Schwarzschild geometry and coordinate horizons

The angular part of the metric makes a symmetry sphere’s area equal to four pi times r squared, defining the areal radius geometrically. It does not make r a radial proper distance. On a constant Schwarzschild-time slice outside the gravitational radius, a small radial proper interval is the coordinate interval divided by the square root of f. Static observers follow constant spatial coordinates only where f is positive; the temporal Killing vector is then timelike.

Birkhoff’s result makes a spherical vacuum region locally Schwarzschild under the stated zero-Lambda hypotheses, without requiring a static stellar interior. It does not show that an arbitrarily chosen exterior region contains an event horizon. A material surface outside twice m removes that coordinate radius from the vacuum exterior altogether. A future event horizon is defined by the global causal structure, such as which events can send signals to future null infinity in an asymptotically flat black-hole completion.

At twice m, the Schwarzschild temporal and radial coefficients become degenerate or divergent, while a scalar formed from the full curvature remains finite. The Ricci scalar alone is an inadequate diagnostic because it vanishes throughout this vacuum solution even near its true singularity. The Kretschmann scalar instead diverges at zero radius and stays finite at the horizon for nonzero mass. Finiteness of one scalar is not a universal proof of regularity, so an explicit regular coordinate chart supplies the complementary check here.

Introduce an ingoing null time using the tortoise coordinate. The metric then has a finite mixed radial-time term and no divergent radial coefficient at the future horizon. This chart extends through that horizon, though it does not by itself cover every region of the maximal eternal solution. A freely falling observer can cross a sufficiently large black hole’s horizon without encountering infinite local curvature. The infinite acceleration needed to remain static there instead diagnoses the failure of that observer family.

A static observer’s proper time is the square root of f times coordinate time. For a photon, the energy associated with the temporal Killing vector is conserved along the null geodesic. Dividing that constant by the local static clock normalization gives the measured frequency. The resulting ratio depends on both endpoint radii and the fact that both observers are static. Radial motion adds a Doppler contribution, so a statement about gravitational redshift without observer specifications is incomplete.

A static emitter deeper in the exterior sends a lower frequency to a static receiver farther out. This comparison concerns successive crests measured with actual proper clocks, not an intrinsic loss of the conserved Killing energy along the ray. The limiting receiver at infinity uses the chosen time normalization. Neither the static formula nor the stationary observer worldline extends as a physical prescription through the horizon. The flat-spacetime lab illustrates coordinate-dependent components with an invariant interval, while carrying no curved metric or horizon.