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Take $u(r)>0$ to be the inward radial speed and select the regular critical solution for a gas reservoir of density ${\rho }_{\infty }$ and adiabatic sound speed $c_{\infty }$. Other mathematically subsonic steady solutions need a different inner boundary condition. For the polytropic equation of state, let $\gamma =1+1/n$, soMass conservation and the Bernoulli equation giveThe gas is at rest at infinity, so its kinetic and gravitational terms vanish there.

Differentiate these two conserved quantities, using $d\log \rho /dr=-2/r-u^{\prime }/u$. Eliminating the density derivative yields the polytropic Bondi accretion equationAt a regular sonic point, both sides vanish:The Bernoulli equation at that point reduces to $n{c}_{\infty }^2=(n-3/2)c_c^2$. For $n>3/2$, equivalently $1<\gamma <5/3$,Since $c_s^2\propto {\rho }^{1/n}$, the critical density is ${\rho }_c={\rho }_{\infty }[2n/(2n-3){]}^n$. Evaluate the conserved mass accretion rate at the sonic point:whereThe isothermal limit $\gamma \to 1$ gives $\lambda =e^{3/2}/4$, agreeing with Isothermal Bondi accretion.

For $\gamma =5/3$, $n=3/2$ and the Newtonian critical radius tends to zero while $c_c$ diverges. The limiting critical solution approaches Mach number one only as $r\to 0$; it has no finite-radius Newtonian sonic transition. This is a failure of the Newtonian inner approximation, not an obstruction to accretion onto a black hole. A physical Schwarzschild black hole has a finite event horizon, and the regular relativistic inflow becomes supersonic outside it.

The singular-looking factor in the rate has a finite limit. Writing $ϵ=n-3/2$, its logarithm is $ϵ\log [n/ϵ]\to 0$, so the critical Bondi accretion rate for gamma equals five thirds isThis is also the leading cold-reservoir limit of relativistic spherical accretion; an exact rate for relativistically hot gas requires the relativistic equations, not this Newtonian expression.

For completeness, the finite relativistic sonic point can be seen without assigning it an arbitrary inner radius. Let $b=c_s^2/c^2$ denote the relativistic squared sound speed and $g=\gamma -1$. For an isentropic relativistic gas with $p=K{\rho }_{\mathrm{rest}}^{\gamma }$, its dimensionless specific enthalpy is $h=g/(g-b)$. Conservation of rest-mass flux and relativistic Bernoulli energy gives the critical relationsFor $g=2/3$ and $c_{\infty }\ll c$, expansion gives $b_{\infty }\simeq (9/4)b_c^2$, henceHere the asymptotic relativistic and Newtonian sound speeds agree to leading order. This finite-radius transition explains why the gas can reach the horizon despite the Newtonian $r_c=0$ result.

Work in the rest frame of the hole and neglect stellar velocities compared with $v$. In addition to $v\gg {\sigma }_{\ast }$, assume the galactic encounter speed is nonrelativistic, $v\ll c$. Treat a star as a test particle with specific angular momentum $\ell =bv$. At its pericentre $r_{\min }$, its radial velocity vanishes. Conservation of energy and conservation of angular momentum giveThe second term is gravitational focusing. For the rough Newtonian estimate in the question, assume capture when the pericentre reaches the Schwarzschild radius, $r_S=2G{M}_{\mathrm{BH}}/c^2$. It follows thatThe approximation keeps the focusing term because $v\ll c$.

Let ${\rho }_{\ast }=m{n}_{\ast }$ be the stellar mass density. Multiplying the geometric collision cross-section by the incoming mass flux gives the collisionless black-hole capture rateFor a supersonic gas with negligible sound speed, Bondi--Hoyle--Lyttleton accretion has capture radius $r_a=2G{M}_{\mathrm{BH}}/v^2$ and rateConsequentlyFor comparable densities and $v=200\mathrm{km}{\mathrm{s}}^{-1}$, the suppression is approximately $4\times 10^{-7}$. Gas can collide, dissipate orbital energy, and be captured from a region of size $GM/v^2$; collisionless particles generally leave on an unbound orbit unless their angular momentum is small enough for direct capture. Chandrasekhar dynamical friction can slow the hole by transferring energy to a stellar wake, but does not mean that the stars producing that wake are swallowed.

