Finsler pp-waves and the Penrose limit

We extend the notion of a Lorentzian pp-wave to that of Finsler spacetimes by providing a coordinate-independent definition of a Finsler pp-wave with respect to the Chern connection; our definition also includes the special case of a plane wave. This treatment introduces suitable lightlike coordinates, in analogy with the Lorentzian case, and utilizes the anisotropic calculus recently developed by one of the authors. We then extend Penrose’s “plane wave limit” to the setting of Finsler spacetimes. New examples of such Finsler pp-waves are also presented.


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 limitit 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.

Preliminaries
Let M be a (connected) manifold, T M its tangent bundle, and π : T M → M the natural projection.Let us consider a connected open subset A ⊂ T M \0 which is conic, namely, satisfying the property that λv ∈ A for all v ∈ A and λ > 0. Further assume that A has smooth boundary, that each A p = A ∩ T p M is nonempty for all p ∈ M , and denote Ā its closure in T M \ 0. Given a function L : Ā ⊂ T M → [0, ∞), 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 Āp . .= T p M ∩ Ā, its fundamental tensor Observe that in this case, Āp is convex and salient for every p ∈ M and the indicatrix Σ = L −1 (1) is a strongly convex hypersurface when restricted to each tangent space, namely, each Σ p = Σ ∩ 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 T M .Observe that as explained in [BJS20], for an observer v ∈ Σ 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 for any v ∈ Ā and u, w, z ∈ T π(v) M .This symmetric anisotropic tensor is what makes different Finsler spacetimes from the classical Lorentzian geometry.It is straightforward to check that for any v ∈ Ā and u, w ∈ T π(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 ∈ Ā and X, Y X(M ), one obtains a different value ∇ v X Y ∈ T π(v) M .In particular, given a chart (U , ϕ), the Christoffel symbols are functions Γ k ij : Ā ∩ T U → R which are homogeneous of degree zero and where ∂ 1 , . . ., ∂ n are the partial vector fields of the chart.If we fix a vector field V ∈ X(U ) which is Ā-admissible, that is to say, taking values in Ā, we obtain then an affine connection ∇ V in U with Christoffel symbols Γ i jk • V .The Levi-Civita-Chern connection can be characterized in terms of the associated affine connections ∇ V .Indeed, it is the only one such for which where X, Y, Z ∈ X(U ) and g V and C 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 ∇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 (4) Observe that when V is parallel, namely, ∇ V X V = 0 for all X ∈ X(U ), then the above Koszul formula coincides with the Koszul formula for g V .This means that when V is parallel, the Levi-Civita-Chern connection ∇ V of L coincides with the Levi-Civita connection of g V .Even if it is not always possible to choose a parallel extension of any vector v ∈ Ā, one can find a pointwise parallel vector field, namely, if p = π(v), there exists an extension V ∈ X(U ) for a certain neighborhood U ⊂ M of p such that (∇ V X V ) p = 0 for all X ∈ X(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 ∈ Ā and X, Y, Z ∈ X(M ), let us choose a pointwise parallel extension , where R V is the curvature tensor of ∇ V .This is because in the general expression of the Chern curvature tensor computed using ∇ V , apart from R V there are some additional tensorial terms evaluated in (∇ 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 V coincides with the Chern curvature tensor R V evaluated at V .A function f : M → R admits a gradient with respect to L, denoted ∇f , if there exists a vector field metrically equivalent to df , namely, which satisfies for all X ∈ X(M ).Moreover, in this case the Hessian of f is defined as the anisotropic tensor (5) This follows from the above formula, because using the almost-compatibility of ∇ with g, (the Cartan term vanishes because homogeneity (3)).
Lemma 1.A smooth function f : M → 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 ∇f is lightlike at p ∈ M , this means that the kernel of df p is tangent to C p .As A p is convex, it remains on one side of the hyperplane ker(df p ), and as g ∇f is a Lorentziantype metric (with index n − 1) it follows that df p | A > 0, because the lightlike cone of g ∇f remains on the same side of ker(df p ).If ∇f is timelike at p ∈ M , then ker(df p ) does not touch Āp and as For the implication to the left, we know that ker(df Then it is easy to see that and therefore ∇f = λv.If ker(df p ) is not tangent to C p , then there exists v ∈ Σ p where the minimum distance between ker(df p ) and Σ p is attained.Then (6) holds for some λ.This also shows that the gradient is unique, because the strict convexity of Σ p implies that the distance is attained in a unique point.
Lemma 2. If a function f : M → 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(∇f )) = 0 and then ), and as this holds for all X ∈ X(M ), it follows that ∇ ∇f ∇f ∇f = 0 and ∇f is geodesic.

