# Finsler pp-Waves and the Penrose Limit

Amir Babak Aazami, Miguel Ángel Javaloyes, Marcus C. Werner  Amir Babak Aazami
Department of Mathematics Clark University Worcester, MA 01610, USA
<EMAIL_ADDRESS>Miguel Ángel Javaloyes Department of Mathematics University
of Murcia 30100 Murcia, Spain<EMAIL_ADDRESS>Marcus C. Werner Zu Chongzhi
Center for Mathematics and Computational Sciences Duke Kunshan University
Kunshan, Jiangsu 215316, China<EMAIL_ADDRESS>
###### Abstract.

The Penrose plane wave limit is a remarkable property of Lorentzian
spacetimes. Here, we discuss its extension to Finsler spacetimes by
introducing suitable lightlike coordinates and adapting the Lorentzian
definition of pp-waves. New examples of such Finsler pp-waves are also
presented.

##  1\. Introduction

Among R. Penrose’s many accomplishments in general relativity are two
foundational results on _plane waves_ , a distinguished class of spacetimes
modeling radiation propagating at the speed of light. These metrics have a
long and rich history within the field of general relativity; they sit inside
the more general family of _pp-wave_ spacetimes introduced by J. Ehlers and W.
Kundt [EK62], which themselves comprise an important subclass of the family of
so called _Brinkmann spacetimes_ due to H. W. Brinkmann [Bri25], which are
spacetimes containing a parallel (covariantly constant) lightlike vector
field. A very comprehensive recent survey of plane waves can be found in
[SHN+17]; pp-waves are now also studied purely in a mathematical context (see,
e.g., [FS06, GL16, LS16, FS20]), in addition to their continued usage in
gravitational physics (see, e.g., [BFOP02, Bla11]). Of their many properties
both physical and mathematical, two of the most noteworthy were discovered by
Penrose himself. The first of these, [Pen65], is that plane waves are never
globally hyperbolic: this is “remarkable”, to borrow Penrose’s own
description, all the more so since plane waves are known to be geodesically
complete. The second result, [Pen76], no less remarkable, is that every
spacetime has a plane wave in a certain well defined limit, a local
construction now known as Penrose’s “plane wave limit,” perhaps the most
important realization of a more general notion of “spacetime limit” due to R.
Geroch [Ger69].
Given the distinguished position that plane waves occupy — and especially in
light of Penrose’s result that every Lorentzian metric admits one as a limit —
it is worthwhile to ask whether there exist analogues of them in _other_
geometries of indefinite signature, and, if so, whether Penrose’s plane wave
limit carries over to such settings as well. A natural direction in which to
take this question is that of Finsler geometry of Lorentzian signature, the
setting of so called _Finsler spacetimes_ , especially in light of their many
recent physical applications; see, e.g., [BJS20, HPV20, JS20, KRT12, EK18,
PW12, LP18] and the references therein. Indeed, there are already examples of
“Finsler pp-waves” in the literature (see, e.g., [FP16]). In pursuing our
question, we are less motivated by the connection between pp-waves,
gravitational radiation, and Einstein’s equation — indeed, there is no agreed
upon analogue of the latter in Finsler geometry — and more by the fact that,
via Penrose’s limit, plane waves play a central role in Lorentzian geometry
per se. What role may they play in Finsler geometry?
In this paper we attempt an answer to this question by first introducing a
general definition of Finsler pp-wave, a definition that subsumes the isolated
examples of Finsler pp-waves already in the literature; indeed, in Section 6
below we introduce additional examples of Finsler pp-waves in accord with our
definition. Second, we show that there does exist a notion of “plane wave
limit” in the Finslerian setting. These two facts, we argue, make them worthy
of study as Finslerian objects. Throughout our paper, at every step of our
construction, we carefully present the modifications required to pass from
Lorentzian to Finslerian geometry: this includes presenting the Finsler
analogue of lightlike coordinates (see, e.g., [Pen72]), the Finsler analogue
of the invariant definition of pp-wave in terms of the Riemann curvature
tensor (see, e.g., [GL16]), and, finally, the Finsler analogue of the
construction of Penrose’s plane wave limit itself.

##  2\. Preliminaries

Let $M$ be a (connected) manifold, $TM$ its tangent bundle, and $\pi\colon
TM\rightarrow M$ the natural projection. Let us consider a connected open
subset $A\subset TM\setminus\bf 0$ which is conic, namely, satisfying the
property that $\lambda v\in A$ for all $v\in A$ and $\lambda>0$. Further
assume that $A$ has smooth boundary, that each $A_{p}=A\cap T_{p}M$ is
nonempty for all $p\in M$, and denote $\bar{A}$ its closure in
$TM\setminus\mathbf{0}$. Given a function $L\colon\bar{A}\subset
TM\rightarrow[0,\infty)$, we will say that $(M,L)$ is a Finsler spacetime if
$L$ is a smooth function which is positive homogeneous of degree $2$ when
restricted to each
$\bar{A}_{p}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=T_{p}M\cap\bar{A}$,
its fundamental tensor

$g_{v}(u,w)=\frac{1}{2}\left.\frac{\partial^{2}}{\partial s\partial
t}L(v+tu+sw)\right|_{t=s=0}$ (1)

for $v\in\bar{A}$ and $u,w\in T_{\pi(v)}M$ has signature $(+-\dots-)$ and the
boundary of $\bar{A}$ in $TM\setminus\\{\mathbf{0}\\}$ coincides with
$\mathcal{C}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=L^{-1}(0)$,
which is called the lightcone of $L$. Observe that in this case, $\bar{A}_{p}$
is convex and salient for every $p\in M$ and the indicatrix $\Sigma=L^{-1}(1)$
is a strongly convex hypersurface when restricted to each tangent space,
namely, each $\Sigma_{p}=\Sigma\cap T_{p}M$ is strongly convex.

Observe that there are quite a few subtleties in the definition of a Finsler
spacetime, but we will adopt the one firstly considered in [JS14]. Amongst the
subtetlies to bear in mind we can point out that

  * •

in some definitions, beginning with that of Beem [Bee70], $L$ is defined in
the whole $TM$. Observe that as explained in [BJS20], for an observer
$v\in\Sigma$ one can consider $g_{v}$ as a positive definite metric in its
restspace, without any need of considering $L$ defined away from the causal
cone.

  * •

In others, some non-smooth directions are allowed [LP18, AJ16, CS18, Min17] or
they must be smooth up to some power [PW12] (relevant in the lightlike
directions),

  * •

there are others that do not consider the lightlike directions as a part of
the model [Asa85],

  * •

there are some models which appear in other contexts with slightly different
properties [KRT12, EK18].

Associated with the Lorentz-Finsler metric $L$, there is another anisotropic
tensor, usually called the Cartan tensor, defined as

$C_{v}(u,w,z)=\frac{1}{4}\left.\frac{\partial^{3}}{\partial r\partial
s\partial t}L(v+tu+sw+rz)\right|_{t=s=r=0},$ (2)

for any $v\in\bar{A}$ and $u,w,z\in T_{\pi(v)}M$. This symmetric anisotropic
tensor is what makes different Finsler spacetimes from the classical
Lorentzian geometry. It is straightforward to check that

$C_{v}(v,u,w)=C_{v}(u,v,w)=C_{v}(u,w,v)=0,$ (3)

for any $v\in\bar{A}$ and $u,w\in T_{\pi(v)}M$. Moreover, the Levi-
Civita–Chern anisotropic connection is a very useful tool for the study of
Finsler spacetimes (see [Jav19, Jav20, JSVn21] for more details about
anisotropic connections and calculus). Recall that an anistropic connection
can be thought as a connection which depends on directions. This means that
for every $v\in\bar{A}$ and $X,Y\mathfrak{X}(M)$, one obtains a different
value $\nabla^{v}_{X}Y\in T_{\pi(v)}M$. In particular, given a chart
$(\mathscr{U},\varphi)$, the Christoffel symbols are functions
$\Gamma^{k}_{ij}\colon\bar{A}\cap T\mathscr{U}\rightarrow\mathbb{R}$ which are
homogeneous of degree zero and
$\nabla^{v}_{\partial_{i}}\partial_{j}=\Gamma^{k}_{ij}(v)\partial_{k}$, where
$\partial_{1},\ldots,\partial_{n}$ are the partial vector fields of the chart.
If we fix a vector field $V\in{\mathfrak{X}}(\mathscr{U})$ which is _$\bar{A}$
-admissible_, that is to say, taking values in $\bar{A}$, we obtain then an
affine connection $\nabla^{\scalebox{0.5}{\emph{V}}}$ in $\mathscr{U}$ with
Christoffel symbols $\Gamma^{i}_{jk}\circ V$. The Levi-Civita–Chern connection
can be characterized in terms of the associated affine connections
$\nabla^{\scalebox{0.5}{\emph{V}}}$. Indeed, it is the only one such for which

  1. 1.

$\nabla^{\scalebox{0.5}{\emph{V}}}$ is torsion-free, namely,
$\nabla^{\scalebox{0.5}{\emph{V}}}_{X}Y-\nabla^{\scalebox{0.5}{\emph{V}}}_{Y}X=[X,Y]$
for all $X,Y\in{\mathfrak{X}}(\mathscr{U})$,

  2. 2.

$\nabla^{\scalebox{0.5}{\emph{V}}}$ is almost $g$-compatible, namely

$X(g_{\scalebox{0.5}{\emph{V}}}(Y,Z))=g_{\scalebox{0.5}{\emph{V}}}(\nabla^{\scalebox{0.5}{\emph{V}}}_{X}Y,Z)+g_{\scalebox{0.5}{\emph{V}}}(Y,\nabla^{\scalebox{0.5}{\emph{V}}}_{X}Z)+2C_{\scalebox{0.5}{\emph{V}}}(\nabla^{\scalebox{0.5}{\emph{V}}}_{X}V,Y,Z),$

where $X,Y,Z\in{\mathfrak{X}}(\mathscr{U})$ and $g_{\scalebox{0.5}{\emph{V}}}$
and $C_{\scalebox{0.5}{\emph{V}}}$ are the classical tensors obtained when (1)
and (2) are evaluated in the vector field $V$.

