Another Approach to Test Gravity around a Black Hole

Pulsars orbiting around the black hole at our galactic center provide us a unique testing site for gravity. In this work, we propose an approach to probe the gravity around the black hole introducing two phenomenological parameters which characterize deviation from the vacuum Einstein theory. The two phenomenological parameters are associated with the energy momentum tensor in the framework of the Einstein theory. Therefore, our approach can be regarded as the complement to the parametrized post-Newtonian framework in which phenomenological parameters are introduced for deviation of gravitational theories from general relativity. In our formulation, we take the possibility of existence of a relativistic and exotic matter component into account. Since the pulsars can be regarded as test particles, as the first step, we consider geodesic motion in the system composed of a central black hole and a perfect fluid whose distribution is static and spherically symmetric. It is found that the mass density of the fluid and a parameter of the equation of state can be determined with precision with $0.1\%$ if the density on the pulsar orbit is larger than $10^{-9}~{\rm g/cm^3}$.