The Light Ray transform on Lorentzian manifolds
Matti Lassas, Lauri Oksanen, Plamen Stefanov, Gunther Uhlmann
TTHE LIGHT RAY TRANSFORM ON LORENTZIAN MANIFOLDS
MATTI LASSAS, LAURI OKSANEN, PLAMEN STEFANOV, AND GUNTHER UHLMANN
Abstract.
We study the weighted light ray transform L of integrating functions on aLorentzian manifold over lightlike geodesics. We analyze L as a Fourier Integral Operatorand show that if there are no conjugate points, one can recover the spacelike singularitiesof a function f from its the weighted light ray transform Lf by a suitable filtered back-projection. Introduction
Let g be a Lorentzian metric with signature ( − , + , . . . , +) on the manifold M of dimen-sion 1 + n , n ≥
2. We study the weighted Light Ray Transform(1.1) L κ f ( γ ) = (cid:90) R κ ( γ ( s ) , ˙ γ ( s )) f ( γ ( s )) d s, of functions (or distributions) over light-like geodesics γ ( s ), known also as null geodesics.There is no canonical unit speed parameterization as in the Riemannian case as discussedbelow, and we have some freedom to chose parameterizations locally by smooth changes ofthe variables. We are interested in microlocal invertibility of L κ , that is, the descriptionof which part of the singularities of the function f can be reconstructed in a stable saywhen L κ f is given. Observe that this property does not depend on the parameterization.Here κ is a weight function, positively homogeneous in its second variable of degree zero,which makes it parameterization independent. When κ = 1, we use the notation L . Thistransform appears in the study of hyperbolic equations when we want to recover a potentialterm, or other coefficients of the equation, from boundary or scattering information, see,e.g., [34, 32, 31, 43, 44, 33, 1, 20, 23, 41, 21, 22, 3] for time dependent coefficients orin Lorentzian setting, and also [2, 25] for time-independent ones. This problem arises inmedical ultrasound tomography (see Section 5 on applications for the details). The tensorialversion of inverse problem for the weighted Light Ray Transform arises in the recovery offirst order perturbations [41] and in linearized problem of recovery a Lorentzian metricfrom remote measurements [23]. The latter is motivated by the problem of recovering thetopological defects in the early stages of the Universe from the red shift data of the cosmic Date : July 5, 2019.ML partly supported by Academy of Finland, grants 273979, 284715, 312110, 314879 and the AtMathproject of UH.LO partly supported by EPSRC Grant EP/P01593X/1 and EP/R002207/1.PS partly supported by NSF Grant DMS-1600327.GU was partly supported by NSF. a r X i v : . [ m a t h . A P ] J u l LASSAS, OKSANEN, P. STEFANOV, AND G. UHLMANN background radiation collected by the Max Planck satellite. The light tray transform L belongs to the class of the restricted X-ray transforms since the complex of geodesics isrestricted to the lower dimensional manifold g ( ˙ γ, ˙ γ ) = 0.The goal of the paper is to study the microlocal invertibility of L κ under some geometricconditions. Injectivity of L on functions in the Minkowski case was proved in [34]. Supporttheorems for analytic metrics and weights were proven in [35], see also [30] for a supporttheorem of L on one-forms in the Minkowski case. Those results in particular imply injec-tivity under some geometric conditions. Microlocal invertibility or the lack of it howeveris important in order to understand the stability of that inversion. It is fairly obviousthat L κ f cannot “see” the wave front set WF( f ), of the function f , in the timelike conebecause L κ is smoothing there. This just follows from the inspection of the wave frontof the Schwartz kernel of L κ , see also Theorem 2.1 for the Minkowski case. Microlocalinvertibility for Minkowski metrics was studied in [23, 42]. We show that in the generalLorentzian setting, one can recover WF( f ) in the spacelike cone if there are no conjugatepoints. In relativistic setting, this roughly speaking means that given L κ f , one can deter-mine the discontinuities (or the other singularities) of f that move slower than the speed oflight. Some restrictions are needed even in the Riemannian case. One possible approach isto analyze the normal operator L (cid:48) κ L κ as in [9, 10, 12]. That operator is a Fourier IntegralOperator (FIO) associated with two intersecting Lagrangians, see [11] and the referencesthere for that class and the I p,l calculus of such operators. The analysis of L (cid:48) κ L κ in theMinkowski case for n = 2 is presented in [9, 10, 12] as an example illustrating a muchmore general theory. Applying the I p,l calculus to get more refined microlocal results how-ever requires the cone condition which cannot be expected to hold on general Lorentzianmanifolds due to the lack of symmetry, as pointed out in [12]. We analyze L κ as an FIOand show that given any conically compact set K in the spacelike cone, one can choose asuitable pseudodifferential operator (ΨDO) cutoff Q so that L (cid:48) QL is a ΨDO elliptic in aneighborhood of K ; therefore we can recover the singularities of f from X κ f in K .The paper is organized as follows. In section 2, we analyze the flat Minkowski casewhere the formulas are more explicit. The Lorentzian case is studied in section 3, whichcontains our main results. In section 4, we show that when n = 2, singularities can actuallycancel each other over pairs of conjugate points, similarly to the Riemannian case [24]. Insection 5, we present two applications where the light ray transform appears naturally andour results can be applied: recovery of a time dependent potential in a wave equation inLorentzian geometry and recovery of a linearization of a time dependent sound speed neara background stationary one. 2. The Minkowski case
Let g = − d t + (d x ) + · · · + (d x n ) be the Minkowski metric in R n . Future pointinglightlike geodesics (lines) are given by(2.1) (cid:96) z,θ ( s ) = ( s, z + sθ )with z ∈ R n and | θ | = 1. This definition is based on parameterization of the lightlikegeodesics by their point of intersection with the spacelike hypersurface t = 0 and direction HE LIGHT RAY TRANSFORM 3 (1 , θ ). The parameterization ( z, θ ) defines a natural topology and a manifold structure ofthe set of the future pointing lightlike geodesics, which we denote by M below. We definethe light ray transform(2.2) Lf ( z, θ ) = (cid:90) R f ( s, z + sθ ) d s, z ∈ R n , θ ∈ S n − . The lightlike geodesics can be reparameterized by shifting and rescaling s . Our choice isbased on having a unit orthogonal projection θ on t = 0 but if we choose another spacelikehyperplane of hypersurface, this changes. Therefore, there is no canonical choice of theparameter along the lightlike lines. Note also that the notion of unit projection θ is notinvariantly defined under Lorentzian transformations, but in a fixed coordinate system, thescaling parameter 1 (i.e., d t/ds = 1) is a convenient choice. More generally, we could usea parameterization locally near a lightlike geodesic γ , by choosing initial points on anyhypersurface S transversal to γ , and initial lightlike directions; and we can identify thelatter with their projections onto S . We will use such a choice in Section 3 below when weconsider more general Lorentzian manifolds.Given a weight κ ∈ C ∞ ( R × R n × S n − ), we can define the weighted version L κ of L by L κ f ( z, θ ) = (cid:90) R κ ( s, z + sθ, θ ) f ( s, z + sθ ) d s, z ∈ R n , θ ∈ S n − . Under a smooth change of the parameterization s (cid:55)→ α ( z, θ ) s with some α >
0, the weightis transformed into a new one: κ/α , and the microlocal properties we study remain un-changed.In the terminology of relativity theory, vectors v = ( v , v (cid:48) ) satisfying | v | < | v (cid:48) | (i.e., g ( v, v ) >
0) are called spacelike . The simplest example are vectors (0 , v (cid:48) ), v (cid:48) (cid:54) = 0. Vectorswith | v | > | v (cid:48) | (i.e., g ( v, v ) <
0) are timelike ; an example is (1 ,
0) which points alongthe time axis.
Lightlike vectors are those for which we have equality: g ( v, v ) = 0. Forcovectors, the definition is the same but we replace g by g − , which is consistent with theoperation of raising and lowering the indices. Of course, in the Minkowski case g and g − coincide. We say that a hypersurface is timelike, respectively spacelike, if its normal (whichis a covector) is spacelike, respectively timelike.We introduce the following three microlocal regions of T ∗ R n \ s = { ( t, x ; τ, ξ ); | τ | < | ξ |} ;lightlike cone, Σ l = { ( t, x ; τ, ξ ); | τ | = | ξ |} ;timelike cone, Σ t = { ( t, x ; τ, ξ ); | τ | > | ξ |} .In the Minkowski case, we can think of them as products of R n and the correspondingcones in the dual space R n .2.1. Fourier Transform analysis.
By the Fourier Slice Theorem, knowing the X-raytransform for some direction ω recovers uniquely the Fourier transform ˆ f , of function f ,on ω ⊥ if, say, f is compactly supported. More precisely, the Fourier Slice Theorem in our LASSAS, OKSANEN, P. STEFANOV, AND G. UHLMANN θzt
Figure 1.
Knowing Lf ( z, θ ) for all z and a fixed θ recovers the FourierTransform of f in codirections conormal to the lightlike lines in that set.Knowing it near some ( z, θ ) recovers WF( f ) near those codirections alongthe line.case can be written as(2.3) ˆ f | τ + ξ · θ =0 = ˆ f ( − θ · ξ, ξ ) = (cid:90) R n e − iz · ξ Lf ( z, θ ) d z, ∀ θ ∈ S n − . The proof is immediate, and is in fact a consequence of the Fourier Slice Theorem in R n for lines restricted to lightlike ones. The union of all (1 , θ ) ⊥ for all unit θ is {| τ | ≤ | ξ |} (theunion Σ s ∪ Σ l of the spacelike and the lightlike cones), as is easy to see. This correlateswell with the theorems below. In particular, we see that knowing Lf for a distribution f for which Lf is well defined, and so is its Fourier transform, recovers ˆ f in the spacelikecone Σ s uniquely and in a stable way. Under the assumption that supp f is contained inthe cylinder | x | ≤ R for some R (and temperate w.r.t. t ), one can use the analyticity ofthe partial Fourier transform of f w.r.t. x to extend ˆ f analytically to the timelike cone,as well. This is how it has been shown in [34] that L is injective on such f . More generalsupport theorems and injectivity results, including such for analytic Lorentzian metrics,can be found in [35].2.2. The normal operator L (cid:48) L . We formulate here a theorem about the Schwartz kernelof the normal operator N = L (cid:48) L , where L (cid:48) is the transpose in terms of distributions (thesame as the L adjoint L ∗ because the kernel of L is real). The measure on R n × S n − isthe standard product one. One way to prove the theorem is to think of L as a weightedversion of the X-ray transform L with a distributional weight 2 √ | ξ | δ ( τ − | ξ | ) and usethe results about the weighted X-ray transform, see e.g. [37], and allow a singular weightthere. See also [36, 23]. Theorem 2.1.