Observe that almost $g$-compatibility is equivalent to $\nabla g=0$ when this
tensor derivative is computed using the anisotropic calculus developed in
[Jav19, Jav20]. Moreover, there is also a Koszul formula that determines
$\nabla^{\scalebox{0.5}{\emph{V}}}$:

$2g_{\scalebox{0.5}{\emph{V}}}(\nabla^{\scalebox{0.5}{\emph{V}}}_{X}Y,Z)=X(g_{\scalebox{0.5}{\emph{V}}}(Y,Z))-Z(g_{\scalebox{0.5}{\emph{V}}}(X,Y))+Y(g_{\scalebox{0.5}{\emph{V}}}(Z,X)\\\
+g_{\scalebox{0.5}{\emph{V}}}([X,Y],Z)+g_{\scalebox{0.5}{\emph{V}}}([Z,X],Y)-g_{\scalebox{0.5}{\emph{V}}}([Y,Z],X)\\\
2\big{(}\\!-C_{\scalebox{0.5}{\emph{V}}}(\nabla^{\scalebox{0.5}{\emph{V}}}_{X}V,Y,Z)-C_{\scalebox{0.5}{\emph{V}}}(\nabla^{\scalebox{0.5}{\emph{V}}}_{Y}V,Z,X)+C_{\scalebox{0.5}{\emph{V}}}(\nabla^{\scalebox{0.5}{\emph{V}}}_{Z}V,X,Y)\big{)}.$
(4)

Observe that when $V$ is parallel, namely,
$\nabla^{\scalebox{0.5}{\emph{V}}}_{X}V=0$ for all
$X\in\mathfrak{X}(\mathscr{U})$, then the above Koszul formula coincides with
the Koszul formula for $g_{\scalebox{0.5}{\emph{V}}}$. This means that when
$V$ is parallel, the Levi-Civita–Chern connection
$\nabla^{\scalebox{0.5}{\emph{V}}}$ of $L$ coincides with the Levi-Civita
connection of $g_{\scalebox{0.5}{\emph{V}}}$. Even if it is not always
possible to choose a parallel extension of any vector $v\in\bar{A}$, one can
find a pointwise parallel vector field, namely, if $p=\pi(v)$, there exists an
extension $V\in\mathfrak{X}(\mathscr{U})$ for a certain neighborhood
$\mathscr{U}\subset M$ of $p$ such that
$(\nabla^{\scalebox{0.5}{\emph{V}}}_{X}V)_{p}=0$ for all
$X\in\mathfrak{X}(\mathscr{U})$ (see [Jav20, Prop. 2.13]). Indeed, with these
extensions we can compute the Chern curvature tensor of $L$ very easily, If
$R_{v}(X,Y)Z$ is the value of the Chern curvature tensor (which is an
anisotropic tensor) for $v\in\bar{A}$ and $X,Y,Z\in\mathfrak{X}(M)$, let us
choose a pointwise parallel extension $V$ of $v$, then

$R_{v}(X,Y)Z=R^{\scalebox{0.5}{\emph{V}}}_{p}(X,Y)Z,$

where $R^{\scalebox{0.5}{\emph{V}}}$ is the curvature tensor of
$\nabla^{\scalebox{0.5}{\emph{V}}}$. This is because in the general expression
of the Chern curvature tensor computed using
$\nabla^{\scalebox{0.5}{\emph{V}}}$, apart from $R^{\scalebox{0.5}{\emph{V}}}$
there are some additional tensorial terms evaluated in
$(\nabla^{\scalebox{0.5}{\emph{V}}}_{X}V)_{p}$ (see [Jav20, Prop. 2.5]). In
particular, when $V$ is parallel, the curvature tensor of the Levi-Civita
connection of $g_{\scalebox{0.5}{\emph{V}}}$ coincides with the Chern
curvature tensor $R_{\scalebox{0.5}{\emph{V}}}$ evaluated at $V$.

A function $f\colon M\rightarrow\mathbb{R}$ admits a gradient with respect to
$L$, denoted $\nabla\\!f$, if there exists a vector field metrically
equivalent to $df$, namely, which satisfies

$g_{\scalebox{0.5}{\emph{$\nabla\\!f$}}}(\nabla\\!f,X)=df(X)$

for all $X\in{\mathfrak{X}}(M)$. Moreover, in this case the Hessian of $f$ is
defined as the anisotropic tensor $H^{f}(X,Y)=\nabla_{\\!X}{df}(Y)$, where
$\nabla$ is the Levi-Civita–Chern connection of $L$. Observe that there is a
dependence on $v\in A$ in the sense that

$H^{f}_{v}(X,Y)=(\nabla_{\\!X}{df})_{v}(Y)=X(df(Y))-df(\nabla^{\scalebox{0.5}{\emph{V}}}_{X}Y).$

Let us observe that $H^{f}$ is symmetric as

$H^{f}_{v}(X,Y)=X(df(Y))-df(\nabla^{\scalebox{0.5}{\emph{V}}}_{X}Y)=X(Y(f))(\pi(v))-\nabla^{\scalebox{0.5}{\emph{V}}}_{X}Y(f),$

and
$\nabla^{\scalebox{0.5}{\emph{V}}}_{X}Y-\nabla^{\scalebox{0.5}{\emph{V}}}_{Y}X=[X,Y]$
(the Levi-Civita–Chern connection is torsion-free), and

$H^{f}_{{\scalebox{0.5}{\emph{$\nabla\\!f$}}}}(X,Y)=g_{\scalebox{0.5}{\emph{$\nabla\\!f$}}}(\nabla^{\nabla\\!f}_{\\!X}(\nabla\\!f),Y).$
(5)

This follows from the above formula, because using the almost-compatibility of
$\nabla$ with $g$,

$\displaystyle
g_{\scalebox{0.5}{\emph{$\nabla\\!f$}}}(\nabla^{\nabla\\!f}_{X}(\nabla\\!f),Y)=$
$\displaystyle
X(g_{\scalebox{0.5}{\emph{$\nabla\\!f$}}}(\nabla\\!f,Y))-g_{\scalebox{0.5}{\emph{$\nabla\\!f$}}}(\nabla\\!f,\nabla^{\nabla\\!f}_{X}Y)$
$\displaystyle=$ $\displaystyle
X(Y(f))-\nabla^{\nabla\\!f}_{X}Y(f)=H^{f}_{\nabla\\!f}(X,Y),$

(the Cartan term vanishes because homogeneity (3)).

######  Lemma 1.

A smooth function $f\colon M\rightarrow\mathbb{R}$ admits a gradient with
respect to a Finsler spacetime $(M,L)$ if and only if $df|_{A}>0$. Moreover,
in this case the gradient is unique.

###### Proof.

For the implication to the right, observe that if $\nabla\\!f$ is lightlike at
$p\in M$, this means that the kernel of $df_{p}$ is tangent to
$\mathcal{C}_{p}$. As $A_{p}$ is convex, it remains on one side of the
hyperplane $\ker(df_{p})$, and as $g_{\scalebox{0.5}{\emph{$\nabla\\!f$}}}$ is
a Lorentzian-type metric (with index $n-1$) it follows that $df_{p}|_{A}>0$,
because the lightlike cone of $g_{\scalebox{0.5}{\emph{$\nabla\\!f$}}}$
remains on the same side of $\ker(df_{p})$. If $\nabla\\!f$ is timelike at
$p\in M$, then $\ker(df_{p})$ does not touch $\bar{A}_{p}$ and as
$df_{p}(\nabla\\!f)=g_{\scalebox{0.5}{\emph{$\nabla\\!f$}}}(\nabla\\!f,\nabla\\!f)=L(\nabla\\!f)>0$,
it follows that $df_{p}|_{A}>0$.

For the implication to the left, we know that $\ker(df)\cap A=\emptyset$. If
$\ker(df_{p})$ is tangent to $\mathcal{C}_{p}$, then there exists
$v\in\mathcal{C}_{p}\cap\ker(df)$. Then it is easy to see that

$df_{p}=\lambda g_{v}(v,\cdot)=g_{\lambda v}(\lambda v,\cdot)$ (6)

and therefore $\nabla\\!f=\lambda v$. If $\ker(df_{p})$ is not tangent to
$\mathcal{C}_{p}$, then there exists $v\in\Sigma_{p}$ where the minimum
distance between $\ker(df_{p})$ and $\Sigma_{p}$ is attained. Then (6) holds
for some $\lambda$. This also shows that the gradient is unique, because the
strict convexity of $\Sigma_{p}$ implies that the distance is attained in a
unique point. ∎

######  Lemma 2.

If a function $f\colon M\rightarrow\mathbb{R}$ has a gradient field with
constant $L$-norm, then its flow is given by geodesics.

###### Proof.

As the gradient field has constant $L$-norm, it follows that
$X(L(\nabla\\!f))=0$ and then

$0=2g_{\scalebox{0.5}{\emph{$\nabla\\!f$}}}(\nabla^{\nabla\\!f}_{X}\nabla\\!f,\nabla\\!f)=2H^{f}_{\scalebox{0.5}{\emph{$\nabla\\!f$}}}(X,\nabla\\!f)=2H^{f}_{\scalebox{0.5}{\emph{$\nabla\\!f$}}}(\nabla\\!f,X)=2g_{\scalebox{0.5}{\emph{$\nabla\\!f$}}}(\nabla^{\nabla\\!f}_{\nabla\\!f}\nabla\\!f,X),$

and as this holds for all $X\in{\mathfrak{X}}(M)$, it follows that
$\nabla^{\scalebox{0.5}{\emph{$\nabla\\!f$}}}_{\\!\nabla\\!f}\nabla\\!f=0$ and
$\nabla\\!f$ is geodesic. ∎

##  3\. Lightlike coordinates and their properties

Although lightlike coordinates exist in all dimensions, we present them here
in dimension 4, for convenience. In the following, when a chart
$(\mathscr{U},\varphi)$ is fixed, we will denote by $g_{ij}\colon\bar{A}\cap
T\mathscr{U}\rightarrow\mathbb{R}$ the coordinates of the fundamental tensor
$g$ in (1), namely, $g_{ij}(v)=g_{v}(\partial_{i},\partial_{j})$.

