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For the opposite internal torques on a black hole and its accretion disk, the total angular momentum ${\mathbf{J}}_t$ and the black-hole spin magnitude $J_h$ are constant. If $A=K_2|{\mathbf{J}}_h\times {\mathbf{J}}_d{|}^2$, then $d({\mathbf{J}}_h\cdot {\mathbf{J}}_t)/dt=A$ and $d{J}_d^2/dt=-2A$. The $K_1$ term produces precession; the positive $K_2$ term increases the alignment of the hole with the total vector while decreasing the disk angular-momentum magnitude. The exact direction equation is $\dot{c}=K_2{J}_h{J}_t(1-c^2)$ for $c=\cos \angle ({\mathbf{J}}_h,{\mathbf{J}}_t)$.

The black-hole sphere of influence is a characteristic radius where the central black hole appreciably changes surrounding stellar dynamics. Equating its Newtonian gravitational potential scale $G{M}_{\bullet }/r$ to the ambient squared velocity dispersion gives $r_{\inf }\simeq G{M}_{\bullet }/{\sigma }^2$. Dynamical mass measurements depend on angular resolution $r_{\inf }/D$, tracer brightness, and degeneracies in the orbital distribution, not solely on black hole mass.

Coherent prograde gas accretion increases the angular momentum of a black hole. For a zero-torque thin disc, the rest mass $d{m}_0$ entering from the innermost stable circular orbit changes the hole by $dM=E_{\mathrm{ISCO}}d{m}_0$ and $dJ={\ell }_{\mathrm{ISCO}}d{m}_0$. Its dimensionless spin obeys $d{a}_{\ast }/d\log M=c{\ell }_{\mathrm{ISCO}}/(GM{E}_{\mathrm{ISCO}})-2a_{\ast }$. Starting from zero spin and neglecting captured radiation, the formal extremal endpoint has $M_f=\sqrt{6}{M}_i$ and requires rest-mass supply approximately $1.8464M_i$. Photon capture leads instead to the Thorne spin limit.

The causal past is the time-reversed causal future. Relative to a specified component ${\mathcal{I}}^{+}$ of future null infinity, the black hole region is the complement of its causal past: $\mathcal{B}=\mathcal{M}\setminus J^{-}({\mathcal{I}}^{+})$.

For $r_{-}<r_{+}$, an ingoing chart with $v=t+r_{\ast }$ has metric $-Vd{v}^2+2dvdr+R^{2}d{\Omega }_2^2$. In its future interior, $\nabla r={\partial }_v+V{\partial }_r$ is future timelike, so every future causal curve decreases $r$ and cannot reach future null infinity. The singularity $r=r_{-}$ is spacelike. A Kruskal–Szekeres coordinates extension has the same qualitative Penrose diagram as Schwarzschild spacetime, with two exterior regions, a future black-hole interior and a past white hole interior. The past interior has the same radial interval but is not a black hole region; specifying the radial interval alone does not identify the future component. For $r_{-}>r_{+}$, the physical exterior $r>r_{-}$ has $V>0$ everywhere and outgoing null rays escape from arbitrarily near a timelike naked singularity.

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.

Collisionless matter approaching a black hole generally cannot dissipate enough orbital energy to remain bound, so direct capture requires sufficiently small angular momentum. For nonrelativistic incident speed $v$, a Schwarzschild black hole has direct test-particle capture cross-section approximately $16\pi G^2{M}^2/(c^2{v}^2)$. A Newtonian pericentre estimate using the Schwarzschild radius instead gives $4\pi G^2{M}^2/(c^2{v}^2)$. Both show that capture at a given mass density is suppressed by order $(v/c{)}^2$ relative to supersonic Bondi--Hoyle--Lyttleton accretion. Tidal disruption events provide a distinct way to accrete some stellar debris.

Bright elliptical galaxies commonly have a shallow central surface-brightness cusp inside a break radius, whereas many less luminous elliptical galaxies have a steep cusp continuing inward. The latter can have many more luminous tracers at a fixed small radius, helping central black hole dynamical measurements. This empirical distinction is not a universal classification by mass alone. Measurements of central galaxy profiles quantify the contrast.

Core collapse can fail to produce an ordinary successful ejecting supernova, leaving a black hole and retaining much of the processed core. Some envelope material can still escape, depending on the event; failure is not equivalent to a universal zero-yield prescription.

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.