For every f ∈ C ∞ ( R n ) ,(a) L (cid:48) Lf = N ∗ f, N ( t, x ) = δ ( t − | x | ) + δ ( t + | x | ) | x | n − . (b) L (cid:48) Lf = C n F − ( | ξ | − τ ) n − + | ξ | n − F f, C n := 2 π | S n − | . HE LIGHT RAY TRANSFORM 5 (c) h ( (cid:3) ) f = C − n | D x | n − (cid:3) − n + L (cid:48) Lf, where h is the Heaviside function, and (cid:3) = ∂ t − ∆ z and F is the Fourier transform. Before proving Theorem 2.1, we make some comments. Above, we used the notation s m + = max( s m ,
0) with the convention that s is the Heaviside function. In particular,when n = 3, we get σ ( L (cid:48) L ) = 4 π | ξ | − h (cid:0) | ξ | − τ (cid:1) . Then h ( (cid:3) ) f = (4 π ) − | D z | L (cid:48) Lf.
As we can expect, there is a conormal singularity of the symbol even away from ξ = 0 livingon the characteristic cone. Moreover, L (cid:48) L is elliptic in the spacelike cone, and only there.This shows that L (cid:48) L is a formal ΨDO with a singular symbol having singularities conormalto the light cone τ = | ξ | , i.e., it is an FIO corresponding to two intersecting Lagrangians.This is one of the main examples in [12]. The theorem shows that “singularities travelingslower than light” can be recovered stably from Lf known globally. The ones travelingfaster cannot. Proof of Theorem 2.1.
To compute the dual L (cid:48) of L , write (cid:104) Lf, φ (cid:105) = (cid:90) S n − (cid:90) R n (cid:90) R f ( s, x + sθ ) φ ( x, θ ) d s d x d θ = (cid:90) S n − (cid:90) R n (cid:90) R f ( s, x ) φ ( x − sθ, θ ) d s d x d θ. Therefore,(2.4) L (cid:48) φ ( t, x ) = (cid:90) S n − φ ( x − tθ, θ ) d θ, φ ∈ C ∞ ( R n × S n − ) . In particular, this identity allows us to define L on E (cid:48) ( R n ) by duality.By (1.1) and (2.4), L (cid:48) Lf ( t, x ) = (cid:90) S n − (cid:90) R f ( s, x − tθ + sθ ) d s d θ = (cid:90) S n − (cid:18)(cid:90) s For the second one, we have (cid:90) S n − (cid:90) s>t f ( s, x − tθ + sθ ) d s d θ = (cid:90) S n − (cid:90) ∞ f ( t + σ, x + σθ ) d σ d θ = (cid:90) R n f ( t + | x − x (cid:48) | , x (cid:48) ) | x − x (cid:48) | n − d x (cid:48) . This completes the proof of (a). To prove (b), one can take formally the Fourier transformof N to get(2.6) ˆ N ( τ, ξ ) = 2 π (cid:90) S n − δ ( τ + θ · ξ ) d θ. This representation can also be justified by writing (2.3) in the formˆ f ( − θ · ξ, ξ ) = F z → ξ Lf ( z, θ ) = ⇒ Lf ( z, θ ) = F − ξ → z ˆ f ( − θ · ξ, ξ ) . Then (cid:104) Lf, Lg (cid:105) = (2 π ) − n (cid:90) (cid:90) S n − ˆ f ( − θ · ξ, ξ )ˆ g ( − θ · ξ, ξ ) d θ d ξ = (2 π ) − n (cid:90) (cid:90) S n − δ ( τ + θ · ξ ) ˆ f ( τ, ξ )ˆ g ( τ, ξ ) d θ d τ d ξ. (2.7)Therefore, if we denote for a moment by K the integral in (2.6) but multiplied by (2 π ) − n instead of 2 π , we get (cid:104) L F − ˆ f , L F − ˆ g (cid:105) = (cid:104) ˆ f , K ˆ g (cid:105) ; hence F ∗− L (cid:48) L F − = K . Since F ∗ =(2 π ) n F − , we get (2.6).To compute ˆ N explicitly, take a test function φ ( τ, ξ ) and write (cid:104) ˆ N , φ (cid:105) = 2 π (cid:90) (cid:90) S n − δ ( τ + θ · ξ ) φ ( τ, ξ ) d θ d τ d ξ = 2 π (cid:90) (cid:90) S n − φ ( − θ · ξ, ξ ) d θ d ξ = 2 π (cid:90) (cid:90) F ( s, ξ ) φ ( s, ξ ) d s d ξ with F is the L function in (2.9) below. This proves (b).Part (c) of the lemma follows directly from (b). (cid:3) We used the following lemma. Lemma 2.1. For every ψ ∈ S ( R n ) , (2.8) (cid:90) S n − ψ ( θ · ξ ) d θ = | S n − || ξ | − n (cid:90) R ψ ( s )( | ξ | − s ) n − + d s, ξ (cid:54) = 0 , where | S n − | is the area of S n − if n ≥ ; equal to when n = 2 . Note that the kernel(2.9) F ( s, ξ ) := | S n − || ξ | − n ( | ξ | − s ) n − + is homogeneous of order − − L function. Also, the l.h.s. of(2.8) is a smooth function of ξ everywhere, including at ξ = 0. HE LIGHT RAY TRANSFORM 7 L κ as an FIO. Theorem 2.1(c) implies some recovery of singularities results already.If f ∈ E (cid:48) ( R n ) and L κ f ∈ C ∞ ( R n × S n − ), then WF( f ) does not contain spacelikesingularities (note that this argument requires global knowledge of L κ f ). One the otherhand, one can easily construct functions of distributions with timelike singularities so that L κ f = 0; for example take any non-smooth h ∈ E (cid:48) ( R ) with integral zero, then for any a ∈ R n with | a | < 1, for f = h ( t + x · a ) we have Lf = 0; andWF( f ) = { ( t, x, τ, ξ ) | t = s − x · a, ξ = aτ, ( s, τ ) ∈ WF( h ) } . Then | ξ | = | τ || a | < | τ | is in the timelike cone. In particular, δ (cid:48) ( t ) is in the kernel of L κ andhas timelike singularities only.We can get more precise statements by studying first the Schwartz kernel of L κ . It isgiven by(2.10) L κ ( z, θ, t, x ) = κ ( t, x, θ ) δ ( x − z − tθ ) . In other words, L κ = κδ X , where X = { ( z, θ, t, x ) | x = z + tθ } is the point-line relation. We write M = R n = R nt,x and let M ∼ = R nz × S n − θ be themanifold of the lines in M . Clearly, X is a 2 n -dimensional submanifold of the product M × M ∼ = R nz × S n − θ × R nt,x which itself is 3 n -dimensional. Its conormal bundle is givenby N ∗ X = (cid:110) (( z, θ, t, x ) , ( ζ, ˆ θ, τ, ξ )) (cid:12)(cid:12) x = z + tθ, ξ = − ζ, τ = − θ · ξ, ˆ θ = t ( − ξ + ( ξ · θ ) θ ) (cid:111) with ˆ θ conormal to S n − at θ . We consider N ∗ X as a subset of T ∗ ( M × M ) \ 0. This is aconical Lagrangian manifold which coincides with the wave front set of the kernel L κ when κ is nowhere vanishing; and includes the latter for general κ .Note that ( τ, ξ ) is space or light-like on N ∗ X and it is the latter if and only if ζ (cid:107) θ. (2.11)Indeed, | τ | = | ξ | is equivalent with | θ · ζ | = | ζ | on N ∗ X . As will be explained below, therelation (2.11) allows us to choose a microlocal cutoff on M so that when applied to L κ f ,it cuts away the singularities in WF( f ) near Σ l . This will be useful in view of the singularbehavior near Σ l , as illustrated for L (cid:48) L in Theorem 2.1.Let us also mention that | τ | = | ξ | is equivalent with − ξ + ( ξ · θ ) θ = 0 on N ∗ X . Inparticular, ˆ θ = 0 in this case. We will show, see Lemma 3.4 below, that on generalLorentzian manifolds, ( τ, ξ ) being lightlike on N ∗ X is equivalent with ζ (cid:107) θ and ˆ θ = 0 (orrather its suitable reformulation in the more general context).The canonical relation associated to L κ is given by C := N ∗ X (cid:48) = (cid:110) (( z, θ, ζ, ˆ θ ) , ( t, x, τ, ξ )) ∈ T ∗ ( M × M ) \ (cid:12)(cid:12) x = z + tθ, τ = − θ · ξ, ζ = ξ, ˆ θ = t ( ξ − ( ξ · θ ) θ ) (cid:111) . (2.12) LASSAS, OKSANEN, P. STEFANOV, AND G. UHLMANN Here we rearranged the variables to comply with the notational convention in [18]. Ifone of the covectors ( ζ, ˆ θ ) and ( τ, ξ ) vanishes, then the other one does, too. Therefore, C is a (homogeneous) canonical relation from T ∗ M \ T ∗ M \ T ∗ ( M × M ) \ 0. Therefore, this, and the fact that its kernel is aconormal distribution, show that L κ is an FIO with the canonical relation C , see [18,Chapter XXV.25.2]. In particular,(2.13) WF( L κ f ) ⊂ C ◦ WF( f ) , a statement independent of the FIO theory. In order to compute the order of L κ , we canwrite its Schwartz kernel as the oscillatory integral κ ( t, x, θ )(2 π ) n (cid:90) R n e i ( tθ · ξ +( z − x ) · ξ ) d ξ, see (2.10). Then the order m of L κ satisfies, see [16, Def. 3.2.2],0 = m + (dim( M ) + dim( M ) − n ) / , that is, m = − n/ , (2.14)because M = R n and M ∼ = R n × S n − .The relation C also allows the following interpretation: it consists of points and lightlikelines through them; next, ( τ, ξ ) is conormal to (1 , θ ), i.e., to each such line (cid:96) z,θ ; and thedual variables ( ζ, ˆ θ ) can be interpreted as projections of Jacobi fields along the line (cid:96) z,θ to its conormal bundle. This interpretation is discussed further in Section 3 below in thecontext of general Lorentzian manifolds, see (3.8).Let π M , π M be the natural projections of C onto T ∗ M and T ∗ M , respectively.