######  Lemma 3 (Lightlike coordinates).

Let $(M,L)$ be a Finsler spacetime and $f$ a smooth function defined on an
open subset $\mathscr{U}\subseteq M$. If
$N\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\nabla\\!f$ is a
lightlike vector field, then there exist coordinates about any point in
$\mathscr{U}$ in which the metric $g_{\scalebox{0.5}{N}}$ has components

$\displaystyle(g_{ij}(N))=\left(\begin{array}[]{cccc}0&1&0&0\\\
1&g_{11}(N)&g_{12}(N)&g_{13}(N)\\\ 0&g_{21}(N)&h_{22}&h_{23}\\\
0&g_{31}(N)&h_{32}&h_{33}\end{array}\right),$ (11)

with $(h_{ij})$ a positive-definite $2\times 2$ matrix.

###### Proof.

Of course, “lightlike” means that $N$ is nowhere vanishing yet satisfies
$L(N)=0$; the former implies that the level sets
$\mathcal{S}_{c}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=f^{-1}(c)$
are embedded hypersurfaces, while the latter implies that $N$ has geodesic
flow, $\nabla^{\scalebox{0.5}{\emph{N}}}_{N}{N}=0$, where
$\nabla^{\scalebox{0.5}{\emph{N}}}$ is the Levi-Civita–Chern connection of $L$
(recall Lemma 2). At any $p\in\mathcal{S}_{c}$,
$g_{\scalebox{0.5}{$N_{p}$}}(N_{p},N_{p})=0$ implies that the induced metric
by $g_{\scalebox{0.5}{$N_{p}$}}$ on $\mathcal{S}_{c}$ is degenerate. Let
$(x^{0},x^{1},x^{2},x^{3})$ denote coordinates within $\mathscr{U}$ in which
$N=\partial_{0}$. Then by the almost compatibility of $g$ and $\nabla$,

$N\big{(}g_{\scalebox{0.5}{\emph{N}}}(N,{\partial_{i}}\big{)}=g_{\scalebox{0.5}{\emph{N}}}{\underbrace{\nabla^{\scalebox{0.5}{\emph{N}}}_{\\!N}N}_{0}}{\partial_{i}}+g_{\scalebox{0.5}{\emph{N}}}(N,N){\underbrace{\nabla^{\scalebox{0.5}{\emph{N}}}_{\\!N}\partial_{i}}_{\nabla_{\\!\partial_{i}}{N}}\,}=0,$

so that each
$c_{i}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{0i}(N)$,
$i=0,\ldots,3$, is independent of $x^{0}$. Of course, at least one of these
$c_{i}$’s must be _nonzero_ , otherwise each $N_{p}^{\perp}$ would be four-
dimensional. Let us assume that $c_{1}\neq 0$. Thus at the moment our metric
$g_{\scalebox{0.5}{\emph{N}}}$ in the coordinates $(x^{0},x^{1},x^{2},x^{3})$
takes the form

$\displaystyle(g_{ij}(N))=\left(\begin{array}[]{cccc}0&c_{1}&c_{2}&c_{3}\\\
c_{1}&g_{11}(N)&g_{12}(N)&g_{13}(N)\\\ c_{2}&g_{21}(N)&g_{22}(N)&g_{23}(N)\\\
c_{3}&g_{31}(N)&g_{32}(N)&g_{33}(N)\end{array}\right)\hskip 14.45377pt,\hskip
14.45377ptc_{i}=c_{i}(x^{1},x^{2},x^{3}),$ (16)

with the function $f$ satisfying

$\displaystyle\frac{\partial f}{\partial
x^{0}}=g_{\scalebox{0.5}{\emph{N}}}(\nabla\\!f,\partial_{0})=0\hskip
14.45377pt,\hskip 14.45377pt\frac{\partial f}{\partial
x^{i}}=g_{\scalebox{0.5}{\emph{N}}}(\nabla\\!f,\partial_{i})=c_{i}.$ (17)

These coordinates, however, are not slice coordinates for the level sets
$\mathcal{S}_{c}$; to make them so, simply define new coordinates
$(\tilde{x}^{i})$ by

$\displaystyle\left\\{\begin{array}[]{ll}\tilde{x}^{0}=x^{0},\\\
\tilde{x}^{1}=f(x^{1},x^{2},x^{3}),\\\ \tilde{x}^{2}=x^{2},\\\
\tilde{x}^{3}=x^{3}.\end{array}\right.$

These new coordinates satisfy
$\nabla\tilde{x}^{1}=\partial/\partial\tilde{x}^{0}$, and they are indeed
slice coordinates for $\mathcal{S}_{c}$:

$\mathcal{S}_{c}=\left\\{q\in\mathscr{U}\,:\,\tilde{x}(q)=\left(\tilde{x}^{0}(q),c,\tilde{x}^{2}(q),\tilde{x}^{3}(q)\right)\right\\}\cdot$

Thus each
$N_{q}^{\perp}=T_{q}\mathcal{S}_{c}=\text{span}\,\\{\partial/\partial\tilde{x}^{0}|_{q},\partial/\partial\tilde{x}^{2}|_{q},\partial/\partial\tilde{x}^{3}|_{q}\\}$,
with

$\frac{\partial}{\partial\tilde{x}^{0}}=\frac{\partial}{\partial x^{0}}\hskip
14.45377pt,\hskip
14.45377pt\frac{\partial}{\partial\tilde{x}^{1}}\overset{\eqref{eqn:c}}{=}\frac{1}{c_{1}}\frac{\partial}{\partial
x^{1}}\hskip 14.45377pt,\hskip
14.45377pt\frac{\partial}{\partial\tilde{x}^{i}}\overset{\eqref{eqn:c}}{=}-\frac{c_{i}}{c_{1}}\,\frac{\partial}{\partial
x^{1}}+\frac{\partial}{\partial x^{i}},$

and $\partial/\partial\tilde{x}^{0}$ has a nice relationship with the other
coordinate basis vectors:

$g_{\scalebox{0.5}{\emph{N}}}(\partial/\partial\tilde{x}^{0},\partial/\partial\tilde{x}^{1})=1\hskip
14.45377pt,\hskip
14.45377ptg_{\scalebox{0.5}{\emph{N}}}(\partial/\partial\tilde{x}^{0},\partial/\partial\tilde{x}^{i})=-c_{i}+c_{i}=0.$

The metric in the new coordinates $(\tilde{x}^{i})$ is now precisely in the
form of (11). Finally, note that since $\partial/\partial\tilde{x}^{2}$ and
$\partial/\partial\tilde{x}^{3}$ are both orthogonal to the lightlike vector
$\partial/\partial\tilde{x}^{0}$, they must satisfy $g_{22}(N)>0,g_{33}(N)>0$.
It follows that each embedded 2-submanifold defined by

$\displaystyle\Lambda_{\scalebox{0.5}{\emph{b,c}}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left\\{q\in\mathscr{U}\,:\,\tilde{x}(q)=\left(b,c,\tilde{x}^{2}(q),\tilde{x}^{3}(q)\right)\right\\}\subseteq\mathcal{S}_{c},$
(19)

with induced metric
$h_{\scalebox{0.5}{\emph{b,c}}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{i=2}^{3}g_{ij}(b,c,\tilde{x^{2}},\tilde{x}^{3},N)d\tilde{x}^{i}\otimes
d\tilde{x}^{j}$, is Riemannian; indeed, since $g_{\scalebox{0.5}{\emph{N}}}$
has negative determinant,

$\displaystyle\text{det}\,g_{\scalebox{0.5}{\emph{N}}}=-(g_{22}g_{33}-g_{23}^{2})<0\hskip
14.45377pt\Rightarrow\hskip 14.45377ptg_{22}g_{33}-g_{23}^{2}>0.$ (20)

Together with the fact that $g_{22}>0$, it follows that the two leading
submatrices of the $2\times 2$ matrix $\begin{pmatrix}g_{22}&g_{23}\\\
g_{32}&g_{33}\end{pmatrix}$ have positive determinant; thus this submatrix is
positive definite. Renaming each $g_{ij}(N)$ to $h_{ij}$ for $i,j=2,3$, the
proof is complete. ∎

######  Lemma 4.

Given and arbitrary lightlike vector $v_{0}$ of a Finsler spacetime $(M,L)$,
it is possible to extend it to a lightlike gradient vector field in a certain
neighborhood of $\pi(v_{0})$.

###### Proof.

First, consider a local splitting $I\times B$ of $M$ with $I\subset\mathbb{R}$
in such a way that $\\{t_{0}\\}\times B$ is spacelike for all $t_{0}\in I$ and
$\partial_{t}$ is timelike. Then consider a surface $S_{0}$ in $B$
diffeomorphic to $S^{2}$ which contains $\pi(v_{0})$ and is orthogonal to
$v_{0}$, with $v_{0}$ pointing to the exterior region of $I\times S_{0}$.
Observe that the cylinder $I\times S_{0}$ is a hypersurface and that it admits
a smooth lightlike vector field $N$ along it with the following property: for
any $(t_{0},p_{0})\in I\times S_{0}$, $N_{(t_{0},p_{0})}$ is orthogonal to
$\\{t_{0}\\}\times S_{0}$. By the Inverse Function Theorem, the exponential
map restricted to the bundle generated by $N$ along $S_{0}$ is a local
diffeomorphism. In particular, one can construct local coordinates around
$\pi(v_{0})$ using product coordinates in $I\times S_{0}$ and then the one of
the exponential map $s\rightarrow\exp_{(t,p)}(sN)$. All this together implies
that the projection onto $I$ in these coordinates provides a function
$f\colon\mathscr{U}\subset M\rightarrow\mathbb{R}$ whose level sets
$f^{-1}(t_{0})$ are the hypersurfaces obtained as the union of all the
lightlike geodesics passing through the points $(t_{0},p)$ with $p\in S_{0}$
and with velocity $N_{(t_{0},p)}$. Reducing $\mathscr{U}$ if necessary, we can
assume that these hypersurfaces coincide with the horismos
$E^{+}(\\{t_{0}\\}\times S_{0})\cup E^{-}(\\{t_{0}\\}\times S_{0})$.
Therefore, they are degenerate and the gradient vector field $\nabla\\!f$ must
be lightlike. ∎

Observe that the last Lemma can be interpreted in the following way. The
lightlike gradient vector field can be thought of as the vector field tangent
to the light rays departing orthogonally from a given spacelike surface and
its temporal cylinder of a given interval of a universal time.

