The Light Ray Transform in Stationary and Static Lorentzian Geometries

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ The Light Ray Transform in Stationary and Static Lorentzian Geometries © The Authors 2020 Published version Feizmohammadi, Ali; Ilmavirta, Joonas; Oksanen, Lauri Feizmohammadi, A., Ilmavirta, J., & Oksanen, L. (2021). The Light Ray Transform in Stationary and Static Lorentzian Geometries. Journal of Geometric Analysis, 31(4), 3656-3682. https://doi.org/10.1007/s12220-020-00409-y 2021 The Journal of Geometric Analysis https://doi.org/10.1007/s12220-020-00409-y The Light Ray Transform in Stationary and Static Lorentzian Geometries Ali Feizmohammadi1 ·Joonas Ilmavirta2 ·Lauri Oksanen1 Received: 15 November 2019 © The Author(s) 2020 Abstract Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar function on a globally hyperbolic stationary Lorentzian manifold and prove injectivity holds if either a convex foliation condition is satisfied on a Cauchy surface on the manifold or the manifold is real analytic and null geodesics do not have cut points. Next, we consider the light ray transform on tensor fields of arbitrary rank in the more restrictive class of static Lorentzian manifolds and show that if the geodesic ray transform on tensors defined on the spatial part of the manifold is injective up to the natural gauge, then the light ray transform on tensors is also injective up to its natural gauge. Finally, we provide applications of our results to some inverse problems about recovery of coefficients for hyperbolic partial differential equations from boundary data. Keywords Inverse problems ·Light ray transform ·Wave equation Mathematics Subject Classification 53C65 BLauri Oksanen [email protected] Ali Feizmohammadi [email protected] Joonas Ilmavirta [email protected] 1Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK 2Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), Jyväskylä, Finland 123 A. Feizmohammadi 1 Introduction Let (N,¯g)be a smooth Lorentzian manifold of dimension 1+n,n≥2, with signature (−,+,...,+). We assume that (N,¯g)is oriented, connected and time oriented, and satisfies the strong causality condition. Here, by strong causality we mean that given any p∈Nand any neighborhood Uof p, there exists a neighborhood V⊂Uof psuch that any causal curve segment with endpoints in Vlies entirely in U.Weare interested in studying the injectivity of the so-called light ray transform on functions and tensors over such Lorentzian manifolds. To formulate the problem precisely we introduce some notations. For each m= 0,1,...,letSm=Sm(N)denote the vector bundle of symmetric tensors of rank m on N. In local coordinates each α∈C∞ c(N;Sm)can be written as α(y,w)=αj1... jm(y)w j1...wjm,∀(y,w)∈TN, where we are using the Einstein summation convention. Next, let βbe a maximal null geodesic in (N,¯g), namely an inextendible geodesic whose tangent vector at each point is lightlike: ∇¯g ˙ β(s)˙ β(s)=0,and ¯g(˙ β(s), ˙ β(s)) =0.(1) Observe that Eq.(1) defines the parametrization of βuniquely up to a group of affine reparametrizations. We also note that such parametrizations can depend on the null geodesic itself. Given any choice of such parametrization along β, we define the light ray transform of α∈C∞ c(N;Sm)along βas follows: Lβα=R α(β(s), ˙ β(s)) ds.(2) Note that, by the strong causality condition the null geodesic β(s)will lie outside of any compact set K⊂N,for|s|large enough (see [21, Lemma 13, p. 408]), and therefore the integral in (2) is well defined for compactly supported α. Note also that the domain of integration in (2) is also justified even when βis not complete since α is compactly supported. Let us observe that an affine reparametrization of βresults in the integral (2)to be scaled. Together with the linearity of the map L, this implies that the choice of the parametrization in (1) is of no significance provided that we are concerned with injectivity of the light ray transform on N. 1.1 The Case of Stationary Geometries The first result in our paper is concerned with the injectivity of the light ray transform on scalar functions under the additional assumption that (N,¯g)is both globally hyperbolic and stationary. For the purposes of this paper, it suffices