Black hole

In an asymptotically predictable spacetime, a black hole is the region that cannot send a causal signal to future null infinity. Its future boundary is the event horizon.

The stationary axisymmetric Einstein–Maxwell family of asymptotically flat spacetimes extends the Kerr metric by including electric charge. Its horizon polynomial is the displayed $\Delta$; for $M^2>Q^2+a^2$ the roots are $r_{\pm }=M\pm \sqrt{M^2-Q^2-a^2}$. The outer Killing horizon has area $4\pi (r_{+}^2+a^2)$, angular velocity $a/(r_{+}^2+a^2)$ and electric potential $Q{r}_{+}/(r_{+}^2+a^2)$.

Differentiate $M=(r_{+}^2+a^2+Q^2)/(2r_{+})$, $J=Ma$ and $A=4\pi (r_{+}^2+a^2)$. Eliminating $d{r}_{+}$ and $da$ gives the displayed relation, with ${\Omega }_H=a/(r_{+}^2+a^2)$ and ${\Phi }_H=Q{r}_{+}/(r_{+}^2+a^2)$. Matching Hawking temperature to this First law of black-hole mechanics fixes the Bekenstein-Hawking entropy as $A/4$.

The outer-horizon surface gravity is $(r_{+}-r_{-})/(2(r_{+}^2+a^2))$. Hawking radiation or nondegenerate Euclidean black-hole regularity condition gives $T_H=\kappa /(2\pi )$ in units $G=\hslash =c=k_B=1$. The zero-temperature extremal value is the nonextremal limit; conical regularity alone does not fix a Euclidean period at a degenerate horizon.

The Oppenheimer-Snyder model describes a homogeneous ball of pressureless matter collapsing from rest, with a closed FLRW metric inside and a Schwarzschild metric outside. In geometric units its interior is $d{s}^2=-d{\tau }^2+a^2(d{\chi }^2+{\sin }^2\chi d{\Omega }^2)$, with $a=a_{\max }(1+\cos \eta )/2$ and $\tau =a_{\max }(\eta +\sin \eta )/2$. The boundary is a fixed comoving angle. Density conservation gives $\rho a^3$ constant, and the Friedmann equation at the initial turning point gives $8\pi {\rho }_0{a}_{\max }^2/3=1$.

The comoving dust ball $0\le \chi \le {\chi }_0<\pi /2$ in the Oppenheimer-Snyder model is an Oppenheimer-Snyder cloud. Its pressureless matter has uniform density on each constant-proper time slice and reaches a simultaneous final singularity in the ideal homogeneous model.

Radial outgoing null geodesics in the interior of Oppenheimer-Snyder collapse obey $d\chi /d\eta =1$. Trace the ray crossing the boundary at $({\eta }_H,{\chi }_0)$ backward: it starts at the centre at $\eta ={\eta }_H-{\chi }_0$. This ray is the event horizon; it can exist inside the cloud before the surface crosses $R=2M$. An event horizon is determined by global escape to future null infinity, rather than by a local stopwatch signal.

A boundary clock in Oppenheimer-Snyder collapse reads $\tau =a_{\max }(\eta +\sin \eta )/2$. Its event horizon crossing occurs at $R=2M$, giving the displayed angle, and the final curvature singularity occurs at $\eta =\pi$. The remaining proper time is $a_{\max }(\pi -{\eta }_H-\sin {\eta }_H)/2$. For $R_0\gg 2M$ this approaches $4M/3$, but it is not exactly that value for finite release radius.

At the boundary of an Oppenheimer-Snyder cloud, its areal radius is $R=a\sin {\chi }_0$. Matching to a Schwarzschild metric gives $M=4\pi \rho R^3/3=a_{\max }{\sin }^3{\chi }_0/2$. Thus ${\sin }^2{\chi }_0=2M/R_0$ and $a_{\max }=\sqrt{R_0^3/(2M)}$. The boundary is a radial timelike geodesic with specific Killing energy $\sqrt{1-2M/R_0}=\cos {\chi }_0$. Its radial equation is ${\dot{R}}^2=2M/R-2M/R_0$. The induced metric and the angular extrinsic curvature agree on both sides, so no surface layer is required.

