review the relation between the two coordin-
ate systems here.
Rosen coordinates can be introduced for the subclass of pp-waves known as plane waves.
These can be characterized by the property that the curvature tensor does not change (i.e. is
covariantly constant) along the Euclidean ‘wave surfaces’ given in Brinkmann coordinates by
du=dv=0, i.e.
∇∂x1Rρσµν =∇∂x2Rρσµν =0.(21)
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
We note that ∇∂vRρσµν =0 always holds, identically, so invariance along du=dv=0 is actu-
ally equivalent to invariance along the hypersurfaces du=0. The conditions (21) are equival-
ent to the statement that ∂a∂b∂cH=0 in Brinkmann coordinates (19), i.e. that H(u,x)is a
second order polynomial in xa. In that case there always exists a coordinate transformation
that removes the linear and constant terms (section 24.5 in [57]) so that the metric can be
written as
A=−2dudv+Aab(u)xaxbdu2+δab dxadxb.(22)
This is the standard expression for a plane-wave metric in Brinkmann form. Moreover, an
argument very similar to the one given in the previous subsection, shows that we may assume
without loss of generality that the 1-form β=duremains unchanged under this transformation.
Any such plane wave metric can also be written in Rosen form
ds2=−2dUdV+hij(U)dyidyj,(23)
where hij is a two-dimensional Riemannian metric. And conversely, any metric of Rosen
form (23) can be cast in the form (22). The two coordinate systems are related via
U=u,V=v−1
2˙
EaiEibxaxb,xa=Eaiyi,(24)
where Aab =¨
EaiEiband Eaiis a vielbein for hij in the sense that hij =EaiEbjδab , satisfying
the additional symmetry condition ˙
EaiEib=˙
EbiEia. Such a vielbein can always be chosen.
For details we recommend the lecture notes [60] by Matthias Blau and references therein
(see also the appendix of [61]). Note that we have momentarily labelled the y-coordinates
by indices i,j,k,... so as to distinguish them from indices a,b,c,... in order that we may
apply the usual notation with regards to the vielbein indices: Eiarepresents the (matrix) inverse
of Eaiand indices a,b,c... are raised and lowered with δab, whereas indices i,j,k,... are
raised and lowered with hij. The dot that appears sometimes above the vielbein represents a U-
derivative. Since the vielbein depends only on U, this derivative is equivalent to a u-derivative,
and moreover the raising and lowering of the a,b,c,... indices commutes with taking such a
derivative of the vielbein.
It is again the case that, after relabeling U,V7→ u,v, the 1-form β=du=dUremains
unchanged under this transformation, which in this case is easy to see. After also relabelling
y7→ x, we conclude that we can express any Class 2 solution of plane-wave type in Rosen
coordinates as follows,
F=αϕ(β/α),A=−2dudv+hij(u)dxidxj, β =du,(25)
where α=p|A|. And conversely, for any choice of ϕ, hij (u), this is a vacuum solution to
the eld equations in Finsler gravity if Ais a vacuum solution to Einstein’s eld equation.
The resulting geometry is of Berwald type with afne connection identical to the Levi-Civita
connection of α.
3.4. Linearized gravitational wave solutions
The exact vacuum eld equation for plane-wave metrics does not have a particularly nice
expression in Rosen coordinates (25). The linearized eld equation, however, turns out to be
very simple. So let us consider the scenario that the pseudo-Riemannian metric αis very close
12
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
to the Minkowski metric. In this case we may write hij(u) = δij +εfij(u)with ε1. The lin-
earized eld equations (i.e. to rst order in ε) for αthen simply read5
f′′
11(u) + f′ ′
22(u) = 0.(26)
Hence f11 and −f22 must be equal up to an afne function of u. Here we will focus on the case
where f11 =−f22, which can always be achieved by means of the transverse traceless gauge6.
Conventionally one writes the subscripts as f11 =−f22 =: f+and f12 =: f×, denoting the plus
and cross polarization of the gravitational wave, so we will stick to that notation from here
onwards. That brings us to the following expression that describes Finslerian gravitational
waves of (α,β)-type:
F=αϕ(β/α),
A=−2dudv+ (1+εf+(u))dx2+ (1−εf+(u))dy2
+2εf×(u)dxdy
β=du
.(27)
Note that if we substitute u= (t−z)/√2 and v= (t+z)/√2, then Areduces to the standard
expression for a gravitational wave metric in GR, i.e.
F=αϕ(β/α),
A=−dt2+ (1+εf+(t−z))dx2+ (1−εf+(t−z)dy2
+2εf×(t−z)dxdy+dz2
β=1
√2(dt−dz)
,(28)
for any choice of the function ϕ.
3.5. Linearized (α,β)-metrics are Randers metrics
It is natural to linearlize not only in ε, characterizing the departure from atness, but to also use
a perturbative expansion in the ‘size’ of the 1-form, characterizing the departure from GR and
pseudo-Riemannian geometry. The physical intuition here is that, seeing how well GR works
in most regimes, the most interesting class of Finsler spacetimes consists of those ones that are
very close to GR spacetimes. The purpose of this section is to highlight that any (α,β)-metric
is perturbatively equivalent to a Randers metric, to rst order, so that from the physics point
of view, Randers metrics are actually quite a bit more general than they might seem at rst
glance. After pointing this out we will turn our focus exclusively to Randers metrics for the
remainder of the article.
So consider an (α,β)-metric constructed form a pseudo-Riemannian metric αand a 1-form
βsuch that β1. To see what happens in such a scenario, we replace βwith λβ and expand
to rst order in λ. Then we obtain
F=αϕλβ
α≈αϕ(0) + λϕ′(0)β
α=αϕ(0) + λϕ′(0)β= ˜α+˜
β. (29)
5The full linearized vacuum eld equation (11) for Fis more complicated in general, but as discussed extensively
above, if the vacuum eld equation for αis satised then so is the eld equation for F. In the case of Randers metrics,
to which we will turn momentarily, the eld equation for Fis even equivalent to the eld equation for α. Hence for
our present purposes the eld equations for αsufce.
6We leave open the question whether the form of the 1-form β=dualways remains invariant under such a trans-
formation to the transverse traceless gauge.
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
Hence to rst order in λ, any (α,β)-metric is indeed equivalent to a Randers metric7.
Consequently, by replacing duby λduin (27), which technically can be achieved by a coordin-
ate transformation that scales uby λand vby 1/λ, it follows that to rst order in λthe Finsler
metric of the (α,β)-type gravitational waves takes the form,
F=α+β,
A=−2dudv+ (1+εf+(t−z))dx2+ (1−εf+(t−z)dy2
+2εf×(t−z)dxdy
β=λdu
.(30)
The parameter λthen characterizes the departure from GR and pseudo-Riemannian geo-
metry. We will assume without loss of generality that λ > 0. Finally, replacing also uand v
by tand z, according to u= (t−z)/p(2)and v= (t+z)/√2, we can write the metric in the
following way, which we will take as the starting point for the calculation of the radar distance
in section 5.