to recall that global 123 The Light Ray Transform in Lorentzian Geometries hyperbolicity is equivalent with (N,¯g)having a smooth spacelike Cauchy hypersurface, that is, a hypersurface which is intersected by every maximal timelike curve exactly once, see, e.g., [26, Corollary 11.19] and [21, Corollary 39, p. 422]. Also, by stationary, we mean that there exists a smooth complete timelike Killing vector field. Let N⊂Ndenote a fixed, smooth, spacelike Cauchy hypersurface in N, write g=¯g|Nand observe that (N,g)is a Riemannian manifold. It is well known (for example [14, Lemma 3.3]) that the manifold (N,¯g)admits an isometric embedding Φ:R×N→N such that Φ∗¯g=−cdt2+dt⊗η+η⊗dt+g,(3) where cis a smooth positive function on Nand ηis a smooth covector field on N.For the convenience of the reader we show this in Sect. 2.1. In the more restrictive case where the one-form ηin (3) vanishes identically in N, the manifold Nis said to be static. Let us remark that given a spacelike Cauchy surface Nand a timelike Killing vector field E, it is possible to fix the parametrization for the null geodesics in a natural way as follows. Given any maximal null geodesic βthere is a unique point of intersection between βand N(see [21, Lemmas 29 and 42, on pp. 415 and 425]). We can then fix the affine parametrization by requiring that β(0)∈Nand ¯g(˙ β(0), E)=−1. Note that the quantity ¯g(˙ β(s), E)is in fact a constant of motion along null geodesics and as such it can be fixed at any arbitrary point along β. Before formulating our injectivity results in the setting of stationary globally hyperbolic Lorentzian manifolds (N,¯g), we give injectivity results in the model setting (M,¯g), where M=R×M,Mis an n-dimensional manifold with a smooth boundary, and ¯ghas the form (3)onR×M. In this setting, we prove injectivity of the light ray transform on scalar functions under one of the two hypotheses that we will formulate next. To state the first hypothesis, we recall some concepts from Lorentzian geometry, namely, the notion of time-separation and null cut locus. The time-separation function, τ(p,q), between two points pand qis defined as the supremum of the semi-Riemannian length of all future-pointing causal curves connecting pto q, and zero if there is no such path. Next, let p∈M,letβ:I→Mbe a future-pointing null geodesic with β(0)=p, and set s0=sup{s∈I|τ(p,β(s)) =0}. If s0∈Iint, we call β(s0)the future null cut point of palong β(see [3, Sect. 9.2]). Finally, the null cut locus C+ N(p)is then defined as the set of all future null cut points of p. 123 A. Feizmohammadi The geodesic βis the only causal path from pto β(s)for s∈(0,s0), see, e.g., [3, Lemma 9.13]. On the other hand, for any s>s0there is a timelike curve from pto β(s). As an example, let us consider the ultrastatic case M=R×Mand ¯g=−dt2+g where (M,g)is a Riemannian manifold with boundary. If γ(s)is a geodesic on M and γ(s0)is a cut point along γin the Riemannian sense, then β(s0)is the future null cut point of β(0)along β(s)=(s,γ(s)). Hypothesis 1 Suppose that M, g, c and ηare real analytic and that the metric ¯g given by (3)on R×M has empty null cut locus. Before stating the second hypothesis, we need to make more definitions. We introduce the conformally scaled metric ¯gcon Mby ¯gc=−dt2+dt⊗ηc+ηc⊗dt+gc,(4) where ηc=c−1η,gc=c−1g. Next, we define Gto be the set of smooth curves bon Mthat satisfy the following ordinary differential equation: ∇gc ˙ b˙ b=G(b,˙ b), (5) subject to the initial data (b(0), ˙ b(0)) ∈TM. The function G(z,v)is defined for each (z,v)∈TM as follows: G(z,v)=−c c+|η|2 g(∇gc vηc)vη c−(ηcv+(ηcv)2+|v|2 gc)F(z,v). (6) Here η cis the vector dual to ηcwith respect to gcand the terms (∇gc vηc)v and ηcv denote the natural pairing between the one-forms ∇gc vηcand ηcwith the vector v, respectively. All the terms in (6) are evaluated at the point z∈M. Finally, the term F(z,v)is the vector field defined through F(z,v)=dηc(·,v) −c c+|η|2 gdηc(η c,v)η  c. As we will later see in Sect. 3, projections of null geodesics on Msatisfy Eq. (5). This is due to the fact the null geodesics are translation invariant along the flow of the Killing field given by ∂t. The second hypothesis relies on a notion of foliation by a family of strictly convex hypersurfaces with respect to curves in G, and can be stated as follows. Hypothesis 2 The dimension n of M satisfies n ≥3, and there is a function ρ:M→ [0,l], so that the following conditions hold: (i) dρ= 0when ρ>0,ρ−1(l)=∂M and ρ−1(0)has empty interior. (ii) For any b ∈G,if d dtρ(b(t)) =0, then d2 dt2ρ(b(t)) > 0. Our main theorem in the model case can now be stated as follows. 