Finsler gravitational waves of $(\alpha,\beta)$-type and their observational signature

Abstract

We introduce a new class of (α,β)(\alpha,\beta)-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 special cases. The linearized versions of these solutions may be interpretted as Finslerian gravitational waves, and we investigate the physical effect of such waves. More precisely, we compute the Finslerian correction to the radar distance along an nterferometer 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 modification of the Randers metric and prove that it has some very interesting properties.

Full text

Finsler gravitational waves of (α, β)-type and their observational signature
Sjors Heefer∗
Department of Mathematics and Computer Science,
Eindhoven University of Technology, Eindhoven, The Netherlands
Andrea Fuster†
Department of Mathematics and Computer Science,
Eindhoven University of Technology, Eindhoven, The Netherlands
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 special cases. The linearized versions of these solutions may be interpretted as
Finslerian gravitational waves, and we investigate the physical effect of such waves. More precisely,
we compute the Finslerian correction to the radar distance 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
modification of the Randers metric and prove that it has some very interesting properties.
∗s.j.heefer@tue.nl
†a.fuster@tue.nl
arXiv:2302.08334v1 [gr-qc] 16 Feb 2023

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CONTENTS
I. Introduction 3
A. Structure of this article 4
II. Finsler gravity 5
A. Finsler spaces of positive definite signature 5
B. Berwald spaces and the Riemannian limit 7
C. Finsler spacetimes 8
D. A note about causal structure and physical interpretation 9
E. The field equations 9
III. (α, β )-metrics 10
A. (α, β)-metrics – basic definitions 10
B. Exact (α, β )-metric solutions in Finsler gravity 11
C. Plane wave solutions in Brinkman and Rosen coordinates 13
D. Linearized gravitational wave solutions 14
E. Linearized (α, β )-metrics are Randers metrics 15
IV. Modified Randers metrics 16
A. Motivation and definition 16
B. Causal structure 18
C. Regularity and signature 19
V. Radar distance for a Finsler gravitational wave 22
A. Geodesics 23
B. Null geodesics and radar distance - GR case 24
C. Null geodesics and radar distance - Finsler case 26
VI. Discussion 30
Acknowledgments 31
A. Some properties of the metric aµν +bµbν31
1. Proof of Lorentzian signature 31
2. Affine structure 32
B. Determinant of a not necessarily positive definite (α, β)-metric 33
References 34

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I. 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 experimen-
tal 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, Pirani and
Schild (EPS) [4] is compatible with Finsler geometry, a natural 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
priori be excluded from our theories and motivates the study for 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 frame-
work 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 defined as the length of its worldline connecting these
events, in this case the Finslerian length rather than 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 definition of proper time has to be revised [8].
Further motivation for the study of Finsler spacetime geometry comes from quantum gravity phenomenol-
ogy [9]. Inspired by various approaches to quantum gravity, a generic feature of phenomenological or effective
quantum gravity models is the presence of Planck-scale modified dispersion relations (MDR), related to de-
parture from (local) Lorentz symmetry [9–11], which may manifest either in the sense of Lorentz invariance
violation (LIV) or in the sense of deformed Lorentz symmetry. It turns out that such MDRs generically induce
a Finsler geometry on spacetime [12]. The mathematical details of this were investigated in [13,14]; see e.g.
[15–17] for applications to specific 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 field equation is replaced by Pfeifer and
Wohlfarth’s field equation. For (pseudo-)Riemannian spacetimes the latter reduces to the former. Although
any solution to the field 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 solu-
tions 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 flat 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 distinguished from
their general relativistic counterparts. To answer this question we consider the linearlized versions of our
exact solutions, which may be interpretted 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

4
general relativistic gravitational wave. The relevant observable measured in interferometer experiments is
essentially the radar distance, so we first recall the calculation of this radar distance in the case of a standard
GR gravitational wave, reproducing the known results [23]. Then we repeat the calculation in the case of a
Finslerian gravitational wave. Although at first sight the expression for the Finsler radar length looks different
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 significance of this. To the best of our knowledge this is the first
time an explicit expression for the Finslerian Radar length has been obtained in the case of finite spacetime
separations, and as such our work may be seen as a proof of concept. In contrast, the radar length for
infinitesimal 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 sufficiently
small, so that a certain perturbative expansion is valid. This nevertheless seems physically justified. We
argue in a heuristic manner that up to first order in λ, any physically viable (α, β)-metric can be equivalently
described by a slightly modified 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 modified 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 defining 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 affine 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 worldline 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.
A. Structure of this article
The paper is organized as follows. We begin in Section II with a discussion of Finsler geometry and the
core ideas behind Finsler gravity. Then in Section III we introduce (α, β)-metrics, and in particular Randers
metrics and discuss their relevance to Finsler gravity. We then introduce our new solutions to the field
equations and show that after linearization these solutions may be interpretted as Finslerian gravitational
waves.
Next, in section IV we propose our modification of the standard Randers metric and prove that it has very
satisfactory properties with respect to its causal structure, affine structure, Lorentzian signature, etc.
Section Vis devoted to the calculation of the radar distance at the moment a Finsler gravitational wave
passes, say, the Earth. We start by recalling the analogous calculation for a standard gravitational wave in
general relativity, and consequently we compute the radar distance in the Finsler setting, clearly pointing out

5
the differences with the general relativity case.
We conclude in section VI.
II. FINSLER GRAVITY
Before we introduce the basic notions in Finsler geometry, some remarks about notation are in order. We will
usually work in local coordinates, i.e., given a smooth manifold Mwe assume that some chart φ:U⊂M→Rn
is provided, and we identify any p∈Uwith its image φ(p)∈Rn. For p∈Ueach Y∈TpM(the tangent
spaces to M) can be written as Y=yi∂ip, where the tangent vectors ∂i≡∂
∂xifurnish the chart-induced basis
of TpM. This provides natural local coordinates on the tangent bundle T M via the chart
˜
φ:˜
U→Rn×Rn,˜
U=[
p∈U{p} × TpM⊂T M, ˜
φ(p, Y )=(φ(p), y1, . . . , yn) =:(x, y).(1)
These local coordinates on T M in turn provide a natural basis of its tangent spaces T(x,y)T M , namely
∂
∂xi=∂i,∂
∂yi=¯
∂i.(2)
Below we start by introducing the basic notions of Finsler geometry in the positive definite case. The
generalization to Lorentzian signature is contains some technicalities and will be introduced subsequently.
Our spacetime signature convention is (−,+,+,+).
A. Finsler spaces of positive definite signature
Before discussing Finsler spacetimes, we first introduce the theory in the positive definite case, which is
certainly simpler and cleaner. In section II C we discuss what is different in the case of Finsler spacetimes
with Lorentzian signature.
A Finsler space is a pair (M, F ), where Mis a smooth manifold and F, the so-called Finsler function, is a
map F:T M →[0,∞)that satisfies the following axioms:
•Fis (positively) homogeneous of degree one with respect to y:
F(x, λy) = λF (x, y),∀λ > 0 ; (3)
•The fundamental tensor, with components gij =¯
∂i¯
∂j1
2F2, is positive definite.
For each x∈Mthe map y7→ F(x, y)is what is known as a Minkowski norm1on TxM. The homogeneity
condition ensures that the length of any curve γ, defined as
L(γ) = ZF( ˙γ)dλ=ZF(x, ˙x)dλ, ˙γ=dγ
dλ,(4)
1Not to be confused with the flat Lorentzian Minkowski metric.