For $D\ge 4$, the static spherical vacuum metric $d{s}^2=-fd{t}^2+f^{-1}d{r}^2+r^{2}d{\Omega }_{D-2}^2$ has an event horizon at $r_h={\mu }^{1/(D-3)}$. With time normalized at infinity its ADM mass is $M=(D-2){\Omega }_{D-2}\mu /(16\pi G_D)$, where ${\Omega }_{D-2}$ is the unit-sphere area. For $D=5$ and $\mu =2m$, $r_h=\sqrt{2m}$ and $M=3\pi m/(4G_5)$. The spherical orbits are three-spheres, even though the radial-time Penrose diagram has the same causal pattern as its four-dimensional counterpart.

The tortoise coordinate is $r_{\ast }=r+(r_h/2)\log |(r-r_h)/(r+r_h)|$. Set $u=t-r_{\ast }$ and $v=t+r_{\ast }$, then $U=-e^{-u/r_h}$ and $V=e^{v/r_h}$ in the right exterior. The extended radial metric is $-r_h^2(r+r_h{)}^2{e}^{-2r/r_h}dUdV/r^2$, regular at $r=r_h$. The horizons are $UV=0$, while $r=0$ gives $UV=1$ and is a spacelike curvature singularity. Compactification by $\arctan U,\arctan V$ displays two exteriors, a future black hole and a past white hole.

The Penrose inequality compares asymptotic energy and an appropriate horizon or enclosing area in geometrized units. Its physical motivation uses censorship, positive radiated energy, settling to a Kerr black hole, and Hawking's area theorem. The area variable requires care: the area of an arbitrary apparent horizon on non-time-symmetric data does not give a universally valid inequality. The Riemannian Penrose inequality uses the relevant outermost minimal surface under nonnegative scalar curvature.

For complete asymptotically flat three-dimensional Riemannian initial data with nonnegative scalar curvature and an outermost compact minimal surface boundary of area $A$, the Riemannian Penrose inequality gives the displayed bound. Equality is realized by the spatial exterior of Schwarzschild spacetime. Time-symmetric gravitational initial data have this form when the Einstein field equations and energy hypothesis give the required nonnegative curvature. This theorem does not justify substituting arbitrary non-time-symmetric apparent horizon area into the same formula.

The heuristic derivation of the Penrose inequality from Hawking's area theorem needs an initial area comparison, in addition to enclosure of the apparent horizon by the event horizon. Set inclusion alone is not an area inequality. An outer area-minimizing surface or appropriate enclosing-area hypotheses supply the needed comparison in formulations where it is valid; arbitrary slice-dependent apparent-horizon area need not do so.

The four-dimensional charged spherically symmetric electrovacuum solution has radial function $f=1-2M/r+Q^2/r^2$. For $0<|Q|<M$, its outer and inner horizon radii are $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. A complete two-ended spatial bridge has a maximal Cauchy development ending at smoothly extendible inner Cauchy horizons in the exact solution.

For $f=(r-r_{+})(r-r_{-})/r^2$, near-horizon Euclidean geometry has polar form $d{\rho }^2+[f^{\prime }(r_{+})/2{]}^2{\rho }^{2}d{\tau }^2$. Smoothness gives inverse temperature $4\pi /f^{\prime }(r_{+})$ and the displayed formula. This assumes a nondegenerate Killing horizon and time normalized at infinity. Its nonextremal limit is zero at $M=|Q|$.

In the ideal thermal-cutoff model where charged particles of mass $m$ are emitted only for $T>m$, a sufficiently large initial mass keeps charge fixed throughout evaporation exactly when $m$ is at least the fixed-charge Reissner-Nordstrom temperature maximum. Neutral radiation then approaches an extremal charged remnant of area $4\pi Q^2$. This conclusion does not include nonthermal electric-field discharge, nor establish finite-time attainment.

Set $y=r_{+}/|Q|\ge 1$. Then $M/|Q|=(y+y^{-1})/2$ and $|Q|T=(y^{-1}-y^{-3})/(4\pi )$. Differentiation gives a unique maximum at $y=\sqrt{3}$, proving the displayed mass and temperature. The curve is zero at extremality and asymptotic to $1/(8\pi M)$ at large mass.

A neutral radial timelike geodesic of an extremal black hole has conserved specific energy $E=(1-M/r{)}^2\dot{t}$, giving the displayed radial equation. An infalling geodesic reaches $r=M$ with finite nonzero $\dot{r}=-E$ and finite proper time. In ingoing coordinates, outgoing radial null curves have $dr/dv=(1-M/r{)}^2/2$; those starting inside approach $r=M$ only at infinite $v$ and cannot return to the same exterior future null infinity. This distinguishes causal access from length on a static slice, and supplies an operational event horizon test even without strictly trapped round spheres.

