Some aspects of modified Hawking mass for two-spheres in Schwarzschild and Reissner-Nordström spacetime

In this paper, we study several aspects of modified Hawking mass (MHM) for two-spheres in Schwarzschild and Reissner-Nordström spacetime. Our purpose is to find the MHM value and its critical points and monotonicity in Schwarzschild and Reissner-Nordström spacetime using conventional calculation methods. The calculations show that MHM is equal to M in Schwarzschild spacetime and is not greater than M in Reissner-Nordström spacetime. In addition, we also find that in Schwarzschild spacetime, MHM has a critical point and is monotone increasing. In contrast, in Reissner-Nordström spacetime, MHM has no critical point and is strictly increasing. These results are in line with our expectations.