123 The Light Ray Transform in Lorentzian Geometries Theorem 1 Let M be a smooth compact manifold with boundary and consider a Lorentzian metric ¯gofform(3)on M=R×M. Suppose one of the following: (i) there is a metric ¯gof form (3)on Msuch that ¯gsatisfies Hypothesis 1and ¯gisin a small enough C3-neighborhood of ¯g,(ii) (M,¯g)satisfies Hypothesis 2. Then the light ray transform in (M,¯g)is injective on scalar functions. In other words, given any f ∈C∞ c(M), there holds Lβf=0for all maximal βin M⇒ f≡0. As applications of Theorem 1, we state the following two perturbative examples: (1) Suppose that (M,g)is a compact real-analytic Riemannian manifold with boundary without cut points. If cis close to 1, ηis close to 0 and gis close to g, then the light ray transform is injective on (R×M,¯g)with ¯gas in (3). (2) Suppose that (M,g)is a compact Riemannian manifold with strictly convex boundary and suppose that there is a strictly convex function on (M,g).Ifcis close to 1, ηis close to 0 and gis close to g, then the light ray transform is injective on (R×M,¯g)with ¯gas in (3). We will now give a corollary of Theorem 1in the setting of globally hyperbolic stationary Lorentzian manifolds (N,¯g). Corollary 1 Let (N,¯g)be a globally hyperbolic stationary Lorentzian manifold, and let Nhave a non-compact Cauchy hypersurface N. Suppose one of the following: (i) The Lorentzian manifold Nand the Cauchy hypersurface N are real analytic, and there is a real-analytic Killing vector field. Moreover, N has empty null cut locus. (ii) There exists a function ρ:N→[0,∞), such that Hypothesis 2holds on each Ml={ρ≤l}with respect to the function ρ|Ml. Then the light ray transform in (N,¯g)is injective on C∞ c(N). We note that under the assumption (i), the corollary follows immediately from Theorem 1, since the embedding Φis also analytic in this case, see Sect. 2.1. In the case that (ii) holds, we note that due to the non-compactness assumption on N, given any scalar function fon Nwith compact support, there exists a large enough lsuch that supp f⊂Φ(R×M)with M={ρ≤l}. The corollary then follows from Theorem 1 since Msatisfies Hypothesis 2with ρ|M. 1.2 The Case of Static Geometries Given a static globally hyperbolic Lorentzian manifold there exists an embedding Φ:R×N→Nsuch that (3) holds with η≡0. To simplify the statement of our results, we define M=Φ(R×M)where M⊂Nis a compact manifold of dimension nwith smooth boundary and study the injectivity of the light ray transform on tensors of arbitrary rank mover M. 123 A. Feizmohammadi Before presenting the main result, we need to recall the definition of the geodesic ray transform on tensors in (M,gc). To this end, suppose that γis a unit-speed geodesic in (M,gc). We define the bundle ∂inSM ={(x,v)∈TM|x∈∂M,v∈TxM,|v|gc=1,gc(v, ν) < 0}, where νdenotes the unit outward pointing normal vector to ∂Mat the point x.For each (x,v) ∈∂inSM, we consider the unique geodesic γwith initial data (x,v)and define τ+(x,v)=inf{r>0|γ(r;x,v)∈∂M,˙γ(r;x,v) /∈Tγ(r;x,v)∂M}. We assume that the manifold (M,gc)is non-trapping, that is, for all unit-speed geodesics γ(·; x,v) with (x,v) ∈∂inSM, there holds τ+(x,v) < ∞. Finally, let Sm=Sm(M)denote the bundle of symmetric tensors of rank mon M(not to be confused with Sm, the corresponding bundle on N) and define the geodesic ray transform of ω∈C∞ c(M;Sm)along γin Mas follows: Iω(x,v):= τ+(x,v) 0 ω(γ(τ;x,v), ˙γ(τ;x,v))dτ. (7) Here, analogously to the Lorentzian case, we have in local coordinates ω(y,w)=ωj1... jm(y)w j1...wjm,∀(y,w)∈TM. We require the following hypothesis to hold: Hypothesis 3 The geodesic ray transform on (M,gc)is solenoidally injective. In other words, for any ω∈C∞ c(M;Sm), there holds Iω(x,v)=0∀(x,v)∈∂inSM ⇒ ∃ θsuch that ω=dsθ, θ|∂M=0, where dsdenotes the symmetrized covariant derivative on (M,gc). The study of the solenoidal injectivity of the geodesic ray transform on tensors of arbitrary rank has a rich literature. For example, Hypothesis 3with a fixed m=0,1is known to be true when (M,gc)is a simple manifold [1,19,20] or has strictly convex boundary and admits a foliation by strictly convex hypersurfaces [36]. Under the latter condition it was later proved that Hypothesis 3holds for all m=0,1,2[31], and subsequently that it holds for all m=0,1,2,...