For $M=|Q|>0$ and the exterior areal radius $r>M$, integrate $d\rho /\rho =dr/(r-M)$ and choose $\rho =r-M$. The Reissner-Nordstrom metric becomes the displayed form. Constant-time proper distance to $\rho =0$ diverges, but these exterior coordinates do not cover the horizon; ingoing null coordinates are regular there. Infinite spatial throat length therefore does not establish absence of an event horizon.

For $M>0$ and $M\ge |Q|$, put $r_{\pm }=M\pm \sqrt{M^2-Q^2}$. The stationary Killing vector field normalized at infinity generates the outer Killing horizon. In an ingoing null chart its covector is $dr$ on that horizon; the Killing equation then gives ${\nabla }_{k}k=\frac{1}{2}{f}^{\prime }(r_{+})k$, where $f=(r-r_{+})(r-r_{-})/r^2$. Thus the positive nonextremal surface gravity is $(r_{+}-r_{-})/(2r_{+}^2)$. It tends to zero when $M=|Q|$, where the horizon becomes a degenerate Killing horizon. The past horizon branch has the opposite signed nonaffinity for the same time-translation field.

For $0<|Q|<M$, a Reissner-Nordstrom spacetime has an outer event horizon and an inner Cauchy horizon. Between them $r$ is timelike; inside the inner horizon $r$ is spacelike and the $r=0$ curvature singularity is timelike. The exact analytic extension contains repeated exterior blocks beyond the Cauchy horizons. Small perturbations produce severe inner-horizon blueshift and mass inflation, so that exact continuation is not a robust prediction for collapse. At $|Q|=M$ the horizons merge into a degenerate Killing horizon; for $|Q|>M$ the spherical electrovacuum solution instead has a naked singularity.

In ingoing Reissner-Nordstrom spacetime, orient $-{\partial }_r$ to the future and normalize the other radial null vector to ${\partial }_v+(f/2){\partial }_r$. The spherical null expansion in ingoing coordinates gives $-2/r$ and $f/r$. Hence the strictly future trapped surfaces among the round spheres occur exactly between the distinct horizons. At either horizon the outgoing null expansion vanishes. Inside the inner horizon the same round spheres are not future trapped.

A white hole is the time reverse of a black hole: causal trajectories can emerge from its past singularity, while exterior future-directed trajectories cannot enter through its past horizon. The maximal Schwarzschild spacetime contains both a future black-hole interior and a past white-hole interior. Both have the same areal-radius range, illustrating why a global time orientation and choice of extension are essential in causal statements.

In four-dimensional asymptotically flat general relativity, a regular stationary vacuum black hole is a Kerr black hole under the standard connectedness and analyticity hypotheses. The electrovacuum extension is the Kerr--Newman family.

For a regular analytic stationary asymptotically flat vacuum black hole with a connected nondegenerate horizon, the stationary Killing vector field either generates the horizon or the spacetime admits an additional commuting axial Killing vector field. In the rotating case the horizon generator is $K+{\Omega }_H\Phi$, where $\Phi$ has closed rotational orbits. Together with the appropriate staticity and black-hole uniqueness theorem, this identifies an equilibrium uncharged collapse remnant with the Kerr metric. The theorem concerns an equilibrium geometry; it does not itself prove that arbitrary dynamical collapse settles to equilibrium.

The black-hole no-hair principle says that radiation carries away the higher multipole structure of a collapsing isolated body, leaving a stationary black hole specified by conserved asymptotic charges such as mass, angular momentum, and electric charge.

An extremal black hole lies at the parameter bound where its inner and outer horizons coincide. Its surface gravity and hence its semiclassical Hawking temperature vanish.

At a degenerate static horizon the Euclidean proper radial distance diverges logarithmically, so the horizon is an infinite end rather than a finite polar origin. Its time circle is not contracted at a conical tip, and local cone regularity imposes no finite period. The vanishing limiting surface gravity determines zero Hawking temperature; the simple-horizon periodicity argument must not be applied unchanged at the degenerate root.

A naked singularity is a spacetime singularity visible to distant observers because it is not hidden behind an event horizon.

The weak cosmic censorship conjecture predicts, under appropriate generic isolated initial-data and matter assumptions, that collapse singularities are hidden from distant observers by an event horizon. It motivates exterior predictability in the physical argument for the Penrose inequality. It is distinct from the strong cosmic censorship conjecture, which concerns inextendibility of the maximal Cauchy development.