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is independent of its parameterization. A fundamental result that is essential for doing computations in Finsler
geometry is Euler’s theorem for homogeneous functions. It says that if f:Rn→Ris (positively) homogeneous
of degree r, i.e., f(λy) = λrf(y)for all λ > 0, then yi∂f
∂yi(y) = rf (y). In particular, this implies the identity
gij (x, y)yiyj=F(x, y)2.(5)
Hence the length of curves is formally identical to the length in Riemannian geometry, the difference being
that now the metric tensor may depend on the direction in addition to position.
The fundamental theorem of Riemannian geometry says that any Riemannian manifold admits a unique
torsion-free affine connection that is compatible with the metric, the Levi-Civita connection. A similar result
is true in Finsler geometry, and this is sometimes called the fundamental lemma of Finsler geometry: it states
that any Finsler space can be endowed with a canonical connection. An essential difference with Riemannian
geometry is that the connection on a Finsler space is in general not a linear one. Let us therefore briefly recall
the notion of a non-linear connection. A non-linear (or Ehresmann) connection is a smooth decomposition of
TTM into a horizontal and a vertical subbundle,
TTM =H T M ⊕V T M , (6)
where ⊕denotes the Whitney sum of vector bundles. This provides the most general notion of parallel
transport of vectors between tangent spaces, and, in particular, it allows one to define whether a curve
γ:I= (a, b)→Mis autoparallel (‘straight’). Intuitively, we would like to call a curve straight whenever the
velocity ˙γ:I→T M is ‘constant’. However, there is no unique way to say, a priori, what ‘constant’ means
in this context, as each image point of ˙γlies in a different tangent space. As a matter of fact, as ˙γ, living
in the tangent bundle, also contains all information about the base point γ, it could never be truly constant.
Indeed, all we can ask is that ˙γchange only ‘parallel to M’, and not in the direction of the fibres of T M .
The rate of change of ˙γ, i.e. ¨γ, is an element of TTM. Therefore, in order to be able to say what we mean
by a straight line we should split the directions in TTM into a space H T M of directions parallel to Mand
a space of directions V T M along the fibers of T M . We then say that a curve γ:I→Mis autoparallel if
¨γ(λ)∈H˙γ(λ)T M for all λ∈I. The vertical subbundle V T M is canonically defined on any smooth manifold,
namely
V T M =span ¯
∂i.(7)
However, there is in general no canonical choice of the horizontal subbundle. In order to be able to speak
about straight curves, in the most general sense, one thus needs to select one. In order to do so, a set of
functions Ni
j(x, y), the connection coefficients, may be specified, leading to the following horizontal subbundle
of TTM.
H T M =span nδi≡∂i−Nj
i¯
∂jo.(8)
Parallel transport of a vector Valong γis then characterized by the parallel transport equation2
˙
Vi+Ni
j(γ, V ) ˙γj= 0 ,(9)
2Note that the parallel transport map is in general nonlinear. Some authors (e.g. [27]) choose to define parallel transport
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. [28], where parallel transport
of a vector is defined via its unique horizontal lift along a given curve. In this case parallel transport is linear if and only if the
connection is linear.

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and consequently, autoparallels are precisely the curves that satisfy
¨γi+Ni
j(γ, ˙γ) ˙γj= 0 .(10)
As mentioned, on generic smooth manifold there is no canonical choice of the connection3Ni
j, but any Finsler
metric induces one, the Cartan non-linear connection. This is the unique homogeneous (non-linear) connection
on T M that is (smooth on T M \{0},) torsion-free and compatible with F. Torsion-freeness is the property that
¯
∂iNk
j=¯
∂jNk
i, and metric-compatibility is the property that δiF2= 0, in terms of the horizontal derivative
induced by the connection, δi≡∂i−Nj
i¯
∂j. Alternatively, metric compatibility can be defined as the property
that ∇gij ≡ykδkgij −Nk
igkj −Nk
jgki = 0, in terms of the so-called dynamical covariant derivative ∇. For
torsion-free homogeneous connections the latter definition of metric-compatibility is equivalent to the former.
This Cartan non-linear connection is given in terms of the Finsler function Fby
Ni
j(x, y) = 1
4¯
∂jgikyl∂l¯
∂kF2−∂kF2(11)
and may be viewed as a generalization of the Levi-Civita connection to Finsler spaces. The autoparallel
curves of the non-linear connection coincide with the geodesics (locally length-minimizing curves) of F. The
curvature tensor, curvature scalar and the Finsler Ricci tensor of (M, F )are defined, respectively, as
Rijk (x, y) = −[δj, δk]i=δjNi
k(x, y)−δkNi
j(x, y),Ric(x, y) = Riij (x, y )yj, Rij(x, y) = 1
2¯
∂i¯
∂jRic.(12)
B. Berwald spaces and the Riemannian limit
A Berwald space is a Finsler space (M, F )for which the Cartan non-linear connection is in fact a linear
connection on T M .4What this means is that the connection coefficients are of the form
Ni
j(x, y) = Γi
jk (x)yk(13)
for a set of functions Γi
jk :M→R. From the transformation behavior of Ni
jit can be inferred that the
functions Γi
jk have the correct transformation behavior to be the Christoffel symbols of a (torsion-free) affine
connection on M. We will refer to this affine connection as the associated affine connection, or simply the
affine connection on the Berwald space. The parallel transport (9) and autoparallel equations (10) reduce in
this case to the familiar equations
˙
Vi+ Γi
jk (γ) ˙γjVk= 0,¨γi+ Γi
jk (γ) ˙γj˙γk= 0 (14)
in terms of the Christoffel symbols. A straightforward calculation reveals that the curvature tensors of a
Berwald space can be written as follows
Rjkl =¯
Rijkl(x)yi,Ric =¯
Rij (x)yiyj, Rij =1
2¯
Rij (x) + ¯
Rji (x),(15)
3From now on we will refer to the connection coefficients Ni
jsimply as the connection.
4See [29] for an overview of the various equivalent characterizations of Berwald spaces and [30] for a more recent equivalent
characterization.

8
in terms of ¯
Rlijk = 2∂[jΓi
k]l+ 2Γi
m[jΓm
k]land ¯
Rlk =¯
Rliik, the Riemann tensor and Ricci tensor, respectively,
of the associated affine connection5, defined in the usual way. In fact, for positive definite 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 [31].
Finsler geometry reduces to Riemannian geometry when the fundamental tensor gij(x, y ) = gij(x)is inde-
pendent of the direction y, i.e., if the fundamental tensor is a Riemannian metric. Equivalently, the space is
Riemannian if F2is quadratic in the y-coordinates. In this case the non-linear connection is actually linear,
so that, in particular, any Riemannian manifold is Berwald. In fact, the associated linear connection is in this
case just the Levi-Civita connection of the Riemannian metric.
C. Finsler spacetimes
The generalization of positive definite Finsler geometry to indefinite signatures is by no means a trivial
matter, and there is as of yet no consensus on what the proper definition should be. To see the basic issue,
note that if the fundamental tensor gµν has Lorentzian signature then there will be (non-zero) null vectors
v∈TxMfor which gµνvµvν= 0. Then F(x, v) = √gµν vµvν=√0, even though v6= 0, so Fcan never be
smooth everywhere on T M \0, one of the axioms introduced in the previous section. Moreover, for spacelike
(or timelike, depending on the convention) directions w,F(x, w)would even be imaginary. Thus some things
clearly need to be modified in order to give an acceptable definition of a Finsler spacetime, and various
approaches are possible. One classical approach [32] is to work with L=F2instead of F. Another is to
restrict the domain of definition of F, for instance to those (x, y)for which F(x, y)2=gµν (x, y)yµyν>0[33].
Several combinations of the two approaches and other variations have been proposed [18,34–37].
Which of these general definitions should be the ‘correct’ one is not terribly relevant for our present pur-
poses. Therefore our approach here will be to simply replace the subbundle T M \0⊂T M by a generic conic
subbundle A ⊂ T M \0, i.e. a conic6open subset of T M \0. Furthermore we will not restrict Fto have only
positive values. Thus we will be using the following definition.
A Finsler spacetime is a triple (M, A, F ), where Mis a smooth manifold, Ais a conic subbundle of T M \0
(with ‘non-empty’ fibers) and F, the so-called Finsler function, is a map F:A → Rthat satisfies the following
axioms:
•Fis (positively) homogeneous of degree one with respect to y:
F(x, λy) = λF (x, y),∀λ > 0 ; (16)
•The fundamental tensor, with components gµν =¯
∂µ¯
∂ν1
2F2, has Lorentzian signature on A.
The discussion and results (for the connection, curvature tensors, etc.) treated in sections II A and II B
apply verbatim for Finsler spacetimes, with the understanding that we only consider points (x, y)∈ A.
Throughout the article we will also assume that the spacetime dimension is 1+3.
5We use the notations T[ij]=1
2(Tij −Tji )and T(ij)=1
2(Tij +Tji )for (anti-)symmetrization.
6The property of being conic means that if (x, y)∈ A then also (x, λy)∈ A, for any λ > 0.