(2.15) CT ∗ M T ∗ M π M π M The dimensions from left to right are 4 n − ≥ n ≥ n + 2. The difference between twoconsecutive terms is n − n = 2. The manifold Z can beparameterized by ( z, θ, t ). Then C can be parameterized by C = { ( z, θ, t, ξ ) ∈ R n × S n − × R × ( R n \ } . . We have(2.16) π M (( z, θ, ζ, ˆ θ ) , ( t, x, τ, ξ )) = ( z, θ, ζ, ˆ θ ) = ( z, θ, ξ, t ( ξ − ( ξ · θ ) θ )) . This is a map from the 3 n dimensional C to the 4 n − T ∗ M . If n = 2, π M is a local diffeomorphism when C is restricted to spacelike ( τ, ξ ). Indeed, we recallthat in that case ξ − ( ξ · θ ) θ (cid:54) = 0. Therefore, the equation ˆ θ = t ( ξ − ( ξ · θ ) θ ) can besolved for t . When n ≥ 3, d π M has full rank 3 n away from the light-like cone, i.e., thedefect is (4 n − − n = n − π M isalso injective, therefore, it is an immersion (on the spacelike cone). Next, there is t ∈ R HE LIGHT RAY TRANSFORM 9 such that ˆ θ = t ( ξ − ( ξ · θ ) θ ) if and only if ˆ θ is colinear with the projection of ζ = ξ to θ ⊥ = { ξ | ξ · θ = 0 } , which describes the range of π M for ( τ, ξ ) spacelike.If ( τ, ξ ) is lightlike, then the right-hand side of (2.16) reduces to ( z, θ, ξ, τ j , ξ j ), j = 1 , 2, the equation π M (( z, θ, ζ, ˆ θ ) , ( t , x , τ , ξ )) = π M (( z, θ, ζ, ˆ θ ) , ( t , x , τ , ξ ))(2.17)is equivalent with ( τ , ξ ) = ( τ , ξ ) and both ( t j , x j ), j = 1 , 2, lying on the line (cid:96) z,θ .For the second projection π M in (2.15) we get π M (( z, θ, ζ, ˆ θ ) , ( t, x, τ, ξ )) = ( t, x, τ, ξ ) = ( t, z + tθ, − θ · ξ, ξ ) . (2.18)Its differential has full rank for spacelike ( τ, ξ ). The projection π M is surjective onto thespacelike cone, as well. Indeed, given ( t, x, τ, ξ ), we need to solve the equation given bythe second equality above for the parameters ( z, t, θ, ξ ). The variables t and ξ are obtainedtrivially, and we need to solve x = z + tθ and τ = − θ · ξ for z and θ . For unit θ , the latterequation has an ( n − 2) dimensional sphere of solutions (the intersection of the unit spherewith that plane in the θ space) when ( τ, ξ ) is spacelike. For each solution θ , we obtain z bysolving z + tθ = x . We can choose a locally smooth solution, which in particular shows thatthe differential has full rank. If ( τ, ξ ) is lightlike, i.e., if | τ | = | ξ | , the equation τ = − θ · ξ has a unique solution for θ given by θ = − sgn( τ ) ξ/ | ξ | . If ( τ, ξ ) is timelike, there are nosolutions.If n = 2, π M is a local diffeomorphism and it is 2-to-1 in the spacelike cone because τ = − θ · ξ has two solutions for θ ∈ S : θ ± = ± (cid:112) − τ / | ξ | ( ξ ⊥ / | ξ | ) − τ ξ/ | ξ | for spacelike( τ, ξ ) with some fixed choice of the rotation by π/ ξ ⊥ . This describes the non-uniqueness class of π M .We summarize the properties of the projections π M and π M as follows. Lemma 2.2. The differential d π M is injective and the differential d π M is surjective at ( z, θ, t, ξ ) ∈ C , with ξ spacelike. The projection π M is injective on the set of points ( z, θ, t, ξ ) ∈ C , with ξ spacelike. The projection π M is surjective onto Σ s . Let us also summarize the properties of C considered as a relation (a multi-valued map C = π M ◦ π − M ). Lemma 2.3. C has domain Σ s ∪ Σ l . For every ( t, x, τ, ξ ) ∈ Σ s , C ( t, x, τ, ξ ) is the set ofall ( x − tθ, θ, ξ, t ( ξ − ( ξ · θ ) θ )) with θ ∈ S n − a solution of ( τ, ξ ) · (1 , θ ) = 0 . (a) If n = 2 , then C is a local diffeomorphism from Σ s to T ∗ M \ , and a -to- mapglobally on Σ s . (b) If n ≥ , for every ( t, x, τ, ξ ) ∈ Σ s , C ( t, x, τ, ξ ) is diffeomorphic to S n − .For every ( t, x, τ, ξ ) ∈ Σ l , C ( t, x, τ, ξ ) = ( x − tθ, θ, ξ, , where θ = − sgn( τ ) ξ/ | ξ | . Inparticular, C (Σ l ) = (cid:8) ( z, θ, ζ, ˆ θ ) (cid:12)(cid:12) ζ (cid:107) θ, ˆ θ = 0 (cid:9) \ . In particular, this proposition says that WF( f ) in the spacelike cone may affect WF( L κ f )at all lightlike lines (cid:96) through the base point and normal to its the covector there.The properties of C − are summarized as follows. Lemma 2.4. C − has domain in T ∗ M \ consisting of all ( z, θ, ζ, ˆ θ ) so that ˆ θ is colinearwith the projection of ζ to θ ⊥ . Its range is Σ s ∪ Σ l . The points mapped to Σ l are the oneswith θ (cid:107) ζ . For every ( z, θ, ζ, ˆ θ ) in the domain with θ (cid:54) (cid:107) ζ , we have C − ( z, θ, ζ, ˆ θ ) = ( t, z + tθ, − θ · ζ, ζ ) , where t is the unique solution to ˆ θ = t ( ξ − ( ξ · θ ) θ ) . When n = 2, the colinearity condition is automatically satisfied. Indeed, the space θ ⊥ is one dimensional then and therefore any ˆ θ ∈ T ∗ θ S ∼ = θ ⊥ is colinear with the projectionof ζ to θ ⊥ . When n ≥ 3, unlike C , the relation C − is a map away from θ (cid:107) ζ . It is notinjective by Lemma 2.3.Most importantly for the purposes of the present paper, the composition C − ◦ C reducesto the identity on Σ s . This can be deduced directly from Lemma 2.2, as will be done inthe proof of Lemma 3.11 in the general Lorentzian context, however, we will give here aproof based on Lemmas 2.3 and 2.4. Lemma 2.5. For every ( t, x, τ, ξ ) ∈ Σ s it holds that ( C − ◦ C )( t, x, τ, ξ ) = ( t, x, τ, ξ ) .Proof. From Lemma 2.3, C ( t, x, τ, ξ ) = { ( z, θ, ξ, ˆ θ ) | z = x − tθ, ˆ θ = t ( ξ − ( ξ · θ ) θ ) , θ ∈ S n − , − θ · ξ = − τ } , and from Lemma 2.4, for ( z, θ, ξ, ˆ θ ) ∈ C ( t, x, τ, ξ ), C − ( z, θ, ξ, ˆ θ ) = ( t, x, τ, ξ ) . (cid:3) Recovery of spacelike singularities. Lemma 2.5 suggests that the composition of L κ with its transpose L (cid:48) κ could be a pseudodifferential operator when restricted on Σ s . Onthe other hand, by Propositions 2.3 and 2.4, C maps the lightlike cone to { ζ (cid:107) θ } , and C − maps the latter to the former. As anticipated above, this suggests that we could cut thedata L κ f microlocally near { ζ (cid:107) θ } to apply a cutoff to WF( f ) near Σ l . This is not anautomatic application of Egorov’s theorem however because C and C − are singular nearthe lightlike cone (and its image under C ) and L κ is not a classical FIO there, in sensethat the associated canonical relation is not a canonical graph. Next theorem gives localrecovery of space like singularities from local data. It is similar to Proposition 11.4 in ourprevious paper [23]. Theorem 2.2. Let Q = q ( z, θ, D z ) be a Ψ DO in M with a symbol q ( z, θ, ζ ) of order zero(independent of ˆ θ ) supported in {| θ · ζ | < | ζ |} . Then L (cid:48) κ QL κ is a Ψ DO in M of order − with essential support in the spacelike cone.Suppose, moreover, that κ is nowhere vanishing. Let U ⊂ R n × S n − be a neighborhoodof ( z , θ ) ∈ R n × S n − , and let ( t , x , τ , ξ ) ∈ Σ s ∩ N ∗ (cid:96) z ,θ . Then Q can be chosen sothat its essential support is contained also in U × ( R n \ and that L (cid:48) κ QL κ is elliptic at ( t , x , τ , ξ ) . HE LIGHT RAY TRANSFORM 11 Proof. The Schwartz kernel L κ of L κ has a wave front set N ∗ Z and C is its relation, seealso (2.13). We can always assume that the essential support ess-sup( Q ) of Q is conicallycompact. The twisted wave front set of the Schwartz kernel of Q as a relation is identityrestricted to ess-sup( Q ). Then its composition with the relation C is C again with itsimage restricted by Q to ess-sup( Q ) which is contained in the conic set {| θ · ζ | < | ζ |} . By(2.12), this implies | τ | < | ξ | near the wave front of the kernel of QL κ . Therefore, QL κ issmoothing in a conic neighborhood of Σ l ∪ Σ t , and so is L (cid:48) κ QL κ .The composition L (cid:48) κ QL κ can be analyzed by using the the transversal intersection cal-culus in the case n = 2, and the clean intersection calculus in the case n = 3. As thecomposition C − ◦ C is the identity on Σ s , the calculi imply that L (cid:48) κ QL κ is a ΨDO of order − 1. We will focus on the more complicated case n ≥ 3, and justify the application of theclean intersection calculus in the next section.Writing σ [ · ] for the principal symbol map, it holds that σ [ L (cid:48) κ QL κ ] is obtained from σ [ L (cid:48) κ ] = κ , σ [ Q ] and σ [ L κ ] = κ by an integration reducing the excess, see [17, Theo-rem 25.2.3]. We will choose Q so that σ [ Q ] is non-negative. As κ is nowhere vanishing, σ [ L (cid:48) κ QL κ ] is positive at ( t , x , τ , ξ ) if and only if the integral of σ [ Q ] does not vanishover the fiber C ( t , x , τ , ξ ).We set ζ = ξ and choose Q so that σ [ Q ]( z , θ , ζ ) > 0. It holds that( z , θ , ζ , ˆ θ ) ∈ C ( t , x , τ , ξ ) , where ˆ θ = t ( ξ − ( ξ · θ ) θ ). Indeed, this