Parametrices for the light ray transform on Minkowski spacetime

We consider restricted light ray transforms arising from an inverse problem of finding cosmic strings. We construct a relative left parametrix for the transform on two tensors, which recovers the space-like and some light-like singularities of the two tensor.


Introduction
Let (M, g) be a smooth Lorentzian manifold. A smooth curve γ : R → M is called a light ray if g(γ(t),γ(t)) = 0, t ∈ R, whereγ denotes the covariant derivative along γ. Let C be the set of light rays on M and Sym 2 denote the bundle of symmetric 2-tensors on M . For f ∈ C ∞ 0 (M ; Sym 2 ), we consider the light ray transform Hereafter, the Einstein summation convention is used i.e. summation is over repeated indices. On 2 + 1 dimensional Minkowski space-time, this transform was studied by Guillemin [10] and it was envisioned to have applications in "cosmological X-ray tomography", see the concluding remarks [10, Section 17]. Recently in [14], such transforms naturally arise from an inverse problem of detecting singularities of the Lorentzian metric of the Universe using Cosmic Microwave Background (CMB) radiation measurements. In particular, let (M, g) be a Friedmann-Lemaître-Robertson-Walker (FLRW) type model for the Universe. For a small parameter , consider a family of metrics on M : representing small perturbations of g. In [14], it is demonstrated that one can obtain a restricted light ray transform of f from the linearization of the CMB measurements. Then it is proved in Theorem 4.4 that one can recover the space-like singularities of f . However, as already noted in [14], light-like singularities are of great interest as they correspond to gravitational waves which may be caused for example by cosmic strings. We address this problem in this note. For restricted geodesic ray transforms on functions (the light ray transform being an example), there is a microlocal framework developed by 5,6,7,8] to understand their mapping properties. We combine it with some calculations in [14] to show that the normal operator of the light ray transform is a paired Lagrangian distribution and construct parametrices on the elliptic part. This allows us to obtain a relative left parametrix for the restricted light ray transform, which among other things recovers the space-like and some light-like singularities of the metric perturbations. We remark that time-like singularities are in the kernel of L C and there is a good physical explanation for one not being able to determine them from the light ray transform, see [10, Page 188], [14] and the interesting work of Stefanov [17] on support theorems of the light ray transform.
The paper is organized as follows. In Section 2, we state the main results after setting up the problem. We show in Section 3 that the normal operator is a paired Lagrangian distribution and we construct the parametrix in Section 4.