F=α+β,
A=−dt2+ (1+εf+(t−z))dx2+ (1−εf+(t−z)dy2
+2εf×(t−z)dxdy+dz2
β=λ
√2(dt−dz)
.(31)
4. Modified Randers metrics
Motivated by the argument above we will now turn our focus to the simplest properly Finslerian
(α,β)-metric, the Randers metric, conventionally dened as F=α+β. We will argue that in
order to have a physically natural causal structure (including, for instance, both a forward and
a backward light cone, details follow below), the conventional denition must be modied
slightly. It might seem to the reader that modifying the Randers metric would be in conict
with the spirit of the previous section, since to rst order any (α,β)-metric should reduce to
a Randers metric. It is important to note, however, that there is in principle the possibility
that to different regions of the tangent bundle could correspond different Randers metrics.
More precisely, we could dene one (α,β)-metric F1on a conic subbundle A1⊂TM \0 and
another (α,β)-metric, F2, on a different conic subbundle A2⊂TM \0. If the two subbundles
do not overlap then this denes a perfectly valid (α,β)-type Finsler spacetime on the union
A=A1∪A2. To rst order in the deviation from (pseudo-)Riemannian geometry this Finsler
metric would reduce to a certain Randers metric on A1and to a different Randers metric on A2.
On Aas a whole, however, the resulting linearized metric might not be expressible as a single
standard Randers metric. This is what we have in mind, and our modication of the Randers
metric, introduced below, is therefore completely consistent with the previous results. While
this paper was in review, a similar procedure was employed in [62] to improve the causal
structure of cosmological unicorn (i.e. non-Berwaldian Landsberg) solutions, based on the
ideas proposed in this section.
After a heuristic argument that motivates the desired modication, we show that our pro-
posed version of the modied Randers metric has a very satisfactory causal structure. As a
result of this a clear (future and past) timelike cone can be identied and within these timelike
cones the signature of the Fundamental tensor is Lorentzian everywhere. The only constraint
is that b2≡aµν bµbν>−1, which, interestingly, is in some sense the opposite of the condi-
tion b2<1 that appears in the well-known positive denite case, see e.g. [63]. In one were
7Actually this is not true for all (α,β)-metrics but only those which allow an expansion around s=β/α =0. This
excludes Kropina metrics, for instance, because they are not well-behaved in the limit β→0.
14
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
to adopt the opposite signature convention to ours, however, the constraint in the Lorentzian
case would also turn out to be b2<1, matching the positive denite case.
4.1. Motivation and denition
First of all, let us review why the denition of a Randers metric is not as clear in Lorentzian
signature as it is in Euclidean signature. The original denition of a Randers metric, in positive
denite Finsler geometry, is just F=α+β, with α=paijyiyja Riemannian metric and β=
biyiany 1-form8. This is well-dened as long as αis positive-denite, because in that case
A≡aijyiyjis always positive. If we allow aij to be a Lorentzian metric, however, the quantity
Acan become negative, in which case √Ais ill-dened, as we want Fto be a real function. One
way to remedy this, at least at a technical level, is to restrict the conic subbundle A ⊂ TM \0 to
those vectors for which aijyiyj>0. This was the approach in e.g. [22], where it was shown that
if Ais dened as the forward timecone9corresponding to α, then under certain conditions on
the 1-form β, such a Randers metric satises all axioms of a Finsler spacetime. The fact that A
is restricted in this way, however, leads to issues when it comes to the physical interpretation.
Here we take a different approach.
The obvious rst alternative to restricting Ato vectors with positive norm is to simply
replace Aby |A|and dene α=p|A|, as we have done throughout this article. In that case
there’s no need to restrict Ato the timecone anymore. This leads to a Randers metric of the
form F=p|A|+β. An undesirable consequence of this denition, however, is that light rays
can only propagate into one half of the tangent space, namely the half given by β < 0, which
follows immediately from the null condition F=0 (see also [51]). In fact, the light cone sep-
arates the tangent space into only two connected components10 and there is consequently
not a straightforward interpretation in terms of timelike, spacelike and lightlike directions,
at least not in the conventional way11. We therefore take the viewpoint that outside of the half
plane β < 0 in each tangent space, this version of the Randers metric cannot be valid, and
we need to modify it in that region. It is possible to remove the condition β⩽0, extending
the lightcone to the other half-plane β > 0, by changing Fto F=sgn(A)p|A|+sgn(β)β=
sgn(A)p|A|+|β|. The result of this is that, under some mild assumptions (details will fol-
low below) the single lightcone (from the β < 0 half space) is mirrored to the complementary
(β⩾0) half space, whereas in the original half space intersected with the original cone of
denition consisting of α-timelike vectors, Freduces to the standard Randers metric with an
overall minus sign, F=−(α+β). This minus sign is not of any relevance, though, as the geo-
metry is essentially determined by F2. In particular, Fis now reversible, i.e. invariant under
y→ −y. Notice also that we could have chosen a minus sign instead of a plus sign in the
modied denition of F, but it turns out that in that case the resulting Finsler metric would
not be guaranteed to have Lorentzian signature everywhere inside of the timelike cones12. The
present metric does have this property as long as b2>−1, and we discuss this in detail below.
8In order to satisfy all the axioms of a Finsler space, the 1-form must satisfy |b|2<1, see e.g. [63].
9We note that the signature convention in [22] is the opposite as the one employed here, so in that case the condition
aijyiyj>0 precisely select the timelike, not spacelike, vectors.
10 This can be checked easily in suitable coordinates adapted to β.
11 We note that in the approach by Javaloyes and Sánchez [31,32] a single, future pointing (by denition) cone is
sufcient, though.
12 In case one employs the opposite signature convention (+,−,−,−)the converse would be true. In that case the
preferable choice would be F=sgn(A)α− |β|rather than F=sgn(A)α+|β|.
15
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
Definition 4. Motivated by the preceding heuristic argument we dene the modied Randers
metric as follows,
F=sgn(A)α+|β|,(32)
where we recall for completeness that α=p|A|,A=aijyiyjβ=biyi.
Both αand Awill sometimes be referred to as the (pseudo-)Riemannian metric, by a slight
abuse of language, but it should always be clear from context what is meant.