The static spherically symmetric electrically charged black hole in five spacetime dimensions has $f(r)=1-2M/r^2+Q^2/r^4$. When $M>Q$ it has outer and inner horizon radii $r_{\pm }^2=M\pm \sqrt{M^2-Q^2}$; for $M=Q$ it is an extremal black hole, while $M<Q$ exposes a naked singularity.

For $V=1-2M/r^2+Q^2/r^4$, an orthonormal-frame curvature calculation gives $(V^{\prime \prime }{)}^2+6(V^{\prime }/r{)}^2+12((1-V)/r^2{)}^2$. Expansion gives the displayed invariant, diverging at zero radius. It is finite at every positive root of V. Thus horizon coordinate singularities are distinct from the central curvature singularity.

In the regular metric $d{s}^2=-fd{v}^2+2dvdr+r^{2}d{\Omega }^2$, a future causal tangent has $\dot{v}\ge 0$. For $\dot{v}>0$, causality gives $\dot{r}\le f\dot{v}/2-r^2|\dot{\Omega }{|}^2/(2\dot{v})$. If $f<0$ between two horizons, the areal radius must decrease. An inner event cannot reach the original exterior future null infinity without traversing that band in the forbidden direction.

For distinct horizon radii $a>b>0$, integrating $d{r}_{\star }/dr=f^{-1}$ gives $r_{\star }=r+\frac{a^3}{2(a^2-b^2)}\log |(r-a)/(r+a)|-\frac{b^3}{2(a^2-b^2)}\log |(r-b)/(r+b)|+C$. The tortoise coordinate defines constant retarded or advanced coordinates on radial null geodesics.

For $f=1-s/r^2+q/r^4$ and the spacetime orientation $-dt\wedge dr\wedge d{\Omega }_3$, the Hodge star operator gives ${\int }_{S^3}\star d{k}^{♭}=2{\pi }^2{r}^3{f}^{\prime }\to 4{\pi }^{2}s$. A prefactor $1/(16\pi )$ produces $\pi s/4$, whereas the dimension-adjusted ADM mass normalization produces $3\pi s/8$. The normalization and orientation must be specified rather than imported from four dimensions.

An event horizon is the null boundary of the causal past of future null infinity. Its global definition makes it respond teleologically to matter that falls in later.

Every event in complete Minkowski spacetime emits a future-directed null geodesic ending on future null infinity. Therefore its black-hole region $M\setminus J^{-}({\mathcal{I}}^{+})$ is empty and there is no intrinsic event horizon. The null edges of the Minkowski conformal compactification represent infinity, not hidden regions inside the physical spacetime. An accelerated observer can instead have an observer-dependent Rindler horizon.

Strong asymptotic predictability requires the exterior region visible from future null infinity to lie inside a globally hyperbolic neighborhood of the conformal completion. It is the cosmic-censorship-type hypothesis used in classical black-hole area and topology arguments.

In a strongly asymptotically predictable spacetime, the causal future on a later Cauchy surface of one connected component of an earlier black-hole region lies inside one connected component of the later black-hole region. A classical black hole can merge with another component, but it cannot bifurcate into causally disconnected black holes.

A Myers-Perry black hole is the higher-dimensional asymptotically flat rotating vacuum generalization of the Kerr black hole. In six dimensions, a solution with one nonzero rotation parameter $a$ and horizon radius $r_{+}$ has mass parameter $\mu =r_{+}(r_{+}^2+a^2)$.

A five-dimensional Myers-Perry black hole with equal rotational parameters has a metric depending on a single radial coordinate. In a fibered coordinate system, its radial functions can be written $f=1-r_0^2/r^2+{\alpha }^2{r}_0^2/r^4$, $h=1+{\alpha }^2{r}_0^2/r^4$, and $\Omega =\alpha r_0^2/(r^{4}h)$. For $0<\alpha <r_0/2$ there are outer and inner Killing horizons; at equality they form a degenerate Killing horizon. With the standard sphere identifications and $G_5=1$, its Komar mass is $3\pi r_0^2/8$.

For the singly rotating six-dimensional Myers-Perry solution with $G=1$ and $\mu =r_{+}(r_{+}^2+a^2)$,and the horizon area is

The BTZ black hole is a quotient of three-dimensional Anti-de Sitter spacetime. In lapse-and-shift coordinates its outer and inner horizon radii are $r_{+}$ and $r_{-}$, and its outer-horizon surface gravity is

For the nonrotating BTZ black hole family with $M<0$ and angular period $2\pi$, the circumference of a small circle divided by its proper radius tends to $2\pi \sqrt{-M}$. A smooth axis requires this ratio to be $2\pi$, so only $M=-1$ gives regular Anti-de Sitter spacetime. Other values have a conical singularity, with deficit angle $\delta$ or an angular excess when $\delta <0$. The lapse $r^2/{\ell }^2-M$ is strictly positive, so outward radial null geodesics escape from arbitrarily near the axis: the defect is a naked singularity, despite finite local curvature away from it.