A high-mass X-ray binary has a massive donor star supplying an accreting neutron star or black hole. A strong stellar wind often feeds it, although Roche-lobe overflow and donor equatorial outflows also occur. The short lifetime of the donor associates these systems with young stellar populations.

For a homologous black hole-dominated cluster, write $E=-bG{M}_{H}Nm/(2R)$, and let a captured star's specific orbital energy be ${ϵ}_{\mathrm{cap}}=\xi E/(Nm)$. Removing negative orbital energy raises the remaining cluster energy, while growth ${\dot{M}}_H=m\Gamma$ deepens its potential. The energy budget gives the displayed radius response. Preferentially removing strongly bound stars can drive expansion; representative-energy removal instead gives the adiabatic contraction due to black-hole growth. The sign is not fixed by the loss cone angle alone.

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 Kruskal spacetime is the maximal analytic extension of the positive-mass vacuum Schwarzschild spacetime. It contains two exterior regions, a black hole, and a white hole. Kruskal–Szekeres coordinates remove both horizon coordinate singularities, while $r=0$ remains a curvature singularity.

A loss cone consists of low-angular momentum stellar trajectories whose next pericentre enters a central absorbing region. For capture radius $r_c$ in a black hole-dominated potential, the Newtonian near-parabolic estimate is $J_{\mathrm{lc}}^2\simeq 2G{M}_H{r}_c$. At radius $r$ and speed $v$, its angular half-width is $J_{\mathrm{lc}}/(rv)$. Two-body relaxation can refill this depleted part of phase space.

In a low-mass X-ray binary, the donor star is relatively low mass. Roche-lobe overflow usually feeds an accretion disk around a neutron star or black hole. The low-mass label concerns the donor rather than the total binary mass. Accumulated fuel on a neutron star can produce a Type I X-ray burst.

The endpoint is set chiefly by the final core mass and composition, with envelope loss and binary interaction modifying the relation to initial mass. A star that never sustains hydrogen fusion is a brown dwarf, which cools; it is not a post-main-sequence remnant of a hydrogen-burning star. Very low-mass hydrogen-burning stars have such long lifetimes that their predicted eventual helium-white-dwarf endpoints need not yet be represented by isolated evolved examples.

Low- and intermediate-mass stars generally lose their giant envelopes and leave white dwarfs supported by electron degeneracy pressure. Most have carbon-oxygen cores; more massive degenerate cores may be oxygen-neon, while a helium white dwarf commonly results from binary stripping before helium ignition. The exposed hot core can ionize the expelled gas as a planetary nebula. The remnant subsequently cools without sustained central fusion. The Chandrasekhar limit constrains a cold nonrotating electron-supported core; it is not a universal initial-star mass cutoff.

More massive stars can proceed through advanced burning to an iron-group core. Further fusion of such nuclei does not provide energy, while electron captures and photodisintegration reduce pressure support and drain energy. Collapse can produce a core-collapse supernova and a neutron star, or a black hole if a sufficiently massive core collapses directly or gains substantial fallback. The remnant outcome depends on final core structure and the success of ejecting the mantle, rather than only on the original mass. Some oxygen-neon cores instead undergo an electron-capture supernova.

Binary evolution introduces additional branches: an accreting white dwarf may undergo classical novae, a thermonuclear Type Ia supernova leaving no ordinary white-dwarf remnant, or collapse in an appropriate core-composition regime. Extremely massive pair-unstable cores can be disrupted entirely by a pair-instability supernova. These possibilities are distinct from simple isolated-star cooling or ordinary core collapse. The broad white-dwarf/neutron-star/black-hole distinction follows the remnant support mechanism and surviving core mass; NASA's stellar overview and stellar-type account describe the corresponding observational classes.

Use $G=\hslash =c_{\mathrm{light}}=k_B=1$ and normalize the stationary time at infinity. For a regular stationary, asymptotically flat electrovacuum black hole, the Kerr-Newman metric hasInitially assume $M>0$ and $d>0$, so the outer Killing horizon is nondegenerate. Its surface gravity, angular velocity, electric potential and area areThese are horizon quantities for the generator $\chi ={\partial }_t+{\Omega }_H{\partial }_{\varphi }$.

The laws of black-hole mechanics provide the first reason to regard $\kappa$ and area as thermodynamic variables. The Zeroth law of black-hole mechanics makes $\kappa$ constant on an equilibrium horizon. The first law for the Kerr-Newman family readsFor example, write the horizon relation as $M=(r_{+}^2+a^2+Q^2)/(2r_{+})$ and use $J=Ma$; differentiating eliminates $d{r}_{+}$ and $da$ to give this law. The last two terms are rotational and electric work. The classical second law of black-hole mechanics says that area cannot decrease under the null energy condition and appropriate global horizon assumptions. The third law of black-hole mechanics concerns unattainability of a zero-surface-gravity regular horizon by a finite physical process.