4.2. Causal structure
Next we will show that the modied Randers metric (32) indeed has very nice properties. By
denition, the light cone is given by
F=0⇔sgn(A)⩽0&|A|=|β|2⇔A=−β2.(33)
It therefore follows that
F=0⇔(aµν +bµbν)dxµdxν=0,(34)
meaning that the light cone of Fis just the light cone of the auxilliary Lorentzian met-
ric ˜
aµν (x) = aµν +bµbν. Indeed, the matrix determinant lemma guarantees that as long as
b2=aµν bµbν>−1 the metric aµν +bµbνhas Lorentzian signature, provided that aµν has
Lorentzian signature. (For a proof see appendix A.) This shows that as long as b2>−1 the
light cone separates the tangent space at each point into three connected components, which
we can naturally interpret in the usual manner as the forward time cone, backward timecone,
and the remainder consisting of spacelike vectors. Coincidentally we note that
F<0⇔(aµν +bµbν)yµyν<0,(35)
and hence it also follows that
F>0⇔(aµν +bµbν)yµyν>0.(36)
This leads to the additional convenience that F-timelike vectors are precisely given by F<0,
and F-spacelike vectors by F>0, in addition to the null vectors being given, by denition, by
F=0. We summarize these results in the following proposition.
Proposition 5. As long as b2>−1, the causal structure of the modied Randers metric F =
sgn(A)α+|β|is identical to the causal structure of the Lorentzian metric aµν +bµbν, with
null vectors given by F =0, timelike vectors given by F <0, and spacelike vectors by F >0.
As a result of these nice features of the causal structure of the modied Randers metric, it
is possible to dene time orientations in the usual manner, by means of a nowhere vanishing
timelike vector eld T. Such Tselects one of the two timelike cones as the ‘forward’ one,
namely the one that contains T. Then another timelike vector yis future oriented (i.e. lies in
the same forward cone as T) if and only if (aµν +bµbν)Tµyν<0. We note that similar char-
acterizations of time orientations in Finsler spacetimes (although not in terms of an auxilliary
pseudo-Riemannian metric, like aµν +bµbν) have been discussed in [64].
In the special case that βis covariantly constant with respect to αwe have even more
satisfactory results. In that case not only the causal structure but also the afne structure of F
can be understood in terms of aµν +bµbν.
Proposition 6. If βis covariantly constant with respect to αand satises b2>−1then the
causal structure and the afne structure of the modied Randers metric F =sgn(A)α+|β|
16
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
are identical to those of the Lorentzian metric ˜
aµν =aµν +bµbν. In other words, the timelike,
spacelike and null geodesics of F coincide with the timelike, spacelike and null geodesics of
˜
aµν .
Proof. The discussion above indicates that the causal structures coincide. It remains to show
that also the afne structures coincide in the case of a covariantly constant 1-form. This is
again a result of the properties of ˜
aµν . It can be shown (see appendix A) that the Christoffel
symbols of ˜
aµν can be expressed in terms of the Christoffel symbols of aµν as
e
Γρ
µν = Γρ
µν +1
1+b2bρ∇(µbν)−aρλ −1
1+b2bρbλbµ∇[λbν]+bν∇[λbµ].(37)
Hence it follows immediately that if bµis covariantly constant then e
Γρ
µν = Γρ
µν and the afne
structure of ˜
aµν is the same as that of aµν . We also known, by proposition 1, that the afne
structure of Fis the same as that of aµν . Hence the afne structure of Fis the same as that of
˜
aµν .
From this it follows immediately that the existence of radar neighborhoods is guaranteed
[26]. More precisely, given an observer and any spacetime event sufciently close to the
observer’s worldline, there is (at least locally) exactly one future pointing light ray and one past
pointing light ray that connect the event to the worldline of the observer. This is of essential
importance in our work, because what it essentially says is that the radar distance, calculated
in section 5, is a well-dened notion.
4.3. Regularity and signature
Given an (α,β )-metric of the form F=αϕ(s), with s=β/α and α=p|A|, it can be shown
that the determinant of the fundamental tensor is given by
detgij =ϕn+1(ϕ−sϕ′)n−2(ϕ−sϕ′+ (sgn(A)b2−s2)ϕ′′)detaij.(38)
The proof can be found in appendix B. Because of the appearance of sgn(A)the expression
is slightly different from the well-known positive denite analogue, to which it reduces when
A>0, i.e. sgn(A) = 1. For a modied Randers metric of the form F=sgn(A)α+|β|the func-
tion ϕis given by ϕ(s) = sgn(A) + |s|, so this reduces to
detg
deta=sgn(A)n−1(sgn(A) + |s|)n+1=sgn(A)F
αn+1
.(39)
Assuming the spacetime dimension nis even, this means that ghas Lorentzian signature13
if and only if sgn(A)F>0. Let us see what this entails. First note that F<0 trivially implies
A<0. Hence F<0 implies Lorentzian signature. Before we move on, we should point out that
this is a very satisfactory result. It means that within the entire timelike cone of F, the signa-
ture of the fundamental tensor is Lorentzian. Similarly, A>0 implies F>0. Hence A>0 also
implies Lorentzian signature. What remains is the region where A⩽0 and F⩾0. Equivalently,
A⩽0 and A+β2⩾0. In this region, the determinant of the fundamental tensor is either
undened, is positive, or vanishes, so in any case the signature is not Lorentzian. But as this
region lies outside the timelike cone, this is not a problem, as argued in section 2.3.
It is helpful to think in terms of both the light cone of the metric aij and the light cone of
the metric aij +bibj(i.e. that of F). As mentioned previously, as long as b2>−1, the latter
13 The argument is the same as in the positive denitivecase, using the same methods as those employed in appendix A.
17
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
metric is Lorentzian, provided the former is. That means its light cone is just a conventional
one that we are familiar with from GR, just like the light cone of aij. The only region where the
signature is not Lorentzian, is precisely the region in between these two lightcones. Note that
since F<0 implies A<0, the F-lightcone can never reach outside of the aij-light cone. The
details depend on the causal character of the 1-form βand are listed below. These properties
can be checked easily by noting we may always choose coordinates such that at a given point
x∈Mthe metric Ahas the form of the Minkowski metric and the 1-form βhas only one
component (in the timelike or spacelike case) or two components (in the null case)14.
•If βis null it is easily seen that the two lightcones intersect only for yµthat are multiples of
bµ. Thus their intersection spans a single line in the tangent space.
•If βis timelike and b2>−1 then the light cones do not intersect (apart from the trivial
intersection in the origin).
•If βis spacelike (and assuming dimM>2), then aij induces a Lorentzian metric on the
(dimM−1)-dimensional hypersurface dened by β=0. In this case the two light cones
intersect along the light cone of this induced Lorentzian metric.
•If b2=−1 there is only a single cone, namely the one corresponding to α. The ‘light cone’
corresponding to F=0 is now in fact a line, consisting all of multiples of bµ. This case
therefore does not have a viable physical interpretation.
•If b2<−1 there is only a single cone, namely the one corresponding to α. The ‘light cone’
corresponding to F=0 is now non-existent, as F=0 has no solutions. This case therefore
does not have a viable physical interpretation either.
To get a better idea, gure 1displays the lightcones and the regions (in green) where the
signature of the fundamental tensor is Lorentzian, for the modied Randers metric
F=sgn(A)α+|β|,A=−(dx0)2+ (dx1)2+ (dx2)2,
β=
ρdx0if timelike
ρ(dx0+dx1)if null
ρdx1if spacelike
,(40)
for a number of representative values of the parameter ρ. In each subgure, the inner lightcone
is that of Fand the outer lightcone that of A. Note that for any aij and bi, it is always possible at
any given point x∈M, to choose coordinates in such a way that Aand βhave the above form
(or rather their analog in the relevant spacetime dimensionality). The following proposition
summarizes these results.
