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The radial first integral for an equatorial null geodesic in Schwarzschild spacetime isSince $L=r^2\dot{\varphi }$, regarding $r$ as a function of $\varphi$ givesSet $y=1/r$. Then $dr/d\varphi =-y^{\prime }/y^2$ and therefore $\dot{r}=-L{y}^{\prime }$. Division by $L^2$ gives the equatorial null orbit in Schwarzschild spacetimewhere $b=L/E$ is the impact parameter. Differentiating with respect to $\varphi$ givesBy continuity this yields

Substituting the forms from part (b) into Cartan's second structure equation gives the independent mixed-index curvature 2-formsEvery form is proportional to its corresponding basis 2-form, so the curvature operator is diagonal. The diagonal Ricci tensor components are contractions of these sectional curvature coefficients. In each case the coefficients cancel in the pattern $2-1-1=0$, with the Lorentzian sign included when the time direction is contracted. HenceThis is consistent with Schwarzschild spacetime being a vacuum solution away from $r=0$.

WriteFor a radial null ray, the Schwarzschild spacetime line element givesOutside the horizon an ingoing ray has $dr/dt<0$, so it takes the minus sign. Since the Schwarzschild tortoise coordinate obeys $d{r}_{\ast }/dr=f^{-1}$,It follows thatis constant along every ingoing radial light ray.

A photon sphere is a spherical set of circular null geodesics. In Schwarzschild spacetime it occurs at $r=3M$ and is unstable.

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.

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.

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.