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 (U , ϕ) is fixed, we will denote by g ij : Ā∩T U → R the coordinates of the fundamental tensor g in (1), namely,

Lemma 3 (Lightlike coordinates). Let (M, L) be a Finsler spacetime and f a smooth function defined on an open subset
. = ∇f is a lightlike vector field, then there exist coordinates about any point in U in which the metric g N has components Proof.Of course, "lightlike" means that N is nowhere vanishing yet satisfies L(N ) = 0; the former implies that the level sets S c . .= f −1 (c) are embedded hypersurfaces, while the latter implies that N has geodesic flow, ∇ N N N = 0, where ∇ N is the Levi-Civita-Chern connection of L (recall Lemma 2).At any p ∈ S c , g Np (N p , N p ) = 0 implies that the induced metric by g Np on S c is degenerate.Let (x 0 , x 1 , x 2 , x 3 ) denote coordinates within U in which N = ∂ 0 .Then by the almost compatibility of g and ∇, Of course, at least one of these c i 's must be nonzero, otherwise each N ⊥ p would be fourdimensional.Let us assume that c 1 = 0. Thus at the moment our metric g N in the coordinates (x 0 , x 1 , x 2 , x 3 ) takes the form These coordinates, however, are not slice coordinates for the level sets S c ; to make them so, simply define new coordinates (x i ) by These new coordinates satisfy ∇x 1 = ∂/∂ x0 , and they are indeed slice coordinates for S c : and ∂/∂ x0 has a nice relationship with the other coordinate basis vectors: The metric in the new coordinates (x i ) is now precisely in the form of (7).Finally, note that since ∂/∂ x2 and ∂/∂ x3 are both orthogonal to the lightlike vector ∂/∂ x0 , they must satisfy g 22 (N ) > 0, g 33 (N ) > 0. It follows that each embedded 2-submanifold defined by Λ b,c . .= q ∈ U : x(q) = b, c, x2 (q), x3 (q) ⊆ S c , (9) Together with the fact that g 22 > 0, it follows that the two leading submatrices of the 2 × 2 matrix g 22 g 23 g 32 g 33 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 π(v 0 ).
Proof.First, consider a local splitting I × B of M with I ⊂ R in such a way that {t 0 } × B is spacelike for all t 0 ∈ I and ∂ t is timelike.Then consider a surface S 0 in B diffeomorphic to S 2 which contains π(v 0 ) and is orthogonal to v 0 , with v 0 pointing to the exterior region of I × S 0 .Observe that the cylinder I × 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 ) ∈ I × S 0 , N (t 0 ,p 0 ) is orthogonal to {t 0 } × 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 π(v 0 ) using product coordinates in I ×S 0 and then the one of the exponential map s → exp (t,p) (sN ).All this together implies that the projection onto I in these coordinates provides a function f : U ⊂ M → 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 ∈ S 0 and with velocity N (t 0 ,p) .Reducing U if necessary, we can assume that these hypersurfaces coincide with the horismos E + ({t 0 } × S 0 ) ∪ E − ({t 0 } × S 0 ).Therefore, they are degenerate and the gradient vector field ∇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 ∆ 4 . .= −det g ij (N ) and , and observe that ∆ 4 = ∆.Then the relationship is as follows: Lemma 5.In lightlike coordinates, a point p ∈ U is a focal point of (10) along the geodesic integral curve of N through p if and only if ∆| p = 0.
Proof.Let (x 0 , x 1 , x 2 , x 3 ) be lightlike coordinates as in the Lemma 3, with N = ∂ 0 , and let Λ be a Riemannian 2-submanifold as in (10).We begin by observing that ∂ 0 , ∂ 1 , ∂ 2 , ∂ 3 are all Jacobi fields along any geodesic integral curve γ(x 0 ) of N starting in Λ.Indeed, setting J . .= ∂ i , and noting that [J, N ] = 0 and ∇ N N N = 0, we have that where R N is the (1, 3)-curvature tensor.Now suppose that ∆| p = 0 at a point p along γ.Then ∂ 2 | p , ∂ 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 γ, then J will be a Jacobi field.In fact it is a Λ-Jacobi field, since J(0) ∈ T γ(0) Λ and since tan Λ J ′ (0) = tan Λ (∇ N J(0) N ); we thus conclude that p is a focal point of Λ along γ.Now for the converse.Suppose that p is a focal point of Λ along the geodesic integral curve γ(x 0 ) of N , with γ(b) = p; let J : [0, b] −→ U denote the corresponding Λ-Jacobi field, with J(b) its (zero) value at p: (In fact f 1 is identically zero, since J is orthogonal to γ.)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 ∂ i | p 's equalling zero, which can happen only if ∆ 4 | p = ∆| p = 0. Thus, assume that each f i (b) = 0, in which case where at least one of ḟ i (b) is nonzero, for otherwise J(b) = J ′ (b) = 0 and so J would have to be trivial.Next, because each ∂ i is a Jacobi field, and because any two Λ-Jacobi fields V, W along a geodesic satisfy g N (V ′ , W ) = g N (V, W ′ ) (see, e.g., [JS15, Prop.3.18]), we have that the latter because J(b) = 0. Thus we've arrived at the system of equations with the ḟ i (b)'s not all zero.This can only happen if ∆ 4 | p = 0, in which case ∆ p = 0 and the proof is complete.

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 = ∇f is parallel if and only if all components of the metric (7) are independent of x0 .
Proof.Assume that ∇ N X N = 0 for all X ∈ X(M ).Consider ∂ i , ∂ j with i, j = 1, . . ., n and observe that by (7), we know that g N (N, ∂ i ) = 0 if 2 ≤ i ≤ n and g N (N, ∂ 1 ) = 1.By the Koszul formula (4), taking into account that N = ∂ 0 and all the Lie Brackets will vanish, ), which implies that N (g N (∂ i , ∂ j )) = 0 for all i, j = 1, . . ., 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 = 0 for any j = 0, . . ., n, which implies that ∇ N N N = 0. Using this identity and the Koszul formula once more, 0, for any i, j = 0, . . ., n, which implies that ∇ N ∂ i N = 0.In the following, given an Ā-admissible N defined in some open subset U ⊂ M , we will use the notation Γ(N ⊥ ) for the space of vector fields X ∈ X(U ) such that g N (N, X) = 0.
Theorem 1 (pp-waves; [GL16]).If N is a parallel lightlike gradient vector field, then the curvature endomorphism R N satisfies for all X, Y, Z ∈ Γ(N ⊥ ), if and only if there exist local coordinates (v, u, x, y) in which (7) takes the form where g uu (N ) . .= 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 Λ b,c given by (10), with its corresponding induced Riemannian metric (Since by Lemma 6 each g ij (N ) is independent of x0 , the components h ij are x0 -independent.)By the Gauss Equation ([Lee18, Theorem 8.5, p. 230]), we know that the (single) component Rm c (∂ 2 , ∂ 3 , ∂ 2 , ∂ 3 ) of the curvature tensor of h b,c is related to that of g by 0, by (11) where II b,c is the second fundamental form of h b,c ; observe that as N is parallel, the curvature tensor of g N coincides with R N .But when N = ∂ 0 is parallel, this equation simplifies considerably, because in such a case the vector field for some smooth functions α, β, γ (here use that parallel and the compatibility of ∇ and g).Thus, since where, as N is parallel, we observe that ∇ N is the Levi-Civita connection of g N ; likewise for II b,c (∂ 2 , ∂ 2 ) and II b,c (∂ 3 , ∂ 3 ). 1 Thus, as ∂ 0 is lightlike, ( 13 At each x1 = c, we thus have the triple (c, r c , s c ); considering −s c if necessary, we may also assume that each {∂ rc , ∂ sc } is positively oriented.We now put these together to form a smooth coordinate chart (x 0 , x1 , x, y), as follows.First, at each point (b, c, 0, 0 ∂ 2 on the submanifold Σ . .= {(x 0 , x1 , 0, 0)}.As each ∂ sc is orthogonal to its corresponding ∂ rc and since an orientation has been fixed, the ∂ sc 's thus also comprise a smooth vector field W on Σ, namely, the unique unit length vector field orthogonal to V and such that {V, W } is positively oriented.Next, at each p ∈ Σ ∩ Λ b,c , parallel transport V, W along the integral curve γ (p) of ∂ rc through p, via the 1 Since the subspace S ..= span{∂2, ∂3} is spacelike, its orthogonal complement S ⊥ is timelike, and each tangent space TpU is the direct sum Sp ⊕S ⊥ p (see, e.g., [O'N83, p. 141]).Therefore, α∂0 ∈ S ⊥ is the (unique) normal component of connection ∇ b,c compatible with the induced metric h b,c ; then, at each point along γ (p) , parallel transport along the integral curve of ∂ sc on Λ b,c ; by abuse of notation, let V, W denote the resulting vector fields, now smoothly defined on U .By the Gauss Formula (see, e.g., [Lee18, Theorem 8.2, p. 228]) and the flatness condition (16), we have, on each (Λ b,c , h b,c ), that , and ∇ N W W .But just as in (15), each II b,c (•, •) must point solely in the direction of ∂ 0 .Finally, define the orthonormal pair which pair is now orthogonal to ∂ 0 , ∂ 1 .It follows that all of the Lie brackets of the frame {∂ 0 , ∂ 1 , X, Y } will have only a ∂ 0 -component, which collectively yields that X ♭ = g(X, •) and Y ♭ = g(Y, •) will be closed; indeed, for any pair of vector fields likewise with dY ♭ .By the Poincaré Lemma, both 1-forms are locally exact: X ♭ = dx and Y ♭ = dy, for some smooth functions x, y.With respect to the coordinate chart (x 0 , x1 , x, y), the ambient Lorentzian metric g thus takes the form which is precisely (12).(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 = ∂ v = ∇u.Then the nonvanishing Christoffel symbols of such a metric are this is precisely the curvature condition (11).Actually something else vanishes, too: 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 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].