Proposition 7. As long as b2>−1, the signature of the fundamental tensor of F =sgn(A)α+
|β|is Lorentzian within the entire timelike cone, which is given by F <0. Immediately outside
of the timelike cone there is a region that does not have Lorentzian signature, and further away
(namely when A >0) the signature becomes Lorentzian again. When b2⩽−1the timelike cone
is the empty set, so this case is not physically interesting.
14 We recall that this can be seen as follows. First, since aij is Lorentzian, it is always possible to choose coordinates
such that Ais just the Minkowski metric at a given point x∈M. Writing bµ= (b0,b1,...bn−1)in these coordinates,
we may do a spatial rotation on the coordinates b1,...,bn−1, such that they are transformed into (b1,0,...,0), leaving
the metric at xunchanged. Then bµ= (b0,b1,0,...,0). Now we separate the three cases. If b2=0, it follows that
b1=±b0and by applying if necessary a spatial reection in the x1direction we may choose either sign. If b2<0
then we may go to the local rest frame by a Lorentz transformation, making b1=0. If on the other hand b2>0 we
may perform a Lorentz transformation making b0=0.
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
Figure 1. The gures show the lightcone and the signature of the fundamental tensor of
F=sgn(A)α+|β|, where A=−(y0)2+ (y1)2+ (y2)2+ (y3)2and β=ρ(y0+y1)(in
the null case) or β=ρy0(in the timelike case) or β=ρy1(in the spacelike case) for
several representative values of ρ, shown in the tangent space TxMat any point x∈M, at
y3=0. Green regions correspond to Lorentzian signature, red regions to non-Lorentzian
signature. Figures (a)–(i) show the physically reasonable scenarios, where b2>−1. In
that case two cones can be observed. The inner cone is the true light cone of F(i.e. the set
F=0), and the outer cone is the light cone of aij (i.e. the set A=0). The only region with
non-Lorentzian signature is precisely the gap in between the two cones. If on the other
hand b2=−1 (gure (j)) then the light ‘cone’ of Fis the line y1=y2=y3=0. And if
b2<−1 (gure (k)) then the light ‘cone’ of Fconsists only of the origin. Therefore we
deem the latter two cases not physically interesting.
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
Since we only require Lorentzian signature within the timelike cone, these results are very
satisfactory. We point out in particular that this is true even when the 1-form is spacelike. On
the other hand, the classical Randers metric can only be considered a physically reasonable
Finsler spacetime if the 1-form is either null or timelike [51]. Finally, regarding the regularity
of F, clearly Fis smooth everywhere except when A=0 or β=0. In particular, the set where
Fis not smooth has measure zero.
5. Radar distance for a Finsler gravitational wave
Now we are nally in the position to analyze the physical effects of a passing Finslerian grav-
itation wave of (α,β)-type. We have seen in section 3.5 that, to rst order, an (α,β)-metrics
is equivalent to a Randers metric. And we have argued in section 4that this should not be
the standard Randers metric but rather our modied Randers metric. Thus our starting point
will be the linearized gravitational wave solution of modied-Randers type. That is, we are
interested in the solution (31) but with the conventional Randers metric F=α+βreplaced
by the modied Randers metric F=sgn(A)α+|β|, where α=p|A|,A=aijyiyj. Note that
this modication does not change any of the results pertaining to classication of solutions
to the eld equations, by the argument given at the beginning of section 4that a modied
Randers metric is ‘locally’ equivalent to a standard Randers metric, in a certain precise sense.
The relevant Finsler metric is therefore given by
F=sgn(A)α+|β|,
A=−dt2+ (1+εf+(t−z))dx2+ (1−εf+(t−z))dy2
+2εf×(t−z)dxdy+dz2
β=λ
√2(dt−dz)
.(41)
Since actual gravitational wave measurements are done with interferometers, that effectively
measure the radar distance, the aim of this section is to compute that radar distance during the
passing of a gravitational wave of the form (41).
The setup is as follows. A light ray is emitted from some spacetime location with coordin-
ates (t0,x0,y0,z0), travels to another location in spacetime with coordinates (t0+ ∆t,x0+
∆x,y0+ ∆y,z0+ ∆z), where it is reected and after which it travels back to the original (spa-
tial) location, with spacetime coordinates (t0+ ∆ttot,x0,y0,z0), being received there again. We
are interested in the amount of proper time that passes between emission and reception of the
light ray, as measured by an ‘inertial’ observer15 located at spatial coordinates (x0,y0,z0).
Because light travels forwards and backwards during this time interval, one half of the time
interval is usually called the radar distance between the two spacetime points (sometimes
the value is multiplied by the velocity of light, c, which we have set to 1, so that it has the
dimensions of distance). In other words, the radar distance can be expressed as R= ∆τ/2.
The expression for the radar distance of a standard gravitational wave in GR has been
obtained in [23] and our calculation below follows essentially the same methods. At each step
of the calculation we will clearly point out the differences with corresponding situation in GR,
so that it is clear where each of the Finslerian effects (three separate effects can be identied)
enters precisely. In addition to linearizing in ε, we also use a perturbative expansion in λ, as
argued for at the end of section 3.4. In fact, instead of working to rst order in λ, we will work
15 In this context, we say that an observer is intertial if it would considered an intertial observer in the absence of
the wave (i.e. when f+=f×=0). In other words, thinking of the gravitational wave as having a nite duration as it
passes the Earth, an observer is inertial precisely if it is inertial before and after the wave passes.
20
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
to second order in the Finslerian parameter, as certain important Finslerian effects only enter
at second order, as we will see. We also neglect terms of combined order ελ2and higher.
