Green operators for low regularity spacetimes

In this paper we define and construct advanced and retarded Green operators for the wave operator on spacetimes with low regularity. In order to do so we require that the spacetime satisfies the condition of generalised hyperbolicity which is equivalent to well- posedness of the classical inhomogeneous problem with zero initial data where weak solutions are properly supported. Moreover, we provide an explicit formula for the kernel of the Green operators in terms of an arbitrary eigenbasis of H 1 and a suitable Green matrix that solves a system of second order ODEs.


Introduction
The existence of Green operators for normally hyperbolic operators is well understood for globally hyperbolic spacetimes where the spacetime metric is a smooth [1]. However, there are two important motivations for analysing spacetime metrics with finite differentiability. Firstly, there are several models of physical phenomena that require finite metric regularity. These include impulsive gravitational waves, stars with well-defined surfaces, general relativistic fluids and cosmic strings. Secondly, Einstein's equations, viewed as a system of hyperbolic PDEs, can be naturally formulated in function spaces with finite regularity. When the spacetime is of finite regularity the existence of Green operators for the Klein-Gordon equation has been explored using semigroup techniques in the Hamiltonian formalism [2]. We complement such an analysis by providing a definition of Green operators subject only to the well posedness of the classical problem. Furthermore we show how to explicitly construct both the advanced and retarded the Green operators needed in order to formulate quantum field theory on a non-smooth background. In the first section of the paper we give the definition of the advanced and retarded Green operators in the smooth case and define the concept of generalised hyperbolicity. In the second section of the paper we extend the definitions to the non-smooth setting and prove the existence of the Green operators for low regularity spacetimes. In the third section of the paper we give an explicit formula for the advanced Green operator in the non-smooth setting.

The general setting
We will consider a manifold M of the form M = (0, T ) × Σ and its closure M = [0, T ] × Σ where Σ is a closed compact manifold. Given a smooth Lorentzian metric g ab of the form ds 2 = N 2 dt 2 − γ ij (dx i + β i dt)(dx j + β j dt) (2.1) the wave operator g acting on a scalar function u is given by where N is the lapse function, β i the shift, γ ij the metric on Σ and √ γdx the induced volume form on Σ. We use a, b, c, d etc. to denote spacetime indices and i, j, k etc. to denote purely spatial indices. The advanced zero initial data inhomogeneous problem for the wave equation on M is given by where f : M → R is a smooth function. Similarly the retarded zero initial data inhomogeneous problem for the wave equation on M is given by where f : M → R is a smooth function. If the spacetime (M, g ab ) is globally hyperbolic the advanced and retarded zero initial data inhomogeneous problems are both well posed i.e. there exist unique solutions u ∈ C ∞ (M ) which depend continuously on f [1]. Moreover, in the globally hyperbolic case well-posedness is equivalent to the existence of unique advanced and retarded Green operators [1]. For convenience we recall the definition of these [1].
is called an advanced Green operator for g . The retarded Green operators E − are defined in a similar fashion.
For a smooth globally hyperbolic spacetime the splitting theorem says that the metric may be written in the form and this remains true even in low regularity [10]. The aim of the paper is to define such advanced and retarded Green operators for the metrics of the above form where the lapse function N and the induced Riemannian metric γ ij are no longer smooth. Notation. We denote the derivative of a function u with respect t by u t or ∂ t u and u i or ∂ i u if it is with respect to the other x i -coordinates. The space of smooth functions of compact support will be denoted by D(M ). A function f on an open set U of R n is said to be Lipschitz if there is some constant K such that for each pair of points p, q ∈ U, |f (p) − f (q)| ≤ K|p − q|, where |p| denotes the usual Euclidean distance. We denote by C k,1 those C k−1 functions where the k-derivative is a Lipschitz function. A function f on M is said to be Lipschitz or C k,1 if f is Lipschitz or C k,1 in some coordinate chart.
In the analysis below we will be working with spaces such as L 2 (Σ), H 1 (Σ) H −1 (Σ) which are defined with respect to a smooth background Riemannian metric h ij on Σ with ν h the corresponding volume form. We then define L 2 (Σ) to be the space of real valued functions g on Σ such that Σ g 2 ν h < ∞ and we denote the associated inner product by (f, g) The space H k (Σ) are the real valued functions g such that their first k derivatives are in L 2 (Σ) and the space H −k (Σ) are the bounded linear functionals on H k (Σ). We also define the space L 2 (M, g) to be the space of real valued functions φ on M such that M φ 2 ν g < ∞ which is defined with respect the volume form given by the metric (2.9) and L 2 (Σ t , γ) to be the space of real valued functions ψ on Σ t = {t} × Σ for 0 ≤ t < T such that Σt ψ 2 ν γ < ∞ which is defined with respect the volume form given by the Riemannian metric γ ij that appears in (2.9) . We denote in a similar way the Sobolev space H k (M, g) and H k (Σ, γ). We will also often think of a function v(t, x) as a map from [0, T ] to a function v(t)(·) in some Hilbert space When thinking of v in this way, we will denote the time derivative byv. We will also use the spaces C k ([0, T ]; X(Σ)) which is the space of functions v : [0, T ] → X(Σ) (2.13) t → v(t) (2.14) such that v(t) ∈ X(Σ) and max where ∂ t l v(t) is the l-derivative with respect time of v(t).

