  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 $10^{-9}~{\rm g/cm^3}$.