follows from Lemma 2.3 since the assumption( t , x , τ , ξ ) ∈ N ∗ (cid:96) z ,θ implies that x = z + t θ and ( τ , ξ ) · (1 , θ ) = 0. Therefore, σ [ Q ] does not vanish identically on C ( t , x , τ , ξ ).It still remains to show that the choice σ [ Q ]( z , θ , ζ ) > Q ) ⊂ {| θ · ζ | < | ζ |} . This follows, since together with ζ = ξ and( t , x , τ , ξ ) ∈ Σ s , the orthogonality ( τ , ξ ) · (1 , θ ) = 0 implies that | θ · ζ | = | τ | < | ξ | = | ζ | . (cid:3) As a corollary, we have the following global result saying that the space like singularitiescan be recovered. Corollary 2.1. Let L κ f ∈ C ∞ ( M ) and assume that κ vanishes nowhere. Then it holdsthat WF( f ) ∩ Σ s = ∅ .Proof. For any ( t , x , τ , ξ ) ∈ Σ s we can choose a lightlike line (cid:96) z ,θ such that ( t , x , τ , ξ )is in N ∗ (cid:96) z ,θ . Then the previous corollary implies that ( t , x , τ , ξ ) / ∈ WF( f ). (cid:3) By combining Theorem 2.2 with a microlocal partition of unity, we can recover, not onlyWF( f ), but a smoothened version of f with the singularities cut off (in a smooth way) inany predetermined neighborhood of Σ t ∪ Σ l . This can be viewed as a regularized inversionof L κ with the regularization cutting away from the ill posed region Σ t and its boundaryΣ l . Let us also give a more explicit construction as follows. We can choose φ ∈ C ∞ ( R ) suchthat φ = 1 on [0 , − ε ] and φ = 0 on [1 − ε/ , ∞ ). Let Q be the zeroth order ΨDO withsymbol φ ( | θ · ζ | / | ζ | ) cut off smoothly near the origin (which is actually not needed). Then L (cid:48) κ QL κ is a ΨDO of order − 1, elliptic away from a neighborhood of Σ t ∪ Σ l determinedby (cid:15) . When κ = 1, one can compute L (cid:48) QL directly. Since | θ · ζ | / | ζ | is a Fourier multiplierw.r.t. z only, it is enough to express QLf by taking the Fourier transform of Lf w.r.t. z only. Then from (2.7) we get L (cid:48) QL = F − φ ( | τ | / | ξ | ) | ξ | (cid:0) | − τ / | ξ | (cid:1) n − F . Therefore, φ ( | D t | / | D x | ) f = | D x | (cid:0) − D t / | D x | (cid:1) n − + L (cid:48) QLf. The clean intersection calculus. We assume that n ≥ 3, and show here thatthe clean intersection calculus can be applied to L (cid:48) κ QL κ in Theorem 2.2. The traditionalformulation of this calculus considers the composition A A of two properly supportedFourier integral operators A and A such that the composition C ◦ C of their canonicalrelations C ⊂ ( T ∗ X \ × ( T ∗ Y \ , C ⊂ ( T ∗ Y \ × ( T ∗ Z \ , is clean, proper and connected [18, Th. 25.2.3]. Here X , Y and Z are smooth manifolds.The operators A = L (cid:48) κ and A = QL κ do not quite satisfy the assumptions of the calculus,since the composition C − ◦ C is clean only away from Σ l . Also, as a canonical relation, C must be closed in T ∗ ( M × M ) \ 0, and we can not simply apply the calculus with C replaced by C \ ( T ∗ M × Σ l ).The proof of [18, Th. 25.2.3] uses a microlocal partition of unity, subordinate to a coverΓ j , j = 1 , , . . . , of the intersection X ∩ Y where X = C × C , Y = T ∗ X × diag( T ∗ Y ) × T ∗ Z, and diag( T ∗ Y ) = { ( p, p ); p ∈ T ∗ Y } . We write K for the Schwartz kernel of A , and recallthat the essential support of A is given byess-sup( A ) = WF (cid:48) ( K ) = { ( x, ξ, y, − η ) | ( x, ξ, y, η ) ∈ WF( K ) } . For the local step of the proof, it is enough to assume that the composition C ◦ C is cleanin each Γ j that intersect the product ess-sup( A ) × ess-sup( A ). The composition C ◦ C being clean in Γ j means that X ∩ Y ∩ Γ j is a smooth manifold and that T p ( X ∩ Y ) = T p X ∩ T p Y , p ∈ X ∩ Y ∩ Γ j . (2.19)The local step implies that ( C ◦ C ) (cid:48) is locally a conic Lagrangian manifold, how-ever, global assumptions are needed, for example, to guarantee that it does not haveself-intersections. The assumptions that C ◦ C is proper and connected are used in theproof [18, Th. 25.2.3] to show that C ◦ C is an embedded submanifold of T ∗ ( X × Z ) \ HE LIGHT RAY TRANSFORM 13 In our case, A = L (cid:48) κ and A = QL κ , X = C − × C, Y = T ∗ M × diag( T ∗ M ) × T ∗ M, and ess-sup( A ) ⊂ Σ s due to the microlocal cutoff Q . Thus we need to consider thecondition (2.19) only for Γ j ⊂ T ∗ M × diag( T ∗ M ) × Σ s (2.20)As for the global structure of C − ◦ C , we already know that A A is smoothing in a conicneighborhood of Σ l ∪ Σ t , and that C − ◦ C is the identity on Σ s . In particular, C − ◦ C is an embedded submanifold of T ∗ ( M × M ) \ A A ), andclosed as its subset.Let us now show that (2.19) holds for (2.20). We write C = X ∩ Y ∩ Γ j and ˜ C = C ∩ ˜Γ j ,where ˜Γ j is the projection of Γ j on T ∗ M× Σ s . Also, we use ∼ = to denote that two manifoldsor vector spaces are isomorphic. Let (˜ t, ˜ x, ˜ τ , ˜ ξ ; z, θ, ζ, ˆ θ ; z, θ, ζ, ˆ θ ; t, x, τ, ξ ) ∈ C . Since(˜ t, ˜ x, ˜ τ , ˜ ξ ) = C − ( z, θ, ζ, ˆ θ ) = ( t, x, τ, ξ )and ( τ, ξ ) is spacelike, Lemma 2.4 implies that (˜ t, ˜ x, ˜ τ , ˜ ξ ) = ( t, x, τ, ξ ). This again impliesthat C ∼ = diag( ˜ C ) ∼ = ˜ C , in particular, C is a smooth manifold. Moreover, X ∼ = C . Let p ∈ ˜ C and observe that for ( δp, δq ) in T ( p,p ) X ∼ = T p C × T p C it holds that ( δp, δq ) ∈ T ( p,p ) Y if and only if d π M δp = d π M δq. Since d π M is injective (again due to ( τ, ξ ) being spacelike), we have δp = δq for all ( δp, δq )in T ( p,p ) X ∩ T ( p,p ) Y . Therefore T ( p,p ) X ∩ T ( p,p ) Y ∼ = T p ˜ C ∼ = T ( p,p ) C . Keeping track of the diffeomorphisms used above, this shows (2.19). We have shownthat the clean intersection calculus applies, and therefore L (cid:48) κ QL κ is a pseudodifferentialoperator.To establish that L (cid:48) κ QL κ has order − 1, we need to verify also that the order m = − n/ L κ and the excess e of the clean intersection satisfy 2 m + e/ − 1. We writeΠ : C → T ∗ ( M × M ) \ C γ = Π − ( { γ } ) for its fibers. The excess e coincides withdim( C γ ), and using again the identification C ∼ = ˜ C , we see that for all γ ∈ Π( C ) there is( t, x, τ, ξ ) ∈ Σ s such that C γ ∼ = { ( t, x, τ, ξ ; z, θ, ζ, ˆ θ ); ( z, θ, ζ, ˆ θ ) ∈ C ( t, x, τ, ξ ) } ∼ = C ( t, x, τ, ξ ) ∼ = S n − , where the last identification is given by part (b) of Lemma 2.3. Hence e = n − m + e/ − C ◦ C being con-nected means that the fibers C γ are connected (when C is taken to be the whole intersection X ∩ Y ). As we are assuming that n > 2, the fibers C γ are connected in our particular case.With a suitable cutoff, this can be arranged also in the more general Lorentzian context considered next, however, analogously to the above discussion, such connectedness is notessential. Even when not connected, the fibers C γ are smooth manifolds, since the projec-tion Π has constant rank by [17, Th. 21.2.14].3. The Lorentzian case Our aim is to prove an analogue of Theorem 2.2 in a more general Lorentzian context.Toward this end, we will consider the light ray transform on a Lorentzian manifold ( M, g ),localized near a lightlike geodesic segment γ : [0 , (cid:96) ] → M , the analogue of (cid:96) z ,θ in Theo-rem 2.2. We parameterize lightlike geodesics near γ by choosing a spacelike hypersurface H containing γ (0) and semigeodesic coordinates associated to H ,(3.1) ( t, z ) ∈ ( − T, T ) × Z, Z ⊂ R n , so that in the coordinates H = { t = 0 } and g = − dt + g (cid:48) , with g (cid:48) = g (cid:48) ( t, z ) a Riemannianmetric on Z that depends smoothly on t . Moreover, the coordinates are chosen so that γ (0) = 0 and ˙ γ (0) = (1 , θ ) where, writing h = g (cid:48) (0 , · ), it holds that ( θ , θ ) h = 1. Thenwe choose local coordinates of the form(3.2) ( z, a ) ∈ Z × A, A ⊂ R n − , on the unit sphere bundle SZ with respect to h , so that writing Z × A (cid:51) ( z, a ) (cid:55)→ ( z, θ ( z, a )) ∈ SZ (3.3)for the coordinate map, it holds that θ = θ (0 , γ z,a for the geodesic γ satisfying γ (0) = (0 , z ) and ˙ γ (0) = (1 , θ ( z, a )), and use also the notation M = Z × A . Analogouslyto (2.1), this parametrization gives the smooth manifold structure in the space of lightlikegeodesics near γ .Let Ω ⊂ M be open and relatively compact, and suppose that the end points γ (0) and γ ( (cid:96) ) are outside Ω. By making Z and A smaller, we suppose without loss of