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The definition of a Finsler spacetime given above is a very weak one in the sense that most other definitions
appearing in the literature are more restrictive. Accordingly, our definition 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.
D. A note about causal structure and physical interpretation
Given a Finsler spacetime geometry, it is natural to postulate, in analogy with GR, that matter trav-
els 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 define what one means a by timelike vector. It certainly does not suffice to define
them as positive length vectors. We do not discuss this issue any further in its full generality here. Only in
the specific case of the Randers metric, in Section IV, 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 definition only slightly, the causal structure of such a modified 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 V, where we compute the radar distance for a Finslerian
gravitational wave of (modified) 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 geometry is well-defined for all timelike
infinitesimal spacetime separations. This property is satisfied by our modified 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.
E. The field equations
In the context of Finsler gravity, arguably the simplest and cleanest proposal for a vacuum field 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 field 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-defined sense [39]) turns out to be the field equation that was proposed by Pfeifer and Wohlfarth

10
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.
Although several other proposals have been made as well [40–48], we consider the Pfeifer-Wohlfarth equa-
tion7[18] to be by far the most promising, and from here onwards we will refer to it simply as the vacuum field
equation in Finsler gravity. We do not show the field 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 −3yµyνRµν = 0 ,(17)
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 sufficient condition for a Berwald spacetime to be
a solution to Eq. (17). In general it is not a necessary condition, except in specific cases, like for Randers
metrics. Indeed for Randers metrics of Berwald type the field equations reduce to Rutz’s equation [22], or
equivalently, to the vanishing of the Finsler Ricci tensor,
Rµν = 0.(18)
III. (α, β )-METRICS
A. (α, β)-metrics – basic definitions
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 defined 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;
•Generalized Kropina (or m-Kropina) metrics F=α1+mβ−mwith msome real number.
For each of these types of (α, β)-metrics certain conditions need to be fulfilled in order to satisfy the definition
of a Finsler space(time).
7In the positive definite setting a similar field equation has been obtained by Chen and Shen [49].

11
B. 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 field equations to properly Finslerian vacuum solutions in Finsler
gravity. This procedure is possible whenever such a solution admits a covariantly constant vector field, 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
solution to the Finslerian field equations. To see why this is true, we first recall the following well-known
result (see e.g. section 6.3.2. in [50]):
Proposition 1. Let Fbe an (α, β)-metric. If βis covariantly constant with respect to αthen Fis of Berwald
type and the affine connection of Fcoincides with the Levi-Civita connection of α.
If the affine connection of Fis the same as the connection of α, the associated curvature tensors and (affine)
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 affine Ricci tensor of Fvanishes as well, which implies, by eq. (17), that
Fis a vacuum solution to Pfeifer and Wohlfarth’s field equation in Finsler gravity. We may summarize this
result in the following theorem.
Theorem 2. Let Fbe any (α, β)-metric such that αsolves the classical vacuum Einstein equations and βis
covariantly constant with respect to α. Then Fis a vacuum solution to the field equation in Finsler gravity.
In this way (α, β)-metrics provide a mechanism to Finslerize any vacuum solution to Einstein’s field equa-
tions, as long as the solution admits a covariantly 1-form, or equivalently a covariantly constant vector field.
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 arbitrary 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 field equations admits a covariantly constant 1-form β, then either αis flat, or βis necessarily null
[51] (see also [52,53]). 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 first of these possibilities, where αis flat, 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,(19)
where bµ=const. The any (α, β)-metric constructed from α=p|A|and βis a vacuum solution
to the field equations in Finsler gravity. The resulting geometry is of Berwald type with all affine
connection coefficients vanishing identically in these coordinates.
Right below Eq. (19) we have used the notation α=p|A|=p|aij dxidxj|. This should be understood
pointwise, i.e.
α=α(y) = q|aij dxidxj|(y) = q|aij dxi(y)dxj(y)|=q|aij yiyj|.(20)

12
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 interesting. In this
case αis CCNV spacetime metric, meaning that it admits a covariantly constant null vector (CCNV), namely
in this case β, or rather its vector equivalent via the isomorphism 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
[51,54] (see section 24.5 in [55] for a summary).
It is an elementary result that by choosing suitable coordinates (u, v, x1, x2), such αand βcan always be
expressed in the form
A=−2du(dv+H(u, x)du+Wa(u, x)dxa) + hab(u, x)dxadxb,(21)
β=du, (22)
where xa=x1, x2and hab is a two-dimensional Riemannian metric. This holds irrespective of whether αis
a solution to Einstein’s field equations or not. If αis additionally assumed to be a vacuum solution, as in
Theorem 2, it turns out that the expression (21) for Acan be simplified even more without changing the form
(22)of β. To see this, we first consider only the metric A. Since Ais a vacuum solution to Einstein’s field
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 [55]). The metric then takes the form
A=−2du(dv+H(u, x)du) + δabdxadxb.(23)
We are, however, not only interested in the transformation behaviour 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)(24)
that leaves the generic form of the metric (21) invariant, but in general changing the expressions for the metric
functions H, Wa, hab 7→ ¯
H, ¯
Wa,¯
hab, has the specific property that u=φ(¯u)for some function φdepending on
¯ualone (see section 31.2 in [55]). This applies in particular to the transformation that relates (21) and (23).
We can therefore express the 1-form as β=du=φ0(¯u)d¯u, or equivalently ¯
bµ=φ0(¯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
(23) with upper index uvanish identically, however. Hence
¯
∇¯u¯
b¯u=∂¯ub¯u−¯
Γu
uuφ0(¯u) = φ00(¯u)!
= 0.(25)
It follows that φ0(¯u) = C=constant, i.e. β=Cd¯u. In this case it is easily seen that scaling ¯uby Cand scaling
¯vby 1/C leaves the metric (23) 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 (23) is a vacuum solution to Einstein’s field equations if and only if (∂2
x1+∂2
x2)H= 0.
We may therefore characterize the second class of solutions in the following way.

