Buchdahl bound, photon ring, ISCO and radial acceleration in Einstein-\ae{}ther theory

TL;DR

This paper revisits Einstein-e6ther theory, deriving solutions and bounds that differ from general relativity, with implications for astrophysical phenomena like the photon ring and ISCO.

Contribution

It provides new vacuum solutions, derives Buchdahl's bound, and analyzes the photon ring, ISCO, and acceleration relations in Einstein-e6ther theory.

Findings

Photon ring and ISCO are larger than in GR.

Upper bound on compactness is lower than in GR.

Radial acceleration relations are parallel in low pressure limit.

Abstract

Spherically symmetric Einstein-{\ae}ther (E{\AE}) theory with a Maxwell-like kinetic term is revisited. We consider a general choice of the metric and the \ae{}ther field, finding that:~(i) there is a gauge freedom allowing one always to use a diagonal metric; and~(ii) the nature of the Maxwell equation forces the \ae{}ther field to be time-like in the coordinate basis. We derive the vacuum solution and confirm that the innermost stable circular orbit (ISCO) and photon ring are enlarged relative to general relativity (GR). Buchdahl's theorem in E\AE{} theory is derived. For a uniform physical density, we find that the upper bound on compactness is always lower than in GR. Additionally, we observe that the Newtonian and E\AE{} radial acceleration relations run parallel in the low pressure limit. Our analysis of E\AE{} theory may offer novel insights into its interesting phenomenological generalization: \AE{}ther--scalar--tensor theory ({\AE}ST).