The discussion focuses on the effects of Hawking Radiation on black holes, specifically how mass and temperature change over time. The Bekenstein–Hawking formula, $$S=\frac{kAc^3}{4G\hbar}$$, is central to understanding these changes. For a Schwarzschild black hole, mass can be calculated using $$M^2=\frac{\hbar c^3}{4\pi k_{\rm B} G}S$$, while temperature is derived from the relationship $$T=\frac{{\rm d} M}{{\rm d} S}=\frac{\hbar c^3}{8\pi k_{\rm B}G}\frac{1}{M}$$. The Stefan-Boltzmann law $$\frac{{\rm d}M}{{\rm d}t}=-\sigma AT^4$$ further illustrates the exponential decay of mass and the corresponding increase in temperature over time.
PREREQUISITESStudents of physics, astrophysicists, and researchers interested in black hole thermodynamics and the implications of Hawking Radiation on mass and temperature dynamics.
According to the Bekenstein–Hawking formula in black hole thermodynamics, $$S=\frac{kAc^3}{4G\hbar}$$ Consider a Schwarzschild black hole, we know that $$r_{\rm g}=\frac{2GM}{c^2},\ A=4\pi r_{\rm g}^2$$ Therefore, we can get $$M^2=\frac{\hbar c^3}{4\pi k_{\rm B} G}S$$ From thermodynamics, we learn that ##{\rm d}M=T{\rm d}S##, so we have $$T=\frac{{\rm d} M}{{\rm d} S}=\frac{\hbar c^3}{8\pi k_{\rm B}G}\frac{1}{M}$$I believe this is what you want.James Way said:how the mass and temperature change over time and how to calculate them