13
(α, β)-metric solutions (Class 2). Let α=p|A|and βbe given by
A=−2du(dv+H(u, x)du) + δabdxadxb,(26)
β=du, (27)
such that δab∂a∂bH= 0. Then any (α, β)-metric constructed from the pair (α, β) is a vacuum
solution to the field equations in Finsler gravity. The resulting geometry is of Berwald type with
affine 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 specific
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 (17) must be of the type described in
theorem 2, that is, αis necessarily a vacuum solution in Einstein gravity and βis 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 theorem 2have
been obtained in the context of Very General Relativity (VGR) [21].
•Any pseudo-Riemannian Finsler metric F=αis trivially a vacuum solution in Finsler gravity 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.
C. Plane wave solutions in Brinkman and Rosen coordinates
Eq. (26) expresses the pp-wave metric in Brinkmann form [56]. For the description of the physical effects of
(plane) gravitational waves in general relativity, it is sometimes more convenient to use a different coordinate
system, known as Rosen coordinates [57]. This remains true in the Finsler case. When we compute the effect
on the radar distance of a passing Randers gravitational wave in section V, our starting point will be the
expression for the gravitational wave in Rosen coordinates. Therefore we briefly review the relation between
the two coordinate 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.(28)

14
We note that ∇∂vRρσµν = 0 always holds, identically, so it would be equivalent to require invariance along
the surfaces du= 0. The conditions (28) are equivalent to the statement that ∂a∂b∂cH= 0 in Brinkmann
coordinates (26), 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 [55]) so that the metric
can be written as
A=−2dudv+Aab(u)xaxbdu2+δab dxadxb(29)
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,(30)
where hij is a two-dimensional Riemannian metric. And conversely, any metric of Rosen form (30) can be
cast in the form (29). The two coordinate systems are related via
U=u, V =v−1
2˙
EaiEibxaxb, xa=Eaiyi,(31)
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 [58] by Matthias Blau and references therein (see also the Appendix of [59]). 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, V 7→ 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=α φ(β/α), A =−2dudv+hij (u)dxidxj, β =du, (32)
where α=p|A|. And conversely, for any choice of φ, hij (u), this is a vacuum solution to the field equations
in Finsler gravity if Ais a vacuum solution to Einstein’s field equation. The resulting geometry is of Berwald
type with affine connection identical to the Levi-Civita connection of α.
D. Linearized gravitational wave solutions
The exact vacuum field equation for plane-wave metrics does not have a particularly nice expression in
Rosen coordinates (32). The linearlized field equation, however, turns out to be very simple. So let’s consider
the scenario that the pseudo-Riemannian metric αis very close to the Minkowski metric. In this case we may

15
write hij (u) = δij +εfij (u)with ε1. The linearlized field equations (i.e. to first order in ε) for αthen
simply read8
f00
11(u) + f00
22(u)=0.(33)
Hence f11 and −f22 must be equal up to an affine function of u. Here we will focus on the case where
f11 =−f22, which can always be achieved by means of the transverse traceless gauge9. 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=α φ(β/α),A=−2dudv+ (1 + εf+(u))dx2+ (1 −εf+(u))dy2+ 2εf×(u)dxdy
β=du(34)
Note that if we substitute u= (t−z)/√2and v= (t+z)/√2, then Areduces to the standard expression
for a gravitational wave metric in GR, i.e.
F=α φ(β/α),A=−dt2+ (1 + εf+(t−z))dx2+ (1 −εf+(t−z)dy2+ 2εf×(t−z)dxdy+dz2
β=1
√2(dt−dz),
(35)
for any choice of the function φ.
E. Linearized (α, β)-metrics are Randers metrics
It is natural to linearlize not only in ε, characterizing the departure from flatness, but to also use a pertur-
bative 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 constists 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 first order,
so that from the physics point of view, Randers metrics are actually quite a bit more general than they might
seem at first 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
β1. To see what happens in such a scenario, we replace βwith λβ and expand to first order in λ. Then
we obtain
F=αφ λβ
α≈αφ(0) + λφ0(0) β
α=αφ(0) + λφ0(0)β= ˜α+˜
β. (36)
8The full linearlized vacuum field equation (17) for Fis more complicated in general, but as discussed extensively above, if the
vacuum field equation for αis satisfied then so is the field equation for F. In the case of Randers metrics, to which we will
turn momentarily, the field equation for Fis even equivalent to the field equation for α. Hence for our present purposes the
field equations for αsuffice.
9We leave open the question whether the form of the 1-form β=dualways remains invariant under such a transformation to
the transverse traceless gauge.

16
Hence to first order in λ, any (α, β)-metric is indeed equivalent to a Randers metric10. Consequently, by
replacing duby λduin (34), which technically can be achieved by a coordinate transformation that scales u
by λand vby 1/λ, it follows that to first 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.(37)
The parameter λthen characterizes the departure from GR and pseudo-Riemannian geometry. We will
assume without loss of generality that λ > 0. Finally, replacing also uand vby 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 V.
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)(38)
IV. 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 defined as F=α+β. We will argue that in order to have a
physically acceptable causal structure, the conventional definition must be modified slightly. It might seem
to the reader that modifying the Randers metric would be in conflict with the spirit of the previous section,
since to first 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 define one (α, β )-metric F1on a conic subbundle A1⊂TM \0and
another (α, β)-metric, F2, on a different conic subbundle A2⊂T M \0. If the two subbundles do not overlap
then this defines a perfectly valid (α, β)-type Finsler spacetime on the union A=A1∪ A2. To first order in
the deviation from (pseudo-)Riemannian geometry this Finsler metric would reduce to certain Randers metric
on A1and to a different Randers metric on A2. Our modification of the Randers metric, introduced below,
is therefore completely consistent with the previous results.
After a heuristic argument that motivates the desired modification, we show that our proposed version of
the modified Randers metric has a very satisfactory causal structure. As a result of this a clear (future and
past) timelike cone can be identified 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 condition b2<1that appears in the well-known positive definite case, see e.g. [60].
In one were 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 definite case.
A. Motivation and definition
First of all, let us review why the definition of a Randers metric is not as clear in Lorentzian signature as
it is in Euclidean signature. The original definition of a Randers metric, in positive definite Finsler geometry,
10 Actually 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.

17
is just F=α+β, with α=paij yiyja Riemannian metric and β=biyiany 1-form11. This is well-defined
as long as αis positive-definite, because in that case A≡aij yiyjis always positive. If we allow aij to be a
Lorentzian metric, however, the quantity Acan become negative, in which case √Ais ill-defined, 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 ⊂ T M \0to those vectors for which aijyiyj>0. This was the approach in e.g. [22], where it was shown
that if Ais defined as the forward timecone12 corresponding to α, then under certain conditions on the 1-form
β, such a Randers spacetime satisfies all axioms of a Finsler spacetime. The fact that Ais restricted in this
way, however, leads to issues when it comes to the physical interpretation. Here we take a different approach.
The obvious first alternative to restricting Ato vectors with positive norm is to simply replace Aby |A|
and define 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 definition, 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. In fact,
the light cone separates the tangent space into only two connected components13 and there is consequently
not a straightforward interpretation in terms of timelike, spacelike and lightlike directions, at least not in the
conventional way14. We therefore take the viewpoint that outside of the half plane β < 0in 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 chang-
ing 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 follow below) the single lightcone (from the β < 0half space) is mirrored to the
complementary (β≥0) half space, whereas in the original half space intersected with the original cone of
definition 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 relevence, though, as the geometry 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 modified definition 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 time-
like cones15. The present metric does have this property as long as b2>−1, and we discuss this in detail below.
Definition 4. Motivated by the preceding heuristic argument we define the modified Randers metric as follows,
F=sgn(A)α+|β|,(39)
where we recall for completeness that α=p|A|,A=aij yiyjβ=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.
11 In order to satisfy all the axioms of a Finsler space, the 1-form must satisfy |b|2<1, see e.g. [60].
12 We 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.
13 This can be checked easily in suitable coordinates adapted to β.
14 We note that in the approach by Javaloyes and Sánchez [36,37] a single, future pointing (by definition) cone is sufficient,
though.
15 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|.