Generalised hyperbolicity
To prove the existence of Green operators in the non-smooth setting we will require that the spacetime satisfy the condition of generalised hyperbolicity which is essentially the requirement that solutions to the wave equation are well-posed [3]. In the low regularity setting the pointwise equation (2.3) may not make sense and therefore a notion of a weak solution is required. Taking into account the form of the metric (2.9) and the geometric conditions that appear in Theorem 2.8 (see below) the wave operator may be written as where −L is an elliptic operator in divergence form given by: (2.17) We can associate with the operator −L the following bilinear form given by: Using this bilinear form, we make the following definition of an advanced weak solution.
The definition of the retarded weak solution is analogous. Remark 2.3. The regularity needed to make sense of Equation (2.20) for all v ∈ L 2 ([0, T ]; H 1 (Σ)) and therefore it make sense to denote the pairing g u, v L 2 H −1 ,L 2 H 1 .
We say an advanced weak solution is regular if it satisfies a suitable energy estimate.

Definition 2.4. (Regular Weak Solution)
We say an advanced weak solution is regular if u satisfies the energy estimate We will also require the weak solutions to satisfy certain support properties. In the smooth case the condition of global hyperbolicity is such that a solution u of the inhomogeneous problem With the obvious modifications we define the condition of retarded generalised hyperbolicity. If a spacetime satisfies the advanced and retarded condition of generalised hyperbolicity we say the spacetime satisfies the condition of generalised hyperbolicity.
Remark 2.7. In the case of a smooth metric, global hyperbolicity is sufficient to guarantee that the condition of generalised hyperbolicity in the above sense is also satisfied. The following series of theorems give sufficient conditions to ensure such that a low regularity spacetime (M, g) satisfies the condition of generalised hyperbolicity. For simplicity, we only deal with the condition of advanced generalised hyperbolicity.
Theorem 2.8. Let the metric g ab given by Equation 2.9 on M satisfy the following conditions

The scalar coefficient of the volume form given by
√ γ for the induced metric γ ij is bounded from below by a positive real number, i.e., | √ γ| > η for some η ∈ R +