There is a direct relationship between the domain of validity of lightlike
coordinates and the existence of focal points along the geodesic integral
curves of $N$. First, define
$\Delta_{4}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sqrt{-\text{det}\,g_{ij}(N)}$
and

$\displaystyle\Delta\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left|\begin{array}[]{cc}h_{22}&h_{23}\\\
h_{32}&h_{33}\end{array}\right|^{\frac{1}{2}},$ (23)

and observe that $\Delta_{4}=\Delta$. Then the relationship is as follows:

######  Lemma 5.

In lightlike coordinates, a point $p\in\mathscr{U}$ is a focal point of (20)
along the geodesic integral curve of $N$ through $p$ if and only if
$\Delta|_{p}=0$.

###### Proof.

Let $(x^{0},x^{1},x^{2},x^{3})$ be lightlike coordinates as in the Lemma 3,
with $N=\partial_{0}$, and let $\Lambda$ be a Riemannian 2-submanifold as in
(20). We begin by observing that
$\partial_{0},\partial_{1},\partial_{2},\partial_{3}$ are all Jacobi fields
along any geodesic integral curve $\gamma(x^{0})$ of $N$ starting in
$\Lambda$. Indeed, setting
$J\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\partial_{i}$, and
noting that $[J,N]=0$ and $\nabla^{\scalebox{0.5}{\emph{N}}}_{N}N=0$, we have
that

$J^{\prime\prime}=\nabla^{\scalebox{0.5}{\emph{N}}}_{N}\nabla^{\scalebox{0.5}{\emph{N}}}_{N}J=\nabla^{\scalebox{0.5}{\emph{N}}}_{N}\nabla^{\scalebox{0.5}{\emph{N}}}_{J}N=R_{\scriptscriptstyle
N}(N,J)N=-R_{\scriptscriptstyle N}(J,N)N,$

where $R_{\scriptscriptstyle N}$ is the $(1,3)$-curvature tensor. Now suppose
that $\Delta|_{p}=0$ at a point $p$ along $\gamma$. Then
$\partial_{2}|_{p},\partial_{3}|_{p}$ must be linearly dependent, hence some
nontrivial linear combination of the two gives the zero vector at $p$. If we
extend this linear combination as is to a vector field $J(x^{0})$ along
$\gamma$, then $J$ will be a Jacobi field. In fact it is a $\Lambda$-Jacobi
field, since $J(0)\in T_{\gamma(0)}\Lambda$ and since
$\text{tan}_{\Lambda}J^{\prime}(0)=\text{tan}_{\Lambda}(\nabla^{\scalebox{0.5}{\emph{N}}}_{J(0)}N)$;
we thus conclude that $p$ is a focal point of $\Lambda$ along $\gamma$. Now
for the converse. Suppose that $p$ is a focal point of $\Lambda$ along the
geodesic integral curve $\gamma(x^{0})$ of $N$, with $\gamma(b)=p$; let
$J\colon[0,b]\longrightarrow\mathscr{U}$ denote the corresponding
$\Lambda$-Jacobi field, with $J(b)$ its (zero) value at $p$:

$J(b)=\sum_{i=0}^{3}f^{i}(b)\partial_{i}\big{|}_{p}=0.$

(In fact $f^{1}$ is identically zero, since $J$ is orthogonal to $\gamma$.) If
the $f^{i}(b)$’s are not all zero, then we are done, since we would thus have
a nontrivial linear combination of the $\partial_{i}|_{p}$’s equalling zero,
which can happen only if $\Delta_{4}|_{p}=\Delta|_{p}=0$. Thus, assume that
each $f^{i}(b)=0$, in which case

$J^{\prime}(b)=\dot{f}^{i}(b)\partial_{i}|_{p}+\cancelto{0}{f^{i}(b)\nabla_{N(b)}^{N(b)}\partial_{i}},$

where at least one of $\dot{f}^{i}(b)$ is nonzero, for otherwise
$J(b)=J^{\prime}(b)=0$ and so $J$ would have to be trivial. Next, because each
$\partial_{i}$ is a Jacobi field, and because any two $\Lambda$-Jacobi fields
$V,W$ along a geodesic satisfy
$g_{\scalebox{0.5}{\emph{N}}}(V^{\prime},W)=g_{\scalebox{0.5}{\emph{N}}}(V,W^{\prime})$
(see, e.g., [JS15, Prop. 3.18]), we have that

$\underbrace{\,g_{\scalebox{0.5}{\emph{N}}}(J^{\prime},\partial_{j})|_{p}\,}_{\dot{f}^{i}(b)g_{ij}(N)}=g_{\scalebox{0.5}{\emph{N}}}(J,\partial_{j}^{\prime})|_{p}=0,$

the latter because $J(b)=0$. Thus we’ve arrived at the system of equations

$g_{ji}(N_{p})\dot{f}^{i}(b)=0,$

with the $\dot{f}^{i}(b)$’s not all zero. This can only happen if
$\Delta_{4}|_{p}=0$, in which case $\Delta_{p}=0$ and the proof is complete. ∎

##  4\. Finsler pp-waves and Brinkmann coordinates

Before proceeding to Finsler pp-waves, we need to establish when our lightlike
gradient vector field is parallel. This is achieved by the following lemma.

######  Lemma 6.

A lightlike gradient vector field $N=\nabla\\!f$ is parallel if and only if
all components of the metric (11) are independent of $\tilde{x}^{0}$.

###### Proof.

Assume that $\nabla^{\scalebox{0.5}{\emph{N}}}_{X}N=0$ for all
$X\in\mathfrak{X}(M)$. Consider $\partial_{i},\partial_{j}$ with
$i,j=1,\ldots,n$ and observe that by (11), we know that
$g_{\scalebox{0.5}{\emph{N}}}(N,\partial_{i})=0$ if $2\leq i\leq n$ and
$g_{\scalebox{0.5}{\emph{N}}}(N,\partial_{1})=1$. By the Koszul formula (4),
taking into account that $N=\partial_{0}$ and all the Lie Brackets will
vanish,

$0=2g_{\scalebox{0.5}{\emph{N}}}(\nabla^{\scalebox{0.5}{\emph{N}}}_{\partial_{i}}N,\partial_{j})=\partial_{i}(g_{\scalebox{0.5}{\emph{N}}}(N,\partial_{j}))-\partial_{j}(g_{\scalebox{0.5}{\emph{N}}}(N,\partial_{i}))+N(g_{\scalebox{0.5}{\emph{N}}}(\partial_{i},\partial_{j})),$

which implies that
$N(g_{\scalebox{0.5}{\emph{N}}}(\partial_{i},\partial_{j}))=0$ for all
$i,j=1,\ldots,n$, namely, the coefficients $g_{ij}(N)$ do not depend on the
coordinate $x^{0}$.

Assume now that all the $g_{ij}(N)$ do not depend on $x^{0}$. Let us see first
that $\nabla^{\scalebox{0.5}{\emph{N}}}_{N}N=0$. Using the Koszul formula (4),

$2g_{\scalebox{0.5}{\emph{N}}}(\nabla^{\scalebox{0.5}{\emph{N}}}_{N}N,\partial_{j})=N(g_{\scalebox{0.5}{\emph{N}}}(N,\partial_{j}))-\partial_{j}(g_{\scalebox{0.5}{\emph{N}}}(N,N))+N(g_{\scalebox{0.5}{\emph{N}}}(N,\partial_{j}))=0$

for any $j=0,\ldots,n$, which implies that
$\nabla^{\scalebox{0.5}{\emph{N}}}_{N}N=0$. Using this identity and the Koszul
formula once more,

$2g_{\scalebox{0.5}{\emph{N}}}(\nabla^{\scalebox{0.5}{\emph{N}}}_{\partial_{i}}N,\partial_{j})=\partial_{i}(g_{\scalebox{0.5}{\emph{N}}}(N,\partial_{j}))-\partial_{j}(g_{\scalebox{0.5}{\emph{N}}}(N,\partial_{i}))+N(g_{\scalebox{0.5}{\emph{N}}}(\partial_{i},\partial_{j}))=0,$

for any $i,j=0,\ldots,n$, which implies that
$\nabla^{\scalebox{0.5}{\emph{N}}}_{\partial_{i}}N=0$. ∎

In the following, given an $\bar{A}$-admissible $N$ defined in some open
subset $\mathscr{U}\subset M$, we will use the notation $\Gamma(N^{\perp})$
for the space of vector fields $X\in\mathfrak{X}(\mathscr{U})$ such that
$g_{\scalebox{0.5}{\emph{N}}}(N,X)=0$.

######  Theorem 1 (pp-waves; [GL16]).

If $N$ is a parallel lightlike gradient vector field, then the curvature
endomorphism $R_{\scriptscriptstyle N}$ satisfies

$\displaystyle R_{\scriptscriptstyle N}(X,Y)Z=0,$ (24)

for all $X,Y,Z\in\Gamma(N^{\perp})$, if and only if there exist local
coordinates $(v,u,x,y)$ in which (11) takes the form

$\displaystyle(g_{ij}(N))=\left(\begin{array}[]{cccc}0&1&0&0\\\ 1&H&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{array}\right),$ (29)

where
$g_{uu}(N)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=H(u,x,y)$.
Such a metric is called a _Finsler pp-wave (_ expressed in so-called
_Brinkmann coordinates[Bri25])_.