18
B. Causal structure
Next we will show that the modified Randers metric (39) indeed has very nice properties. By definition,
the light cone is given by
F= 0 ⇔sgn(A)≤0 & |A|=|β|2⇔A=−β2.(40)
It therefore follows that
F= 0 ⇔(aµν +bµbν)dxµdxν= 0,(41)
meaning that the light cone of Fis just the light cone of the auxilliary Lorentzian metric ˜aµν (x) = aµν +bµbν.
Indeed, the matrix determinant lemma guarantees that as long as b2=aµνbµbν>−1the metric aµν +bµbν
has Lorentzian signature, provided that aµν has Lorentzian signature. (For a proof see appendix A.) This
shows that as long as b2>−1the 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,(42)
and hence it also follows that
F > 0⇔(aµν +bµbν)yµyν>0.(43)
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 definition, by F= 0. We summarize these
results in the following proposition.
Proposition 5. As long as b2>−1, the causal structure of the modified 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 modified Randers metric, it is possible to
define time orientations in the usual manner, by means of a nowhere vanishing timelike vector field T. This
is not possible for Finsler spacetimes in general. 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.
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 affine structure of Fcan be understood in terms of
aµν +bµbν.
Proposition 6. If βis covariantly constant with respect to αand satisfies b2>−1then the causal structure
and the affine structure of the modified Randers metric F=sgn(A)α+|β|are identical to those of the
Lorentzian metric ˜aµν =aµν +bµbν. In other words, the timelike, spacelike and null geodesics of Fcoincide
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
affine structures concide in the case of a covariantly constant 1-form. This is again a result of the properties

19
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 + b2bρbλbµ∇[λbν]+bν∇[λbµ].(44)
Hence it follows immediately that if bµis covariantly constant then e
Γρ
µν = Γρ
µν and the affine structure of ˜aµν
is the same as that of aµν . We also known, by Prop. 1, that the affine structure of Fis the same as that of
aµν . Hence the affine structure of Fis the same as that of ˜aµν.
From this it immediately follows that the existence of radar neighborhoods is guaranteed [26]. More precisely,
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 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 V,
is a well-defined notion.
C. 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
det gij =φn+1(φ−sφ0)n−2(φ−sφ0+ (sgn(A)b2−s2)φ00) det aij .(45)
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 definite analogue, to which it reduces when A > 0, i.e. sgn(A)=1. For a modified
Randers metric of the form F=sgn(A)α+|β|the function φis given by φ(s) = sgn(A) + |s|, so this reduces
to
det g
det a=sgn(A)n−1(sgn(A) + |s|)n+1 =sgn(A)F
αn+1
.(46)
Assuming the spacetime dimension nis even, this means that ghas Lorentzian signature16 if and only if
sgn(A)F > 0. Let us see what this entails. First note that F < 0trivially implies A < 0. Hence F < 0implies
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 signature of the fundamental tensor is Lorentzian. Similarly,
A > 0implies F > 0. Hence A > 0also implies Lorentzian signature. What remains is the region where
A≤0and F≥0. Equivalently, A≤0and A+β2≥0. In this region, the determinant of the fundamental
tensor is either undefined, 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 II D.
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 metric is Lorentzian,
provided the former is. That means its light cone is just a conventional one that we’re 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
16 The argument is the same as in the positive definitive case, using the same methods as those employed in Appendix A.

20
between these two lightcones. Note that since F < 0implies 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∈M
the 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)17.
•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>−1then the light cones do not intersect (apart from the trivial intersection in
the origin).
•If βis spacelike (and assuming dim M > 2), then aij induces a Lorentzian metric on the (dim M−1)-
dimensional hypersurface defined by β= 0. In this case the two light cones intersect along the light cone
of this induced Lorentzian metric.
•If b2=−1there 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<−1there 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, Fig. 1displays the lightcones and the regions (in green) where the signature of the
fundamental tensor is Lorentzian, for the modified Randers metric
F=sgn(A)α+|β|, A =−(dx0)2+ (dx1)2+ (dx2)2, β =


ρdx0if timelike
ρ(dx0+dx1)if null
ρdx1if spacelike
,(47)
for a number of representative values of the parameter ρ. In each subfigure, the inner lightcone is that of F
and 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.
Since we only require Lorentzian signature within the timelike cone, these results are very satisfactory.
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.
17 We recall that this can be seen as follows. First, since aij is Lorentzian, it is always possible to choose coordinates such that A
is 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 reflection in the x1direction we may choose either sign. If b2<0then we may go to the local rest frame by a Lorentz
transformation, making b1= 0. If on the other hand b2>0we may perform a Lorentz transformation making b0= 0.

21
(a) Null 1-form with ρ= 0.6(b) Null 1-form with ρ= 1 (c) Null 1-form with ρ= 1.4
(d) Timelike 1-form with
ρ= 0.65
(e) Timelike 1-form with
ρ= 0.8
(f) Timelike 1-form with
ρ= 0.9
(g) Spacelike 1-form with
ρ= 0.8
(h) Spacelike 1-form with
ρ= 1.4
(i) Spacelike 1-form with
ρ= 2
(j) Timelike 1-form with
b2=−1.
(k) Timelike 1-form with
b2<−1.
FIG. 1: The figures 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=ρ y0(in the timelike
case) or β=ρ y1(in the spacelike case) for several representative values of ρ, shown in the tangent tangent
space TxMat any point x∈M, at y3= 0. Green regions correspond to Lorentzian signature, red regions to
non-Lorentzian signature. Fig. 1a -1i 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(Fig. 1j) then the light ‘cone’ of Fis the line
y1=y2=y3= 0. And if b2<−1(Fig. 1k) then the light ‘cone’ of Fconsists only of the origin. Therefore
we deem the latter two cases not physically interesting.

22
V. RADAR DISTANCE FOR A FINSLER GRAVITATIONAL WAVE
Now we are finally in the position to analyze the physical effects of a passing Finslerian gravitation wave
of (α, β)-type. We have seen in Section III E that, to first order, an (α, β)-metrics is equivalent to a Randers
metric. And we have argued in section IV that this should not be the standard Randers metric but rather
our modified Randers metric. Thus our starting point will be the linearlized gravitational wave solution of
modified-Randers type. That is, we are interested in the solution (38) but with the conventional Randers
metric F=α+βreplaced by the modified Randers metric F=sgn(A)α+|β|, where α=p|A|,A=aij yiyj.
Note that this modification does not change any of the results pertaining to classification of solutions to the
field equations, by the argument given at the beginning of section IV that a modified 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)
(48)
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 (48).
The setup is as follows. A light ray is emitted from some spacetime location with coordinates (t0, x0, y0, z0),
travels to another location in spacetime with coodinates (t0+ ∆t, x0+ ∆x, y0+ ∆y, z0+ ∆z), where it
is reflected and after which it travels back to the original (spatial) 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’ observer18 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.
We will compute the radar distance first for a classical GR gravitational wave and then we repeat the
calculation for the Randers gravitational wave, so that it is clear where each of the Finslerian effects enters
precisely. In Section V A we derive the explicit form of the geodesics. Conveniently, the geodesics in the Finsler
setting are the same as those in the GR setting, so these results are general. (Null geodesics are different,
though, because the null conditions are not the same.) Then, using the form of the geodesics, we first recall
the calculation of the radar distance in the GR setting [23] in Section V B, and then in Section V C we compute
the radar distance in the Finsler setting. Remarkably the results, when interpreted correctly, turn out to be
identical to the ones for GR.
18 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 finite duration as it passes the Earth, an
observer is inertial precisely if it is inertial before and after the wave passes.

