New test on the Einstein equivalence principle through the photon ring
  of black holes
  Einstein equivalence principle (EEP), as one of the foundations of general
relativity, is a fundamental test of gravity theories. In this paper, we
propose a new method to test the EEP of electromagnetic interactions through
observations of black hole photon rings, which naturally extends the scale of
Newtonian and post-Newtoian gravity where the EEP violation through a variable
fine structure constant has been well constrained to that of stronger gravity.
We start from a general form of Lagrangian that violates EEP, where a specific
EEP violation model could be regarded as one of the cases of this Lagrangian.
Within the geometrical optical approximation, we find that the dispersion
relation of photons is modified: for photons moving in circular orbit, the
dispersion relation simplifies, and behaves such that photons with different
linear polarizations perceive different gravitational potentials. This makes
the size of black hole photon ring depend on polarization. Further assuming
that the EEP violation is small, we derive an approximate analytic expression
for spherical black holes showing that the change in size of the photon ring is
proportional to the violation parameters. We also discuss several cases of this
analytic expression for specific models. Finally, we explore the effects of
black hole rotation and derive a modified proportionality relation between the
change in size of photon ring and the violation parameters. The numerical and
analytic results show that the influence of black hole rotation on the
constraints of EEP violation is relatively weak for small magnitude of EEP
violation and small rotation speed of black holes.