Forthe transformations $dv=dt+dr/f$ and $d\chi =d\varphi +\Omega dr/f$ givewhich is regular at every horizon where $f$ vanishes while $\Omega$ remains finite.

For the nonrotating BTZ metric in ingoing coordinates, future null normals to a circle of fixed $v,r$ may be chosen as $\ell ={\partial }_v+(f/2){\partial }_r$ and $n=-{\partial }_r$. Their expansions are ${\theta }_{(\ell )}=f/(2r)$ and ${\theta }_{(n)}=-1/r$, so both are negative inside the outer horizon.

The BTZ black hole can be null-geodesically incomplete at $r=0$ even though every local curvature invariant is finite there. The endpoint comes from the global anti-de Sitter quotient rather than curvature blowup.

In four dimensions the Schwarzschild--anti-de Sitter black hole hasWriting the horizon radius as $r_{+}$ gives $M=r_{+}(1+r_{+}^2/L^2)/2$ and

The spherical five-dimensional Schwarzschild--anti-de Sitter metric has $f(r)=1+r^2/L^2-\mu /r^2$. Its horizon satisfies $\mu =r_H^2(1+r_H^2/L^2)$ and has area $2{\pi }^2{r}_H^3$.

The Schwarzschild--anti-de Sitter heat capacity has the sign of $d{T}_H/d{r}_{+}$. It is positive precisely for $r_{+}>L/\sqrt{3}$, so these large black holes are locally stable in a fixed-temperature reservoir.

A black string is a higher-dimensional black object whose horizon extends along one spatial direction. The direct product of a $d$-dimensional static black-hole metric with a line or circle gives a uniform black string in $d+1$ spacetime dimensions.

A Penrose diagram conformally compactifies a spacetime while preserving causal directions, placing null infinity, timelike infinity, spacelike infinity, horizons, and singularities at finite coordinate locations.

For the Kruskal–Szekeres coordinates of a positive-mass Schwarzschild spacetime, $UV=(1-r/(2M))e^{r/(2M)}$. The displayed conformal compactification makes radial null geodesics lines of slope $\pm 1$. The Killing horizons are $T=\pm X$, and the spacelike curvature singularities $r=0$ lie at $T=\pm \pi /4$, with $|X|<\pi /4$. Two asymptotically flat exterior regions each have past null infinity, null infinity to the future, timelike infinity and spacelike infinity. The two exterior regions cannot communicate by any causal curve; future-directed curves that cross a future horizon instead end at the black-hole singularity.

Spacelike infinity is the ideal endpoint of spatial directions in the Minkowski conformal compactification, represented by $(T,R)=(0,\pi )$. Unlike null infinity, it is a vertex of the radial Penrose diagram.

Future and past timelike infinity are ideal endpoints of asymptotically escaping timelike trajectories in a conformal completion. For the Minkowski conformal compactification, they are the vertices $(T,R)=(\pi ,0)$ and $(-\pi ,0)$; these vertices are not physical spacetime events.

This is the past endpoint of complete inertial timelike geodesics in the Minkowski conformal compactification, at $(T,R)=(-\pi ,0)$. It is the time reversal of future timelike infinity and is reached at infinite physical proper time to the past.

This is the future endpoint of complete inertial timelike geodesics in the Minkowski conformal compactification, at $(T,R)=(\pi ,0)$. The endpoint has finite unphysical coordinate time but infinite physical proper time. Accelerated timelike observers need not have this endpoint; a Rindler horizon can accompany an observer asymptotic to a null direction instead.

Null infinity is the ideal boundary reached by asymptotically escaping null geodesics in a suitable conformal completion. It has future and past components. A finite location on a Penrose diagram need not represent a finite affine parameter.

Past null infinity is the ideal boundary from which incoming null geodesics arrive in an asymptotically flat spacetime. In the Minkowski conformal compactification it is $T-R=-\pi$, with the retarded coordinate tending to $-\infty$ and the advanced coordinate finite.