[8]. For more related results we refer the reader to [5,22–24,30] and the review article [13]. We can now state our main theorem for the injectivity of the light ray transform on tensors. Theorem 2 Let (N,¯g)be a static globally hyperbolic Lorentzian manifold of dimension 1+n. Let Φbe an embedding satisfying (3)with η=0and let M=Φ(R×M) where M is a compact n dimensional submanifold of N with smooth boundary such 123 The Light Ray Transform in Lorentzian Geometries that Hypothesis 3holds. Let α∈C∞ c(M;Sm). The following injectivity result holds for the light ray transform on (M,¯g): Lβα=0 for all maximalβin M⇒ ∃ T,Us.tα≡¯ dsT+U¯g, where ¯ dsdenotes the symmetrized1covariant derivative, T ∈C∞ c(M;Sm−1),U ∈ C∞ c(M;Sm−2)and U ¯g denotes the symmetrized tensor product of the tensors U and ¯g. Let us emphasize that the gauge appearing in the statement of Theorem 2is the natural one since the light ray transform of any tensor of the form ¯ dsT+U¯gwith T,Ucompactly supported in M, vanishes. We refer the reader to Lemma 1for the details. Observe also that, akin to Corollary 1, the result of Theorem 2can be formulated for compactly supported tensor fields on a suitable non-compact Lorentzian manifold. Finally we mention that Theorem 2extends analogous results obtained in [10, Proposition 1.4], where only the cases m=0,1 were considered. To our knowledge, there are no results on tensor tomography along general flows of the form given by (5). For this reason, we leave the case of higher rank tensors on stationary spacetimes as a topic of future work. 1.3 Applications and Examples We discuss some applications of our main results in general relativity that builds on the perturbative examples (1) and (2) in Sect. 1.1. Indeed, Theorem 1can be applied in the context of the Kerr black hole spacetime as discussed next, following the notations in [2, Sect. 5]. Recall that the Kerr geometry (R4,¯gKerr )is an exact solution to the Einstein field equations in general relativity and describes the geometry of vacuum spacetime around an axially symmetric black hole with a so-called quasi-spherical event horizon. The metric has the following form: ¯gKerr =−dt2+λdr2 +dθ2+(r2+a2)sin2θdφ2+2mr λ(asin2θdφ−dt)2. (8) Here, (r,θ,φ)are the usual spherical coordinates in R3,mdenotes the mass and ma is the angular momentum as measured from infinity. The parameters and λare defined through λ=λ(θ, r)=r2+a2cos2θand =(r)=r2−2mr +a2. It is easy to see that both vector fields ∂tand ∂θare Killing fields. Moreover, ∂tis timelike outside the ergosphere, i.e., the region {(r,θ,φ)∈R×R+×S2:r>m+m2−a2cos2θ}. 1We refer the reader to expressions (12)–(13) in Sect. 2.2 for the definition of the symmetrized covariant derivative and symmetrized tensor product in local coordinates. 123 A. Feizmohammadi Clearly the metric ¯gKerr is real analytic in this region, since is non-vanishing there. It follows that given any M⊂R3that is a sufficiently small compact submanifold with boundary outside the ergosphere, then the submanifold (R×M,¯gKerr)satisfies the conditions of Hypothesis 1. In fact, the further away Mis from the origin of R3, the larger it can be since the Kerr geometry is close to the Minkowski geometry far away from the center of the black hole. Let us emphasize that this application is using the full strength of Theorem 1in the sense that ηis non-vanishing in (3). Only the case a=0, corresponding to the Schwarzschild black hole, gives η≡0. Theorem 1and Corollary 1have applications to the recovery of zeroth order timedependent coefficients for the wave equation from boundary data. It is well known that the canonical wave equation is well-posed on a globally hyperbolic Lorentzian manifold, see, e.g., [26]. If the manifold is also stationary, there is a rich theory on the solution space of the wave equation. This space is used for instance in the context of quantum