5.1. Finslerian null geodesics
The rst important observation here is that the geodesics in a Randers gravitational wave space-
time with Finsler metric F=sgn(A)α+|β|as in equation (41) coincide with the geodesics of
the GR spacetime with metric ds2=A, because the afne connection of Fcoincides with
the Levi-Civita connection of A, by proposition 1. For the derivation of the general form of
geodesics, we may therefore assume the geometry is given by ds2=A. Thus our point of
departure is the metric
ds2=A=−dt2+ (1+εf+(t−z))dx2(1−εf+(t−z))dy2
+2εf×(t−z)dxdy+dz2, ε 1,(42)
and we can essentially follow [23]. Using the coordinates u= (t−z)/√2 and v= (t+z)/√2
the geodesic equations to rst order in εcan be written as
−˙
u=: pv=const,(43)
(1+εf+(u))˙
x+εf×(u)˙
y=: px=const,(44)
(1−εf+(u))˙
y+εf×(u)˙
x=: py=const,(45)
¨
v+1
2ε˙
x2−˙
y2f′
+(u) + εf′
×(u)˙
x˙
y=0,(46)
where (43)–(45) are obtained as rst integrals, using the fact that the Lagrangian corresponding
to Ais independent of the coordinates v,x,y. The rst three equations can be rewritten to rst
order as
˙
u=−pv,˙
x= (1−εf+(u))px−εf×(u)py,˙
y= (1+εf(u))py−εf×(u)px,(47)
and can be integrated (with respect to an afne parameter σ, chosen without loss of generality
such that ˙
u=1) to
u=u0+σ, x=x0+σ1−ε¯
f+(σ)px−ε¯
f×(u)py,
y=y0+σ1+ε¯
f+(σ)py−ε¯
f×(u)px,(48)
where
¯
f+,×(σ)≡1
σˆσ
0
f+,×(u0+σ)dσ(49)
is the averaged value of f+,×. The equation for vcan be integrated to
˙
v=−˜
pu0−1
2εp2
x−p2
yf+(u0+σ)−εf×(u0+σ)pxpy(50)
where ˜
pu0=pu0−ε
2(p2
x−p2
y)f(u0)−εf×(u0)pxpy,pu=−˙
v(not necessarily constant) and pu0
is its initial value at σ=0. Integrating once again, we obtain
v=v0−˜
pu0σ−1
2ε(p2
x−p2
y)σ¯
f+(σ)−ε¯σf×(σ)pxpy.(51)
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
Any geodesic emanating from a given point xµ
0can thus be described by the following
parameterized path, for certain values of px,pyand ˜
pu0:
u(σ) = u0+σ, (52)
x(σ) = x0+σ1−ε¯
f+(σ)px−ε¯
f×(σ)py,(53)
y(σ) = y0+σ1+ε¯
f+(σ)py−ε¯
f×(σ)px,(54)
v(σ) = v0−˜
pu0σ−1
2ε(p2
x−p2
y)σ¯
f+(σ)−εσ¯
f×(σ)pxpy.(55)
We need to know the specic expression for null geodesics, however. The modied null
condition or MDR for massless particles, F=0, is the rst place where the Finslerian character
of the gravitational wave enters. According to section 4.2, the condition F=0 is equivalent to
A=−β2, i.e.
−2˙
u˙
v+ (1+εf+(u))˙
x2+ (1−εf+(u))˙
y2+2εf×(u)˙
x˙
y=−β2=−λ2˙
u2,(56)
which, after substituting (52)–(55), becomes 2˜
pu0+p2
x+p2
y=−λ2. We may therefore elim-
inate ˜
pu0and directly substitute this into the expression (55) for v(σ). A null geodesic starting
at (u0,x0,y0,v0)at σ=0 can therefore be described by the following parameterized path,
u=u0+σ, (57)
x=x0+σ1−ε¯
f+(σ)px−ε¯
f×(u)py,(58)
y=y0+σ1+ε¯
f+(σ)py−ε¯
f×(u)px,(59)
v=v0+σ
2(p2
x+p2
y+λ2)−1
2ε(p2
x−p2
y)σ¯
f+(σ)−εσ¯
f×(σ)pxpy.(60)
Here we can make two important observations:
1. The effect due to the MDR or modied null condition enters at order λ2;
2. In the limit λ→0 we recover the null geodesics for a standard gravitational wave in GR
[23].
5.2. Radar distance
Next we plug in the boundary conditions at the receiving point, (u0+ ∆u,x0+ ∆x,y0+
∆y,v0+ ∆v). Note that σ= ∆uat that point, and hence from the middle two equations we
infer that
px=∆x
∆u1+ε¯
f+(∆u)+ε¯
f×(∆u)∆y
∆u,py=∆y
∆u1−ε¯
f+(∆u)+ε¯
f×(∆u)∆x
∆u.(61)
Plugging this into the vequation yields
2∆u∆v= ∆x2(1+ε¯
f+(∆u)) + ∆y2(1−ε¯
f+(∆u)) + 2ε¯
f×∆x∆y+λ2∆u2,(62)
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
or equivalently,
1−λ2
2∆t2=1+ε¯
f(∆u)∆x2+1−ε¯
f(∆u)∆y2+2ε¯
f×(∆u)∆x∆y
+1+λ2
2∆z2−λ2∆z∆t,(63)
where we have used that −2∆u∆v=−∆t2+ ∆z2. This equation is solved to rst order in ε
and λ2(neglecting ελ2terms) by16
∆t= ∆ℓ+∆x2−∆y2
2∆ℓε¯
f+(∆u) + ∆x∆y
∆ℓε¯
f×(∆u)
+1
2∆x2+ ∆y2+2∆z2
2∆ℓ−∆zλ2,(64)
where ∆ℓ≡p∆x2+ ∆y2+ ∆z2.
The right hand side in principle still depends on tthough, via ¯
f(∆u), so this is not yet a
closed formula for ∆t. However, since ¯
fonly appears together with ε, and since we are only
interested in the rst order expression for ∆t, any zeroth order expression for¯
fsufces in this
formula. We have
¯
f(∆u) = 1
∆uˆ∆u
0
f(u0+σ)dσ=√2
∆t−∆zˆ(∆t−∆z)/√2
0
f(u0+σ)dσ(65)
=√2
∆ℓ−∆zˆ(∆ℓ−∆z)/√2
0
f(u0+σ)dσ+O(ε)(66)
=√2
∆ℓ−∆zˆ(∆ℓ−∆z)/√2
0
f1
√2(t0−z0) + σdσ+O(ε)(67)
since ∆t= ∆ℓ+O(ε). We introduce another symbol for this expression, namely
¯
f(∆ℓ,∆z,t0−z0)≡√2
∆ℓ−∆zˆ(∆ℓ−∆z)/√2
0
f1
√2(t0−z0) + σdσ, (68)
where the explicit display of the arguments serves to remind us that¯
fdepends only on ∆ℓ,∆z
and the initial value of t−z. Since ε¯
f(∆u) = ε¯
f(∆ℓ,∆z,t0−z0) + O(ε2), it follows that we
can rewrite equation (64), to rst order in εand λ2, as
∆t= ∆ℓ+∆x2−∆y2
2∆ℓε¯
f+(∆ℓ,∆z,t0−z0) + ∆x∆y
∆ℓε¯
f×(∆ℓ,∆z,t0−z0)(69)
+1
2∆x2+ ∆y2+2∆z2
2∆ℓ−∆zλ2,(70)
which is a closed expression for the elapsed coordinate time ∆tinterval for a light ray traveling
a certain spatial coordinate distance, in terms of the spatial coordinate separations and the
initial value of t−z.
16 In addition to this solution there is, formally, another solution to the equation. However, this other solution has the
wrong zeroth order term, namely a negative one, which renders it physically irrelevant.
