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It is not the case that Weyl's 'rest field' condition smuggles in a symmetry assumption (global homogeneity) that is not derivable from the absence of matter alone.
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Reasons For
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Reason for
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1.
Rest field condition derives from physical requirement that vacuum geometry be stationary, not arbitrary symmetry imposition.
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2.
Regularity and boundary conditions at infinity naturally select homogeneous solutions without assuming symmetry a priori.
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3.
Global homogeneity is the unique solution respecting isotropy of background—a weaker, empirically justified principle, not hidden assumption.
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Reasons Against
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Reason against
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1.
Absence of matter constrains only energy-momentum tensor, leaving metric degrees of freedom underdetermined by field equations alone.
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2.
Global homogeneity is a contingent symmetry choice, not entailed by vacuum solutions to Einstein equations without additional assumptions.
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3.
Weyl's condition assumes isotropic rest frame exists everywhere, which presupposes preferred foliation absent from background-independent formulations.
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