generalitythat the end points γ z,a (0) and γ z,a ( (cid:96) ) are outside Ω for all ( z, a ) ∈ M . In what follows weconsider the local version of the light ray transform defined as follows L κ f ( γ ) = (cid:90) (cid:96) κ ( γ ( s ) , ˙ γ ( s )) f ( γ ( s )) d s, f ∈ C ∞ (Ω) , γ = γ z,a , ( z, a ) ∈ M . (3.4)Observe that, given a geodesic γ : R → M , the integral (cid:82) R f ( γ ( s )) d s may not be well-defined even for f ∈ C ∞ (Ω) if γ returns to Ω infinitely often. We note that if ( M, g ) isglobally hyperbolic, L κ f ( γ ) can be defined for all f ∈ C ∞ ( M ). However, in this paperwe consider only the local version (3.4) in order to avoid making global assumptions on( M, g ).Note that the coordinates (3.1) are valid locally only; and we cannot use them in ouranalysis of the contributions of possible conjugate points on the geodesics γ z,a . They areused only to parametrize these geodesics. Moreover, the parametrization and, in particular,the normalization of ˙ γ (0), is not invariant. It depends on the choice of H and the coordi-nates (3.1)–(3.2). On the other hand, if ˜ H is another spacelike hypersurface intersecting γ , then the lightlike geodesic flow provides a natural map from T H to T ˜ H , however, the HE LIGHT RAY TRANSFORM 15 projections of the tangents of the geodesics γ z,a onto T ˜ H may not be of unit length. Ifthe geodesics γ z,a are re-parameterized so that the projections are unit, then the weight κ is multiplied by a smooth Jacobian. While this would change L κ , it would not change itsmicrolocal properties. We will use this fact later to choose H in a convenient way. θ ( z, a ) γ z,a H Kz ∂∂t ˙ γ z,a (0) Figure 2. Parameterization of lightlike geodesics near γ .3.1. Point-geodesic relation. The point-geodesic relation(3.5) X = { ( z, a, x ) ∈ M × Ω | x = γ z,a ( s ) for some s ∈ (0 , (cid:96) ) } is a smooth 2 n dimensional submanifold of the 3 n dimensional M × Ω, parameterized bythe map ( z, a, s ) (cid:55)→ ( z, a, γ z,a ( s )). Writing x = γ z,a ( s ), this map has differential Id 0 00 Id 0 ∂x/∂z ∂x/∂a ˙ γ z,a ( s ) d z d a d s which has maximal rank 2 n . The conormal bundle N ∗ X at any point is the space conormalto the range of that differential; that is, it is described by the kernel of its adjoint. Therefore,the canonical relation C := N ∗ X (cid:48) ⊂ T ∗ ( M × Ω) \ C = (cid:110)(cid:0) ( z, a, ζ, α ) , ( x, ξ ) (cid:1)(cid:12)(cid:12) x = γ z,a ( s ) , (cid:104) ξ, ˙ γ z,a ( s ) (cid:105) = 0 , ζ j = (cid:104) ξ, ∂ z j γ z,a ( s ) (cid:105) ,j = 1 , . . . , n, α k = (cid:104) ξ, ∂ a k γ z,a ( s ) (cid:105) , k = 1 , . . . , n − , s ∈ (0 , (cid:96) ) (cid:111) . (3.6)Clearly ζ = 0 and α = 0 if ξ = 0. It follows from Lemma 3.1 below that also the converseholds. Therefore C is closed in T ∗ ( M × Ω) \ 0, and L κ is a Fourier integral operator. TheSchwartz kernel of L κ is a conormal distribution on X with the (un-reduced) symbol κ ,and by [17, Th. 18.2.8], the order m of L κ is satisfies0 = m + (3 n − n ) / , that is m = − n/ . As in the Minkowski case, the covector ξ must be lightlike or spacelike at x as a con-sequence of (cid:104) ξ, ˙ γ z,a ( s ) (cid:105) = 0. Relation (2.13) holds in this case as well and it shows thattimelike singularities of f do not affect WF( L κ f ), that is, they are invisible. Moreover,the dimensions of the manifolds in the diagram (2.15) are unchanged from the Minkowskicase. The canonical relation C is parameterized by C = { ( z, a, s, ξ ) ∈ M × (0 , (cid:96) ) × T ∗ γ z,a ( s ) Ω | ξ (cid:54) = 0 , (cid:104) ξ, ˙ γ z,a ( s ) (cid:105) = 0 } . More precisely, C is a 3 n -dimensional smooth manifold and, in view of the definition (3.6),there is a diffeomorphism between C and C .3.2. Variations of the geodesics γ z,a . Let us consider the Jacobi fields associated tothe variations through the geodesics γ z,a , ( z, a ) ∈ Z × A ,(3.7) M j ( s ; z, a ) = ∂ z j γ z,a ( s ) , j = 1 , . . . , n, J k ( s ; z, a ) = ∂ a k γ z,a ( s ) , k = 1 , . . . , n − . Observe that by (3.6), it holds on C that(3.8) ζ j = (cid:104) ξ, M j ( s ; z, a ) (cid:105) , α k = (cid:104) ξ, J k ( s ; z, a ) (cid:105) , that is, the dual variables ζ and α are given by projections of the Jacobi fields M j and J k to ξ .For a vector field J along a curve γ , we use the shorthand notation J (cid:48) ( s ) = ∇ s J ( s ) forthe covariant derivative ∇ s = ∇ ˙ γ along γ . We write also˙ γ ( s ) ⊥ = { v ∈ T γ ( s ) M | ( v, ˙ γ ( s )) g = 0 } . Since every Jacobi field along a null geodesic is a certain variation of the latter, the lemmabelow in particular characterizes the Cauchy data ( J, J (cid:48) ) of such fields at any point. Lemma 3.1. Let ( z, a ) ∈ M and write γ = γ z,a . Write Γ( s ) = s ˙ γ ( s ) , and consider theJacobi fields J := span { M , . . . , M n , J , . . . , J n − , ˙ γ, Γ } along γ . Then for any s ∈ [0 , (cid:96) ] itholds that { ( J ( s ) , J (cid:48) ( s )) | J ∈ J } = { ( V, W ) ∈ ( T γ ( s ) M ) | W ∈ ˙ γ ( s ) ⊥ } . In particular, { J ( s ) | J ∈ J } = T γ ( s ) M .Proof. Let us begin by showing that ( J (cid:48) (0) , ˙ γ (0)) g = 0 for J ∈ J . Consider the curve r (cid:55)→ (0 , z + re j ) in coordinates (3.1), where z ∈ Z is fixed and e j is the n -dimensionalvector with 1 in the j th position, all other entries zero, and denote by ∇ z j the covariantderivative along this curve. Using the symmetry property ∇ s ∂ z j γ = ∇ z j ∂ s γ , we see that(3.9) (cid:0) M j (0) , M (cid:48) j (0) (cid:1) = ((0 , e j ) , (0 , ∇ z j θ )) , (cid:0) J k (0) , J (cid:48) k (0) (cid:1) = (cid:0) (0 , , (0 , ∂ a k θ ) (cid:1) , where θ is the map defined by (3.3). Hence ( M (cid:48) j (0) , ˙ γ (0)) g = ( ∇ z j θ, θ ) h = ∂ z j ( θ, θ ) h / J (cid:48) k (0) , ˙ γ (0)) g = 0. Finally, as ˙ γ (cid:48) = 0 and Γ (cid:48) = ˙ γ , we have shown that( J (cid:48) (0) , ˙ γ (0)) g = 0 for J ∈ J .Recall that ∂ s ( J (cid:48) ( s ) , ˙ γ ( s )) g = 0 for any Jacobi field J along γ . Therefore ( J (cid:48) , ˙ γ ) g = 0identically on [0 , (cid:96) ] for J ∈ J . In particular,(3.10) J ( s ) := { ( J ( s ) , J (cid:48) ( s )) | J ∈ J } ⊂ { ( V, W ) ∈ ( T γ ( s ) M ) | W ∈ ˙ γ ( s ) ⊥ } . The vectors ( J (0) , J (cid:48) (0)), J ∈ J , are linearly independent, as can be seen from (3.9)and from ( ˙ γ (0) , γ (cid:48)(cid:48) (0)) = ((1 , θ ) , , (Γ(0) , Γ (cid:48) (0)) = (0 , (1 , θ )) . HE LIGHT RAY TRANSFORM 17 As Jacobi fields satisfy a linear second order differential equation, it follows that the di-mension of J is 2 n + 1 and that the same is true for J ( s ), s ∈ [0 , (cid:96) ]. The claim follows from(3.10) since both the spaces there have the same dimension. (cid:3) For fixed s , s ∈ [0 , (cid:96) ], consider the spaces(3.11) J s = { J ∈ J | J ( s ) = 0 } , J (cid:48) s ,s = { J (cid:48) | J ∈ J s ∩ J s } , and set for every s ∈ [0 , (cid:96) ](3.12) J s ( s ) = { J ( s ) | J ∈ J s } , J (cid:48) s ,s ( s ) = { J (cid:48) ( s ) | J (cid:48) ∈ J (cid:48) s ,s } . Lemma 3.1 implies that J (cid:48) s ,s ( s ) ⊂ ˙ γ ( s ) ⊥ . The same is true for J s ( s ) since ∂ s ( J, ˙ γ ) g = ( J (cid:48) , ˙ γ ) g = 0 , J ∈ J , and ( J ( s ) , ˙ γ ( s )) g = 0 for J ∈ J s . To summarize, for every s ∈ [0 , (cid:96) ],(3.13) J s ( s ) ∪ J (cid:48) s ,s ( s ) ⊂ ˙ γ ( s ) ⊥ , and in particular, both spaces consist of spacelike or lightlike vectors. Furthermore, ˙ γ ( s ) ∈J s ( s ) if and only s (cid:54) = s , because ( s − s ) ˙ γ ( s ) ∈ J s . On the other hand, ˙ γ ( s ) ∈ J (cid:48) s ,s ( s )if and only s = s = s .We will need below the following simple lemma. Lemma 3.2. If two lightlike vectors v, w ∈ T x M satisfy ( v, w ) g = 0 then they are parallel.Proof. We can choose local coordinates so that g coincides with the Minkowski metric at x . Then v and w are parallel with vectors of the form (1 , θ ) and (1 , ω ) with θ and ω unitvectors. Now ( v, w ) g = 0 implies that ω · θ = 1, and thus ω and θ must be parallel. (cid:3) We will need the following property: for any Jacobi fields I, J along a geodesic γ , theWronskian(3.14) ( I, J (cid:48) ) g − ( I (cid:48) , J ) g is constant along γ, see e.g. [29, p. 274]. Lemma 3.3. Let ( z, a ) ∈ M and write γ = γ z,a . Then for every s , s ∈ [0 , (cid:96) ] , we have (i) J s ( s ) and J (cid:48) s ,s ( s ) are mutually orthogonal with respect to g , (ii) J s ( s ) ∩ J (cid:48) s ,s ( s ) = { } , (iii) J s ( s ) + J (cid:48) s ,s ( s ) = ˙ γ ( s ) ⊥ .Proof. Note first that if s = s , then J s ( s ) = { } and J (cid:48) s ,s ( s ) = ˙ γ ( s ) ⊥ by Lemma 3.1.Therefore the lemma holds in this case, and we can assume s (cid:54) = s in what follows.For w ∈ T γ ( s ) M with w ∈ J (cid:48) s ,s ( s ), let I ∈ J s be the Jacobi field with Cauchy data(0 , w ) at s = s . (If I (cid:54) = 0, then γ ( s ) and γ ( s ) are conjugate along γ .) By (3.14), forevery J ∈ J s , we get ( w, J ( s )) g = ( I (cid:48) ( s ) , J ( s )) g − ( I ( s ) , J (cid:48) ( s )) g = 0, therefore, w isorthogonal to J s ( s ). This proves (i).To prove (ii), assume that w ∈ J s ( s ) ∩ J (cid:48) s ,s ( s ). Then w is orthogonal to itself by(i), therefore it is lightlike. By (3.13) it is also perpendicular to the lightlike vector ˙ γ ( s ), and Lemma 3.2 implies that w must be parallel to ˙ γ ( s ). That is, w = λ ˙ γ ( s ) with some λ ∈ R . Since w ∈ J (cid:48) s ( s ), then there is J ∈ J s with Cauchy data (0 , λ ˙ γ ( s )) at s = s ;but then J ( s ) = λ ( s − s ) ˙ γ ( s ). Now J ∈ J s and s (cid:54) = s imply λ = 0, hence J = 0 andalso w = 0.Consider now (iii). We write W = J s ( s ) + J (cid:48) s ,s ( s ). As W ⊂ ˙ γ ( s ) ⊥ by (3.13),it remains to show the opposite inclusion. We will establish this by showing that W ⊥ iscontained in R ˙ γ ( s ) := { λ ˙ γ ( s ) | λ ∈ R } . Then ˙ γ ( s ) ⊥ ⊂ ( W ⊥ ) ⊥ and (iii) follows from( W ⊥ ) ⊥ = W , see e.g. [29, Lemma 22, p. 49] for the latter fact.Let w ∈ W ⊥ and let I be the Jacobi field with Cauchy data (0 , w ) at s = s . As w is inparticular orthogonal to J s ( s ), by using (3.14) we get for every J ∈ J s ,( I ( s ) , J (cid:48) ( s )) g = ( I ( s ) , J (cid:48) ( s )) g − ( I (cid:48) ( s ) , J ( s )) g = 0 . Recall that by Lemma 3.1, { J (cid:48) ( s ) | J ∈ J s } = J (cid:48) s ,s ( s ) = ˙ γ ( s ) ⊥ . Therefore I ( s ) is in( ˙ γ ( s ) ⊥ ) ⊥ = R ˙ γ ( s ) and we write I ( s ) = λ ˙ γ ( s ). Then for the Jacobi field K ( s ) = I ( s ) + λ s − s s − s ˙ γ ( s )it holds that K ( s ) = 0 and K ( s ) = 0. Writing u = K (cid:48) ( s ) and µ = λ ( s − s ) − , we have u ∈ J (cid:48) s ,s ( s ) and u = w + µ ˙ γ ( s ).Let us now use the fact that w is orthogonal to the whole W . It follows from (3.13) that R ˙ γ ( s ) ⊂ W ⊥ and therefore also u = w + µ ˙ γ ( s ) ∈ W ⊥ . But u ∈ J (cid:48) s ,s ( s ) ⊂ W , and u must be lightlike. Lemma 3.1 implies that ( u, ˙ γ ( s )) g = 0 and then u ∈ R ˙ γ ( s ) by Lemma3.2. Hence also w ∈ R ˙ γ ( s ). (cid:3) We will denote by ζ ∗ ∈ T z Z the image of ζ ∈ T ∗ z Z , with z ∈ Z , under the canonicalisomorphism induced by h , i.e., ζ j ∗ = h jk ζ k . Analogously for ξ ∈ T ∗ x M , with x ∈ M , wedenote by ξ ∗ ∈ T x M the vector defined by ξ j ∗ = g jk ξ k .Recall that in the Minkowski case the lightlike covectors on the canonical relation arecharacterized by (2.11), or equivalently by ξ (cid:107) θ . These two characterizations have thefollowing analogues in the present context. Lemma 3.4. Let ( z, a, s, ξ ) ∈ C . Then the following three conditions are equivalent: (i) ξ is lightlike, (ii) ξ ∗ is parallel to ˙ γ z,a ( s ) , (iii) ζ ∗ is parallel to θ ( z, a ) and α = 0 where ζ and α are given by (3.8).Proof. We will suppress ( z, a ) in the notation below. Let us suppose first that ξ ∗ is lightlikeand show that ξ ∗ is parallel to ˙ γ ( s ). As ( ξ ∗ , ˙ γ ( s )) g = (cid:104) ξ, ˙ γ ( s ) (cid:105) = 0, Lemma 3.2 implies that ξ ∗ is parallel to ˙ γ ( s ).Let us now suppose that ξ ∗ = λ ˙ γ ( s ) for some λ ∈ R , and show that ζ ∗ = λθ and α = 0.Lemma 3.1 implies ∂ s ( M j ( s ) , ˙ γ ( s )) g = ( M (cid:48) j ( s ) , ˙ γ ( s )) g = 0 . Hence using also (3.9) ζ j = λ ( ˙ γ ( s ) , M j ( s )) g = λ ( ˙ γ (0) , M j (0)) g = λ ((1 , θ ) , (0 , e j )) g = λθ k h kj . This establishes ζ ∗ = λθ . Analogously, α k = λ ( ˙ γ (0) , J k (0)) g = 0 since J k (0) = 0. HE LIGHT RAY TRANSFORM 19 Let us now suppose that ζ ∗ = λθ and α = 0 and show that ξ ∗ = λ ˙ γ ( s ). The equationsin the previous step imply that( ξ ∗ , M j ( s )) g = ζ j = λ ( ˙ γ ( s ) , M j ( s )) g , ( ξ ∗ , J k ( s )) g = α k = 0 = λ ( ˙ γ (0) , J k (0)) g . Moreover, ( ξ ∗ , ˙ γ ( s )) g = 0 = λ ( ˙ γ ( s ) , ˙ γ ( s )) g . By Lemma 3.1, { J ( s ) | J ∈ J } = T γ ( s ) M , andhence ξ ∗ = λ ˙ γ ( s ). This again implies that ξ ∗ is lightlike. (cid:3) The projection π M . We analyze π M next. We have(3.15) π M (cid:0) ( z, a, ζ, α ) , ( x, ξ ) (cid:1) = ( z, a, ζ, α ) . Since C is parameterized by ( z, a, s, ξ ) ∈ C , we view π M as a function of those parameters.As before, this projection is a map from the 3 n dimensional C to the 4 n − T ∗ M . To see whether π M is injective, let the right-hand side of (3.15) be given. Thismeans in particular that the geodesic γ z,a is fixed. We want to find out whether thedefining equations of C , that is,(3.16) ζ j = (cid:104) ξ, M j ( s ; z, a ) (cid:105) , α k = (cid:104) ξ, J k ( s ; z, a ) (cid:105) , (cid:104) ξ, ˙ γ z,a ( s ) (cid:105) = 0 , have more than one solution for s and ξ . Lemma 3.5. Let ( z, a ) ∈ M , s (cid:54) = s , J (cid:48) ∈ J (cid:48) s ,s , and let λ ∈ R . Then ξ j ∗ = J (cid:48) ( s j ) + λ ˙ γ ( s j ) , j = 1 , , satisfy (3.17) (cid:104) ξ , M j ( s ; z, a ) (cid:105) = (cid:104) ξ , M j ( s ; z, a ) (cid:105) (cid:104) ξ , J k ( s ; z, a ) (cid:105) = (cid:104) ξ , J k ( s ; z, a ) (cid:105) , and (cid:104) ξ j , ˙ γ z,a ( s j ) (cid:105) = 0 , j = 1 , .Proof. The claimed equations are linear, so it is enough to verify that the choices ξ j ∗ = J (cid:48) ( s j )and ξ j ∗ = ˙ γ ( s j ) satisfy them. We begin with the former choice. By (3.14) it holds that (cid:10) ξ , M j ( s ) (cid:11) = ( J (cid:48) ( s ) , M j ( s )) g − ( J ( s ) , M (cid:48) j ( s )) g = ( J (cid:48) ( s ) , M j ( s )) g − ( J ( s ) , M (cid:48) j ( s )) g = (cid:10) ξ , M j ( s ) (cid:11) , and analogously (cid:10) ξ , J k ( s ) (cid:11) = (cid:10) ξ , J k ( s ) (cid:11) . The last equation follows from (3.13). Letus now consider the choice ξ j ∗ = ˙ γ ( s j ). By Lemma 3.1 the scalar products ( M (cid:48) j , ˙ γ ) g and( J (cid:48) k , ˙ γ ) g and vanish identically. Thus ( M j , ˙ γ ) g is constant along γ , and the same holdsfor ( J k , ˙ γ ) g . Therefore ˙ γ ( s ) and ˙ γ ( s ) solve (3.17). The last equation holds since γ islightlike. (cid:3) Lemma 3.6. Let ( z, a, ζ, α ) ∈ T ∗ M and let ( s j , ξ j ) ∈ [0 , (cid:96) ] × T ∗ γ z,a ( s j ) M , j = 1 , , solve(3.16). Then the following hold: (i) Either both ξ and ξ are spacelike or they are both lightlike. (ii) If s = s then ξ = ξ . (iii) If s (cid:54) = s then there are unique J (cid:48) ∈ J (cid:48) s ,s and λ ∈ R such that ξ j ∗ = J (cid:48) ( s j ) + λ ˙ γ ( s j ) , j = 1 , . Moreover, ξ and ξ are spacelike if and only if J (cid:48) (cid:54) = 0 . Let us remark that the case J (cid:48) = 0 in (iii) is the analogue of the fact that in theMinkowski case, equation (2.17) for lightlike ( τ j , ξ j ) is equivalent with ( τ , ξ ) = ( τ , ξ )and both ( t j , x j ), j = 1 , 2, lying on the same line (cid:96) z,θ . Proof. We will again suppress ( z, a ) in the notation below. We will begin by proving (i).Recall that (cid:104) ξ j , ˙ γ ( s j ) (cid:105) = 0 implies that ξ j is lightlike or spacelike. It is enough to showthat ξ being lightlike implies that also ξ is lightlike. So suppose that ξ is lightlike. ThenLemma 3.4 implies that ζ ∗ is parallel to θ and α = 0. Therefore ξ is lightlike by the samelemma.Let us now show (ii). When s = s , equation (3.16) implies that (cid:10) ξ , M j ( s ) (cid:11) = (cid:10) ξ , M j ( s ) (cid:11) , (cid:10) ξ , J k ( s ) (cid:11) = (cid:10) ξ , J k ( s ) (cid:11) , (cid:10) ξ , ˙ γ ( s ) (cid:11) = (cid:10) ξ , ˙ γ ( s ) (cid:11) , and, as { J ( s ) | J ∈ J } = T γ ( s ) M by Lemma 3.1, it holds that ξ = ξ .We turn to (iii). As ξ ∗ ∈ ˙ γ ( s ) ⊥ , there are unique u ∈ J s ( s ) and w ∈ J (cid:48) s ,s ( s ) suchthat ξ ∗ = u + w by Lemma 3.3. As ( s j , ξ j ) solve (3.16), it holds that for all J ∈ J s that0 = ( ξ ∗ , J ( s )) g = ( ξ ∗ , J ( s )) g . In other words, ξ ∗ ∈ J s ( s ) ⊥ . By (i) of Lemma 3.3, also w ∈ J s ( s ) ⊥ . Therefore u = ξ ∗ − w ∈ J s ( s ) ∩ J s ( s ) ⊥ and u must be lightlike. As u is orthogonal to ˙ γ ( s ) by (3.13), it follows from Lemma 3.2that u = λ ˙ γ ( s ) for some λ ∈ R . Let J be the Jacobi field with Cauchy data (0 , w ) at s = s . Then J ( s ) = 0 since w ∈ J (cid:48) s ,s ( s ). Setting ˜ ξ ∗ = J (cid:48) ( s ) + λ ˙ γ ( s ), the covectors ˜ ξ and ξ give a solution to (3.17) by Lemma 3.5. It then follows from part (ii) that ξ = ˜ ξ .Clearly both ξ j , j = 1 , 2, are lightlike if J (cid:48) = 0. On the other hand, if ξ j , j = 1 , ξ j ∗ ∈ ˙ γ ( s j ) ⊥ , applying Lemma 3.2, implies that J (cid:48) ( s j ) = µ ˙ γ ( s j ) for some µ ∈ R . Now J ( s ) = 0 and J (cid:48) ( s ) = µ ˙ γ ( s ) imply that J ( s ) = µ ( s − s ) ˙ γ ( s ), and J ( s ) = 0implies that µ = 0. (cid:3) The above lemma says in particular that if there are two distinct solutions ( s j , ξ j ), j = 1 , 2, to (3.16) and if ξ is spacelike then γ z,a ( s ) and γ z,a ( s ) are conjugate along γ z,a .By Lemma 3.5 the converse holds as well. Indeed, if γ z,a ( s ) and γ z,a ( s ) are conjugatealong γ z,a then there is non-zero J (cid:48) ∈ J (cid:48) s ,s and for any λ ∈ R the vectors ξ j ∗ in Lemma 3.5are spacelike solutions to (3.17).The characterization of the pairs ( ξ , ξ ) is related to that in the Riemannian case, see[39, Theorem 4.2] where the conjugate points are assumed to be of fold type; see also [15]for a more general case.We will finish our study of π M by showing that d π M is injective in the spacelike cone. HE LIGHT RAY TRANSFORM 21 Lemma 3.7. Let p := (( z , a , ζ , α ) , ( x , ξ )) ∈ C and suppose that ξ is spacelike.Then d π M is injective at p .Proof. After reparametrization, we can assume x ∈ H , x = (0 , z ) = (0 , 0) in the semi-geodesic coordinates (3.1), and a = 0. In particular, we can consider x = ( x , x (cid:48) ) near x as a point in ( − T, T ) × Z . We write also ξ = ( ξ , ξ (cid:48) ). Then the points ( z, a, s, ξ ) in C near(0 , ξ ) can be parameterized by ( z, a, s, ξ (cid:48) ) ∈ Z × A × R × R n by setting ξ = ( ξ ( z, a, s, ξ (cid:48) ) , ξ (cid:48) )where ξ ( z, a, s, ξ (cid:48) ) is the unique solution to (cid:104) ξ, ˙ γ z,a ( s ) (cid:105) = 0 near ξ . Indeed, this followsfrom the implicit function theorem since ∂ ξ (cid:104) ξ, ˙ γ (0) (cid:105) = 1.Using the above parameterization, we write π M ( z, a, s, ξ (cid:48) ) = ( z, a, ζ, α ) with ζ and α asin (3.8). To show that d π M is injective at p , it is enough to show that ∂ ( ζ, α ) /∂ ( s, ξ (cid:48) ) isinjective at (0 , ξ (cid:48) ). Moreover, using (3.9), we have at (0 , ξ (cid:48) ), ∂ζ∂ξ (cid:48) = M (0) · · · M n (0)... . . . ... M n (0) · · · M nn (0) = Id . As also ∂α/∂ξ (cid:48) = 0 there, it is enough to show that ∂α/∂s (cid:54) = 0. Using once again (3.9), itholds at (0 , ξ (cid:48) ) that ∂α k ∂s = ( ∇ s ξ ∗ , J k ) g + ( ξ ∗ , ∇ s J k ) g = ( ξ ∗(cid:48) , ∂ a k θ ) h . To get a contradiction, suppose that ( ξ ∗(cid:48) , ∂ a k θ ) h = 0, k = 1 , . . . , n − 1. As the vectors ∂ a k θ , k = 1 , . . . , n − 1, span the tangent space of the unit sphere S z Z at θ , the vector ξ ∗(cid:48) must be parallel to θ . But then (cid:10) ξ , ˙ γ (0) (cid:11) = 0 implies that ξ ∗ is parallel to (1 , θ ), acontradiction with ξ being spacelike. (cid:3) The projection π M . As above, we regard the projection π M in (2.15) as a map of C parameterized by ( z, a, s, ξ ) ∈ C to T ∗ M . We have π M (cid:0) ( z, a, ζ, α ) , ( x, ξ ) (cid:1) = ( x, ξ ) = ( γ z,a ( s ) , ξ ) , with ξ conormal to ˙ γ z,a ( s ). It maps the 3 n dimensional C to the 2 n + 2 dimensional T ∗ M .Moreover, π M is surjective in the sense that there are ( z, a, s ) ∈ Z × A × (0 , (cid:96) ) satisfying(3.18) x = γ z,a ( s ) , (cid:104) ξ, ˙ γ z,a ( s ) (cid:105) = 0 , assuming that ( x, ξ ) ∈ Σ s is close to N ∗ γ . Indeed, as in the Minkowski case, solvingfor η ∗ = ˙ γ z,a ( s ) modulo rescaling in the second equation in (3.18), we obtain a ( n − ξ is spacelike; and two distinct vectors when n = 2. Moreover, when x is close to γ ( s ) for some s ∈ (0 , (cid:96) ) and ξ is close to N ∗ γ ( s ) γ , wecan choose η ∗ near ˙ γ ( s ). Then finding z and a is straightforward because H is transversalto γ .It follows from [18, Prop. 25.3.7] that the differential d π M is surjective whenever d π M is injective. Let us, however, show this also directly for a point p as in Lemma 3.7. We re-parametrize again as in Lemma 3.7. Then at (0 , ξ (cid:48) )d π M = ∂ ( x , x (cid:48) , ξ , ξ (cid:48) ) ∂ ( z, a, s, ξ (cid:48) ) = ∂x (cid:48) /∂s ∂ξ /∂z ∂ξ /∂a ∂ξ /∂s ∂ξ /∂ξ (cid:48) , and we see that d π M is surjective if and only if ∂ξ /∂a (cid:54) = 0. It follows from (cid:104) ξ, ˙ γ z,a ( s ) (cid:105) = 0and (3.9) that 0 = ∂ξ ∂a k + (cid:10) ξ , ∂ a k ˙ γ (cid:11) = ∂ξ ∂a k + ( ξ ∗(cid:48) , ∂ a k θ ) h . We showed in Lemma 3.7 that ( ξ ∗(cid:48) , ∂ a k θ ) h can not vanish for all k = 1 , . . . , n − ξ is spacelike. Thus d π M is surjective in this case.3.5. Conclusions. Analogously to Lemma 2.2, we summarize the results above: Lemma 3.8. The differential d π M is injective and the differential d π M is surjective at ( z, a, s, ξ ) ∈ C , with ξ spacelike. The projection π M is injective in a neighborhood of theset of points (0 , , s, ξ ) ∈ C , with ξ spacelike, if and only if there are no conjugate pointson γ . The projection π M is surjective onto a neighborhood of Σ s ∩ N ∗ γ in Σ s . We have also the following partial analogues of Lemmas 2.3 and 2.4, where write again C = π M ◦ π − M . Lemma 3.9. For all ( x, ξ ) in a small enough neighborhood of Σ s ∩ N ∗ γ in Σ s it holdsthat C ( x, ξ ) is the ( n − -dimensional manifold given by { ( z, a, ζ, α ) ∈ T ∗ M | (3.18) and (3.8) hold for some s ∈ (0 , (cid:96) ) } . Proof. If (( z, a, ζ, α ) , ( x, ξ )) ∈ π − M ( x, ξ ), then ( z, a ) satisfies (3.18) for some s ∈ (0 , (cid:96) ). Bythe argument above, the solutions to this equation form a ( n − z, a ), the parameter s is fixed by x = γ z,a ( s ), and then ζ and α are givenby (3.8). (cid:3) Lemma 3.10. Suppose that ( z, a, ζ, α ) ∈ T ∗ M is in the domain of C − and not in the set L = { ( z, a, ζ, α ) ∈ T ∗ M| ζ ∗ || θ ( z, a ) , α = 0 } . (3.19) Suppose, furthermore, that there are no conjugate points on γ . Then C − ( z, a, ζ, α ) =( γ z,a ( s ) , ξ ) where ( s, ξ ) is the unique solution of (3.16).Proof. If (( z, a, ζ, α ) , ( x, ξ )) ∈ π − M ( z, a, ζ, α ), then ξ is spacelike by Lemma 3.4. It followsfrom Lemma 3.6 that (3.16) has a unique solution ( s, ξ ). Finally x = γ z,a ( s ) by (3.6). (cid:3) The analogue of Lemma 2.5 reads: Lemma 3.11. Suppose that there are no conjugate points on γ . For all ( x, ξ ) in a smallenough neighborhood of Σ s ∩ N ∗ γ in Σ s it holds that ( C − ◦ C )( x, ξ ) = ( x, ξ ) .Proof. By Lemma 3.8, the projection π M is injective near the non-empty set π − M ( x, ξ ).Therefore ( C − ◦ C )( x, ξ ) = ( π M ◦ π − M )( x, ξ ) = ( x, ξ ) . (cid:3) HE LIGHT RAY TRANSFORM 23 For a set V ⊂ T ∗ M \ V c the conical set generated by V , that is, V c = { ( x, λξ ) ∈ T ∗ M \ | ( x, ξ ) ∈ V , λ > } . Similarly to Theorem 2.2, we have: Theorem 3.1. Suppose that there are no conjugate points on γ . Then there is a ze-roth order pseudodifferential operator χ on Z × A such that L (cid:48) κ χL κ is a pseudodifferentialoperator of order − with essential support in the spacelike cone.Suppose, moreover, that κ is nowhere vanishing. Then for any ( x , ξ ) ∈ Σ s ∩ N ∗ γ theoperator χ can be chosen so that L (cid:48) κ χL κ is elliptic at ( x , ξ ) .Proof. Let ( x , ξ ) ∈ T ∗ Ω ∩ Σ s ∩ N ∗ γ and let s ∈ (0 , (cid:96) ) satisfy x = γ ( s ). Writing z = 0and a = 0 we have ( z , a , s , ξ ) ∈ C . We define also ζ = ζ and α = α where ζ and α are given by (3.8) with ξ = ξ , s = s , z = z and a = a .Lemma 3.4 implies that ( z , a , ζ , α ) is outside the set L defined by (3.19). We choose aneighborhood U ⊂ T ∗ M of ( z , a , ζ , α ) such that U is compact and U ∩L = ∅ . Moreover,we choose χ so that χ = 1 near ( z , a , ζ , α ) and so that it is essentially supported