###### Proof.

Consider each embedded 2-submanifold $\Lambda_{\scalebox{0.5}{\emph{b,c}}}$
given by (20), with its corresponding induced Riemannian metric

$h_{\scalebox{0.5}{\emph{b,c}}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{i,j=2}^{3}h_{ij}(c,\tilde{x}^{2},\tilde{x}^{3})d\tilde{x}^{i}\otimes
d\tilde{x}^{j}.$

(Since by Lemma 6 each $g_{ij}(N)$ is independent of $\tilde{x}^{0}$, the
components $h_{ij}$ are $\tilde{x}^{0}$-independent.) By the Gauss Equation
([Lee18, Theorem 8.5, p. 230]), we know that the (single) component
$\text{Rm}_{\scalebox{0.5}{\emph{c}}}(\partial_{2},\partial_{3},\partial_{2},\partial_{3})$
of the curvature tensor of $h_{\scalebox{0.5}{\emph{b,c}}}$ is related to that
of $g$ by

$\displaystyle\overbrace{\,\text{Rm}_{N}(\partial_{2},\partial_{3},\partial_{2},\partial_{3})\,}^{\text{0,
by \eqref{eqn:ppwave}}}$ $\displaystyle=$
$\displaystyle\text{Rm}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{2},\partial_{3},\partial_{2},\partial_{3})$
$\displaystyle-\,g_{\scriptscriptstyle
N}(\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{2},\partial_{3}),\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{3},\partial_{2}))+g_{\scriptscriptstyle
N}(\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{2},\partial_{2}),\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{3},\partial_{3})),$

where $\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}$ is the second fundamental
form of $h_{\scalebox{0.5}{\emph{b,c}}}$; observe that as $N$ is parallel, the
curvature tensor of $g_{\scalebox{0.5}{\emph{N}}}$ coincides with
$R_{\scriptscriptstyle N}$. But when $N=\partial_{0}$ is parallel, this
equation simplifies considerably, because in such a case the vector field
$\nabla^{\scalebox{0.5}{\emph{N}}}_{\partial_{2}}{\partial_{3}}$ has no
$\partial_{1}$-component,

$\displaystyle\nabla^{\scalebox{0.5}{\emph{N}}}_{\partial_{2}}{\partial_{3}}=\alpha\partial_{0}+\beta\partial_{2}+\gamma\partial_{3},$
(31)

for some smooth functions $\alpha,\beta,\gamma$ (here use that

$g_{\scalebox{0.5}{\emph{N}}}(\nabla^{\scalebox{0.5}{\emph{N}}}_{\partial_{2}}{\partial_{3}},\partial_{0})=-g_{\scalebox{0.5}{\emph{N}}}(\nabla^{\scalebox{0.5}{\emph{N}}}_{\partial_{2}}{\partial_{0}},\partial_{3})=0$

because $\partial_{0}=N$ is parallel and the compatibility of $\nabla$ and
$g$). Thus, since $\partial_{0}$ is orthogonal to
$\Lambda_{\scalebox{0.5}{\emph{b,c}}}$,

$\displaystyle\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{2},\partial_{3})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(\nabla^{\scalebox{0.5}{\emph{N}}}_{\partial_{2}}{\partial_{3}})^{\perp}=\alpha\partial_{0},$
(32)

where, as $N$ is parallel, we observe that $\nabla^{\scalebox{0.5}{\emph{N}}}$
is the Levi-Civita connection of $g_{\scalebox{0.5}{\emph{N}}}$; likewise for
$\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{2},\partial_{2})$ and
$\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\partial_{3},\partial_{3})$.111Since
the subspace
$S\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\text{span}\\{\partial_{2},\partial_{3}\\}$
is spacelike, its orthogonal complement $S^{\perp}$ is timelike, and each
tangent space $T_{p}\mathscr{U}$ is the direct sum $S_{p}\oplus S_{p}^{\perp}$
(see, e.g., [O’N83, p. 141]). Therefore, $\alpha\partial_{0}\in S^{\perp}$ is
the (unique) normal component of
$\nabla_{\\!\partial_{2}}{\partial_{3}}=\alpha\partial_{0}+\beta\partial_{2}+\gamma\partial_{3}$.
Thus, as $\partial_{0}$ is lightlike, (4) simplifies to

$\displaystyle\text{Rm}_{\scalebox{0.5}{\emph{b,c}}}=0.$ (33)

In other words, each Riemannian submanifold
$(\Lambda_{\scalebox{0.5}{\emph{b,c}}},h_{\scalebox{0.5}{\emph{b,c}}})$ is
flat, which means that there exist local coordinates
$r_{\scalebox{0.5}{\emph{c}}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=r_{\scalebox{0.5}{\emph{c}}}(\tilde{x}^{2},\tilde{x}^{3}),s_{\scalebox{0.5}{\emph{c}}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=s_{\scalebox{0.5}{\emph{c}}}(\tilde{x}^{2},\tilde{x}^{3})$
on each $\Lambda_{\scalebox{0.5}{\emph{b,c}}}$ with respect to which the
induced metric $h_{\scalebox{0.5}{\emph{b,c}}}$ takes the form

$h_{\scalebox{0.5}{\emph{b,c}}}=dr_{\scalebox{0.5}{\emph{c}}}\otimes
dr_{\scalebox{0.5}{\emph{c}}}+ds_{\scalebox{0.5}{\emph{c}}}\otimes
ds_{\scalebox{0.5}{\emph{c}}}.$

At each $\tilde{x}^{1}=c$, we thus have the triple
$(c,r_{\scalebox{0.5}{\emph{c}}},s_{\scalebox{0.5}{\emph{c}}})$; considering
$-s_{\scalebox{0.5}{\emph{c}}}$ if necessary, we may also assume that each
$\\{\partial_{r_{\scalebox{0.5}{\emph{c}}}},\partial_{s_{\scalebox{0.5}{\emph{c}}}}\\}$
is positively oriented. We now put these together to form a smooth coordinate
chart $(\tilde{x}^{0},\tilde{x}^{1},x,y)$, as follows. First, at each point
$(b,c,0,0)\in\Lambda_{\scalebox{0.5}{\emph{b,c}}}$, rotate each
$\\{\partial_{r_{\scalebox{0.5}{\emph{c}}}},\partial_{s_{\scalebox{0.5}{\emph{c}}}}\\}$
so that $\partial_{r_{\scalebox{0.5}{\emph{c}}}}$ points in the direction of
$\partial_{2}$ (i.e.,
$\partial_{r_{\scalebox{0.5}{\emph{c}}}}=\frac{1}{\sqrt{h_{22}}}\partial_{2}\big{|}_{(b,c,0,0)}$);
these rotated $\partial_{r_{\scalebox{0.5}{\emph{c}}}}$’s thus comprise a
smooth vector field
$V\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\frac{1}{\sqrt{h_{22}}}\partial_{2}$
on the submanifold
$\Sigma\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{(\tilde{x}^{0},\tilde{x}^{1},0,0)\\}$.
As each $\partial_{s_{\scalebox{0.5}{\emph{c}}}}$ is orthogonal to its
corresponding $\partial_{r_{\scalebox{0.5}{\emph{c}}}}$ and since an
orientation has been fixed, the $\partial_{s_{\scalebox{0.5}{\emph{c}}}}$’s
thus also comprise a smooth vector field $W$ on $\Sigma$, namely, the unique
unit length vector field orthogonal to $V$ and such that $\\{V,W\\}$ is
positively oriented. Next, at each
$p\in\Sigma\cap\Lambda_{\scalebox{0.5}{\emph{b,c}}}$, parallel transport $V,W$
along the integral curve $\gamma^{(p)}$ of
$\partial_{r_{\scalebox{0.5}{\emph{c}}}}$ through $p$, via the connection
$\nabla^{\scalebox{0.5}{\emph{b,c}}}$ compatible with the induced metric
$h_{\scalebox{0.5}{\emph{b,c}}}$; then, at each point along $\gamma^{(p)}$,
parallel transport along the integral curve of
$\partial_{s_{\scalebox{0.5}{\emph{c}}}}$ on
$\Lambda_{\scalebox{0.5}{\emph{b,c}}}$; by abuse of notation, let $V,W$ denote
the resulting vector fields, now smoothly defined on $\mathscr{U}$. By the
Gauss Formula (see, e.g., [Lee18, Theorem 8.2, p. 228]) and the flatness
condition (33), we have, on each
$(\Lambda_{\scalebox{0.5}{\emph{b,c}}},h_{\scalebox{0.5}{\emph{b,c}}})$, that

$\nabla^{\scalebox{0.5}{\emph{N}}}_{V}V=\cancelto{0}{\nabla^{\scalebox{0.5}{b,c}}_{\\!V}\,V}+\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(V,V);$

similarly for
$\nabla^{\scalebox{0.5}{\emph{N}}}_{V}W,\nabla^{\scalebox{0.5}{\emph{N}}}_{W}V$,
and $\nabla^{\scalebox{0.5}{\emph{N}}}_{W}W$. But just as in (32), each
$\mathrm{II}_{\scalebox{0.5}{\emph{b,c}}}(\cdot,\cdot)$ must point solely in
the direction of $\partial_{0}$. Finally, define the orthonormal pair

$X\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-g(\partial_{1},V)\partial_{0}+V\hskip
14.45377pt,\hskip
14.45377ptY\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=-g(\partial_{1},W)\partial_{0}+W,$

which pair is now orthogonal to $\partial_{0},\partial_{1}$. It follows that
all of the Lie brackets of the frame $\\{\partial_{0},\partial_{1},X,Y\\}$
will have only a $\partial_{0}$-component, which collectively yields that
$X^{\flat}=g(X,\cdot)$ and $Y^{\flat}=g(Y,\cdot)$ will be closed; indeed, for
any pair of vector fields $A,B\in\\{\partial_{0},\partial_{1},X,Y\\}$,

$dX^{\flat}(A,B)=\cancelto{0}{A(g(X,B))}-\cancelto{0}{B(g(X,A))}-\cancelto{0}{g(X,[A,B])}=0;$

likewise with $dY^{\flat}$. By the Poincaré Lemma, both 1-forms are locally
exact: $X^{\flat}=dx$ and $Y^{\flat}=dy$, for some smooth functions $x,y$.
With respect to the coordinate chart $(\tilde{x}^{0},\tilde{x}^{1},x,y)$, the
ambient Lorentzian metric $g$ thus takes the form

$\displaystyle(g_{ij}(N))=\left(\begin{array}[]{cccc}0&1&0&0\\\
1&g_{11}(N)&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{array}\right),$ (38)

which is precisely (29). (This argument generalizes to dimensions $>4$.)
Conversely, suppose that local coordinates $(v,u,x,y)$ exist in which the
metric takes this form, with $N=\partial_{v}=\nabla u$. Then the nonvanishing
Christoffel symbols of such a metric are

