On the Uniqueness of Fock's Harmonic Coordinate Systems in the Presence of Static, Spherically Symmetric Sources

Abstract

Fock has claimed that his "harmonic" coordinate systems in curved space flattening out toward spatial infinity are uniquely determined but for an arbitrary inhomogeneous Lorentz transformation. If this is so, introduction of Fock's harmonic coordinate conditions would provide a natural way of introducing a Lorentz subgroup of the general coordinate transformation group of Einstein's gravitational theory, and of defining a Minkowski metric besides the curved-space metric. This would open the way to close relations between Einstein's gravitational theory on the one hand, and Lorentz-covariant quantum field theory on the other hand. A general proof of the correctness of Fock's claim, for universes satisfying his boundary conditions, has never been given rigorously. Here we extend an earlier proof of this uniqueness for the Schwarzschild field around a single gravitational singularity, to the case of the static and spherically symmetric field generated in some coordinate system by an extended static and spherical distribution of energy and of stresses. The uniqueness (but for the zero point of time and for a spatial rotation) of the harmonic coordinate system, in which this field is spherical and at rest around the spatial origin, is here guaranteed by the condition that there must be a one-to-one correspondence between the points $x$, $y$, $z$, $t$ of the harmonic coordinate system and the points in physical space.