Schwarzschild Horizon Anatomy

Schwarzschild spacetime is the cleanest place to separate a black hole’s invariant anatomy from the failure of a particular coordinate chart. For a positive mass, the surface at areal radius $r=2GM$ is a regular null event horizon and a Killing horizon; the curvature singularity is instead at $r=0$. The distinction is geometric, survives any regular change of coordinates, and provides the benchmark against which rotating and charged horizons should be checked.

Helpful background. Exact Solutions of Einstein’s Equations supplies the metric family and its extensions; Causality and Global Structure supplies event horizons and conformal infinity; The Einstein Equation fixes the vacuum theory; Killing Fields and Isometry Groups supplies static symmetry; and Black-Hole Geometry and Mechanics orients the horizon notions used here.

The invariant Schwarzschild regions

Work in four-dimensional, asymptotically flat vacuum general relativity with $M\gt0$. In the usual exterior chart,

$$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_2^2, \qquad f(r)=1-\frac{2GM}{r}.$$

The coordinate $r$ has invariant meaning: every orbit of spherical symmetry has area $4\pi r^2$. Thus statements formulated with the areal radius do not depend on retaining Schwarzschild coordinates. The connected exterior associated with one asymptotic end is $r\gt2GM$. In the maximally extended eternal spacetime it is the domain of outer communications for that end: the region capable both of receiving signals from its past null infinity and of sending signals to its future null infinity.

The hypersurface $r=2GM$ is null. Its future component is the boundary of the causal past of future null infinity and hence an event horizon. The static Killing field $\xi=\partial_t$, normalized so that $\xi^2\to-1$ at spatial infinity, becomes null there. Consequently the same surface is a Killing horizon generated by the integral curves of $\xi$. These are two different definitions that coincide in this equilibrium solution; their coincidence is not part of the definition of either one.

Visser 2009, §1 and the Schwarzschild limit in §9 emphasizes the need to distinguish the spacetime from a convenient chart. The global event-horizon identification additionally assumes the chosen asymptotically flat extension, not merely the local line element near a zero of $f$.

Why the horizon is not a curvature singularity

In Schwarzschild coordinates, $g_{tt}$ vanishes and $g_{rr}$ diverges at $r=2GM$. Those components diagnose the chart, not the curvature. The Kretschmann scalar is

$$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} =\frac{48G^2M^2}{r^6}.$$

At the horizon it equals $3/(4G^4M^4)$, which is finite for $M\gt0$; it diverges only as $r\to0$. Ingoing Eddington–Finkelstein coordinates make the regularity explicit. With

$$v=t+r_*, \qquad \frac{dr_*}{dr}=\frac{1}{f(r)},$$

the metric becomes

$$ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2d\Omega_2^2,$$

whose coefficients are nonsingular at $r=2GM$. This chart crosses the future horizon but does not by itself create the global extension. Conversely, finding finite metric coefficients is not enough to prove that a surface is an event horizon; the causal boundary must still be identified.

The invariant singular boundary at $r=0$ is spacelike in the black-hole interior of the maximal Schwarzschild solution. It is not a worldline of a point particle sitting at the center. The vacuum metric ends there because curvature becomes unbounded and the classical extension fails.

Horizon generators and the bifurcation sphere

The eternal geometry contains a future horizon $\mathcal H^+$ and a past horizon $\mathcal H^-$. They intersect in a spacelike two-sphere $B$, the bifurcation sphere, on which the horizon-generating Killing field vanishes. Every cross-section of either horizon has the round metric

$$q_{AB}dx^A dx^B=(2GM)^2d\Omega_2^2,$$

so the basic horizon data are

$$r_H=2GM, \qquad A_H=16\pi G^2M^2, \qquad {}^{(2)}R=\frac{2}{r_H^2}=\frac{1}{2G^2M^2}.$$

For the asymptotically normalized generator, the surface gravity follows from $\kappa=f'(r_H)/2$:

$$\kappa=\frac{1}{4GM}.$$

This normalization matters. Replacing $\xi$ by a constant multiple rescales $\kappa$, although the null hypersurface and its intrinsic two-geometry are unchanged. The role of the Killing generator and surface gravity in stationary mechanics is developed in Bardeen, Carter, and Hawking 1973, pp. 162–164.

A black hole formed by collapse has a future event horizon but need not have the past horizon or bifurcation sphere of the eternal extension. The latter are therefore useful equilibrium structures, not compulsory pieces of every physical formation history.

Trapped and untrapped round spheres

For a round two-sphere, each future null expansion is the fractional rate of change of its area along one of the two future null normals:

$$\theta_{(k)}=\frac{1}{A}\frac{dA}{d\lambda_k} =\frac{2}{r}\frac{dr}{d\lambda_k}.$$

In the exterior, outgoing light rays increase $r$ while ingoing rays decrease it, so the two expansions have opposite signs. On a suitable stationary horizon cross-section, the outward expansion vanishes. In the future black-hole region, both future-directed families move toward smaller areal radius, and the round spheres are trapped. The time-reversed white-hole region instead contains anti-trapped round spheres. Saying simply “all $r\lt2GM$ spheres are trapped” would therefore ignore which component of the maximal extension is meant.

These expansion statements are quasi-local properties of chosen spheres. The event horizon remains globally defined. Their agreement at $r=2GM$ in the stationary slicing is a powerful consistency check, but a dynamical spacetime can separate the corresponding surfaces.

Check the complete horizon data

Starting from $f(r)=1-2GM/r$, find the positive root, horizon area, surface gravity, intrinsic Gaussian curvature, and four-dimensional Kretschmann scalar at the root. Which of these quantities proves that the root is an event horizon?

Solution

The root is $r_H=2GM$. Hence $A_H=4\pi r_H^2=16\pi G^2M^2$ and, with the Killing field normalized at infinity, $\kappa=f'(r_H)/2=1/(4GM)$. A round two-sphere of radius $r_H$ has Gaussian curvature $K=1/r_H^2=1/(4G^2M^2)$ and scalar curvature ${}^{(2)}R=2K$. The spacetime invariant is $48G^2M^2/r_H^6=3/(4G^4M^4)$, finite rather than singular.

None of these local or quasi-local numbers alone proves the event-horizon claim. That claim requires the global statement $\mathcal H^+=\partial J^-(\mathscr I^+)$ in the selected asymptotically flat extension. The calculations instead verify that the candidate surface is regular and has the expected stationary horizon data.

Reissner–Nordström Horizon Anatomy next shows how electric charge splits the single nondegenerate Schwarzschild root into outer and inner horizons.

References

• Bardeen, James M., Brandon Carter, and Stephen W. Hawking. “The Four Laws of Black Hole Mechanics.” Communications in Mathematical Physics 31 (1973): 161–170. DOI.
• Visser, Matt. “The Kerr Spacetime: A Brief Introduction.” In The Kerr Spacetime, edited by David L. Wiltshire, Matt Visser, and Susan M. Scott. Cambridge: Cambridge University Press, 2009. arXiv:0706.0622v3.