A Black Hole Is Not Defined by Newtonian Escape Speed

The familiar calculation that sets a Newtonian escape speed equal to $c$ finds the Schwarzschild length scale, but it does not define a black hole. It imports a relativistic limiting speed into a theory with absolute time, instantaneous gravity, and no causal light cones. In general relativity the question is instead whether any future-directed causal signal can reach a declared asymptotic end.

The Newtonian coincidence

Normalize the potential of an isolated point mass by $\Phi(\infty)=0$, so $\Phi(r)=-GM/r$. A unit test mass launched radially with speed $v$ has conserved specific energy

$$\varepsilon=\frac12v^2-\frac{GM}{r}.$$

Marginal escape means arrival at infinity with zero speed, hence $\varepsilon=0$ and

$$v_{\rm esc}^2=\frac{2GM}{r}.$$

This is a Newtonian energy calculation in a stationary point-mass field, as developed in Tong 2013, §§2.2–2.3, pp. 18–26, PDF. Inserting the relativistic speed $c$ gives

$$v_{\rm esc}=c \quad\Longrightarrow\quad r=\frac{2GM}{c^2}.$$

For a nonrotating, uncharged Schwarzschild black hole, this equals the event horizon’s areal radius. The equality is a useful dimensional coincidence, not a Newtonian derivation: Newtonian mechanics neither makes $c$ a limiting speed nor describes the causal propagation of light, and the substitution is made where $GM/(rc^2)=1/2$, outside a weak-field expansion. The full classical calculation and its assumptions belong to The Newtonian Gravitational Field and Potential.

The relativistic criterion

Let $(\mathcal M,g)$ be a time-oriented spacetime with an asymptotically flat conformal completion, and choose one asymptotic end $E$ with future null infinity $\mathscr I_E^+$. With the causal past taken in the completion, the black-hole region relative to that end is

$$\mathcal B_E =\mathcal M\setminus J^-(\mathscr I_E^+), \qquad \mathcal H_E^+=\partial\mathcal B_E.$$

Thus $p\in\mathcal B_E$ precisely when no future-directed causal curve from $p$ reaches $\mathscr I_E^+$. The set depends on the complete future and on which asymptotic end counts as escape; Booth 2005, §2.1, pp. 1075–1078, Open PDF gives this definition and contrasts it with quasi-local horizons.

No local replacement answers the same question:

• A local speed cannot decide eventual escape. Every local inertial observer measures light at $c$, and every massive body at less than $c$, on either side of a regular event horizon.
• Local curvature need not single out the horizon. A sufficiently large black hole can have weak curvature there, while strong curvature alone does not determine whether a signal reaches the chosen infinity.
• A material surface is unnecessary. In gravitational collapse the event horizon is a causal boundary in spacetime, not the surface of the collapsing matter.
• A preferred slicing is unnecessary. Apparent and trapping horizons may depend on a chosen spacelike slice; the event-horizon definition above does not.

The Black-Hole Region and Future Null Infinity develops the end-relative definition, its asymptotic assumptions, and its global consequences.

A horizon without a black hole

In Minkowski spacetime, uniformly accelerated observers confined to the right Rindler wedge $X\gt\lvert cT\rvert$ cannot receive signals from all of the spacetime. The null wedge boundaries $X=\pm cT$ are Rindler horizons for those observers. Nevertheless the Riemann tensor vanishes everywhere, there is no material boundary, and every Minkowski event can send a causal signal to the spacetime’s future null infinity. Relative to that infinity, $\mathcal B=\varnothing$.

The Rindler construction and its inertial extension are explicit in Crispino, Higuchi, and Matsas 2008, §II.B, arXiv printed pp. 6–7, Eqs. (2.31)–(2.38), Open PDF. This counterexample separates three notions that the escape-speed slogan conflates: an observer-dependent causal horizon, a black-hole region, and strong gravitational curvature.

What the estimate is good for

The escape-speed argument remains a quick estimate of the gravitational length scale $GM/c^2$. It warns that an object compressed to a radius of order $2GM/c^2$ requires relativistic treatment, and it reproduces the exact Schwarzschild areal radius after that solution has supplied the missing causal geometry. It does not locate a horizon in a rotating, charged, dynamical, or non-asymptotically-flat spacetime, and it cannot decide from local data whether a black-hole region exists.

References

• Booth, Ivan. “Black-Hole Boundaries.” Canadian Journal of Physics 83 (2005): 1073–1099. DOI. Open PDF.
• Crispino, Luís C. B., Atsushi Higuchi, and George E. A. Matsas. “The Unruh Effect and Its Applications.” Reviews of Modern Physics 80 (2008): 787–838. DOI. Open PDF.
• Tong, David. Dynamics and Relativity. University of Cambridge lecture notes, 2013. Official PDF.