23
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
Now let us consider the complete trip, from xµ
0to xµ
0+ ∆xµand ‘back’. The total coordinate
time elapsed during this trip is the sum of the forward trip and the backward trip time intervals.
Schematically:
∆ttot = ∆t(∆x,∆y,∆z,t0−z0) + ∆t(−∆x,−∆y,−∆z,t0+ ∆t−(z0+ ∆z)),(71)
since the spatial interval on the backward trip is simply minus the forward spatial interval,
and the ‘initial’ value of t−zfor the backward trip is just the nal value t0−z0+ ∆t−∆z
corresponding to the forward trip. Plugging in (70) yields
∆ttot =2∆ℓ+ε∆x2−∆y2
2∆ℓ¯
f+,tot +ε∆x∆y
∆ℓ¯
f×,tot
+1
2λ2∆x2+ ∆y2+2∆z2
∆ℓ,(72)
where ¯
f+,tot =¯
f+,forward +¯
f+,backward and similarly for the ×-polarization, in terms of the for-
ward and backward averaged amplitudes, respectively, given by
¯
f+,×,forward =¯
f+×,(∆ℓ,∆z,t0−z0)(73)
=√2
∆ℓ−∆zˆ(∆ℓ−∆z)/√2
0
f+,×1
√2(t0−z0) + σdσ, (74)
¯
f+,×,backward =¯
f+,×(∆ℓ,−∆z,t0−z0+ ∆t−∆z)(75)
=√2
∆ℓ+ ∆zˆ(∆ℓ+∆z)/√2
0
f+,×1
√2(t0+ ∆ℓ−z0−∆z) + σdσ, (76)
where in the last expression we have replaced ∆tby ∆ℓin the argument of f+,×, because to
zeroth order this makes no difference, and only the zeroth order expression for ¯
f+,backward is
relevant because¯
f+,backward always appears multiplied with εin the expressions we care about,
like ∆ttot.
Equation (72) gives the total coordinate time elapsed during the trip forward and back. The
next step in the calculation of the radar distance R= ∆τ/2 is to convert the coordinate time
interval into the proper time interval measured by the stationary observer local to the emis-
sion and reception of the light ray. This is where a second Finslerian effect enters. For such a
stationary observer we have x=y=z=const and hence the 4-velocity is given by (˙
t,0,0,0),
where we will assume without loss of generality that ˙
t>0. The proper time measured by an
observer is given by the Finslerian length along its worldline ∆τ=−´Fdσ. If we use σ=τ
as our curve parameter, differentiating with respect to it shows that Fshould be normalized as
F=−1. This is the Finsler equivalent of the fact that in GR the worldline of a particle paramet-
erized proper-time should always satisfy gµν ˙
xµ˙
xν=−1 (or +1, depending on the signature
convention). In the case of our observer the condition becomes
F=sgn(A)α+|β|=sgn(−˙
t2)q|˙
t2|+|λ˙
t|
√2=−|˙
t|+|λ˙
t|
√2=−1+λ
√2˙
t!
=−1.(77)
It follows that
∆τ=1−λ
√2∆ttot (78)
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
along the worldine of the stationary observer. Plugging in equations (72) and (78) into
R= ∆τ/2 we conclude that, to rst order in εand second order in λ, the radar distance is
given by
R=1−λ
√2∆ℓ+ε1−λ
√2∆x2−∆y2
4∆ℓ¯
f+,tot
+1−λ
√2∆x∆y
2∆ℓε¯
f×,tot +λ2
4∆ℓ+∆z2
∆ℓ.(79)
This expresses the radar distance as a function of the spatial coordinate distances and the initial
value of t−z(the latter enters the expression via ¯
f+,×,tot). In the limit λ→0 we recover the
expression for the radar distance in the case of a standard gravitational wave in GR [23]:
R= ∆ℓ+ε∆x2−∆y2
4∆ℓ¯
f+,tot +ε∆x∆y
2∆ℓ¯
f×,tot +O(ε2).(80)
Before we move on, let us summarize in what ways the Finslerian parameter λhas entered
our analysis so far:
1. The null trajectories are altered due to the fact the Finsler metric induces a modied null
condition or MDR. As a result, it takes a larger coordinate time interval for a light ray to
travel a given spatial coordinate distance. This effect works in all spatial directions, even
the direction parallel to the propagation direction of the light ray. This effect enters at order
λ2.
2. The ratio of proper time and coordinate time is altered with the result that less proper time
is experienced per unit coordinate time. This effect enters at order λ.
There is, however, a third way in which the parameter enters. Namely in the relation between
the coordinate distance and radar distance in the absence of the wave. For a gravitational wave
in GR these conveniently coincide; in the case of our Randers waves they do not. The formula
for the radar distance derived above refers merely to coordinates. In order to make sense of the
result, we would like to express the right hand side in terms of measurable quantities, like the
radar distances in the various directions in the absence of the wave. Employing equation (79)
we write
∆X=1−λ
√2∆x+λ2
4∆x,(81)
∆Y=1−λ
√2∆y+λ2
4∆y,(82)
∆Z=1−λ
√2∆z+λ2
2∆z,(83)
for the radar distance in the x,yand zdirection in the absence of the wave, and
R0=1−λ
√2∆ℓ+λ2
4∆ℓ+∆z2
∆ℓ,(84)
for the radar distance (79) in the relevant direction in the absence of the wave. Eliminating the
coordinate distances in favor of the physical radar distances by virtue of the inverse transform-
ations, valid to second order in λ,
∆x= ∆X1+λ
√2+λ2
4(85)
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Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
∆y= ∆Y1+λ
√2+λ2
4(86)
∆z= ∆Z1+λ
√2(87)
∆ℓ=R01+λ
√2+3
4λ2−∆z2
4R0
λ2(88)
=R01+λ
√2+λ2
4−∆Z2
4R0
λ2(89)
we can express the radar distance in the presence of the wave as
R=R0+ε∆X2−∆Y2
4R0¯
f+,tot +ε∆X∆Y
2R0¯
f×,tot +O(ε2,λ3, ελ2).(90)
This is a remarkable result. By expressing the radar distance in terms of the physical observ-
ables ∆X,∆Yand R0rather than merely coordinates, all dependence on λhas disappeared to
the desired order and the expression is identical to its GR counterpart, equation (80)! We must
conclude, therefore, that the effect of a Randers gravitational wave on interferometer experi-
ments is virtually indistinguishable from that of a conventional GR gravitational wave.
It is important to remark that by no means this implies that all phenomena in such a Finsler
spacetime are identical to their GR counterparts. It might be possible to detect the presence of
a non-vanishing λby some other means. This is a very interesting and important questions,
however it is beyond the scope of this article and something to explore in future work. Our
results pertain merely to gravitational wave effects as observed by interferometers.
6. Discussion
The main aim of this paper was to study the physical effect of Finslerian gravitational waves
and, in particular, to investigate the question if and how such waves can be distinguished,
observationally, from the classical gravitational waves of general relativity. To this effect we
have derived an expression for the radar distance at the moment a Finsler gravitational passes,
say, the earth. This radar distance is the main observable that is measured by interferometers.