This analogy alone does not determine the temperature scale: if entropy were $\eta A_H$, the first law would only give $T=\kappa /(8\pi \eta )$. The decisive input is quantum field theory in the collapsing or stationary background. In a collapse vacuum that is regular for freely falling observers, the late outgoing retarded time $u$ and a regular affine null coordinate $U$ satisfy the Hawking exponential ray mapAn outgoing mode $e^{-i\omega u}$ therefore behaves as $(-U/C{)}^{i\omega /\kappa }$. Continuing this power through the horizon changes its amplitude by the factor $e^{-\pi \omega /\kappa }$. Its positive- and negative-frequency decomposition consequently has the thermal ratio of Hawking Bogoliubov coefficientsCombining that ratio with bosonic normalization $|\alpha {|}^2-|\beta {|}^2=1$ gives a Planck occupation $1/(e^{2\pi \omega /\kappa }-1)$. The fermionic normalization instead gives the corresponding Fermi factor. Thus Hawking radiation fixesFor rotating charged modes, the horizon energy is $\tilde{\omega }=\omega -m_{\varphi }{\Omega }_H-q{\Phi }_H$, so the emission spectrum contains the same temperature with angular-momentum and charge chemical potentials. Exterior scattering supplies greybody factors; it changes the received flux, not the horizon temperature. Superradiant bosonic modes require the usual signed absorption factor, and a globally regular thermal bath need not exist throughout an asymptotically flat rotating exterior. The collapse-state emission argument is the relevant one.

A complementary check is the Euclidean black-hole regularity condition. The local corotating nondegenerate horizon geometry is Rindler-like:Smoothness at $\rho =0$ requires $\tau$ to have period $2\pi /\kappa$. Imaginary-time periodicity is precisely inverse temperature, agreeing with the radiation calculation. Matching the resulting temperature to the first law fixes the Bekenstein-Hawking entropy to $S=A_H/4$. The generalized second law then uses $S_{\mathrm{outside}}+A_H/4$: the classical area theorem alone does not apply to the negative-energy quantum flux responsible for evaporation.

Substitution gives the Kerr-Newman horizon temperatureIt reduces to $1/(8\pi M)$ for a Schwarzschild black hole. The nonextremal limit towards $d=0$ gives zero temperature. At exact extremality the Euclidean horizon is degenerate, so the elementary conical-period argument does not itself fix a period; zero temperature here is the limiting semiclassical result. The reasoning assumes ordinary Einstein–Maxwell dynamics, a regular horizon, the stated normalization at infinity and a regime where quantum fields on a slowly evolving classical geometry are a useful approximation. The mechanical laws, radiation spectrum and Euclidean regularity agree under these assumptions, which is substantially stronger evidence than the classical analogy alone.

For $M>0$ and $Q^2>2M^2$, the physical exterior starts at $r=b>2M$. Throughout it, $f>0$, so the putative zero of $f$ is not in the exterior and no regular horizon shields the curvature singularity. The normal to $r=b$ has positive limiting squared norm $g^{rr}=f(b)>0$, identifying a timelike singular boundary. Outgoing radial null geodesics obey $dr/dt=f(r)>0$ and escape from arbitrarily close to that boundary. HenceAt $Q^2=2M^2$, the singular boundary coincides with $f=0$ and is null. There is no regular shielding horizon of positive area; this marginal singular extremal case must be separated from both the regular black hole and the supercritical timelike naked singularity. If “naked” is used simply to mean unshielded by a regular horizon, equality belongs on that side of the distinction, but its causal boundary is different.

The usual positive-mass restriction matters. For $M<0$, $f(r)>0$ at every positive radius and there is no positive-radius horizon; the central zero-area boundary at $r=0$ is singular and naked. For $Q=0$ its angular sectional curvature is $2M/r^3$, which diverges there. For $Q\ne 0$, $C=r(r-b)$ with $b<0$, and the angular sectional curvature $[1-f(C^{\prime }{)}^2/(4C)]/C$ also diverges as $r\downarrow 0$. The displayed family is undefined at $M=0$ with nonzero $Q$; its uncharged zero-mass limit is flat.