Hawking Radiation & Black Hole Thermodynamics
Black holes as thermal objects with temperature ~ 1/M and entropy ~ area.
A Schwarzschild black hole of mass $M$ has event horizon at $r_s = 2GM/c^2$ and is characterized by surface gravity $\kappa = c^4/(4GM)$. Hawking (1974) showed quantum vacuum near the horizon emits blackbody radiation at the Hawking temperature
$$T_H = \frac{\hbar c^3}{8\pi G M k_B}.$$For a solar-mass BH this is $\sim 60$ nK — negligible. For a $10^{12}$ kg primordial BH, $T_H \sim 10^{12}$ K, and the BH evaporates explosively.
The four laws of black-hole thermodynamics (Bardeen, Carter, Hawking) are formal analogs of ordinary thermodynamics, identifying surface gravity with temperature and horizon area with entropy:
$$S_{\rm BH} = \frac{k_B c^3 A}{4 G \hbar} \quad \text{(Bekenstein–Hawking)}.$$An "$N = 1$" Schwarzschild BH of mass $M$ has $S \sim 10^{77}$ — vastly more than ordinary matter of the same mass. This vast entropy is the seed of the information paradox: where does the infalling information go when the BH evaporates? Modern resolutions involve entanglement, islands, and the AdS/CFT correspondence.