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The vanishing (artificial) viscosity method has played a fundamental role in the theory of classical shock waves, both by providing a mollified limit that identifies the correct physical (Lax admissible) shock waves, and as a guiding principle in the design of numerical difference schemes for simulating shock waves. However, for relativistic fluid flow, the underlying dissipation mechanism based on the Euclidean Laplace operator violates Lorentz invariance (and hence the speed of light bound) -- the fundamental principle of Special Relativity. In this paper we introduce a simple dissipation mechanism for the relativistic Euler equations which is Lorentz invariant and consistent with the laws of Special Relativity. To establish basic consistency of the model for the study of shock waves, we prove existence and decay of Fourier Laplace mode solutions (implying dissipation), and we prove that 1-D shock profiles (viscous travelling wave approximations) exist if and only if the approximated shock waves are Lax admissible. Our analysis of shock profiles reveals an interesting simplification over classical artificial viscosity, leading to a one dimensional fixed point problem, due to the speed of light bound of Relativity., Comment: 18 pages