field theory, see Sect. 4.3 of [37]. For another example we refer to the recent relativistic generalization of the Gutzwiller–Duistermaat–Guillemin trace formula for the wave group [34,35]. To keep the notation simple, we suppose that (M,¯g)is as in Theorem 1, and consider the following initial boundary value problem: ⎧ ⎨ ⎩ ¯gu+qu=0,on R×M, u=h,on R×∂M, u=0,on (−∞,0)×M, (9) where ¯gdenotes the wave operator on (M,¯g)given in local coordinates by the expression ¯gu=− n  i,j=0 |det ¯g|−1 2∂ ∂xi|det ¯g|1 2¯gij ∂ ∂xju and qis a smooth a priori unknown function with compact support in the set (T,∞)× Mwith large enough T>0. We consider the problem of recovering qfrom the Dirichlet-to-Neumann operator qthat is defined for all hcompactly supported in ∂Mthrough q:h→ ∂¯νu|∂M. It can be shown that the question of unique recovery of qfrom qreduces to the question of injectivity of the light ray transform on (M,¯g)(see for example [33]). As an immediate consequence of Theorem 1, we deduce that qdetermines quniquely, if Hypothesis 1or 2holds. 1.4 Previous Literature The study of injectivity of the light ray transform on tensors of arbitrary rank is motivated in part due to its connection with coefficient determination problems for the wave equation on Lorentzian manifolds from boundary data, as shown for example in 123 The Light Ray Transform in Lorentzian Geometries Let ¯ Γi jk denote the Christoffel symbols on (M,¯gc)and observe that ¯ Γi 00 =0for i=1,...,n. Using this and the definition of a null geodesic, we see that bsatisfies Eq. d2bi ds2+¯ Γi jk(b(s)) ˙ bj˙ bk+2¯ Γi 0k˙a˙ bk=0,(21) for i=1,...,n. We can choose, without loss of generality, the positive sign in Eq.(20). Indeed, suppose that (a+(s), b+(s)) solves (20)–(21) with the positive sign in (20)fors∈I. Then, (a−(s), b−(s)) := (a+(−s), b+(−s)) with s∈−Isolves the same two equations with the negative sign in (20). Hence, the choice of sign corresponds to affine reparametrizations of a fixed null geodesic. For this reason, we will just consider the positive sign in (20). Now, Eq. (21) can be recast in the form ∇gc ˙ b˙ b=G(b,˙ b), (22) where Gi(b,˙ b):= (Γ i jk −¯ Γi jk)˙ bj˙ bk−2¯ Γi 0k(ηc˙ b+(ηc˙ b)2+|˙ b|2)˙ bk,(23) where Γi jk denotes the Christoffel symbols on (M,gc). We will now simplify the latter expression and show that the curves b∈Gare coordinate invariant in M. To see this, we first observe that ¯g−1 c=c c+|η|2 g−1η c (η c)Tc+|η|2 g cg−1 c−η c⊗η c, where η cdenotes the canonical vector that is dual to the one-form ηcand Tdenotes the transposition operation. Now, using the definition of the Christoffel symbols together with the fact that the coefficients of the metric are time-independent, we write Γi jk −¯ Γi jk =− 1 2(¯gc)i0(¯gc)0,j;k+(¯gc)0,k;j   I +1 2((gc)im −(¯gc)im)(gc)mj;k+(gc)jm;k−(gc)jk;m   II , where the term II involves a summation over the index m=1,...,n.ThetermI reduces as follows: I=− 1 2c c+|η|2 g(η c)i(ηc)j;k+(ηc)k;j. 123 A. Feizmohammadi Similarly, the term II reduces as follows: II =1 2((gc)im −(¯gc)im)(gc)mj;k+(gc)jm;k−(gc)jk;m =1 2c c+|η|2 g(η c)i(η c)m(gc)mj;k+(gc)jm;k−(gc)jk;m =1 2c c+|η|2 g(η c)i(ηc)l(gc)ml (gc)mj;k+(gc)jm;k−(gc)jk;m =c c+|η|2 g(η c)i(ηc)lΓl jk. Combining the expressions for Iand II we deduce that (Γ i jk −¯ Γi jk)˙ bj˙ bk=−c c+|η|2 g(∇gc ˙ bηc)˙ b(η c)i. We now consider the last term in the expression for G. Using the definition of the Christoffel symbols again and the expression of the inverse matrix ¯g−1 cabove, this reduces as follows: 2¯ Γi 0k˙ bk=¯gim c((gc)m0;k−(gc)0k;m)˙ bk=¯gim c((ηc)m;k−(ηc)k;m)˙ bk =gim c−c c+|η|2 g (η c)i(η c)m(ηc)m;k−(ηc)k;m˙ bk. Recalling that (dηc)mk =(ηc)m;k−(ηc)k;m, we conclude that Gcan be rewritten as givenbyEq.(6), thus establishing that it is an invariantly defined vector field on M. Let us emphasize that the parametrization of the curve b(s)in Mwith s∈Iis not a unit-speed parametrization and is directly induced by the initial choice of an affine parametrization for the null geodesic βin M. Theorem 3 If Iis injective, then L is also injective. Proof Suppose that Lβf=0 where β:I→Mdenotes any maximal null geodesic in Mwith maximal interval I. Differentiating Eq. (18)ktimes with respect to τand evaluating at τ=0, we obtain 0= k  j=0I (ιa(s))k−j∂j τˆ f(0,b(s)) ds∀b∈G. Setting k=0, we have (Iˆ f(0,·))(b)=0∀b∈G. 123 The Light Ray Transform in Lorentzian Geometries By injectivity of I, it holds that ˆ f(0,·)=0. In a similar manner, by using induction on ktogether with the injectivity of I, we deduce that ∂k τˆ f(0,·)=0,∀k∈N. As f(t,·)is compactly supported in t,ˆ f(τ, ·)is analytic in τ, and thus fvanishes everywhere.  