Theory of relativity

The Schwarzschild solution and black holes

Outside a spherical, nonrotating mass, the Schwarzschild solution predicts an event horizon at the Schwarzschild radius.

r_s=\frac{2GM}{c^2}

Definition: Key quantity

The metric and field tensors have coordinate components, but physical predictions must be invariant. The central relation is $r_s=\frac{2GM}{c^2}$ .

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Reading the equation

Outside a spherical, nonrotating mass, the Schwarzschild solution predicts an event horizon at the Schwarzschild radius.

Example: Limit and interpretation

Consider a weak field, low speeds, and a region far from the source. General relativity should approach the appropriate Newtonian description; near a horizon or in a strong field this approximation fails.

Solution

This is the correspondence principle: the newer theory recovers tested results of the older one in its domain of validity.

Example: Estimating the Schwarzschild radius

The Schwarzschild radius is $r_s=2GM/c^2$ . Estimate it for the Sun using $M_\odot=1.99\times10^{30}\,kg$ , $G=6.67\times10^{-11}\,SI$ , and $c=3.00\times10^8\,m/s$ . Why does this not mean the present Sun is a black hole?

Solution

$r_s\approx2(6.67\times10^{-11})(1.99\times10^{30})/(3.00\times10^8)^2\approx2.95\times10^3\,m$ , about $3\,km$ . The Sun’s radius is about $7\times10^8\,m$ , vastly larger; a horizon forms only if the mass is compressed inside $r_s$ .

In Schwarzschild coordinates, the metric has a time coefficient $1-r_s/r$ and radial coefficient $(1-r_s/r)^{-1}$ . Their coordinate singularity at $r=r_s$ is not divergent curvature: Eddington–Finkelstein or Kruskal coordinates make the metric regular across the horizon. At that radius, even outward-directed light cannot increase its radius enough to escape the interior. The horizon is a global causal boundary, not a material surface. Spherical symmetry and zero spin are essential assumptions; realistic rotating black holes require the Kerr solution.

Under spherical symmetry, Birkhoff’s theorem says the vacuum exterior must be static and Schwarzschild, even if the interior mass distribution is not uniform. It does not describe the collapsing interior or a rotating black hole. A distant observer sees gravitational redshift and increasingly delayed signals, while a freely falling observer crosses the horizon in finite proper time.

Quick check

What does the central expression in this topic describe?

Which is the most appropriate interpretation?

References

Sean Carroll (2019). Spacetime and Geometry
Misner, Thorne, Wheeler (1973). Gravitation