There is an important accuracy limitation in the requested numerical coefficient. Newtonian orbital mechanics evaluated at a relativistic event horizon cannot give the exact capture boundary. In Schwarzschild spacetime, a slowly incoming massive particle has dimensionless energy $\mathcal{E}\simeq 1$ and timelike geodesic effective potentialThe double turning point satisfying $V=1$ and $dV/dr=0$ occurs at $r=4r_g$ and $\ell =4G{M}_{\mathrm{BH}}/c$. The exact low-speed boundary is therefore $b_{\mathrm{cap}}\simeq 4G{M}_{\mathrm{BH}}/(cv)$, twice the rough value; it makes the rate ratio $4({\rho }_{\ast }/{\rho }_g)(v/c{)}^2$. The key suppression scaling is unchanged.

Direct capture of ordinary stars or dark matter is therefore an inefficient route to rapid growth of a supermassive black hole under these conditions. Tidal disruption events can capture stellar debris without swallowing an intact point-like star, and special dense environments can modify the supply. These qualifications do not make collisionless capture equivalent to dissipative gas accretion.

The simple zeros $r_{-}$ and $r_{+}$ are Killing horizons: respectively the black-hole event horizon and the cosmological horizon of the static region.

A gravitational-collapse spacetime is stationary and approximately empty in the remote past and settles to a stationary black-hole exterior in the future, with a dynamical region between them. The same comparison of in- and out-positive-frequency modes therefore applies. Tracing an outgoing late-time mode backward through the collapse produces an exponentially blueshifted mixture of early positive and negative frequencies. Its nonzero Bogoliubov beta coefficient yields the thermal occupation numbers of Hawking radiation, while partner modes pass through the event horizon.

Let a future-directed causal tangent in the ingoing chart beThe chosen time orientation has $\dot{v}\ge 0$; equality occurs only for the ingoing radial null direction, which has $\dot{r}<0$. Causality requiresFor $r_{-}<r<r_{+}$ one has $f<0$. The first and third terms are then nonnegative, so if $\dot{v}>0$ the inequality forces $\dot{r}<0$. If $\dot{v}=0$, it forces $\dot{\chi }=0$, and future direction again gives $\dot{r}<0$. Thus $r$ strictly decreases along every future causal curve in this region. No such curve can cross outward through $r=r_{+}$ into $r>r_{+}$, which proves the one-way causal disconnection characteristic of an event horizon.

The second law of black-hole mechanics states that, assuming the null energy condition and the usual predictability hypotheses, the area of cross-sections of a future event horizon never decreases toward the future.

Indeed, the horizon generators form a hypersurface-orthogonal null congruence, so their twist vanishes. If the expansion were negative anywhere, the null focusing theorem would make it diverge to $-\infty$ within finite affine parameter, provided the generator extends that far. This produces a conjugate point, after which the generator cannot remain on an achronal set such as an event horizon. Future completeness rules out the alternative that the generator simply ends first. Therefore $\theta \ge 0$ everywhere, and

In a collapsing spacetime, positive-frequency in-modes are defined at past null infinity and outgoing modes at future null infinity. Tracing a late outgoing wave packet backward toward the forming event horizon produces an exponential blueshift. Its rapidly varying phase has both positive- and negative-frequency parts relative to the past time coordinate, so the Bogoliubov coefficient $\beta$ is nonzero. The resulting bosonic occupation numbers have the Bose-Einstein distribution atwhere $\kappa$ is the surface gravity. The outgoing quanta form Hawking radiation, while their correlated partners cross the horizon.

The second law of black-hole mechanics states that the area of spatial cross-sections of a future event horizon cannot decrease toward the future, provided the null energy condition and the standard global assumptions hold.

