Abstract. We demonstrate the existence of solutions with shocks for the equations describing a perfect fluid in special relativity, namely, divT= 0, where T ~ = (p + pcZ)ulU j + prl ij is the stress energy tensor for the fluid. Here, p denotes the pressure, u the 4-velocity, p the mass-energy density of the fluid, t/~ the flat Minkowski metric, and c the speed of light. We assume that the equation of state is given by p --a2p, where o -2, the sound speed, is constant. For these equations, we construct bounded weak solutions of the initial value problem in two dimensional Minkowski spacetime, for any initial data of finite total variation. The analysis is based on showing that the total variation of the variable ln(p) is non-increasing on approximate weak solutions generated by Glimm's method, and so this quantity, unique to equations of this type, plays a role similar to an energy function. We also show that the weak solutions (p(x ~ x 1), v(x~ 1)) themselves satisfy the Lorentz invariant estimates Var{ln(p(x~ < Vo and Var fln ~ + v(x~ _ v(xO,.)j < V1 for all t x ~ > 0, where Vo and V~ are Lorentz invariant constants that depend only on the total variation of the initial data, and v is the classical velocity. The equation of state p = (c2/3)p describes a gas of highly relativistic particles in several important general relativistic models which describe the evolution of stars.

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We demonstrate the consistency of the Einstein equations at the level of shock-waves by proving 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. For these solutions, the components of the gravitational metric tensor are only Lipschitz continuous 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 analysis introduces a locally inertial Glimm scheme that exploits the locally flat character of spacetime, and relies on special properties of the relativistic compressible Euler equations when p = σ 2 ρ, σ ≡ const.