23
A. Geodesics
The first important observation here is that the geodesics in a Randers gravitational wave spacetime with
Finsler metric F=sgn(A)α+|β|as in Eq. (48) coincide with the geodesics of the GR spacetime with metric
ds2=A, because the affine connection of Fcoincides with the Levi-Civita connection of A, by Prop. 1. For
the derivation of the general form of geodesics, we may therefore assume the geometry is given by ds2=A.
The results then apply both to the GR scenario as well as the Randers one. 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.(49)
Throughout this section and the next, we essentially follow [23], although our notation and presentation is
sometimes different.
If we use coordinates u= (t−z)/p(2) and v= (t+z)/√2the geodesic equations to first order in εare
given by
−˙u=:pv=const (50)
(1 + εf+(u)) ˙x+εf×(u) ˙y=:px=const (51)
(1 −εf+(u)) ˙y+εf×(u) ˙x=:py=const (52)
¨v+1
2˙x2−˙y2f0
+(u) + εf0
×(u) ˙x˙y= 0 (53)
The first three equations can be rewritten to first order as
˙u=−pv,˙x= (1 −εf+(u))px−εf×(u)py,˙y= (1 + εf (u))py−εf×(u)px,(54)
and can be integrated (with respect to an affine parameter σ, chosen without loss of generality such that
˙u= 1) to
u=u0+σ, x =x0+σ1−ε¯
f+(σ)px−ε¯
f×(u)py, y =y0+σ1 + ε¯
f+(σ)py−ε¯
f×(u)px,(55)
where
¯
f+,×(σ)≡1
σZσ
0
f+,×(u0+σ)dσ(56)
is the averaged value of f+,×. The equation for vcan be integrated to
˙v=−˜pu0−1
2p2
x−p2
yf+(u0+σ)−εf×(u0+σ)pxpy(57)
where ˜pu0=pu0−ε
2(p2
x−p2
y)f(u0)−εf×(u0)pxpy,pu=−˙v(not necessarily constant) and pu0is 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.(58)
Any geodesic emanating from a given point xµ
0can then be described by the following parameterized path,
for certain values of px, pyand ˜pu0:

24
u(σ) = u0+σ, (59)
x(σ) = x0+σ1−ε¯
f+(σ)px−ε¯
f×(σ)py,(60)
y(σ) = y0+σ1 + ε¯
f+(σ)py−ε¯
f×(σ)px,(61)
v(σ) = v0−˜pu0σ−1
2ε(p2
x−p2
y)σ¯
f+(σ)−εσ ¯
f×(σ)pxpy.(62)
B. Null geodesics and radar distance - GR case
To find the radar distance, we need to know the expression for null geodesics. In the GR case, null curves
satisfy −2 ˙u˙v+ (1 + εf(u)) ˙x2+ (1 −εf(u)) ˙y2+ 2εf×˙x˙y= 0. Substituting the general form of u, x, y, v for
geodesics, Eqs. (59)-(62), into the null condition, and eveluating it at u=u0(i.e. σ= 0), the condition
reduces to 2˜pu0+p2
x+p2
y= 0. We may therefore eliminate ˜pu0and directly substitute this into the expression
(62) 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, (63)
x=x0+σ1−ε¯
f+(σ)px−ε¯
f×(u)py,(64)
y=y0+σ1 + ε¯
f+(σ)py−ε¯
f×(u)px,(65)
v=v0+σ
2(p2
x+p2
y)−1
2ε(p2
x−p2
y)σ¯
f+(σ)−εσ ¯
f×(σ)pxpy(66)
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
∆u1 + ε¯
f+(∆u)+ε¯
f×(∆u)∆y
∆u, py=∆y
∆u1−ε¯
f+(∆u)+ε¯
f×(∆u)∆x
∆u.(67)
Plugging this into the vequation yields
2∆u∆v= ∆x2(1 + ε¯
f(∆u)) + ∆y2(1 −ε¯
f(∆u)) + 2ε¯
f×∆x∆y, (68)
or equivalently,
∆t2=1 + ε¯
f(∆u)∆x2+1−ε¯
f(∆u)∆y2+ 2ε¯
f×(∆u)∆x∆y+ ∆z2,(69)
where we have used that −2∆u∆v=−∆t2+ ∆z2. Hence to first order in εwe have
∆t= ∆`+∆x2−∆y2
2∆`ε¯
f+(∆u) + ∆x∆y
∆`ε¯
f×(∆u),(70)
where ∆`≡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 first order
expression for ∆t, any zeroth order expression for ¯
fsuffices in this formula. We have

25
¯
f(∆u) = 1
∆uZ∆u
0
f(u0+σ)dσ=√2
∆t−∆zZ(∆t−∆z)/√2
0
f(u0+σ)dσ(71)
=√2
∆`−∆zZ(∆`−∆z)/√2
0
f(u0+σ)dσ+O(ε)(72)
=√2
∆`−∆zZ(∆`−∆z)/√2
0
f1
√2(t0−z0) + σdσ+O(ε)(73)
since ∆t= ∆`+O(ε). We introduce another symbol for this expression, namely
¯
f(∆`, ∆z, t0−z0)≡√2
∆`−∆zZ(∆`−∆z)/√2
0
f1
√2(t0−z0) + σdσ, (74)
where the explicit display of the arguments serves to remind us that ¯
fdepends only on ∆`, ∆zand the initial
value of t−z. Since ε¯
f(∆u) = ε¯
f(∆`, ∆z, t0−z0) + O(ε2), it follows that we can rewrite Eq. (70), to first
order, as
∆t= ∆`+∆x2−∆y2
2∆`ε¯
f+(∆`, ∆z, t0−z0) + ∆x∆y
∆`ε¯
f×(∆`, ∆z, t0−z0),(75)
which is a closed expression for the elapsed coordinate time ∆tinterval for a light ray traveling a certain
spatial distance, in terms of the spatial coordinate separations and the initial value of t−z.
Now let’s 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)),(76)
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 final value t0−z0+ ∆t−∆zcorresponding to the forward trip.
Plugging in (75) yields
∆ttot = 2∆`+ε∆x2−∆y2
2∆`¯
f+,tot +ε∆x∆y
∆`¯
f×,tot,(77)
where ¯
f+,tot =¯
f+,forward +¯
f+,backward and similarly for the ×-polarization, in terms of the forward and
backward averaged amplitudes, respectively, given by
¯
f+,×,forward =¯
f+×,(∆`, ∆z, t0−z0)(78)
=√2
∆`−∆zZ(∆`−∆z)/√2
0
f+,×1
√2(t0−z0) + σdσ, (79)
¯
f+,×,backward =¯
f+,×(∆`, −∆z, t0−z0+ ∆t−∆z)(80)
=√2
∆`+ ∆zZ(∆`+∆z)/√2
0
f+,×1
√2(t0+ ∆`−z0−∆z) + σdσ, (81)