For $g=q+C(d\psi +A{)}^2$ with $C>0$, projection of every causal curve is causal for $q$, since $q(\dot{\gamma },\dot{\gamma })\le g(\dot{\gamma },\dot{\gamma })\le 0$. Conversely, the lift defined by $d\psi =-A$ along any given base curve has the same norm as that curve. Thus the orbit-space metric $q$ describes the projected causal structure exactly. A Penrose diagram suppressing that circle is a quotient diagram, not the metric of a constant-$\psi$ slice.

A tortoise coordinate straightens radial null directions. For a static radial metric $-F(r)d{t}^2+G(r)d{r}^2$ outside a horizon, choose $d{r}_{\ast }/dr=\sqrt{G/F}$; then $t\pm r_{\ast }$ are constant along the corresponding radial null geodesics. Continuation across horizons requires choosing branches consistently rather than replacing an analytic derivative by its absolute value.

Black-hole thermodynamics relates horizon geometry to thermodynamic laws, assigning temperature and entropy to a stationary black hole.

Black-hole energy extraction transfers part of a black hole's mass, charge, or rotational energy to its exterior. A reversible limiting process preserves the horizon area and saturates the maximum extraction permitted by the First law of black-hole mechanics and Hawking's area theorem.

The laws of black-hole mechanics relate surface gravity, horizon area, mass, angular momentum, and charge in close analogy with the laws of thermodynamics. Quantum theory identifies the analogy physically through Hawking temperature and Bekenstein-Hawking entropy.

The zeroth law states that the surface gravity is constant over a stationary Killing horizon under the standard field-equation and energy assumptions.

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.

In units $G=\hslash =c=k_B=1$, a black hole of horizon area $A$ has entropy $S_{BH}=A/4$.

Changing the Euclidean time period away from its smooth value creates a curvature delta function at the horizon. Applying $(1-\beta {\partial }_{\beta })$ to the Euclidean gravitational partition function isolates this term and gives $S=A/(4G_N)$.

The Gibbons–Hawking–York boundary term makes the Einstein--Hilbert variational problem well posed when the induced boundary metric is fixed. It is the integral of the trace of the boundary's extrinsic curvature with the conventional normalization and sign.

For a nonrotating uncharged stationary black hole, the first law is $\delta M=\kappa \delta A/(8\pi )=T_H\delta S_{BH}$.

The physical-process first law derives the area change caused by a small flux of matter through an initially and finally stationary horizon. Linearized Null Raychaudhuri equation evolution converts the Killing-energy flux into $\Delta M=\kappa \Delta A/(8\pi )$.

For a small perturbation of a stationary horizon, the linearized Null Raychaudhuri equation gives $\delta {\theta }^{\prime }=-8\pi \delta T_{kk}$. With vanishing boundary term $[\lambda \delta \theta ]$, integration by parts gives $\delta A=8\pi \int \lambda \delta T_{kk}d\lambda d{A}_0$. Combining this with affine horizon-generator scaling converts area response to horizon Killing-energy flux.

For horizon generator $\xi =k+{\Omega }_{H}m$, a small matter flux into an initially and finally stationary rotating black hole satisfies

Under the null energy condition and the usual global assumptions excluding naked pathologies, the area of every cross-section of a future event horizon is nondecreasing toward the future. The Null Raychaudhuri equation and future completeness rule out negative expansion of its generators.

Hawking's area theorem states that the total area of classical future event horizons cannot decrease when the null energy condition and the required global assumptions hold. It is the geometric content of the second law of black-hole mechanics.

Two initially well-separated Schwarzschild black holes of mass $m$ each have total horizon area $32\pi G^2{m}^2/c^4$. If they merge into a Schwarzschild black hole, Hawking's area theorem gives final mass $M\ge \sqrt{2}m$. Conserved total energy then bounds escaped radiation by the displayed expression, or $1-1/\sqrt{2}$ of the initial rest energy. This is an upper bound under classical horizon and energy assumptions, not a proof that an actual merger can saturate it.

For future-complete generators of an event horizon, the null convergence condition and hypersurface orthogonality reduce the Null Raychaudhuri equation to ${\theta }^{\prime }\le -{\theta }^2/2$. If ${\theta }_0<0$, integration gives focusing within affine distance $2/|{\theta }_0|$. The null focusing theorem identifies this with a conjugate point to a spacelike surface. A null generator cannot remain on an achronal boundary beyond such a focal point, contradicting future completeness on the horizon. Thus $\theta \ge 0$, and $dA/d\lambda =\theta A\ge 0$ gives the second law of black-hole mechanics. New horizon generators may join toward the future at crease sets, adding area, but generators have no future endpoints on a regular event horizon under these assumptions.