Remarkably, we have found that the expression for the radar distance is indistinguishable from
its non-Finslerian counterpart, leading us to conclude that interferometer experiments would
not be able to distinguish between a general relativistic and a Finslerian gravitational wave, at
least not with regards to the radar distance. This is on the one hand disappointing, since indic-
ates means we cannot use such measurements to test the Finslerian character of our spacetime.
On the other hand, though, it means that the current gravitational wave measurements are all
compatible with the idea that spacetime has a Finslerian nature. To the best of our knowledge
this is the rst time an explicit expression for the Finslerian Radar length has been obtained
for the case of nite spacetime separations, and as such our work may be seen as a proof of
concept. Repeating the analysis for other Finsler spacetime geometries may lead to additional
insight as to the observational signature of Finsler gravity.
It is important to point out that Finslerian effects may also play a role in the generation of
gravitational waves in, say, a black hole merger event. This could lead to a Finslerian correction
to the waveform and this could be measured in interferometer experiments, at least in principle.
In order to be able to investigate this, however, Finslerian black hole solutions need to be better
26
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
understood. A start in this direction has been made in [65], where all 4-dimensional spherically
symmetric Finsler metrics of Berwald type have been classied.
The other parts of the article, leading up to the calculation of the radar length, were more
mathematical in nature. We have introduced a class of exact solutions to the eld equation in
Finsler gravity that have a close resemblance to the well-known general relativistic pp-waves,
and that generalize all of the pp-wave-type solutions currently known in the literature [20–22].
These solutions are (α,β)-metrics, where αis a classical pp-wave and βis its dening covari-
antly constant null 1-form. Consequently our solutions are of Berwald type. Their linearized
versions, we have shown, may be interpreted as Finslerian gravitational waves of modied
Randers type.
Indeed, along the way we have introduced a small modication to the standard denition
the Randers metric, motivated by the observation that the physical interpretation of the causal
structure of the standard Randers metric is not immediately obvious. In contrast, we have
shown that our modied Randers metrics have the nice property that their causal structure is
completely equivalent to the causal structure of some auxiliary (pseudo-)Riemannian metric,
hence leading to a perfectly clear physical interpretation. We stress that this auxilliary met-
ric is different from the dening (pseudo-)Riemannian metric α. In the special case that the
dening 1-form of the Randers metric is covariantly constant (which is the case, for example,
for our solutions) we have even more satisfactory results. In this case not only the causal
structure, but also the afne structure of the Randers metric coincides with that of the auxilli-
ary (pseudo)-Riemannian metric, i.e. the timelike, spacelike and null geodesics of the Finsler
metric can be understood, respectively, as the timelike, spacelike and null geodesics of the
auxiliary (pseudo)-Riemannian metric. A particularly nice consequence of this is the guaran-
teed existence of radar neighborhoods, i.e. that given an observer and any event in spacetime,
there is (at least locally) exactly one future pointing light ray and one past pointing light ray
that connect the worldline of the observer to the event. This is of essential importance in our
work, because without this property it would have not been possible to perform the calculation
of the radar distance in the last part of the article, simply because the notion of radar distance
would not even make sense in that case.
Let us now point out some of the limitations of our investigation. First of all, it is by no
means expected that the Finslerian gravitational waves discussed here should be only possible
ones. Although being much larger than even the complete class of all Lorentzian (i.e. non-
Finslerian) geometries, the class of (α,β)-metrics of Berwald type, to which we have restric-
ted our analysis, is still quite restrictive in the large scheme of (Finsler geometric) things.
Moreover, even within the class of (α,β)-metrics, our analysis is only valid for those metrics
that can be regarded as ‘close’ to a Lorentzian metric, such that they can be approximated
by Randers metrics. So even though our results suggest that there is no observable difference
between the Finslerian gravitational waves discussed in this article and their GR counterparts,
there might be more general types of Finslerian gravitational waves that could be distinguished
observationally from the general relativistic ones by means of interferometer experiments.
Furthermore, radar distance experiments are by no means the only way of probing our space-
time geometry. It might be possible to detect the Finslerian character of spacetime in some
other way. We have not explored this possibility here, but we plan to investigate this in the
future.
Moreover, we have assumed in our calculations that the amplitude of the gravitational waves
as well as the Finslerian deviation from general relativity are sufciently small such that a
perturbative approach to rst order in the former and second order in the latter is valid. It
would be of interest to repeat the calculation to higher order in perturbation theory. We expect
that this would in principle be a straightforward, yet possibly tedious, exercise.
27
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
Data availability statement
No new data were created or analyzed in this study.
Acknowledgments
S H wants to thank Rick Sengers and Nicky van den Berg for fruitful discussions and for their
input with regards to the gures. S H also wants to thank Luc Florack for fruitful discussions,
in particular his suggestions with regards to perturbation theory. We would like to acknow-
ledge networking support by the COST Action CA18108, supported by COST (European
Cooperation in Science and Technology).
Appendix A. Some properties of the metric aµν +bµbν
A.1. Proof of Lorentzian signature
Here we prove that if aµν has Lorentzian signature and aµνbµbν>−1 then ˜
aµν =aµν +bµbν
also has Lorentzian signature. We write b2=aµν bµbν. First, the matrix determinant lemma
says that
det˜
a= (1+b2)deta.(A1)
As long as b2>−1 this implies that det˜
ahas the same sign as deta, so assuming aµν is
Lorentzian, ˜
ahas negative determinant. In 4D this immediately implies that ˜
ais Lorentzian
(although the signs of the eigenvalues might be ipped with respect to aµν). However, let
us assume the dimensionality is arbitrary. Consider the family of 1-forms b(η)
µ=ηbµ, where
η∈[0,1]. For any ηwe have
det g
a(η)=1+b(η)2deta=1+η2b2det a,(A2)
det g
a(η)has the same sign for all values of η. Now since each of the neigenvalues of g
a(η)can be
expressed as continuous function of η, it follows that the respective signs of the neigenvalues
cannot change when we change η. To see why, suppose that the kth eigenvalue is positive for
some η1and negative for some η2. By the intermediate value theorem, there must exist some
ηbetween η1and η2for which the eigenvalue vanishes. In that case the determinant vanishes
for that value of η, which is a contradiction, as the determinant never vanishes as we have just
seen. This argument proves that g
a(η)has the same signature for all values of η, because the
signs of the eigenvalues remain unchanged. In particular, ˜
a=g
a(1)has the same signature as
a=g
a(0). Therefore, if aµν is Lorentzian and b2>−1 then ˜
ais Lorentzian as well.
A.2. Affine structure
Here we derive an explicit formula for the Christoffel symbols of the metric ˜
aµν =aµν +bµbν,
where it is again assumed that b2>−1.