3.1 Proof of Theorem 1 It is clear that Theorem 1follows, once we prove injectivity of the ray transform I along all maximal curves b∈G. We prove this under the assumption that Hypothesis 1 or Hypothesis 2holds. In fact, the transform Ihas been studied for more general vector fields G(z,v)than the one given by expression (6) and invertibility is known to hold under some assumptions. When Hypothesis 2holds on (M,¯g), injectivity of I follows from [36, Theorem 4.2] in the appendix by Hanming Zhou and the remarks immediately following that theorem. To prove Theorem 1under Hypothesis 1,wewilluse[11, Theorems 1–2]. There, injectivity of the map Iis proved under the assumption that the manifold Mis real analytic and Gis in a sufficiently small C2neighborhood of a real-analytic vector field Gand that the curves in Gdo not contain any conjugate points. Hence, to conclude the proof, we need to show that if Hypothesis 1holds in M, the aforementioned assumptions are satisfied. To this end, let us first recall the definition of conjugate points along curves b∈ G(following [11]) and conjugate points along null geodesics βin M. Given any (s,ξ) ∈TM, we define the exponential map expx(s,ξ) =b(s)where b∈Gwith b(0)=xand ˙ b(0)=ξ. Subsequently, we say that the point b(s0)is conjugate to xif (Ds,ξ expx)(s0,ξ 0)has rank less than n, where ξ0=˙ b(0). The conjugate points on M are defined analogously, in terms of the exponential map, exp :TM→TMof the Lorentzian manifold (M,¯gc)along null geodesics (see for example [21, Definition 10.9]). We now return to verifying the assumptions of [11, Theorem 2] Under Hypothesis 1. It follows from (6) that Gand GareinasmallC2neighborhood of each other. To see that the curves in Gare real analytic we note that (M,¯g),cand ηareassumedtobe real analytic. Thus the curves b∈Gare also real analytic as they solve a second-order linear ordinary differential equation (5) with real-analytic coefficients. In order to apply [11, Theorem 2] and deduce the injectivity of the ray transform I, it remains to verify that the curves in Gdo not have conjugate points. This will be proved in the following lemma. Lemma 4 If (M,¯g)has an empty null cut locus, then there are no conjugate points along any curve b ∈G. Proof Suppose for contrary that there exists a curve b∈Gwith a pair of conjugate points b(0)and b(s0). The above definition of conjugate points implies in particular that there exists a one-parameter family of curves brin Gwith rin a small neighborhood of origin, such that 123 A. Feizmohammadi br(0)=b(0), ˙ br(0)=˙ b(0)+rv for some fixed v∈Tb(0)Mand dist(br(s0), b(s0)) ≤Cr2(24) for some uniform constant C>0, where dist(·,·)is the Riemannian distance function on (M,gc)(note that b0≡b). We define the functions ar(s)as the solutions to the following differential equation: dar ds=ηc˙ br±(ηc˙ br)2+|˙ br|2and ar(0)=0, where the sign ±is chosen in order to make the curve (ar(s), br(s)) future-pointing. Observe that the curves βr(s)=(ar(s), br(s)) define a family of maximal null geodesics in (M,¯g)(in what follows, we will drop the subscript rwhen r=0). Next, we observe that there exists a constant δ>0 depending only on ηcand gc, such that if |dist(brk(s0), b(s0)) |<δ|ark(s0)−a(s0)|(25) for a sequence rk→0, then there exists a causal path between β(s0)and βrk(s0)for all ksufficiently large. If (25) does not hold for any sequence rk→0, then in particular it implies that |ar(s0)−a(s0)|<C δr2and all rsufficiently close to zero. But then the first variation of βamong the family of null geodesics βrmust vanish at the point β(s0)and consequently the point β(s0)is a conjugate point to β(0)along β.By[21, Proposition 10.48], there exists a future-pointing timelike curve connecting β(0)to β(s0)and therefore there exists a null cut point on βcorresponding to β(0)which is a contradiction. Thus, we assume that (25) holds, for a sequence rk→0 and