$\displaystyle\nabla^{\scalebox{0.5}{\emph{N}}}_{\partial_{x}}{\partial_{u}}\\!$
$\displaystyle=$
$\displaystyle\\!\nabla^{\scalebox{0.5}{\emph{N}}}_{\partial_{u}}{\partial_{x}}=\frac{H_{x}}{2}\partial_{v},$
$\displaystyle\nabla^{\scalebox{0.5}{\emph{N}}}_{\partial_{y}}{\partial_{u}}\\!$
$\displaystyle=$
$\displaystyle\\!\nabla^{\scalebox{0.5}{\emph{N}}}_{\partial_{u}}{\partial_{y}}=\frac{H_{y}}{2}\partial_{v},$
$\displaystyle\nabla^{\scalebox{0.5}{\emph{N}}}_{\partial_{u}}{\partial_{u}}\\!$
$\displaystyle=$
$\displaystyle\\!\frac{H_{u}}{2}\partial_{v}-\frac{H_{x}}{2}\partial_{x}-\frac{H_{y}}{2}\partial_{y},$

from which it follows that $R_{\scriptscriptstyle
N}(\partial_{x},\partial_{y})\partial_{x}=R_{\scriptscriptstyle
N}(\partial_{x},\partial_{y})\partial_{y}=0$; this is precisely the curvature
condition (24). Actually something else vanishes, too:

$R_{\scriptscriptstyle
N}(\partial_{x},\partial_{y})\partial_{u}=\nabla^{\scalebox{0.5}{\emph{N}}}_{\\!\partial_{x}}\nabla^{\scalebox{0.5}{\emph{N}}}_{\\!\partial_{y}}\partial_{u}-\nabla^{\scalebox{0.5}{\emph{N}}}_{\\!\partial_{y}}\nabla^{\scalebox{0.5}{\emph{N}}}_{\\!\partial_{x}}\partial_{u}=\frac{H_{yx}}{2}\partial_{v}-\frac{H_{xy}}{2}\partial_{v}=0.$

Thus $R_{\scriptscriptstyle N}(X,Y)W=0$ for all $X,Y\in\Gamma(N^{\perp})$ and
$W\in\mathfrak{X}(\mathscr{U})$. ∎

We will give some examples of Finsler pp-waves in Section 6, but let us
observe that the examples of Finsler pp-waves already present in literature
can be included in the definition of Theorem 1 up to some interpretations. In
both cases, [FP16, HPF21], the Lorentz-Finsler metric is not smooth on the
lightlike parallel vector field $N$, but as these Finsler spacetimes are of
Berwald type, there is an available affine connection which makes $N$
parallel. Moreover, the role of $g_{\scalebox{0.5}{\emph{N}}}$-orthogonality
to $N$ can be played by the tangent space to the lightcone at $N$. In both
cases, it coincides with the tangent space to the lightlike cone of a
Lorentzian metric which is already a pp-wave with $N$ as parallel vector
field. Therefore, the conditions of Theorem 1 are satisfied also for the
Berwaldian Finsler pp-waves of [FP16, HPF21].

##  5\. The quotient bundle of a Finsler pp-wave

######  Theorem 2 ([CBS13, LS16]).

Let $(M,L)$ be a Finsler spacetime and $N=\nabla\\!f$ a lightlike, parallel
gradient vector field defined in an open subset $\mathscr{U}\subseteq M$, with
orthogonal complement $N^{\perp}\subseteq T\mathscr{U}$. Then the vector
bundle $N^{\perp}/N$ admits a positive-definite inner product $\bar{g}$,

$\bar{g}([X],[Y])\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=g_{\scalebox{0.5}{N}}(X,Y)\hskip
14.45377pt\text{for all}\hskip 14.45377pt[X],[Y]\in\Gamma(N^{\perp}/N),$

and a corresponding linear connection
$\overline{\nabla}\colon\mathfrak{X}(\mathscr{U})\times\Gamma(N^{\perp}/N)\longrightarrow\Gamma(N^{\perp}/N)$,

$\overline{\nabla}_{\\!W}{[Y]}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=[\nabla^{\scalebox{0.5}{N}}_{W}{Y}]\hskip
14.45377pt\text{for all}\hskip 14.45377ptW\in\mathfrak{X}(\mathscr{U})\hskip
14.45377pt\text{and}\hskip 14.45377pt[Y]\in\Gamma(N^{\perp}/N).$

This connection is flat if and only if $(\mathscr{U},L|_{\mathscr{U}})$ is a
Finsler pp-wave.

###### Proof.

The metric $\bar{g}$ will be well defined, and positive definite, whenever $N$
is lightlike; indeed, every $X\in\Gamma(N^{\perp})$ not proportional to $N$ is
necessarily spacelike, so that $\bar{g}$ is nondegenerate (and positive-
definite), and if $[X]=[X^{\prime}]$ and $[Y]=[Y^{\prime}]$, so that
$X^{\prime}=X+fN$ and $Y^{\prime}=Y+kN$ for some smooth functions $f,k$, then

$\bar{g}([X^{\prime}],[Y^{\prime}])=g_{\scalebox{0.5}{\emph{N}}}(X^{\prime},Y^{\prime})=g_{\scalebox{0.5}{\emph{N}}}(X,Y)=\bar{g}([X],[Y]).$

On the other hand, the connection $\overline{\nabla}$ requires $N$ to be
parallel or else it is not well defined:
$\nabla^{\scalebox{0.5}{\emph{N}}}_{W}{Y}\in\Gamma(N^{\perp})$ if and only if
$N$ is parallel, in which case

$\overline{\nabla}_{\\!W}{[Y^{\prime}]}=[\nabla^{\scalebox{0.5}{\emph{N}}}_{W}{Y^{\prime}}]=[\nabla^{\scalebox{0.5}{\emph{N}}}_{W}{Y}]+\cancelto{0}{[W(k)N]}+\cancelto{0}{[k\nabla^{\scalebox{0.5}{\emph{N}}}_{W}{N}]}\,=\overline{\nabla}_{\\!W}{[Y]}.$

That $\overline{\nabla}$ is indeed a linear connection follows easily. Now, if
this connection is flat, then by definition its curvature endomorphism

$\displaystyle\overline{\text{R}}\colon\mathfrak{X}(\mathscr{U})\times\mathfrak{X}(\mathscr{U})$
$\displaystyle\times$
$\displaystyle\Gamma(N^{\perp}/N)\rightarrow\Gamma(N^{\perp}/N),~{}\text{given
by}$ $\displaystyle\overline{\text{R}}(U,W)[X]$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=$
$\displaystyle\overline{\nabla}_{\\!V}[\overline{\nabla}_{\\!W}[X]]-\overline{\nabla}_{\\!W}[\overline{\nabla}_{\\!U}[X]]-\overline{\nabla}_{\\![U,W]}{[X]}$

vanishes, for any section $[X]\in\Gamma(N^{\perp}/N)$ and vector fields
$U,W\in\mathfrak{X}(\mathscr{U})$. Using the metric $\bar{g}$, this flatness
condition is equivalent to

$\bar{g}(\overline{\text{R}}(U,W)[X],[Y])=0\hskip 7.22743pt\text{for
all}\hskip 7.22743ptU,W\in\mathfrak{X}(\mathscr{U})\hskip 7.22743pt,\hskip
7.22743pt[X],[Y]\in\Gamma(N^{\perp}/N).$

But if we unpack the definitions of $\overline{\nabla}$ and $\bar{g}$, we see
that

$\displaystyle\bar{g}(\overline{\text{R}}(U,W)[X],[Y])=\text{Rm}_{\scalebox{0.5}{\emph{N}}}(U,W,X,Y)=\text{Rm}_{\scalebox{0.5}{\emph{N}}}(X,Y,U,W).$
(39)

It follows that $\overline{\text{R}}=0$ if and only if $R_{\scriptscriptstyle
N}(X,Y)W=0$ for all $X,Y\in\Gamma(N^{\perp})$ and
$W\in\mathfrak{X}(\mathscr{U})$; by (24), this is precisely the condition to
be a Finsler pp-wave. ∎

##  6\. Examples

###  6.1. Parallel lightlike vector field

Let us choose a Finsler metric $F$ on $\mathbb{R}^{2}\times M$ with $v,u$ the
coordinates of $\mathbb{R}^{2}$ and such that $\partial_{v}$ is a Killing
field, namely, $F$ does not depend on $v$. Define a one-form $\omega$ such
that

$\displaystyle\omega(\partial_{v})$ $\displaystyle=F(\partial_{v}),$
$\displaystyle\omega(\partial_{u})$
$\displaystyle=(1+g_{\scalebox{0.5}{\emph{N}}}(\partial_{u},\partial_{v}))/F(\partial_{v}),$
$\displaystyle\omega(\partial_{x})$
$\displaystyle=g_{\scalebox{0.5}{\emph{N}}}(\partial_{v},\partial_{x})/F(\partial_{v}),$

where $\partial_{x}$ is tangent to $M$. Then the Lorentz-Finsler metric
defined by

$L(v)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\omega(v)^{2}-F(v)^{2}$
(40)

admits $N=\partial_{v}$ as a lightlike parallel vector field (here we are
using the fact that if $A=\\{v\in T(\mathbb{R}^{2}\times M):L(v)>0\\}$ is
nonempty, then $L$ determines a Finsler spacetime on $\mathbb{R}^{2}\times M$,
see [JS20, Theorem 4.1]). To check this, observe that the fundamental tensor
$g^{\scalebox{0.5}{\emph{L}}}$ of $L$ is given by

$g^{\scalebox{0.5}{\emph{L}}}_{v}(u,w)=\omega(u)\omega(w)-g_{\scalebox{0.5}{\emph{V}}}(u,w)$

where $g$ is the fundamental tensor of $F$. By the definition of $\omega$, it
follows that the coefficients $g^{\scalebox{0.5}{\emph{L}}}_{ij}(N)$, with
$N=\partial_{v}$, form a matrix as in (11). This implies in particular that
$N=\partial_{v}$ is the $L$-gradient of the function
$f\colon\mathbb{R}^{2}\times M\rightarrow\mathbb{R}$ defined as $f(v,u,p)=u$.
By Lemma 6, recalling that $F$ has been chosen independent of $v$ and that
$\omega$ has been constructed from $F$, we conclude that $N$ is parallel.

