Hawking temperature ($T_H$)

In units $G=\hslash =c=k_B=1$, a stationary black hole with surface gravity $\kappa$ has Hawking temperature $T_H=\kappa /(2\pi )$.

Near a nonextremal Euclidean horizon, the radial--time plane is smooth only when Euclidean time has period ${\beta }_H=4\pi /f^{\prime }(r_H)$. Any other period produces a conical singularity at the horizon.

On the side where V is positive, set $r-r_0=V^{\prime }(r_0){\rho }^2/4$. Its radial Euclidean metric becomes the displayed plane in polar coordinates. The would-be singularity is conical unless the angular variable $|V^{\prime }(r_0)|\tau /2$ has period exactly $2\pi$. This fixes $\beta =4\pi /|V^{\prime }(r_0)|$; a nontrivial integer multiple still has a cone excess at the tip.

For a normalized static metric $d{s}^2=-Vd{t}^2+d{r}^2/V+r^{2}d{\Omega }^2$ with a simple outer zero at $r_0$, set $b=V^{\prime }(r_0)>0$. The Wick rotation and $r-r_0=b{\rho }^2/4$ give $d{\rho }^2+(b\rho /2{)}^{2}d{t}_E^2$ to leading order, so smoothness fixes period $4\pi /b$. In an ingoing null chart, ${\nabla }_{k}k=(b/2)k$ on the future horizon; the outgoing chart gives the opposite sign on the past branch. The Hawking temperature is $b/(4\pi )=|\kappa |/(2\pi )$ and uses the time generator normalized at infinity.

For a Euclidean radial-time metric $C(r)(f(r)d{t}_E^2+d{r}^2/f(r))$, assume $f$ has a simple horizon zero and $C$ is smooth, finite and positive there. A proper radial coordinate $\rho =2\sqrt{C(r_H)(r-r_H)/f^{\prime }(r_H)}$ puts the leading metric in the form $d{\rho }^2+f^{\prime }(r_H{)}^2{\rho }^{2}d{t}_E^2/4$. Smoothness fixes the displayed period independently of $C(r_H)$. When the Lorentzian time is normalized at infinity, the period is the inverse Hawking temperature in natural units. A conformal factor singular at the horizon need not preserve this argument.

// Target: general-relativity.bigb

Hawking radiation is the approximately thermal flux seen at future infinity when outgoing quantum-field modes are traced backward through a collapsing black-hole geometry. The exponential near-horizon redshift produces nonzero Bogoliubov beta coefficients at temperature $T_H=\kappa /(2\pi )$.

In a collapse spacetime, choose the quantum field's incoming state to have no incoming particles at past null infinity and to be regular in freely falling coordinates near the forming event horizon. The late outgoing modes then have the Hawking temperature, while there is no incoming thermal bath. This collapse-state boundary condition differs from an equilibrium state on an eternal black hole. The Hawking exponential ray map and the thermal ratio of Hawking Bogoliubov coefficients determine the leading late-time particle spectrum; greybody factors describe scattering outside the event horizon.

Black-hole evaporation is mass loss through Hawking radiation. For a large isolated Schwarzschild black hole, the leading mass dependence is $dM/dt\propto -M^{-2}$ and the evaporation timescale is proportional to $M^3$. This semiclassical scaling cannot determine the endpoint when the curvature reaches quantum-gravitational scales.

For a large isolated Schwarzschild black hole with a fixed effective set of radiated species, Hawking radiation has $T\propto M^{-1}$ and luminosity $P=\alpha /M^2$ in natural units. Integrating $dM/dt=-\alpha /M^2$ gives the displayed law. A horizon-area blackbody radiation estimate gives $\alpha =1/(15360\pi )$; greybody factors and species change it. With mass-dependent $\alpha$, integrate $t={\int }_{M(t)}^{M_0}{m}^{2}dm/\alpha (m)$ instead. The semiclassical gravity approximation fails near Planck mass, so a formal zero of the cubic law does not settle the final state.

Approximate an isolated Schwarzschild black hole as emitting photon blackbody radiation from an area $16\pi G^2{M}^2/c^4$ at Hawking temperature $\hslash c^3/(8\pi G{k}_{B}M)$. The Stefan–Boltzmann law gives $P=\hslash c^6/(15360\pi G^2{M}^2)$. Integrating $c^{2}dM/dt=-P$ gives the displayed lifetime, about $2\times 10^{67}$ years for a solar mass. Greybody factors and additional emitted species modify the coefficient; the robust large-mass scaling is $M_0^3$, and the semiclassical calculation does not determine the final quantum-gravity endpoint.

// Target: general-relativity.bigb

The late outgoing rays from a nonextremal collapsing black hole have an exponential relation between a regular early affine null coordinate $U$ and late retarded time $u$. Logarithmic divergence of the tortoise coordinate at the horizon gives $U_H-U=A{e}^{-\kappa u}$ up to subleading terms. Backward propagation turns $e^{-i\omega u}$ into $(U_H-U{)}^{i\omega /\kappa }$; its positive- and negative-frequency Fourier parts give a thermal Bogoliubov transformation.

For the Hawking exponential ray map, regulated integrals ${\int }_0^{\infty }{x}^{ia}{e}^{-(ϵ\pm i{\omega }^{\prime })x}dx=\Gamma (1+ia)(ϵ\pm i{\omega }^{\prime }{)}^{-1-ia}$ have squared-modulus ratio $e^{-2\pi a}$ as $ϵ\downarrow 0$, where $a=\omega /\kappa$. Combining this with the canonical identities for a bosonic Bogoliubov transformation gives the bosonic occupation $[e^{2\pi \omega /\kappa }-1{]}^{-1}$. Continuum modes require wave-packet normalization to interpret finite particle counts.

For a Schwarzschild black hole, a greybody factor is a mode's transmission probability through the exterior scattering potential. The spectrum of Hawking radiation at future null infinity is the thermal horizon occupation multiplied by this factor, with a sum over angular modes.

The black hole information paradox is the apparent conflict between unitary quantum evolution and the semiclassical prediction that complete black-hole evaporation leaves only thermal Hawking radiation. A resolution must explain how information is preserved or why one of the assumptions behind the conflict fails.