consequently that there exists a future-pointing causal path connecting β(s0)to βrk(s0),orviceversa, for some k. First, we consider the case where this future-pointing causal curve is from β(s0)to βrk(s0). Then the points β(0)and βrk(s0)can be connected through the concatenation of the curve βthat connects β(0)to β(s0)and the causal curve that connects β(s0)to βrk(s0).By[21, Proposition 10.46], we conclude that τ(β rk(0), βrk(s0)) = 0, which implies that C+ N(β(0)) =∅. In the other case that the future-pointing causal curve connects βrk(s0)to β(s0), we can use a similar argument to conclude that τ(β(0), β(s0)) = 0 and subsequently that C+ N(β(0)) =∅. 4 Proof of Theorem 2 We start by considering an embedding of the form (3) with η≡0 and satisfying Hypothesis 3. Throughout this section and for the sake of brevity of notation we will assume without loss of generality that c≡1 so as to discard the notations ¯gcand gc (see Sect. 2.2). Observe that due to the more restrictive form of the metric (compared 123 The Light Ray Transform in Lorentzian Geometries to the stationary case), null geodesics in (M,¯g)can conveniently be parameterized as β(·; r0,x,v)=(r+r0;γ(r;x, v)), with r0∈R,(x,v)∈∂in SM and γ(·; x,v)denoting a unit-speed geodesic with initial data (x,v)∈∂in SM. Owing to this identification of null geodesics, we can recast the light ray transform on R×Mfor α∈C∞ c(R×M;Sm)as (Lα)(r0,x,v)=τ+(x,v) 0 α((r+r0,γ(r;x, v)), (1,˙γ(r;x, v)))dr, for all (r0,x,v)∈R×∂in SM. 4.1 Notations For symmetric tensors fand h, we denote the symmetrized tensor product simply by fh. In particular, if fand hare 1-forms, then fh(v, w) =1 2(f(v)h(w) +f(w)h(w)), v, w ∈TM. Following [7], we next define three operators. The operator i i i:C∞(M;Sm)→C∞(M;Sm+2) is defined through i i if =fg, where we recall that Smdenotes the bundle of symmetric tensors of rank mon M. Next, the operator j j jis the trace with respect to g, that is, j j j:C∞(M;Sm+2)→C∞(M;Sm) is the adjoint of i i i, and in local coordinates we can write, (j j jf)j1... jn=gjk fjkj1... jn. The composition j j ji i iis self-adjoint and positive definite [7, Lem. 2.3]. In particular, the inverse (j j ji i i)−1exists. Moreover, by the same lemma, the bundle Smhas the orthogonal decomposition into sub-bundles Sm=Ker(j j j)⊕Ran(i i i). Finally, the operator p p p:C∞(M;Sm)→C∞(M;Sm) is defined to be the orthogonal projection from Smto Ker(j j j), and it can be written as p p p=1−i i i(j j ji i i)−1j j j, see [7, Eq. (2.15)]. 123 A. Feizmohammadi 4.2 Helmholtz Decomposition Let us first recall the Helmholtz decomposition as proven in [27, Theorem 3.3.2], that is, given any ω∈C∞(M;Sm), there are unique ωs∈C∞(M;Sm)and h∈ C∞(M;Sm−1)satisfying ω=ωs+dsh,δ sωs=0,h|∂M=0, where δsis the adjoint of ds. We say that ωis solenoidal if ω=ωs. For a family ω∈C∞ c(R;C∞(M;Sm)) we define ωs(t)=(ω(t))s. As the corresponding potential h(t)is obtained by solving the elliptic partial differential equation, δsdsh(t)=δsω(t), h(t)|∂M=0, we see that h∈C∞ c(R;C∞(M;Sm−1)) and ωs∈C∞ c(R;C∞(M;Sm)). We define also the Fourier transform in time by ω(τ) =R e−ιτtω(t)dt. Then dsω(τ) = dsω(τ) and δsω(τ) = δsω(τ). In particular, ω(τ) = ωs(τ) +ds h(τ) and δs ωs(τ) =0. As the Helmholtz decomposition of ω(τ) is unique, we obtain (ω(τ))s= ωs(τ). (26) 4.3 Trace-Free Helmholtz Decomposition We will next recall the trace-free Helmholtz decomposition as discussed for examplein[7]. By [7, Theorem 1.5], for any ω∈C∞(M;Sm)there are unique ωtfs ∈ C∞(M;Sm),h∈C∞(M;Sm−1)and ωt∈C∞(M;Sm−2)satisfying ω=ωtfs +i i iωt+dsh,δ sωtfs =0,h|∂M=0,j j jωtfs =0,j j jh =0.(27) This decomposition is obtained by first solving the following elliptic partial differential equation for h, δsp p pdsh=δsp p pω, h|∂M=0. Then ωt=(j j ji i i)−1j j j(ω −dsh)and ωtfs =ω−i i iωt−dsh. The last equation j j jh =0 in the decomposition (27) is in fact a consequence of the first four equations. That is, if ω=ω0+i i iω1+dsω2,δ sω0=0,ω 2|∂M=0,j j jω0=0,(28) 123 The Light Ray Transform in Lorentzian Geometries then ω0=ωtfs,ω1=ωtand ω2=h. Indeed, writing ω 0=ω0−ωtfs,ω 1=ω1−ωt and ω 2=ω2−h, we obtain ω 1=−(j j ji i i)−1j j jdsω 2. Then p p pdsω 2=−ω 0and ω 2 solves δsp p pdsω 2=0,ω  2|∂M=0. Therefore ω 2=0 and ω2=h.Nowalsoω0=ωtfs and ω1=ωtby [7, Th. 1.5]. We record the following consequence that will be useful in what follows. Remark 1 If w=dshfor some h∈C∞(M;Sm−1)satisfying h|∂M=0, then wtfs =0 and wt=0. Analogously to the previous section, for a family ω∈C∞ c(R;C∞(M;Sm)) we can define ωtfs(t)=(ω(t))tfs,ω t(t)=(ω(t))t, that gives smooth families of tensors that are compactly supported in time. Observe that i i iω(τ) = i i iω(τ) and j j jω(τ) = j j jω(τ), and analogously with (26), we have (ω(τ))tfs = ωtfs(τ), (ω(τ))t= ωt(τ). 