Let $k^a$ be an affinely parametrized horizon generator. It is hypersurface-orthogonal, so its null twist vanishes. The Null Raychaudhuri equation in $D$ dimensions and the Einstein equation giveIf $\theta ({\lambda }_0)<0$, integration implies that $\theta$ diverges to $-\infty$ within affine distance at most $(D-2)/|\theta ({\lambda }_0)|$. The resulting focal point would make the generator leave the achronal boundary that defines the event horizon, contradicting its assumed future completeness. Hence $\theta \ge 0$ everywhere. Since null expansion obeysevery horizon area element, and therefore every complete cross-section area, is nondecreasing.

The Penrose diagram is the maximally extended Schwarzschild diagram modified by an ingoing null shell from the chosen ${\mathcal{I}}^{-}$. To its past the vacuum mass is $M$; to its future the exterior mass is $M+E$. Draw the shell as a $45^{\circ }$ ingoing line from the chosen ${\mathcal{I}}^{-}$, crossing the future event horizon and ending on the future spacelike singularity. In that same exterior, label $i^{-}$ and $i^{+}$ at the lower and upper timelike corners, $i^0$ at the spatial corner, and ${\mathcal{I}}^{-}$ and ${\mathcal{I}}^{+}$ as the past and future null boundaries.

The crucial teleological feature is that the final ${\mathcal{H}}^{+}$, traced backward before the shell, lies outside the old $r=2M$ future horizon. It crosses the old past horizon ${\mathcal{H}}^{-}$ at the bifurcation two-sphere and continues into the white-hole region, ending on the past spacelike singularity. Thus the diagram must not draw the relevant ${\mathcal{H}}^{+}$ as beginning at the old bifurcation sphere.

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.

Relativistic spherical accretion is steady radial gas inflow in Schwarzschild spacetime, with conserved rest-mass flux and relativistic Bernoulli function. A regular solution passes a finite sonic point outside the event horizon. For an isentropic gas with $\gamma =5/3$ and cold asymptotic sound speed $c_{\infty }\ll c$, the critical radius is approximately $3GM/(4c{c}_{\infty })$ and the leading mass accretion rate is $\pi G^2{M}^2{\rho }_{\infty }/c_{\infty }^3$. Relativistic corrections resolve the zero-radius critical point of the Newtonian $\gamma =5/3$ model without eliminating accretion.

A timelike geodesic with unit specific Killing energy can plunge from infinity into a Schwarzschild black hole only if its specific angular momentum satisfies $|h|<4M$. The limiting case $|h|=4M$ asymptotes to the unstable orbit at $r=4M$. A geodesic already inside the angular-momentum barrier can cross the event horizon with larger $|h|$, so the origin-at-infinity condition matters.

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.

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.

For a small bounded mass perturbation $m(v)=M_0+\mu (v)$ that becomes constant in the future, write the event horizon radius as $r_H(v)=2M_0+\rho (v)$. The outgoing radial null geodesics of the Vaidya metric give ${\rho }^{\prime }-\rho /(4M_0)=-\mu /(2M_0)$ to first order. Future stationarity selects the displayed solution and removes the exponentially growing homogeneous solution. If ${\mu }^{\prime }\ge 0$, integration by parts gives $\rho -2\mu =2{\int }_v^{\infty }{e}^{-(s-v)/(4M_0)}{\mu }^{\prime }(s)ds\ge 0$. Thus the event horizon can grow before local matter arrives, lying outside the instantaneous spherical apparent horizon $r=2m(v)$. This is a consequence of the event horizon's global definition, not propagation of a local signal into the past.

A trapping horizon is a hypersurface foliated by marginal two-surfaces with one null expansion zero and the other nonzero. A future outer trapping horizon has ${\theta }_{+}=0$, ${\theta }_{-}<0$ and ${\mathcal{L}}_n{\theta }_{+}<0$ for a future ingoing null normal $n$. Unlike an event horizon, this is a quasi-local definition. Under the null energy condition, a regular future outer horizon is spacelike or null. The ingoing derivative condition distinguishes its outer type from an inner trapping 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.