26
where in the last expression we have replaced ∆tby ∆`in 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 (77) gives the total coordinate time elapsed during the trip forward and back. Recall that the
radar distance is defined as R= ∆τ/2in terms of the proper time measured by the stationary observer local
to the emission and reception of the light ray. For such a stationary observer x=y=z=const, so in fact
the proper time coincides with coordinate time. The radar distance is therefore given by R= ∆ttot/2, that is
R= ∆`+ε∆x2−∆y2
4∆`¯
f+,tot +ε∆x∆y
2∆`¯
f×,tot +O(ε2).(82)
This agrees with the result obtained in [23].
C. Null geodesics and radar distance - Finsler case
Now we consider the full Randers metric (48). The 1-form appearing in the metric is β=λdu, with
0< λ 1. The result (82) can be regarded as the Radar distance in the special case that λ= 0. Our
aim in this section is to find the corresponding expression for non-zero values of λ. As argued at the end of
Section III D, in addition to linearizing in ε, we also use a perturbative expansion in λ. In fact, instead of
working to first order in λ, we will work to second order in the Finslerian parameter, as certain important Fins-
lerian effects only enter at second order, as we will see. We also neglect terms of combined order ελ2and higher.
Recall that the geodesic structure of the Randers metric is equivalent to that of the GR metric characterized
by λ= 0. Hence the expressions (59)-(62) for the explicit form of the geodesics still apply in the current
scenario.
The first place where the Finsler character enters is due to the modified null condition F= 0, that one may
equivalently think of as a modified dispersion relation (MDR) for massless particles. According to Section
IV B, null curves now satisfy A=−β2, i.e. −2 ˙u˙v+ (1 + εf (u)) ˙x2+ (1 −εf (u)) ˙y2+ 2εf×˙x˙y=−β2=−λ2˙u2,
which, after substitution of the form of our geodesics above, becomes 2˜pu0+p2
x+p2
y=−λ2. Here we can
make two important observations:
1. The effect due to the MDR or modified null condition enters at order λ2;
2. In the limit λ→0we recover the standard null condition used in the previous section.
As before, using the null condition, we may eliminate ˜pu0and directly substitute this into the expression (62)
for v(σ). It follows that a null geodesic starting at (u0, x0, y0, v0)at σ= 0 can be described by the following
parameterized path:
u=u0+σ, (83)
x=x0+σ1−ε¯
f+(σ)px−ε¯
f×(u)py,(84)
y=y0+σ1 + ε¯
f+(σ)py−ε¯
f×(u)px,(85)
v=v0+σ
2(p2
x+p2
y+λ2)−1
2ε(p2
x−p2
y)σ¯
f+(σ)−εσ ¯
f×(σ)pxpy(86)

27
We can now use exactly the same methods as we did in the previous section for the GR wave to find the
coordinate time interval between the emission and reception of a light ray between two points. From the
middle two equations it follows again that
px=∆x
∆u1 + ε¯
f+(∆u)+ε¯
f×(∆u)∆y
∆u, py=∆y
∆u1−ε¯
f+(∆u)+ε¯
f×(∆u)∆x
∆u,(87)
and substituting this into the vequation yields
2∆u∆v= ∆x2(1 + ε¯
f(∆u)) + ∆y2(1 −ε¯
f(∆u)) + 2ε¯
f×∆x∆y+λ2∆u2,(88)
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, (89)
where we have used that −2∆u∆v=−∆t2+∆z2. This equation is solved to first order in εand λ2(neglecting
ελ2terms) by19
∆t= ∆`+∆x2−∆y2
2∆`ε¯
f+(∆u) + ∆x∆y
∆`ε¯
f×(∆u) + 1
2∆x2+ ∆y2+ 2∆z2
2∆`−∆zλ2,(90)
where ∆`≡p∆x2+ ∆y2+ ∆z2. Again, as in the GR case, we may replace ¯
fby its zeroth order equivalent,
which is still given by Eq. (74). To see why this is still the case, observe that ¯
fcan be expressed as a zeroth
order term plus an εcorrection, plus a λ2correction. Since ¯
fonly appears in ∆tas the product ε¯
f, the two
correction terms in ¯
fresult in a O(ε2)term and a O(ελ2)term, respectively, both of which we may neglect
in our current perturbative expansion. It follows that, to first order in εand λ2we have the closed expression
∆t= ∆`+∆x2−∆y2
2∆`ε¯
f+(∆`, ∆z, t0−z0) + ∆x∆y
∆`ε¯
f×(∆`, ∆z, t0−z0)(91)
+1
2∆x2+ ∆y2+ 2∆z2
2∆`−∆zλ2,(92)
where the expressions for ¯
f+,×(∆`, ∆z, t0−z0)are identical to their GR counterparts, Eq. (74). This is the
coordinate time interval needed for a light ray to travel a spatial distance (∆x, ∆y, ∆z).
The total coordinate time elapsed during the complete trip from xµ
0to xµ
0+ ∆xµand ‘back’ is the sum of
the forward trip and the backward trip time intervals. Just as in the GR case we can write this schematically
as
∆ttot = ∆t(∆x, ∆y, ∆z, t0−z0)+∆t(−∆x, −∆y, −∆z, t0+ ∆t−(z0+ ∆z)),(93)
Plugging in (92) yields
∆ttot = 2∆`+ε∆x2−∆y2
2∆`¯
f+,tot +ε∆x∆y
∆`¯
f×,tot
+1
2λ2∆x2+ ∆y2+ 2∆z2
∆`,(94)
19 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.

28
where the expressions for ¯
f+,tot =¯
f+,forward +¯
f+,backward and ¯
f×,tot =¯
f×,forward +¯
f×,backward are again
identical to their GR counterparts, Eqs. (79),(81).
The last step in the computation of the radar distance R= ∆τ/2is to convert the coordinate time interval
to a proper time interval. This is where a second Finslerian effect enters. We consider again a ‘stationary’
observer located at the point where the light ray is emitted and later received. Such an observer has a 4-
velocity 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 of its worldline ∆τ=−RFdσ. If we use σ=τas
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 parameterized 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.(95)
It follows that
∆τ=1−λ
√2∆ttot (96)
along the worldine of the stationary observer. Plugging in Eq. (94) we conclude that, to first order in and
second order in λ, the radar distance is given by
R=1−λ
√2∆`+ε1−λ
√2∆x2−∆y2
4∆`¯
f+,tot +1−λ
√2∆x∆y
2∆`ε¯
f×,tot +λ2
4∆`+∆z2
∆`.
(97)
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). Before we move on, let us summarize in what ways the
Finslerian parameter λhas entered in our derivation so far:
1. The null trajectories are altered due to the fact the Finsler metric induces a modified null condition or
MDR. As a result, it takes a larger coordinate time interval for a light ray to travel a given spatial coor-
dinate 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 don’t. 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.

29
Employing Eq. (97) we write
∆X=1−λ
√2∆x+λ2
4∆x, (98)
∆Y=1−λ
√2∆y+λ2
4∆y, (99)
∆Z=1−λ
√2∆z+λ2
2∆z, (100)
for the radar distance in the x, y and zdirection in the absence of the wave, and
R0=1−λ
√2∆`+λ2
4∆`+∆z2
∆`,(101)
for the radar distance (97) in the relevant direction in the absence of the wave. Eliminating the coordinate
distances in favour of the physical radar distances by virtue of the inverse transformations, valid to second
order in λ,
∆x= ∆X1 + λ
√2+λ2
4(102)
∆y= ∆Y1 + λ
√2+λ2
4(103)
∆z= ∆Z1 + λ
√2(104)
∆`=R01 + λ
√2+3
4λ2−∆z2
4R0
λ2(105)
=R01 + λ
√2+λ2
4−∆Z2
4R0
λ2(106)
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(2, λ3, λ2).(107)
This is a remarkable result. By expressing the radar distance in terms of the physical observables ∆X, ∆Y
and R0rather than merely coordinates, all dependence on λhas disappeared to the desired order and the
expression is identical to its GR counterpart, Eq. (82)! We must conclude, therefore, that the effect of a
Randers gravitational wave on interferometer experiments is virtually indistinguishable from that of a con-
ventional 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.

