Testing the Non-circularity of the Spacetime around Sagittarius A* with Orbiting Pulsars

A disformal Kerr black hole solution is a rotating black hole solution in a modified gravity theory which breaks the circular condition of spacetime differently from the case of the Kerr spacetime. In this paper, assuming that Sagittarius A* (Sgr A*) is a disformal Kerr black hole, we examine the potential to test the spacetime geometry with a hypothetical pulsar whose orbital elements are similar to those of the S2/S0-2 star. By numerically solving the equations of motion for the pulsar and photons emitted from it, we calculate the apparent position of the pulsar and the time of arrival (TOA) of the emitted pulse signals. Our analysis shows that the magnitude of the difference in the TOAs reaches the order of $10\>{\rm ms}$ if the deviation from the Kerr spacetime is significant. The time difference is mainly caused by the non-circularity of the spacetime at the $1.5$ post-Newtonian order. The accuracy of the TOA measurement by a future radio telescope named the Square Kilometer Array (SKA) is between about $0.1\>{\rm ms}$ and $10\>{\rm ms}$ for a normal pulsar. Thus, we expect that the SKA can distinguish the disformal Kerr black hole from the Kerr black hole through the non-circularity of the spacetime around Sgr A*.