Gödel-symmetric backgrounds and explicit spacetime symmetry breaking

Abstract

Explicit diffeomorphism breaking in gravitational theories with nondynamical background fields is constrained by the compatibility between the modified Einstein equations and the contracted Bianchi identities. We implement an isometry-based approach in which the background fields are required to be invariant under the Killing symmetries of the spacetime. Applying this construction to the Gödel geometry, we determine the most general form-invariant scalar, symmetric rank-two, and Riemann-like rank-four background tensors and analyze the corresponding modified gravitational field equations. We show that the resulting configurations satisfy the required consistency conditions and derive the associated Noether identities in the presence of nondynamical backgrounds. The background coefficients modify the relations among the cosmological constant, the matter density, and the Gödel rotation parameter while preserving the Gödel metric as an exact solution. We also determine the corresponding critical radius for closed timelike curves and examine how the background fields modify the boundary between the causal and noncausal regions of the spacetime.