in U c .The closed set π − M (Σ l ) is disjoint from the closed set π − M ( U ) by Lemma 3.4. We willshow next that there is a conical neighborhood W of π − M (Σ l ) such that W ∩ π − M ( U ) = ∅ .It is enough to show that π − M ( U ) is bounded. This boils down showing that there is C > z, a, s, ξ ) ∈ π − M ( U ) satisfy | ξ | ≤ C . Consider the map F taking ( z, a, s, ξ ) tothe point in R n with the coordinates (cid:104) ξ, M ( z, a, s ) (cid:105) , . . . , (cid:104) ξ, M n ( z, a, s ) (cid:105) , (cid:104) ξ, J ( z, a, s ) (cid:105) , . . . , (cid:104) ξ, J n − ( z, a, s ) (cid:105) , (cid:104) ξ, ˙ γ z,a ( s ) (cid:105) . Clearly F is homogeneous of degree one in ξ , and by (3.6), F ( π − M ( U )) = { ( ζ, α, | ( z, a, ζ, α ) ∈ U for some ( z, a ) } . But this set is bounded due to U being compact. Therefore also π − M ( U ) is bounded.As d π M is surjective, π M is an open map and π M ( W ) is a neighborhood of Σ l , consideredas a subset of the range π M ( C ). We may choose a pseudodifferential operator ˜ χ so that˜ χ = 1 near Σ l and that is essentially supported in π M ( W ). Then χL κ (1 − ˜ χ ) = χL κ moduloa smoothing operator. Moreover, L κ (1 − ˜ χ ) is smoothing on Σ l .We can now apply the clean intersection calculus: the proof that (2.19) holds for (2.20) isin verbatim the same as in the Minkowski case, except that we invoke Lemma 3.10 insteadof Lemma 2.4. Also C − ◦ C has the same global structure. Furthermore, the order iscomputed as in the Minkowski case, except that Lemma 3.9 is used instead of Lemma 2.3.For the claimed ellipticity, we choose χ so that σ [ χ ] is non-negative. Note that thepoint ( z , a , ζ , α ) is on the fiber C ( x , ξ ). As χ = 1 near ( z , a , ζ , α ), the integralof σ [ χ ] does not vanish over the fiber C ( x , ξ ). The ellipticity follows again from [17,Theorem 25.2.3]. (cid:3) Examples of metrics which do not allow conjugate points along lightlike geodesics includethe Minkowski metric, product type of metrics − d t + g ( x ) with g having no conjugatepoints, the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric − d t + a ( t )d x with a > 0, and in particular the Einstein-de Sitter metric corresponding to a ( t ) = t / ; as well as metric conformal to them and small perturbations of all those examples on compactmanifolds. Of course, any Lorentizan metric is free of conjugate points on small enoughsubset of M . We refer to [23] for the conformal invariance of this problem: the FLRWmetric can be transformed into a ( s )( − d s + d x ) after a change of variables s = s ( t )solving d s/ d t = a − ( t ). Next two metrics conformal to each other have the same lightlikegeodesics as smooth curves, but possibly parameterized differently, which does not changethe property of existence or not of conjugate points. Going back to the original parame-terization would multiply the weight κ by a smooth non-vanishing factor, which would notchange our conclusions.4. Cancellation of singularities in two dimensions Non-detectability and invisibility results have been extensively studied for inverse prob-lems, see [5, 6, 7, 8] and references therein. For the Riemannian geodesic ray transform, itwas shown in [24], see also [14], that in presence of conjugate points, singularities cannot beresolved locally, at least, i.e., knowing the ray transform near a single (directed) geodesic.We will prove an analogous result in the Lorentzian case in 1 + 2 dimensions.We will review some of the results in section 3 emphasizing on the specifics for the n = 2 case. The point-geodesic relation X , see (3.5), is 4-dimensional, and all manifoldsin the diagram (2.15) (valid in the variable curvature case as well) are 6 dimensional. Theprojection π M is a local diffeomorphism in a neighborhood of a point ( γ , ˆ γ , x , ξ ) ∈ C with ( x , ξ ) spacelike (here, ˆ γ is a dual variable to γ = ( z, a )), if and only if there are nopoints on γ conjugate to x . The projection π M is also a local diffeomorphism under thesame non-conjugacy condition. As a result, the canonical relation C = π M ◦ π − M is a localdiffeomorphism from Σ s to its image. The composition as in Theorem 2.2 then followswithout the need to invoke the clean intersection calculus.We take a closer look at the geometry of the conjugate points when n = 2. Two pointsalong a geodesic are conjugate when there exists a non-zero Jacobi field vanishing at thosepoints. This property is invariant under rescaling and shifting of the parameter s of γ ( s ),so we can take s = 0. A basis for J (the Jacobi fields vanishing at 0, see (3.11)), in localcoordinates, is given by J , see (3.7), and s ˙ γ ( s ). Since the second one does not vanish at s (cid:54) = 0, a conjugate point could be at most at of order 1, that is, the Jacobi fields J with J (0) = J (1) = 0 form an one-dimensional linear space, also true pointwise. One the otherhand, at any point γ ( s ), the conormal bundle to γ is two-dimensional; and this is true forits restriction to the spacelike cone as well. Proposition 4.1. C ( x, ξ ) = C ( y, η ) if and only of there is a lightlike geodesic joining x and y , that is, γ (0) = x , γ (1) = y , so that(a) x and y are conjugate to each other on γ ,(b) ξ ∗ = J (cid:48) (0) + λ ˙ γ (0) , η ∗ = J (cid:48) (1) + λ ˙ γ (1) with some λ ∈ R , where J is a Jacobi fieldwith J (0) = J (1) = 0 . The proposition follows from Lemma 3.6. Note that the proposition is consistent withthe observation that at every point of γ , its conormal bundle is two-dimensional: the Jacobi HE LIGHT RAY TRANSFORM 25 field J in the lemma is scaled so that the proposition holds, and λ is responsible for thesecond dimension.Assume that γ : [0 , (cid:96) ] → M is a lightlike geodesic with endpoints outside Ω, where f ∈ E (cid:48) (Ω). Assume that x := γ ( s ) and x := γ ( s ) are conjugate along γ . Let V j , j = 1 , T ∗ M defined as the covectors ( γ ( s ) , ξ ) for s close to s j and ξ any spacelike covectors at γ ( s ). Then C is a diffeomorphism from V j to its image if V j is small enough. We can choose V j so that C ( V ) = C ( V ) =: V ⊂ T ∗ M and so that V j projected to the base M is a neighborhood of x j . Set C j = C | V j , j = 1 , C := C − ◦ C : V → V . Then C is the canonical relation of L − L where L j is L microlocalized to V j , j = 1 , f = f + f with f j supported near x j but away from the endpoints of γ j , j = 1 , Theorem 4.1. Suppose that κ does not vanish near x and x . Let f j ∈ E (cid:48) (Ω) with WF( f j ) ⊂ V j with V j as above and small enough, j = 1 , . Then L ( f + f ) ∈ H s ( V ) if and only if f + L − L f ∈ H s − / ( V ) , where the inverses are microlocal parametrices. The proof is immediate given the properties of L j above which make L j elliptic FIOs oforder − / f with spacelike singularities near x in a neighborhood of the conormal bundle to γ at x , one can also construct f singular near x so that L ( f + f ) is smooth. This statementis symmetric w.r.t. s and s , of course. Therefore, the singularity in the light ray transformthat is produced by f is cancelled by the singularity produced by f . On a manifold thatcontains many conjugate points, Theorem 4.1 can be considered as a cloaking result forthe singularities. For instance, on a Lorentzian manifold ( M, g ), that is conformal to theproduct space ( R × S n , − dt + g Sn ), any space-like element (( t , y ) , ( τ , η )) ∈ WF( f )can be cancelled by a function f that is supported near a point ( t , y ), where t = t + (2 π + 1) m, m ∈ Z , and y is an antipodal point to y . Also, observe that the function f that hides an element of the wave front set of f can be supported either in the futureor in the past of the support of function f . This has similar spirit to results on cloakingfor the Helmholtz equation by anomalous localized resonance [28] and the active cloakingresults [4], where scattered field produced by an object is cancelled by a metamaterialobject or an active source that located near the object.Theorem 4.1 also describes the microlocal kernel of L in V ∪ V .5. Applications We discuss two application we already mentioned in the introduction. A time dependent potential.