The quotient bundle of a Finsler pp-wave
Theorem 2 ([CBS13, LS16]).Let (M, L) be a Finsler spacetime and N = ∇f a lightlike, parallel gradient vector field defined in an open subset U ⊆ M , with orthogonal complement N ⊥ ⊆ T U .Then the vector bundle N ⊥ /N admits a positive-definite inner product ḡ, ).This connection is flat if and only if (U , L| U ) is a Finsler pp-wave.
Proof.The metric ḡ will be well defined, and positive definite, whenever N is lightlike; indeed, every X ∈ Γ(N ⊥ ) not proportional to N is necessarily spacelike, so that ḡ is nondegenerate (and positive-definite), and if On the other hand, the connection ∇ requires N to be parallel or else it is not well defined: That ∇ is indeed a linear connection follows easily.Now, if this connection is flat, then by definition its curvature endomorphism vanishes, for any section [X] ∈ Γ(N ⊥ /N ) and vector fields U, W ∈ X(U ).Using the metric ḡ, this flatness condition is equivalent to But if we unpack the definitions of ∇ and ḡ, we see that It follows that R = 0 if and only if R N (X, Y )W = 0 for all X, Y ∈ Γ(N ⊥ ) and W ∈ X(U ); by (11), 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 R 2 × M with v, u the coordinates of R 2 and such that ∂ v is a Killing field, namely, F does not depend on v. Define a one-form ω such that where ∂ x is tangent to M .Then the Lorentz-Finsler metric defined by admits N = ∂ v as a lightlike parallel vector field (here we are using the fact that if To check this, observe that the fundamental tensor g L of L is given by g L v (u, w) = ω(u)ω(w) − g V (u, w) where g is the fundamental tensor of F .By the definition of ω, it follows that the coefficients g L ij (N ), with N = ∂ v , form a matrix as in (7).This implies in particular that N = ∂ v is the L-gradient of the function f : R 2 × M → R defined as f (v, u, p) = u.By Lemma 6, recalling that F has been chosen independent of v and that ω has been constructed from F , we conclude that N is parallel.6.2.Finsler pp-waves.Let us give a particular example of Finsler ppwave.Choose an arbitrary Finsler metric F on R 2 (depending on (u, x, y)) and define the one-form in R 2 determined by Observe that the matrix of g N , for N = ∂ v , is of the form (12).Moreover, this metric is smooth on A, because it is not smooth in the vectors (0, w) ∈ T (R 2 × M ) with w ∈ T M but in this case, L(0, w) = −w 2 1 − w 2 2 < 0 if w = 0.Even if this metric is not of the same type as (18), it is very similar.Indeed, in this case, the role of the metric F in (18) is played by 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 (19) determines a Finsler spacetime.Moreover, as in the last subsection, one can check that L admits N = ∂ v as a parallel and lightlike vector field, and the coordinates of g N are of the form (12), which concludes by Theorem 1, that L is a Finsler pp-wave.

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 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 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.
, each Riemannian submanifold (Λ b,c , h b,c ) is flat, which means that there exist local coordinates r c . .= r c (x 2 , x3 ), s c . .= s c (x 2 , x3 ) on each Λ b,c with respect to which the induced metric h b,c takes the form h b,c = dr c ⊗ dr c + ds c ⊗ ds c .