Authors: Steven Kenneth Kauffmann
Because the Einstein equation can't uniquely determine the metric, it must be supplemented by additional metric constraints. Since the Einstein equation can be derived in a purely special-relativistic context, those constraints (which can't be generally covariant) should be Lorentz-covariant; moreover, for the effect of the constraints to be natural from the perspective of observational and empirical physical scientists, they should also constrain the general coordinate transformations (which are compatible with the unconstrained Einstein equation) so that the constrained transformations manifest a salient feature of the Lorentz transformations. The little-known Einstein-Schwarzschild coordinate condition, which requires the metric's determinant to have its -1 Minkowski value, thereby constrains coordinate transformations to have unit Jacobian, and for that reason causes tensor densities to transform as true tensors, which is a salient feature of the Lorentz transformations. The Einstein-Schwarzschild coordinate condition also allows the static Schwarzschild solution's singular radius to be exactly zero; though another coordinate condition that allows zero Schwarzschild radius exists, it isn't Lorentz-covariant.
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