The main results
It is known that a FLRW type space-time is conformal to the Minkowski space-time. Since conformal diffeomorphisms preserve light-like geodesics, as discussed in [14], it suffices to consider light ray transforms on where x = (t, x ) = (x 0 , x 1 , x 2 , x 3 ) denotes the coordinates on M . In this case, the light rays are straight lines and we denote by C the set of light rays. As demonstrated in [14,Lemma 4.3], the light ray transform L C defined as (1.1) has a non-trivial null space given by where E (M ) denotes the space of distributions with compact support, d s is the symmetric differential given in local coordinates by with ∇ i the covariant derivative, and Λ 1 denotes the bundle of one forms. Let U be an open set of R 3 and define the line complex i.e. collection of all light rays intersecting U , see Figure 1. We denote by L C 0 = L C | C 0 the restricted light ray transform on C 0 . To describe the null space of L C 0 , we denote L (U ) = {p ∈ M : there exists q ∈ U and a light ray joining p and q}. Then we observe that then we observe that L C 0 is injective on P(U ). The microlocal nature of L C 0 is well-understood. Let be the point-line relation. Then the Schwartz kernel of L C 0 is δ Z the delta distribution on C 0 × M supported on Z. Hence we know from Hömander's theory that L C 0 is a Fourier integral operator of order −3/4 associated with the canonical relation N * Z (see (3.1)). Although we do not explore this point here, the operator should fit into the framework in [8], see also [3]. In Section 3, we use a more direct approach to show that the Schwartz kernel of the normal operator L t C 0 • L C 0 is a paired Lagrangian distribution and we obtain the Sobolev estimate of L C 0 , see Theorem 3.1.
To state the main result, we need to describe the two Lagrangians associated to the normal operator. Let T * M be the cotangent bundle and (x, ξ) be the coordinate for The Hamilton vector field of p denoted by H p is defined through The integral curves of H p in L * M are called null bicharacteristics. It is well known that their projections to M are light-like geodesics. We denote where 0 stands for the zero section. We let Λ be the flow out of Σ meaning Then ∆ and Λ are Lagrangian subamanifolds of T * (M × M ) and they form a pair of cleanly intersecting Lagrangians in the following sense: two Now we briefly recall the notion of Lagrangian and paired Lagrangian distributions. Let Λ be a smooth conic Lagrangian submanifold of T * M \0. We denote by I µ (M ; Λ) the space of Lagrangian distributions of order µ on M associated with Λ. For two Lagrangians Λ 0 , Λ 1 ⊂ T * M \0 intersecting cleanly at a codimension k submanifold, the space of paired Lagrangian distributions associated with (Λ 0 , Λ 1 ) is denoted by I p,l (M ; Λ 0 , Λ 1 ), We use I p,l (Λ 0 , Λ 1 ) when the background manifold is clear. By abuse of notations, we also use I p,l (Λ 0 , Λ 1 ) for section valued distributions in Sym 2 . We know (from e.g. Prop. 3.1 of [4]) that if u ∈ I p,l (Λ 0 , Λ 1 ), then u ∈ I p+l (Λ 0 \Λ 1 ) and u ∈ I p (Λ 1 \Λ 0 ). So u has well-defined symbols on each Lagrangian.
For any subset A of T * M , we let 1 A be the microlocal cut-off defined as where χ A is the characteristic function for A and f ∈ E (M ; Sym 2 ). Our main result is Theorem 2.1. There exists a relative left parametrix A for L C 0 such that Using this result as a reconstruction formula and wave front analysis, we see that for f ∈ P(U ), we can recover the singularities in f on space-like directions and on some light-like directions. One may not be able to recover all light-like singularities due to the error term, see a related example in [7, Section 2]. However, Bf contains singularities on the flow out which can be regarded as artifacts in the reconstruction. As we already mentioned, light-like singularities corresponds to gravitational waves and the artifacts may help us to identify these singularities. Furthermore, we notice that B is an Fourier integral operator associated with the canonical relation Λ . The rank of the projection of Λ to T * M drops by 1. From Hörmander's result on L 2 boundedness of Fourier integral operators [12,Theorem 4.3.2], we conclude that if f ∈ H s (M ; Sym 2 )∩P(U ), then Bf ∈ H s (M ). So the artifacts have the same order of Sobolev regularity as f does. In a different context [16], the problem of reducing and enhancing the artifacts due to a similar mechanism is studied. The same strategy should work here as well.