Proposition 8. The Christoffel symbols of ˜
aµν can be expressed as
e
Γρ
µν = Γρ
µν +1
1+b2bρ∇(µbν)−aρλ −1
1+b2bρbλbµ∇[λbν]+bν∇[λbµ].(A3)
where ∇is the covariant derivative corresponding to aµν
28
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
We prove this below, but rst we point out the following immediate consequence.
Corollary 9. If bµis covariantly constant with respect to aµν , the afne structure of ˜
aµν is the
same as the afne structure of aµν, i.e. e
Γρ
µν = Γρ
µν .
Proof. As long as b2>−1 the formula for the determinant displayed above shows that ˜
aµν =
aµν +bµbνis invertible as a matrix. It can be easily checked that its inverse is given by
˜
aµν =aµν −1
1+b2bµbν.(A4)
Unless otherwise specied (as in the case of e
Γbelow!) indices are raises and lowered with
aµν . Denoting Γλµν =aλρΓρ
µν and e
Γλµν =˜
aλρe
Γρ
µν we rst note that we can express the
latter as
e
Γλµν =1
2(∂µ˜
aλν +∂ν˜
aµλ −∂λ˜
aµν )=Γλµν +bλ∂(µbν)−bµ∂[λbν]−bν∂[λbµ],(A5)
where (µ,ν)denotes symmetrization and [µ,ν]denotes anti-symmetrization. Therefore it fol-
lows that
e
Γρ
µν =˜
aρλe
Γλµν =aρλ −1
1+b2bρbλΓλµν +bλ∂(µbν)−bµ∂[λbν]−bν∂[λbµ](A6)
= Γρ
µν −1
1+b2bρbλΓλ
µν +aρλ −1
1+b2bρbλbλ∂(µbν)
−aρλ −1
1+b2bρbλbµ∂[λbν]+bν∂[λbµ].(A7)
The second and third term add up to
−1
1+b2bρbλΓλ
µν +aρλ −1
1+b2bρbλbλ∂(µbν)(A8)
=−1
1+b2bρbλΓλ
µν +bρ∂(µbν)−b2
1+b2bρ∂(µbν)(A9)
=−1
1+b2bρbλΓλ
µν +1
1+b2bρ∂(µbν)(A10)
=1
1+b2bρ∂(µbν)−bλΓλ
µν (A11)
=1
1+b2bρ∇(µbν).(A12)
This shows that
e
Γρ
µν = Γρ
µν +1
1+b2bρ∇(µbν)−aρλ −1
1+b2bρbλbµ∂[λbν]+bν∂[λbµ].(A13)
Finally, we may replace all partial derivatives with covariant ones because
∇[λbν]=∇λbν−∇νbλ=∂λbν−Γµ
λν bµ−∂νbλ+ Γµ
νλ bµ=∂λbν−∂νbλ=∂[λbν].(A14)
That yields the desired formula.
29
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
Appendix B. Determinant of a not necessarily positive definite (α,β )-metric
Here we derive the formula equation (38) for the determinant of a not necessarily positive
denite (α,β)-metric, generalizing the well-known result from the positive denite case.
More precisely, we consider Finsler metrics the form F=αϕ(s), where s=β/α,α=p|A|=
p|aijyiyj|,A=aij yiyj=sgn(A)α2, and where aij is assumed to be a (pseudo-)Riemannian
metric, i.e. not necessarily Riemannian/positive denite.
In complete analogy with the positive denite case, it can be shown by direct calculation
that the fundamental tensor gij ≡1
2¯
∂i¯
∂jF2is given by
gij =sgn(A)ρaij +ρ0bibj+ρ1(biαj+αibj) + ρ2αiαj,(B1)
where we have dened αi=aijyj/α, and with coefcients given by
ρ=ϕ(ϕ−sϕ′),(B2)
ρ0=ϕϕ′′ +ϕ′ϕ′,(B3)
ρ1=−(sρ0−ϕϕ′) = −[s(ϕ ϕ ′ ′ +ϕ′ϕ′)−ϕϕ ′],(B4)
ρ2=−sρ1=s[s(ϕϕ′′ +ϕ′ϕ′)−ϕϕ ′].(B5)
The only difference here with the positive denite case is the factor sign(A) appearing in the
rst term in equation (B1). Denoting dimM=ncan write this in matrix notation as
g=sgn(A)ρa+UWVT,(B6)
in terms of the three matrices
W=sgn(A)
ρI4×4,U= (
b,
b, α, α),V= (ρ0
b,ρ1α, ρ1
b,ρ2α).(B7)
Uand Vare both n×4 matrices. It is a well-known result (one of the matrix determinant
lemmas, see e.g. [66]) that assuming ais an invertible matrix the determinant of the expression
in brackets is equal to
deta+UWVT=det I4×4+WVTa−1Udet a.(B8)
It follows that
detg=sgn(A)nρndet I4×4+WVTa−1Udet a.(B9)
The matrix product WVTa−1U=sgn(A)
ρVTa−1Ucan be evaluated by explicit computation and
reads
WVTa−1U=sgn(A)
ρ
b2ρ0b2ρ0sgn(A)sρ0sgn(A)sρ0
sgn(A)sρsgn(A)sρsgn(A)ρsgn(A)ρ
b2ρ1b2ρ1sgn(A)sρ1sgn(A)sρ1
sgn(A)sρ2sgn(A)sρ2sgn(A)ρ2sgn(A)ρ2
.(B10)
30
Class. Quantum Grav. 40 (2023) 184002 S Heefer and A Fuster
Hence we obtain
detg
=sgn(A)nρndet
I4×4+sgn(A)
ρ
b2ρ0b2ρ0sgn(A)sρ0sgn(A)sρ0
sgn(A)sρsgn(A)sρsgn(A)ρsgn(A)ρ
b2ρ1b2ρ1sgn(A)sρ1sgn(A)sρ1
sgn(A)sρ2sgn(A)sρ2sgn(A)ρ2sgn(A)ρ2
deta
(B11)
=sgn(A)nρndet
I4×4+1
ρ
sgn(A)b2ρ0sgn(A)b2ρ0sρ0sρ0
sρsρ ρ ρ
sgn(A)b2ρ1sgn(A)b2ρ1sρ1sρ1
sρ2sρ2ρ2ρ2
deta(B12)
=ϕn+1(ϕ−sϕ′)n−2(ϕ−sϕ′+ (sgn(A)b2−s2)ϕ′′)det aij.(B13)
Some useful identities that we have used are: αi=sgn(A)yi/α so that αiαi=sgn(A)and
αibi=sgn(A)s. We conclude that
detgij =ϕn+1(ϕ−sϕ′)n−2(ϕ−sϕ′+ (sgn(A)b2−s2)ϕ′′)detaij.(B14)
In the case that αis positive denite, sign(A) = 1 everywhere, so the formula reduces to
the standard result (see e.g. [63]).
ORCID iD
Sjors Heefer https://orcid.org/0000-0002-2057-6301
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