The lapse function N can be chosen as
then for every f ∈ L 2 (M, g) there are advanced and retarded weak regular solutions Remark 2.9. Condition 3 is chosen for simplicity. However the condition can be weakened to require only that it is a bounded function with a positive lower bound, i.e.
This modification is at the expense of adding additional terms and to avoid undue complications in the formulae we do not pursue this here. Proof The main idea is to use Galerkin's method. We sketch the proof below for a complete proof see [5,6,7].
• Insert the approximate solution u (m) (t, x) = m k=1 d  • Prove that the u (m) satisfy the energy estimate uniformly in m • Apply Banach-Alouglu theorem to extract a subsequence which is a solution to the weak formulation • Use lower semi-continuity of the norm and energy estimates to prove uniqueness, stability and regularity of the weak solution. Proof. The results of [5,6,7] establish well-posedness. The only thing that remains to be shown is that the weak solutions have the correct support. To do so we follow [4] and obtain an energy inequality that determines the support of the solution.
Consider a weak solution u with energy-momentum tensor If we contract this with a Lipschitz timelike vector field, Υ a and use the fact that J − Σt (K)∩J + (Σ 0 ) is compact by causal theory for globally hyperbolic C 1,1 metrics [8,9,10] (where J − Σt (K) denotes the causal past of a compact region K ⊂ Σ t ), then provided we have sufficient regularity that one can apply the divergence theorem we obtain In order for the above equation to be well-defined we require that div(T ab Υ a ) should be integrable with respect the volume form ν g , and this in turn requires that the weak solutions are sufficiently regular. In fact it is enough that the weak solutions have two derivatives in L 2 (M, g) if the metric and the timelike vector field are in the space C 0,1 [12] and this is ensured by the additional regularity of the metric g ∈ C 1,1 and source f ∈ H 1 (M, g) required in the conditions of Theorem 2.10 compared to Theorem 2.8. Moreover, such an improvement in regularity also guarantees that u belongs to the space C 0 ([0, T ], H 1 (Σ)) ∩ C 1 ([0, T ], L 2 (Σ)). See [5] for a proof of these results. The left hand side of 2.24 takes the explicit form: where ∇ b Υ a denotes the covariant derivative of Υ a while the right hand side takes the form: , ν h is the induced volume form on H and n b denotes outward pointing normal vectors on K, Σ 0 and H. Moreover, in the globally hyperbolic C 1,1 case we have that H is a null hypersurface ruled by null geodesics [9] and by the dominant energy condition [4] we have that H T ab Υ a n b ν h ≥ 0 (2.27) For the integral over K we define the energy integral: which is equivalent to a Sobolev type norm on K In a similar way we may relate energy integral E Σ 0 and Sobolev norms˜ u 1 Σ 0 on Σ 0 . We now use the fact that where C is a constant that depends on g ab and the interval [0, t] [11]. Estimating all the terms in (2.24) by using the Cauchy Schwarz inequality together with the regularity of the metric and solutions gives the inequality: where k 0 , k 1 are positive constants that depend on the metric g ab , the vector field Υ a and the covariant derivative ∇ b Υ a . Now rewriting (2.32) using (2.30) and (2.31) we find Using Gronwall's inequality we obtain where K 4 is a positive constant that depends on the chosen finite time t, the metric g ab , the vector field Υ a and the covariant derivative ∇ b Υ a . In term of the Sobolev norms we obtain the expression: for some constant A. By definition we have that the advanced weak solution have vanishing initial data, thereforẽ u 1 Σ 0 = 0 and we obtain the energy estimate This establishes the fact that an advanced solution u of the zero initial data inhomogenous problem g u = f on K ⊂ Σ t is determined by the value of the source f in the region Notice that if p does not belongs to a neighbourhood of J + (supp(f )) then J − (p) does not intersect supp(f ) and therefore we can construct a compact set K ⊂ Σ t ′ that contains p for some t ′ such that J − This implies that the right hand side of (2.36) vanishes and u = 0 on K. Then, taking a possible smaller region K ′ ⊂ K and a small time interval ǫ we can form a neighbourhood U = ǫ × K ′ that contains p and where u vanishes. For example, it is sufficient to take points q, r such that q ∈ J + (K) and r ∈ J − (K) such that q, r does not intersect J + (supp(f )) and consider the region J − (q) ∩ J + (r). In this region we have that u = 0 and this implies that p is not in the support of u which gives the desired result. Remark 2.11. It may be possible to relax the regularity conditions to those of Theorem 2.8 by analysing the causal structure of spacetimes with low regularity as in [8,9,10] and using results on propagation of singularities for differential equations with rough coefficients as in [13].

Green operators in the non-smooth setting
In this section we define the notion of advanced and retarded Green operators in a weak sense. Notice that the Definition 2.1 can not be used in a setting with finite differentiability because weak solutions are in general not smooth so that one cannot expect E + (f ) ∈ C ∞ (M ). We therefore propose the following definitions suitable for settings with finite differentiability.