4.4 Injectivity of the Light Ray Transform on Tensors For the remainder of the paper and for the sake of brevity, we will abuse the notation slightly and identify tensors in Mwith their identical copies in Φ−1(M)without explicitly writing the embedding. Let α∈C∞(M;Sm)and suppose that Lα≡0. As ¯g=−dt2+gand αis symmetric, we write α=fdt+ω+b¯g,(29) where f∈C∞ c(R;C∞(M;Sm−1)) ω ∈C∞ c(R;C∞(M;Sm)) b∈C∞ c(M;Sm−2). We can simplify (29) further by considering the Helmholtz decomposition of f, that we denote by f=fs+dsp. To this end, we begin by writing dspdt=¯ ds(pdt)+∂tp¯g−∂tpg. Note that the term ¯ ds(pdt)+∂tp¯gtakes the form of the gauge (11) and by Lemma 1 lies in the kernel of L, while −∂tpg∈C∞ c(R;C∞(M;Sm)). 123 A. Feizmohammadi In particular, we can replace ωwith ω−∂tpgin (29). As also b¯gis in the kernel of L, we can assume without loss of generality that α=fdt+ω, f=fs.(30) We have the following Fourier slicing lemma. Lemma 5 Suppose that α∈C∞ c(R×M;Sm)is of the form (30). Then for k = 0,1,..., and (x,v)∈∂in SM it holds that ∂k τ Lα(τ, x,v)|τ=0=I(∂k τ f(τ, ·)|τ=0)(x,v)+ k−1  j=0k jRk−j(∂ j τ f(τ, ·)|τ=0)(x,v) +I(∂k τω(τ, ·)|τ=0)(x,v)+ k−1  j=0k jRk−j(∂ j τω(τ, ·)|τ=0)(x,v), (31) where Rjω(x,v)=τ+(x,v) 0 (ιr)jω(γ(r;x,v), ˙γ(r;x,v))dr,ω∈C∞ c(M;Sm). We are now ready to prove the main theorem. Proof of Theorem 2As discussed above, we can write αin the form (30) with f=fs. Now note that for any (x,v)∈∂in SM we have that (y,w)∈∂in SM as well, where y=γ(τ +(x,v);x,v), w =−˙γ(τ +(x,v);x,v). Moreover, we have that Iω(x,v)=Iω(y,w)for any ω∈C∞ c(M;Sm)with even m but Iω(x,v)=−Iω(y,w)for any f∈C∞ c(M;Sm)with odd m. Applying (31) with k=0 and using the above observation implies that I( f(0)) =0,I(ω(0)) =0.(32) Using Hypothesis 3together with f=fsand Remark 1we deduce that  f(0)=0, ωtfs(0)=0, ωt(0)=0. Let us define a0(t,x)=t −∞ ωt(t,x)dt.(33) 123 The Light Ray Transform in Lorentzian Geometries As ωis compactly supported in time, a0(t)vanishes for tsufficiently small. Moreover, for large t, a0(t)=∞ −∞ ωt(t)dt= ωt(0)=0. Thus a0∈C∞ c(R;C∞(M;Sm−2)). Observe also that ∂ta0=ωtand hence ιτa0= ωt. In particular, ∂k τ ωt(0)=ιk∂k−1 τa0(0), k=0,1,.... In what follows, we will write ω=ωtfs +i i iωt+dsa1,a1=as 1+dsh, to denote the trace-free Helmholtz decomposition of ωand the Helmholtz decomposition of a1, respectively. We will use the fact that for any u∈C∞ c(M;Sm), Rj(dsu)=ιjτ+ 0 rjdsu(γ (r), ˙γ(r))dr =ιjτ+ 0 rj∂r(u(γ (r), ˙γ(r)))dr =−ιjRj−1(u). When k=1, Eq.(31) reduces to the fact that I(∂τ f)+I(∂τ ωtfs +ιa0)+R1(dsa1) =I(∂τ f)+I(∂τ ωtfs +ιa0)−ιI(a1) =I(∂τ f−ι as 1)+I(∂τ ωtfs +ιa0g) vanishes at τ=0. Note that ∂τ f−ι as 1is solenoidal and of rank m−1. Moreover, the tensor w:= ∂τ ωtfs +ιa0g is of rank mand satisfies wt=ιa0gand wtfs =∂τ ωtfs. Hence at τ=0, ∂τ f=ι as 1,∂ τ ωtfs =0,a0=0. 123 A. Feizmohammadi We will now proceed with an induction argument to show that for all j∈N,it holds at τ=0 that ∂j τ f=ιj∂j−1 τ as 1,∂ j τ ωtfs =0,∂ j−1 τa0=−ι( j−1)∂ j−2 τ h.(34) Indeed, let us suppose that this hypothesis holds for all j=1,...,k−1. Together with (31) this implies that I(∂k τ f)+ι k−1  j=0k jjRk−j(∂ j−1 τ as 1)+I(∂k τ ωtfs +ιk∂k−1 τa0) + k−1  j=0k jRk−j(ι j∂j−1 τa0+ds∂j τa1) vanishes at τ=0. As a1vanishes on R×∂M,wehave Rk−j(ds∂j τa1)=−ι(k−j)Rk−(j+1)(∂ j τa1) =−ι(k−j)Rk−(j+1)(∂ j τas 1)−ι(k−j)Rk−(j+1)(ds∂j τ h). (35) Next, using the identity −ι k−1  j=0 k! j!(k−j)!(k−j)Rk−(j+1)(∂ j τ as 1)=−ι k  j=1k jjRk−j(∂ j−1 τ as 1), together with (35), we see that I(∂k τ f)+I(∂k τ ωtfs +ιk∂k−1 τa0)−ιkI(∂k−1 τ as 1) + k−1  j=0k jRk−j(ι j∂j−1 τa0)−ι k−1  j=0 k! j!(k−(j+1))!Rk−(j+1)(ds∂j τ h) vanishes at τ=0. As hvanishes on R×∂M,wehavefor j=0,...,k−2, Rk−(j+1)(ds∂j τ h)=−ι(k−(j+1))Rk−(j+2)(∂ j τ h), and for j=k−1, Rk−(j+1)(ds∂j τ h)=I(ds∂j τ h)=0. We rewrite 123