30
VI. DISCUSSION
The main aim of this paper was to study the physical effect of Finslerian gravitational waves and, in partic-
ular, 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 indicates 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 first time an explicit expression for the
Finslerian Radar length has been obtained for the case of finite 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.
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 field 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 defining covariantly 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 modified
Randers type.
Indeed, along the way we have introduced a small modification to the standard definition 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 modified 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 metric is different from the defining (pseudo-)Riemannian metric α. In the special case
that the defining 1-form of the Randers metric is covariantly constant (which is the case, for example, for our
solutions) we have even more satifactory results. In this case not only the causal structure, but also the affine
structure of the Randers 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. A particularly nice consequence
of this is the guaranteed 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 restricted our analysis, is still relatively quite restrictive in the large scheme of
(Finsler geometric) things. 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

31
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 spacetime 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 sufficienty small such that a perturbative approach to first
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.
ACKNOWLEDGMENTS
S.H. wants to thank Rick Sengers and Nicky van den Berg for fruitful discussions and in their input with
regards to the figures. S.H. also wants to thank Luc Florack for fruitful discussions, in particular his suggestions
with regards to perturbation theory. We would like to acknowledge 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
1. Proof of Lorentzian signature
Here we prove that if aµν has Lorentzian signature and aµν bµbν>−1then ˜aµν =aµν +bµbνalso has
Lorentzian signature. We write b2=aµνbµbν. First, the matrix determinant lemma says that
det ˜a= (1 + b2) det a. (A1)
As long as b2>−1this implies that det ˜ahas the same sign as det a, so assuming aµν is Lorentzian, ˜a
has negative determinant. In 4D this immediately implies that ˜ais Lorentzian (although the signs of the
eigenvalues might be flipped with respect to aµν ). However, let’s 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)2det a=1 + η2b2det 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 k-th 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>−1then ˜ais Lorentzian as well.

32
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 + b2bρbλbµ∇[λbν]+bν∇[λbµ].(A3)
where ∇is the covariant derivative corresponding to aµν
We prove this below, but first we point out the following immediate consequence.
Corollary 9. If bµis covariantly constant with respect to aµν, the affine structure of ˜aµν is the same as the
affine structure of aµν, i.e. e
Γρ
µν = Γρ
µν .
Proof. As long as b2>−1the 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 specified (as in the case of e
Γbelow!) indices are raises and lowered with aµν. Denoting
Γλµν =aλρΓρ
µν and e
Γλµν = ˜aλρ e
Γρ
µν we first 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 follows that
e
Γρ
µν = ˜aρλ e
Γλµν =aρλ −1
1 + b2bρbλΓλµν +bλ∂(µbν)−bµ∂[λbν]−bν∂[λbµ](A6)
= Γρ
µν −1
1 + b2bρbλΓλ
µν +aρλ −1
1 + b2bρbλbλ∂(µbν)−aρλ −1
1 + b2bρbλbµ∂[λbν]+bν∂[λbµ].
(A7)
The second and third term add up to
−1
1 + b2bρbλΓλ
µν +aρλ −1
1 + b2bρbλbλ∂(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)

33
This shows that
e
Γρ
µν = Γρ
µν +1
1 + b2bρ∇(µbν)−aρλ −1
1 + b2bρbλbµ∂[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.
Appendix B: Determinant of a not necessarily positive definite (α, β)-metric
Here we derive the formula Eq. (45) for the determinant of a not necessarily positive definite (α, β )-metric,
generalizing the well-known result from the positive definite case. More precisely, we consider Finsler metrics
the form F=αφ(s), where s=β/α,α=p|A|=p|aij yiyj|,A=aij yiyj=sgn(A)α2, and where aij is
assumed to be a (pseudo-)Riemannian metric, i.e. not necessarily Riemannian/positive definite.
In complete analogy with the positive definite case, it can be shown by direct calculation that the funda-
mental tensor gij ≡1
2¯
∂i¯
∂jF2is given by
gij =sgn(A)ρaij +ρ0bibj+ρ1(biαj+αibj) + ρ2αiαj,(B1)
where we have defined αi=aij yj/α, and with coefficients given by
ρ=φ(φ−sφ0),(B2)
ρ0=φφ00 +φ0φ0,(B3)
ρ1=−(sρ0−φφ0) = −[s(φφ00 +φ0φ0)−φφ0],(B4)
ρ2=−sρ1=s[s(φφ00 +φ0φ0)−φφ0].(B5)
The only difference here with the positive definite case is the factor sign(A)appearing in the first term in Eq.
(B1). Denoting dim M=ncan write this in matrix notation as
g=sgn(A)ρa+U W V T,(B6)
in terms of the three matrices
W=sgn(A)
ρ
1
4×4, U = (~
b,~
b, ~α, ~α), V = (ρ0~
b, ρ1~α, ρ1~
b, ρ2~α).(B7)
Uand Vare both n×4matrices. It is a well-known result (one of the matrix determinant lemmas) that
assuming ais an invertible matrix the determinant of the expression in brackets is equal to
det a+U W V T= det 
1
4×4+W V Ta−1Udet a. (B8)
It follows that
det a=sgn(A)nρndet 
1
4×4+W V Ta−1Udet a. (B9)

34
The matrix product W V Ta−1U=sgn(A)
ρVTa−1Ucan be evaluated by explicit computation and reads
W V Ta−1U=sgn(A)
ρ


b2ρ0b2ρ0sgn(A)sρ0sgn(A)sρ0
sgn(A)sρ sgn(A)sρ sgn(A)ρsgn(A)ρ
b2ρ1b2ρ1sgn(A)sρ1sgn(A)sρ1
sgn(A)sρ2sgn(A)sρ2sgn(A)ρ2sgn(A)ρ2


.(B10)
Hence we obtain
det a=sgn(A)nρndet 


1
4×4+sgn(A)
ρ


b2ρ0b2ρ0sgn(A)sρ0sgn(A)sρ0
sgn(A)sρ sgn(A)sρ sgn(A)ρsgn(A)ρ
b2ρ1b2ρ1sgn(A)sρ1sgn(A)sρ1
sgn(A)sρ2sgn(A)sρ2sgn(A)ρ2sgn(A)ρ2




det a(B11)
=sgn(A)nρndet 


1
4×4+1
ρ


sgn(A)b2ρ0sgn(A)b2ρ0sρ0sρ0
sρ sρ ρ ρ
sgn(A)b2ρ1sgn(A)b2ρ1sρ1sρ1
sρ2sρ2ρ2ρ2




det a(B12)
=φn+1(φ−sφ0)n−2(φ−sφ0+ (sgn(A)b2−s2)φ00) 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
det aij =φn+1(φ−sφ0)n−2(φ−sφ0+ (sgn(A)b2−s2)φ00) det aij .(B14)
In the case that αis positive definite, sign(A)=1everywhere, so the formula reduces to the standard result
(see e.g. [60]).
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