Definition 3.1. A bounded linear map E
is an advanced weak Green operator for g . A retarded weak Green operator E + is defined similarly.
Remark 3.2. If g ab is in C 0,1 and for all f ∈ D(M ) we have E ± (f ) in H 2 (M, g) then the first property can be restated as If for all f ∈ D(M ) we have E ± ( g f ) in L 2 (M, g) then the second property can be restated as We now show that the notion of generalised hyperbolicity is sufficient to establish the existence of weak Green operators. Proof. We will only show the existence of the advanced weak Green operator. The existence of the retarded weak Green operator follows from time reversal. Define the linear map E + : L 2 (M, g) → L 2 ([0, T ]; H 1 (Σ)) which send a source function f to the advanced regular weak solution u + of the advanced zero initial data inhomogeneous problem with source f . That such u + exist and is unique is a consequence of generalised hyperbolicity which follows from Theorem 2.10. We first show that this map is well defined and continuous.
Let f = g ∈ L 2 (M, g), then by definition we have that E + (f ) = u + and E + (g) = v + are regular weak solutions of the zero initial data inhomogeneous problem with source function f = g causally supported to the future. By the condition of generalised hyperbolicity which guarantees uniqueness of solutions, we have u + = v + and therefore E + (f ) = E + (g). Therefore the map is well defined. That the map is continuous follows from the regularity of solutions. We have We next show that E + is a suitable right inverse in the sense of Definition 3.1. First notice that the regularity of the weak solutions given by the condition of generalised hyperbolicity allows us to define for every advanced weak solution u + the linear map g u + [·]. See remark 2.3. Therefore given E + (f ) = u + , we have Now we prove a similar equality for E + g f [v] to show that it is a left inverse. First we notice that by definition we have that u + = E + ( g f ) is an advanced weak solution of the inhomogeneous problem with source g f and therefore we have that At the same time using integration by parts we have that so f is also a weak solution. Moreover, using the smoothness of f and the assumed regularity of the metric we can obtain an energy estimate which proves that f is a regular weak solution (see [5]). Then u + and f are weak regular solutions to the zero initial data inhomogeneous problem with the same source. So by uniqueness of the solutions we have that u + = f which guarantees that Finally to analyse the support of E + (f ) we use that the weak solutions are causally supported. We have that supp(E + (f )) = supp(u + ) ⊂ J + (supp(f )) for all f ∈ D(M ) which gives the desired result. We next show that the formal adjoint of E + is E − . Theorem 3.6. Given χ ∈ D(M ) and ϕ ∈ D(M ) we have that In order to show this result we need the following Lemma Proof. The proof follows straightforward from using integration by parts twice and using the boundary conditions given by the hypothesis. Proof of Theorem 3.6. First, notice that because the metric has regularity C 1,1 and χ, ϕ ∈ D(M ) we have that E + (χ), E − (ϕ) ∈ H 2 (M, g). [5]. Hence, where the second and fourth equality follows from remark 3.2 while the third inequality used the initial data given by the advanced and retarded zero initial data inhomogeneous problem and Lemma 3.7.
In the smooth case it is often convenient to consider the Green operators as bi-distributions on the product space. We show that this result extends to the non-smooth setting.
Proof of Theorem 3.8.

Galerkin type approximation of E +
In this section we will provide a construction of the operator E + and give a formula for the kernel. To construct the map E + we will build a sequence of operators E + m that converge in the weak operator topology to E + . We will construct the operators E + m using Galerkin's method following the methods of [5,6,7]. First consider a complete orthogonal basis {w j (x)} j∈J in H 1 (Σ) which is also orthonormal in L 2 (Σ). Then inserting the m-approximate solution u (m) where F kj m (t, s) is the Green matrix for the (regular) ODE problem. The next step is to show that {u m (t, x)} m is bounded uniformly in L 2 ([0, T ]; H 1 (Σ)). This is a consequence of the energy estimates. See [5] for a derivation of the estimates and the uniform bound. Applying the Banach-Alaoglu theorem to the bounded sequence we can obtain a subsequence {u m l (t, x)} l that converges weakly in L 2 ([0, T ]; H 1 (Σ)). i.e. there is a subsequence {E + m l (f ) = u m l } l that converges to a linear bounded operator E + on L 2 ([0, T ]; H 1 (Σ)) and its value on v coincides with u + [v] for all v ∈ L 2 ([0, T ]; H 1 (Σ)). We may therefore write  where the first identity is the weak convergence of the m-approximate solutions E + m (f )(x, t) = u m (x, t) to the weak solution u + (x, t) in L 2 ([0, T ]; H 1 (Σ)) and the second identity follows from equation 4.5. We have therefore shown that we may write the kernel of the advanced Green operator as G + (t, x; s, y) = lim l→∞ j,k F kj m l (t, s)w k (x)w j (y)γ(y, s)γ(x, t). (4.9) Note that in terms of the above representation the fact that the formal adjoint of E + is E − is reflected in the property F (s, t) = F T (t, s) for the Green matrix of a symmetric self-adjoint system of 2nd order ODEs [14].