###  6.2. Finsler pp-waves

Let us give a particular example of Finsler pp-wave. Choose an arbitrary
Finsler metric $F$ on $\mathbb{R}^{2}$ (depending on $(u,x,y)$) and define the
one-form in $\mathbb{R}^{2}$ determined by

$\displaystyle\omega(\partial_{v})$ $\displaystyle=F(\partial_{v}),$
$\displaystyle\omega(\partial_{u})$
$\displaystyle=(1+g_{\scalebox{0.5}{\emph{N}}}(\partial_{u},\partial_{v}))/F(\partial_{v}),$

then

$L=\omega^{2}-F^{2}-dx^{2}-dy^{2}$ (41)

is a Finsler pp-wave defined in the region $A\\{v\in T(\mathbb{R}^{2}\times
M):L(v)\geq 0\\}$. Observe that the matrix of $g_{\scalebox{0.5}{\emph{N}}}$,
for $N=\partial_{v}$, is of the form (29). Moreover, this metric is smooth on
$A$, because it is not smooth in the vectors $(0,w)\in T(\mathbb{R}^{2}\times
M)$ with $w\in TM$ but in this case, $L(0,w)=-w_{1}^{2}-w^{2}_{2}<0$ if
$w\not=0$. Even if this metric is not of the same type as (40), it is very
similar. Indeed, in this case, the role of the metric $F$ in (40) is played by
$\sqrt{F^{2}+dx^{2}+dy^{2}}$, which is not a regular Finsler metric because of
the smoothness issues, but the proof of [JS20, Theorem 4.1]) still works to
guarantee that (41) determines a Finsler spacetime. Moreover, as in the last
subsection, one can check that $L$ admits $N=\partial_{v}$ as a parallel and
lightlike vector field, and the coordinates of $g_{\scalebox{0.5}{\emph{N}}}$
are of the form (29), which concludes by Theorem 1, that $L$ is a Finsler pp-
wave.

##  7\. Penrose’s construction

With the generalizations of lightlike coordinates and pp-waves to Finsler
spacetimes in Sections 3 and 4, respectively, we are now ready to consider the
extension of Penrose’s “plane wave limit” [Pen76]. Again, consider a
4-dimensional Finsler spacetime $(M,L)$ and $N$ a lightlike gradient vector
field such that locally in lightlike coordinates $(x^{0},\dots,x^{3})$,
$g_{\scriptscriptstyle N}$ takes the form

$\displaystyle(g_{ij}(N))\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\begin{pmatrix}0&1&0&0\\\
1&g_{11}(N)&g_{12}(N)&g_{13}(N)\\\ 0&g_{21}(N)&h_{22}&h_{23}\\\
0&g_{31}(N)&h_{32}&h_{33}\end{pmatrix},$

being $N=\frac{\partial}{\partial x^{0}}$. Recall that lightlike gradient
vector fields always exist at least locally (see Lemma 4). Penrose’s
construction now proceeds as follows. Define another coordinate system
$(\tilde{x}^{0},\ldots,\tilde{x}^{3})$ via the diffeomorphism $\varphi$ given
by

$\displaystyle(x^{0},\dots,x^{3})\overset{\varphi}{\mapsto}(x^{0},\Omega^{-2}x^{1},\Omega^{-1}x^{2},\Omega^{-1}x^{3})\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(\tilde{x}^{0},\dots,\tilde{x}^{3}),$

where $\Omega>0$ is a constant. Next, define the following metric
$h_{\scriptscriptstyle\Omega}$ in the new coordinates
$(\tilde{x}^{0},\dots,\tilde{x}^{3})$,

$\displaystyle\big{(}(h_{\scriptscriptstyle\Omega})_{ij}\big{)}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\underbrace{\,\begin{pmatrix}0&1&0&0\\\
1&\Omega^{2}g_{11}(N)&\Omega g_{12}(N)&\Omega g_{13}(N)\\\ 0&\Omega
g_{21}(N)&h_{22}&h_{23}\\\ 0&\Omega
g_{31}(N)&h_{32}&h_{33}\end{pmatrix}\,}_{\text{defined in the
coordinates}~{}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})},$

where each component $(h_{\scriptscriptstyle\Omega})_{ij}$ is (strategically)
defined as follows,

$\displaystyle(h_{\scriptscriptstyle\Omega})_{11}(\tilde{x}^{0},\dots,\tilde{x}^{3})$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=$
$\displaystyle\Omega^{2}\underbrace{\,{g}_{11}(N)(\tilde{x}^{0},\Omega^{2}\tilde{x}^{1},\Omega\,\tilde{x}^{2},\Omega\,\tilde{x}^{3})\,}_{=\,{g}_{11}(N)(x^{0},\dots,x^{3})},$
(43)
$\displaystyle(h_{\scriptscriptstyle\Omega})_{22}(\tilde{x}^{0},\dots,\tilde{x}^{3})$
$\displaystyle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=$
$\displaystyle\underbrace{\,h_{22}(\tilde{x}^{0},\Omega^{2}\tilde{x}^{1},\Omega\,\tilde{x}^{2},\Omega\,\tilde{x}^{3})\,}_{=\,{h}_{22}(x^{0},\dots,x^{3})},$
(44)

and similarly with the others. Note that as $\Omega\to 0$,

$\displaystyle\lim_{\Omega\to 0}(h_{\scriptscriptstyle\Omega})_{11}$
$\displaystyle\overset{\eqref{newcomp}}{=}$ $\displaystyle
0\cdot{g}_{11}(\tilde{x}^{0},0,\dots,0)=0,$ $\displaystyle\lim_{\Omega\to
0}(h_{\scriptscriptstyle\Omega})_{22}$
$\displaystyle\overset{\eqref{newcomp2}}{=}$
$\displaystyle{h}_{22}(\tilde{x}^{0},0,\dots,0);$

similarly with the others. Note also that the metric
$h_{\scriptscriptstyle\Omega}$ is _conformal_ to
$(\varphi^{-1})^{*}g_{\scalebox{0.5}{\emph{N}}}$; to see this, use the fact
that

$((\varphi^{-1})^{*}g_{\scalebox{0.5}{\emph{N}}})(\partial_{\tilde{x}^{i}},\partial_{\tilde{x}^{j}})d\tilde{x}^{i}\otimes
d\tilde{x}^{j}=g_{\scalebox{0.5}{\emph{N}}}(\partial_{x^{i}},\partial_{x^{j}})dx^{i}\otimes
dx^{j},$

as well as (7), to obtain

$\displaystyle dx^{0}\otimes dx^{1}$ $\displaystyle=$
$\displaystyle\Omega^{2}\,d\tilde{x}^{0}\otimes d\tilde{x}^{1},$
$\displaystyle g_{11}(N)(x^{0},x^{1},x^{2},\dots,x^{n})\,dx^{1}\otimes
dx^{1}\\!\\!$ $\displaystyle\overset{\eqref{newcomp}}{=}$
$\displaystyle\\!\\!\Omega^{2}\,(h_{\scriptscriptstyle\Omega})_{11}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})\,d\tilde{x}^{1}\otimes
d\tilde{x}^{1},$ $\displaystyle
g_{12}(N)(x^{0},x^{1},x^{2},\dots,x^{n})\,dx^{1}\otimes dx^{2}$
$\displaystyle=$
$\displaystyle\Omega^{2}\,(h_{\scriptscriptstyle\Omega})_{12}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})\,d\tilde{x}^{1}\otimes
d\tilde{x}^{2},$ $\displaystyle
h_{22}(N)(x^{0},x^{1},x^{2},\dots,x^{n})\,dx^{2}\otimes dx^{2}\\!\\!$
$\displaystyle\overset{\eqref{newcomp2}}{=}$
$\displaystyle\\!\\!\Omega^{2}\,(h_{\scriptscriptstyle\Omega})_{22}(\tilde{x}^{0},\tilde{x}^{1},\tilde{x}^{2},\dots,\tilde{x}^{n})\,d\tilde{x}^{2}\otimes
d\tilde{x}^{2},$

and so on, which clearly yields the relationship

$\displaystyle(\varphi^{-1})^{*}g_{\scalebox{0.5}{\emph{N}}}=\Omega^{2}\,h_{\scriptscriptstyle\Omega}.$
(45)

In particular, setting
$\tilde{g}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=(\varphi^{-1})^{*}g_{\scalebox{0.5}{\emph{N}}}$,
observe that the homothety (45) means that the Levi-Civita connections of
$h_{\scriptscriptstyle\Omega}$ and $\tilde{g}$ are equal:
$\nabla^{\scriptscriptstyle\Omega}=\nabla^{\scriptscriptstyle\tilde{g}}$.
Taking the limit

$\lim_{\Omega\to 0}h_{\scriptscriptstyle\Omega}=\lim_{\Omega\to
0}\frac{(\varphi^{-1})^{*}g_{\scalebox{0.5}{\emph{N}}}}{\Omega^{2}}$

thus yields the metric

$\displaystyle\underbrace{\,\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\
0&0&h_{22}(\tilde{x}^{0},0,\dots,0)&h_{23}(\tilde{x}^{0},0,\dots,0)\\\
0&0&h_{32}(\tilde{x}^{0},0,\dots,0)&h_{33}(\tilde{x}^{0},0,\dots,0)\end{pmatrix}\,}_{\text{defined
in the coordinates}~{}(\tilde{x}^{0},\dots,\tilde{x}^{3})},$

which is clearly a metric in so-called Rosen coordinates of Lorentzian
geometry, also referred to as Baldwin-Jeffery-Rosen coordinates (e.g.
[DGHZ17]) due to the pioneering paper [BJ26] by Baldwin and Jeffery. So,
finally, we need to confirm that this can indeed be interpreted as a Finsler
pp-wave in Brinkmann coordinates as discussed before. Following the standard
coordinate transformation as described e.g. by Blau and O’Loughlin [BO03], we
introduce a matrix $M$ such that

$h_{ij}(\tilde{x}_{0})M^{i}_{k}(\tilde{x}^{0})M^{j}_{l}(\tilde{x}^{0})=\delta_{kl}\,,$

satisfying the symmetry property

$h_{ij}\dot{M}^{i}_{k}M^{j}_{l}=h_{ij}M^{i}_{k}\dot{M}^{j}_{l}\,,$

with $2\leq i\,,j\,,k\leq 3$ labelling the transverse coordinates, and the dot
denoting differentiation with respect to $\tilde{x_{0}}$. Since $h_{ij}$
depends smoothly on $\tilde{x}_{0}$, such a matrix $M$ can always be found for
sufficiently small $\tilde{x}_{0}$ (e.g. [BFOP02]). Then the transformation

$\displaystyle\tilde{x}^{0}$ $\displaystyle=u\,,$ $\displaystyle\tilde{x}^{1}$
$\displaystyle=v-\frac{1}{2}h_{ij}\dot{M}^{i}_{k}M^{j}_{l}x^{k}x^{l}\,,$
$\displaystyle\tilde{x}^{i}$ $\displaystyle=M^{i}_{j}x^{j}\,,$

gives rise to the metric in Brinkmann coordinates $(v\,,u\,,x^{i})$ with

$H=-(h_{ij}\dot{M}^{i}_{k}\dot{)}M^{j}_{l}x^{k}x^{l}$

as the $uu$-component of $g_{\scalebox{0.5}{\emph{N}}}$ in (29).

