2022ApJ...941..131C

We provide a complete general relativistic and fluid dynamical treatment to derive an expression for the relativistic accretion rate in curved spacetime. We exploit the Killing symmetry of stationary axisymmetric spacetime to obtain the conserved current, using which we find an integral relation between mass flux and height of the accretion disk. In comparison with the existing different models of thin- and slim-disk approaches our analysis is self-consistent and mathematically rigorous. We reviewed the previous work and surveyed the recent status in this field. Our formula is different from the existing expression in the literature. The difference arises solely due to the rotation parameter of the compact star. The effect becomes important for slim/thick disks and for rapidly spinning compact stars. Thus one needs to alter the formula $\dot{M}=2\pi {rH}\rho {u}^{r}$ for Newtonian or spherically symmetric spacetime while studying the accretion flows around a Kerr black hole. Our formula can also be used to connect the variability of the vertical structure of the accretion disk through the luminosity variation of X-ray binaries, and can thereby explain the origin of the hysteresis loop of the spectral states.