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These notes present a self contained mathematical treatment of the initial value problem for shock wave solutions of the Einstein equations in General Relativity. The first two chapters provide background for the introduction of a locally inertial Glimm Scheme in Chapter 3, a non-dissipative numerical scheme for approximating shock wave solutions of the Einstein equations in spherically symmetric spacetimes. In Chapter 4 a careful analysis of this scheme provides a proof of the existence of (shock wave) solutions of the spherically symmetric Einstein equations for a perfect fluid, starting from initial density and velocity profiles that are only locally of bounded total variation. To keep the analysis as simple as possible, we assume throughout that the equation of state is of the form p = σ^2ρ, σ = const . For these solutions, the components of the gravitational metric tensor are only Lipschitz continuous functions of the spacetime coordinates at shock waves, and so it follows that these solutions satisfy the Einstein equations, as well as the relativistic compressible Euler equations, only in the weak sense of the theory of distributions. The existence theory presented here establishes the consistency of the initial value problem for the Einstein equations at the weaker level of shock waves, for spherically symmetry spacetimes.

We consider the relativistic equations for a perfect fluid (2.1.1) $$ divT = 0, $$ in flat Minkowski spacetime, (2.1.2) $$ ds^2 = \eta _{ij} dx^i dx^j = - d\left( {ct} \right)^2 + d\left( {x^1 } \right)^2 + d\left( {x^2 } \right)^2 + d\left( {x^3 } \right)^2 , $$ where (2.1.3) $$ T^{ij} = \left( {p + \rho c^2 } \right)w^i w^j + p\eta i^j , $$ denotes the stress-energy tensor for the fluid. Recall that in Minkowski spacetime, (2.1.4) $$ divT \equiv T_{j,i}^i $$ where again we assume summation over repeated up-down indices, “, i ” denotes differentiation with respect to the variable x ^i, and in general all indices run from 0 to 3 with x ^0 = ct . In (2.1.3), c denotes the speed of light, (we take c = 1 when convenient), p the pressure, w = ( w ^0, ..., w ^3) the 4—velocity of the fluid particle, ρ the mass-energy density, and η^ij ≡ η_ij ≡ diag (−1, 1, 1, 1).

In this chapter we show that Einstein equations (1.3.2)–(1.3.5) are weakly equivalent to the system of conservation laws with time dependent sources (1.4.3),(1.4.4), so long as the metric is in the smoothness class C ^0,1, and T is in L ^∞. Inspection of equations (1.3.2)–(1.3.5) shows that it is in general not possible to have metrics smoother than Lipschitz continuous, (that is, smoother than C ^0,1 at shocks), when the metric is written in standard Schwarzschild coordinates. Indeed, at a shock wave, the fluid variables, and hence T , suffer jump discontinuities. At such a discontinuity, (1.3.2)–(1.3.5) imply that Ar, Br and Bt all suffer jump discontinuities as well.

In this chapter, taken from [13], we present a proof that shock wave solutions of (1.3.2)–(1.3.5), (1.2.4) and (1.3.1), defined outside a ball of fixed total mass, exist up until some positive time T > 0, and we prove that the total mass M ∞ = lim_ r →∞ M(r, t) is constant throughout the time interval [0, T ). A local existence theorem is all that we can expect for system (1.3.2)–(1.3.5) in general because black holes are singularities in standard Schwarzschild coordinates, $$ B = \frac{1} {{1 - \tfrac{{2M}} {r}}} \to \infty $$ at a black hole, and black holes can form in finite time. For these solutions, the fluid variables ρ, p and w , and the components of the stress tensor T ^ ij , are discontinuous , and the metric components A and B are Lipschitz continuous, at the shock waves, c.f. (1.3.2) and (1.3.4). Since (1.3.5) involves second derivatives of A and B , it follows that these solutions satisfy (1.3.2)–(1.3.5) only in the weak sense of the theory of distributions. Thus our theorem establishes the consistency of the initial value problem for the Einstein equations at the weaker level of shock waves.