The normal operator
We choose a parametrization of C 0 and find the normal operator in the parametrization. Some of these are done in [14]. Let y ∈ U ⊂ R 3 , v ∈ S 2 .
Then for f ∈ C ∞ 0 (M ; Sym 2 ), we have The point-line relation is parametrized by Therefore, we can find the conormal bundle N * Z and the canonical relation C = N * Z as see (39) of [14]. Now let's consider the double fibration picture If ρ is an injective immersion, the double fibration satisfies the Bolker condition. In this case, the composition L t C 0 • L C 0 belongs to the clean intersection calculus, see Hörmander [13]. However, as demonstrated in [14,Lemma 11.1], ρ fails to be injective on the set L ∩ C where Now let's consider the wave front set of the normal operator. The canonical relation for L t C 0 is C t , so by the calculus of wave front set (see e.g. [13]), we have Here we observed that To show that L t C 0 • L C 0 actually belongs to the paired Lagrangian space I p,l (∆, Λ) and determine p, l, it suffices to show that the symbol belongs to the class of symbols of product type. For convenience, we shall work with χL C 0 for χ ∈ C ∞ 0 (U ) and find the symbol of (χL C 0 ) t • (χL C 0 ). Here we can regard χ(y) as a function χ(y, v) defined on C 0 . Moreover, the analysis below works for any χ ∈ C ∞ 0 (C 0 ). For f, h ∈ C ∞ 0 (M ; Sym 2 ), we compute where we made the change of variable x 0 = s, x = y + sv. We obtain that We can write this as an oscillatory integral using since the integrand is supported on η 0 = −v · η . Therefore, we can write where the symbol is given by The computation in [14,Lemma 8.1] see also [14,Prop. 11.4] showed that a jklm is a locally integrable function and the integral was explicitly evaluated, which we recall now. Consider the set Now we see that on ∆\Λ, a jklm is a symbol of order −1 so that the normal operator is a pseudodifferential operator of order −1 microlocally restricted to ∆\Λ. This was obtained in [14]. Also, a jklm is singular at Σ consisting of light-like vectors η. According to the discussion at the end of Section 2, the symbol a belongs to the class S m,m (M × M ; ∆, Λ) with m = 0. Moreover, we have m = l − 1 2 = 0, p + l = −1 and we solve that p = − 3 2 , l = 1 2 . Therefore, (χL C 0 ) t • (χL C 0 ) ∈ I − 3 2 , 1 2 (∆, Λ). Now we can apply [6, Theorem 3.3] and a duality argument to obtain the Sobolev estimates of χL C 0 . Thus we've proved We remark that the Sobolev estimates can be seen from a more general result of Greenleaf and Seeger 1 . In [3], the authors demonstrated (in Section 4) that in absence of conjugate points, the light ray transform on a general Lorentzian manifold is an FIO associated to a canonical relation where one projection is a submersion with folds, and the mapping properties of such operators are analyzed. We can apply [3, Corollary 4.2] to L C 0 to obtain L C 0 : H s comp (M ) → H s+1/2+ loc (C 0 ) for any > 0. However, one can check the proof of [3, Theorem 1.1] to conclude that the loss does not happen for L C 0 because the Hessian of the submersion with folds in this case is sign-definite.

The parametrix construction
We prove Theorem 2.1. Notice that since we shall consider the operator L C 0 acting on distributions in P(U ) so that L C 0 is injective, we actually have L C 0 = L C so we just need to consider the light ray transform L C . The analysis in Section 3 applies to this case by taking U = R 3 and χ = 1. Notice that ∆\Σ has disjoint components We consider the set where the symbol a in (3.2) (when χ = 1) is elliptic. For (x, η) ∈ Ω + , consider a(x, η) : f lm → a jklm (x, η)f lm as a linear map on Sym 2 x . Since χ = 1 does not vanish identically on S 1 η , we know from [14, Lemma 9.1] that the kernel of the map is given by N x = {cg(x) + η ⊗ w + w ⊗ η; c ∈ R, w ∈ R 4 }, for any x ∈ M.
Therefore, a| Ω + is injective on C ∞ (M ; Sym 2 )\N . In particular, one can find b ijkl (x, η) such that b αβjk (x, η)a jklm (x, η)| Ω + = δ αl δ βm on C ∞ (M ; Sym 2 )\N . Since a jklm (x, η) is a symbol of order −1 on Ω + , we can find b jklm (x, η) a symbol of order 1 on Ω + . Now we use the calculus of paired Lagrangian distribution to construct a parametrix for L C 0 . The argument is quite standard as for elliptic pseudo-differential operators. We will use the symbol calculus [4,Prop. 3.4] and the composition of I p,l for the flow out model [4,Prop. 3.5]. These results can be found in [1,2,11] as well. First, we let A 0 ∈ I