##  8\. Concluding remarks

The first main result of this article is the construction of a lightlike
gradient vector field $N$ in Finsler spacetimes (see Lemma 4) yielding a
particular chart for $g_{\scalebox{0.5}{\emph{N}}}$, as given in Lemma 3.
Secondly, after establishing a condition for $N$ to be parallel, we extend the
notion of pp-waves, as given by [GL16], to Finsler spacetimes in Theorem 1.
New examples of such Finsler pp-waves are found in Section 6, and we also show
their quotient bundle structure in Section 5. Finally, Penrose’s plane wave
limit is adapted to Finsler pp-waves in Section 7.
In closing, let us note that Lorentzian spacetimes with a parallel lightlike
vector field are also known as _Bargmann manifolds_. These have proven useful
for studying kinematical groups, in particular the Carroll group of plane
waves (notably in [DGHZ17]), thus raising the question of the kinematical
group structure associated with Finsler pp-waves. The optical properties of
Finsler pp-waves may offer another worthwhile avenue for future work, as
Lorentzian plane waves are well-known to exhibit some remarkable lensing
effects (e.g., [Har13]). Indeed, it may be interesting to see whether Finsler
pp-waves can also be regarded as members of a Kundt class generalized to
Finsler spacetimes.

## Acknowledgements

MAJ was supported by the project PGC2018-097046-B-I00 funded by MCIN/ AEI
/10.13039/501100011033/ FEDER “Una manera de hacer Europa.”

## References

  * [AJ16] Amir Babak Aazami and Miguel Ángel Javaloyes.  Penrose’s singularity theorem in a Finsler spacetime.  Classical Quantum Gravity, 33(2):025003, 22, 2016.
  * [Asa85] G. S. Asanov.  Finsler geometry, relativity and gauge theories.  Fundamental Theories of Physics. D. Reidel Publishing Co., Dordrecht, 1985\.
  * [Bee70] John K. Beem.  Indefinite Finsler spaces and timelike spaces.  Canadian J. Math., 22:1035–1039, 1970.
  * [BFOP02] Matthias Blau, Jose Figueroa-O’Farrill, and George Papadopoulos.  Penrose limits, supergravity and brane dynamics.  Classical and Quantum Gravity, 19(18):4753, 2002.
  * [BJ26] O. R. Baldwin and G. B. Jeffery.  The relativity theory of plane waves.  Proc. R. Soc. Lond. A, 111:95–104, 1926.
  * [BJS20] Antonio N. Bernal, M. A. Javaloyes, and M. Sánchez.  Foundations of Finsler Spacetimes from the Observers’ Viewpoint.  Universe, 6(4), 2020.
  * [Bla11] Matthias Blau.  Plane waves and Penrose limits.  Lecture Notes for the ICTP School on Mathematics in String and Field Theory (June 2-13 2003), 2011.
  * [BO03] Matthias Blau and Martin O’Loughlin.  Homogeneous plane waves.  Nucl. Phys. B, 654:135–176, 2003.
  * [Bri25] H.W. Brinkmann.  Einstein spaces which are mapped conformally on each other.  Mathematische Annalen, 94(1):119–145, 1925.
  * [CBS13] Miguel Sánchez Caja, Oihane F. Blanco, and José M. M. Senovilla.  Structure of second-order symmetric Lorentzian manifolds.  Journal of the European Mathematical Society, 15(2):595–634, 2013\.
  * [CS18] Erasmo Caponio and Giuseppe Stancarone.  On Finsler spacetimes with a timelike Killing vector field.  Classical Quantum Gravity, 35(8):085007, 28, 2018.
  * [DGHZ17] C. Duval, G. W. Gibbons, P. A. Horvathy, and P. M. Zhang.  Carroll symmetry of plane gravitational waves.  Class. Quant. Grav., 34(17):175003, 2017.
  * [EK62] Jürgen Ehlers and Wolfgang Kundt.  Exact solutions of the gravitational field equations.  In Gravitation: an Introduction to Current Research, pages 49–101. Wiley, 1962.
  * [EK18] Benjamin R. Edwards and V. Alan Kostelecký.  Riemann-Finsler geometry and Lorentz-violating scalar fields.  Phys. Lett. B, 786:319–326, 2018.
  * [FP16] Andrea Fuster and Cornelia Pabst.  Finsler pp-waves.  Phys. Rev. D, 94(10):104072, 2016.
  * [FS06] José L. Flores and Miguel Sánchez.  On the geometry of pp-wave type spacetimes.  In Analytical and Numerical Approaches to Mathematical Relativity, pages 79–98. Springer, 2006.
  * [FS20] José L. Flores and Miguel Sánchez.  The Ehlers-Kundt conjecture about gravitational waves and dynamical systems.  Journal of Differential Equations, 268(12):7505–7534, 2020.
  * [Ger69] Robert Geroch.  Limits of spacetimes.  Communications in Mathematical Physics, 13(3):180–193, 1969.
  * [GL16] Wolfgang Globke and Thomas Leistner.  Locally homogeneous pp-waves.  Journal of Geometry and Physics, 108:83–101, 2016.
  * [Har13] Abraham I. Harte.  Strong lensing, plane gravitational waves and transient flashes.  Class. Quant. Grav., 30:075011, 2013.
  * [HPF21] Sjors Heefer, Christian Pfeifer, and Andrea Fuster.  Randers pp-waves.  Phys. Rev. D, 104(2):Paper No. 024007, 10, 2021.
  * [HPV20] Manuel Hohmann, Christian Pfeifer, and Nicoleta Voicu.  Relativistic kinetic gases as direct sources of gravity.  Phys. Rev. D, 101(2):024062, 13, 2020.
  * [Jav19] Miguel Ángel Javaloyes.  Anisotropic tensor calculus.  Int. J. Geom. Methods Mod. Phys., 16(suppl. 2):1941001, 26, 2019\.
  * [Jav20] Miguel Ángel Javaloyes.  Curvature computations in Finsler geometry using a distinguished class of anisotropic connections.  Mediterranean Journal of Mathematics, 17(4):1–21, 2020.
  * [JS14] Miguel Ángel Javaloyes and Miguel Sánchez.  Finsler metrics and relativistic spacetimes.  Int. J. Geom. Methods Mod. Phys., 11(9):1460032, 15, 2014.
  * [JS15] Miguel Ángel Javaloyes and Bruno Learth Soares.  Geodesics and Jacobi fields of pseudo-Finsler manifolds.  Publ. Math. Debrecen, 87(1-2):57–78, 2015.
  * [JS20] Miguel Ángel Javaloyes and Miguel Sánchez.  On the definition and examples of cones and Finsler spacetimes.  Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114(1):Paper No. 30, 46, 2020.
  * [JSVn21] Miguel Ángel Javaloyes, Miguel Sánchez, and Fidel F. Villaseñor.  Anisotropic connections and parallel transport in finsler spacetimes.  2021\.
  * [KRT12] V. Alan Kostelecký, N. Russell, and R. Tso.  Bipartite Riemann-Finsler geometry and Lorentz violation.  Phys. Lett. B, 716(3-5):470–474, 2012.
  * [Lee18] John M. Lee.  Introduction to Riemannian Manifolds, volume 176.  Springer, $2^{\text{nd}}$ edition, 2018.
  * [LP18] Claus Lämmerzahl and Volker Perlick.  Finsler geometry as a model for relativistic gravity.  Int. J. Geom. Methods Mod. Phys., 15(suppl. 1):1850166, 20, 2018\.
  * [LS16] Thomas Leistner and Daniel Schliebner.  Completeness of compact Lorentzian manifolds with abelian holonomy.  Mathematische Annalen, 364(3-4):1469–1503, 2016.
  * [Min17] Ettore Minguzzi.  Affine sphere relativity.  Comm. Math. Phys., 350(2):749–801, 2017.
  * [O’N83] Barrett O’Neill.  Semi-Riemannian Geometry with Applications to Relativity, volume 103.  Academic press, 1983.
  * [Pen65] Roger Penrose.  A remarkable property of plane waves in general relativity.  Reviews of Modern Physics, 37(1):215, 1965.
  * [Pen72] Roger Penrose.  Techniques of Differential Topology in Relativity.  SIAM, 1972.
  * [Pen76] Roger Penrose.  Any space-time has a plane wave as a limit.  In Differential Geometry and Relativity, pages 271–275. Springer, 1976.
  * [PW12] Christian Pfeifer and Mattias N. R. Wohlfarth.  Beyond the speed of light on Finsler spacetimes.  Phys. Lett. B, 712(3):284–288, 2012.
  * [SHN+17] Christina Sormani, Denson C. Hill, Paveł Nurowski, Lydia Bieri, David Garfinkle, and Nicolás Yunes.  The Mathematics of Gravitational Waves: A Two-Part Feature.  Notices of the AMS, 64(7):684–707, 2017.