  Within the class of nonlinear hyperbolic balance laws posed on a curved
spacetime (endowed with a volume form), we identify a hyperbolic balance law
that enjoys the same Lorentz invariance property as the one satisfied by the
Euler equations of relativistic compressible fluids. This model is unique up to
normalization and converges to the standard inviscid Burgers equation in the
limit of infinite light speed. Furthermore, from the Euler system of
relativistic compressible flows on a curved background, we derive, both, the
standard inviscid Burgers equation and our relativistic generalizations. The
proposed models are referred to as relativistic Burgers equations on curved
spacetimes and provide us with simple models on which numerical methods can be
developed and analyzed. Next, we introduce a finite volume scheme for the
approximation of discontinuous solutions to these relativistic Burgers
equations. Our scheme is formulated geometrically and is consistent with the
natural divergence form of the balance laws under consideration. It applies to
weak solutions containing shock waves and, most importantly, is well-balanced
in the sense that it preserves steady solutions. Numerical experiments are
presented which demonstrate the convergence of the proposed finite volume
scheme and its relevance for computing entropy solutions on a curved
background.
