Black-hole horizons and causal structure

The Schwarzschild chart has $g_{tt}=f c^2$ and $g_{rr}=-1/f$, with $f=1-r_s/r$. At $r=r_s$ the static time coordinate fails. The invariant Kretschmann scalar is $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=12r_s^2/r^6$. It is finite at any horizon with nonzero mass parameter, but diverges as $r\to0$ in the ideal Schwarzschild solution. A coordinate coefficient blowing up and an invariant curvature blow-up are therefore different tests.

A stationary observer at fixed r can exist only outside the horizon in this chart. Holding that observer ever closer to the horizon requires increasing proper acceleration; a freely falling observer is a different worldline and has no such support force. Finite horizon curvature does not mean tides are always mild: their magnitude depends on black-hole mass and on the observer’s size. A sufficiently large black hole has much smaller horizon curvature than a small one.

Introduce the tortoise coordinate $r_*=r+r_s\ln|r/r_s-1|$ and advanced time $v=t+r_*/c$. The ingoing Eddington–Finkelstein metric has no inverse-f divergence at the future horizon. Radial ingoing null rays have constant v, while the other radial null family obeys $dr/dv=cf/2$. Outside, this outgoing family can increase r. At the horizon it stays on the null boundary, and inside the future black-hole region its future-directed branch moves toward smaller r.

This describes a statement about possible causal directions, not a material surface pushing light backward. An infaller crosses the horizon in finite proper time, even though the static coordinate t becomes unsuitable for describing the crossing. Signals emitted successively closer to the horizon reach distant observers increasingly delayed and redshifted in the idealized exterior description. No physical observer obtains infinite proper time merely because a coordinate label diverges.

For an asymptotically flat black-hole spacetime, define the black-hole region as events outside the causal past of future null infinity. Its boundary is the event horizon. This definition asks whether a future-directed light signal can ultimately escape, so it uses the global future geometry. An experiment confined to one tiny neighborhood cannot generally locate an event horizon solely from a local curvature measurement.

A trapped surface instead concerns converging future-directed null congruences on a surface. Apparent horizons use related conditions on a chosen spatial slicing and need not coincide with the event horizon in a dynamical spacetime. The ideal eternal Schwarzschild extension also contains regions absent from a simple collapse history. Use its diagrams to understand causal possibilities, while distinguishing the mathematical extension from an actual model of a star forming a black hole.