Schwarzschild radius

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The Schwarzschild radius or gravitational radius is a characteristic radius associated with every mass. The term is used in physics and astronomy, especially in the theory of gravitation, general relativity. It was found in 1916 by Karl Schwarzschild and results from his discovery of an exact solution for the gravitational field outside a static, spherically symmetric star (see Schwarzschild metric, which is a solution of the Einstein field equations). The Schwarzschild radius is proportional to the mass. The Sun's mass has a Schwarzschild radius of about 3 km, the Earth's about 9 mm.

An object smaller than its Schwarzschild radius is called a black hole. The surface at the Schwarzschild radius acts as an event horizon. Neither light nor particles can escape through this surface from the region inside, hence the name black hole.

Connie Willis's hard science fiction short story "The Schwarzschild Radius" offers both an accessible and accurate explanation of the phenomenon which makes it surprisingly applicable to our not-so-scientific pursuits.

Contents

1 Formula for the Schwarzschild radius

2 Average density within the Schwarschild radius

3 Classification by Schwarzschild radius

3.1 Supermassive black hole
3.2 Stellar black hole
3.3 Primordial black hole

4 See also

Formula for the Schwarzschild radius

The Schwarzschild radius is proportional to the mass, with a proportional constant involving the gravitational constant and the speed of light. The formula for the Schwarzschild radius is

<math>R_\mbox{sch} = \frac{2GM}{c^2} \approx 1.5 \times 10^{-27} M<math>

where

R_sch is the Schwarzschild radius

<math>G<math> is the gravitational constant, that is 6.67 × 10^-11 N m² / kg²;

M is the mass of the black hole; and

c² is the speed of light squared, that is (299,792,458 m/s)² = 8.98755 × 10¹⁶ m²/s².

Average density within the Schwarschild radius

It is interesting to see what the average density of an amount of matter of mass <math>M<math> is, if squeezed inside a volume with radius <math>R<math> equal to the Schwarzschild radius.

The volume for a sphere of radius <math>R<math> scales as the third power of the radius, <math>R^3<math>. The Schwarzschild radius is proportional to the mass <math>M<math>, so the volume inside the Schwarzschild radius scales as <math>M^3<math>. The average density <math>\overline{\rho}<math> scales as

<math>\overline{\rho} = \frac{mass}{volume} \sim \frac{M}{M^3}

\sim \frac{1}{M^2}<math> so the larger the mass the smaller the average density of matter is, if squeezed to within its Schwarzschild radius.

Classification by Schwarzschild radius

Supermassive black hole

If one accumulates matter of normal density (say 1000 kg/m³, such as water, which also happens to be about the same as the average density of the Sun) up to about 300,000 times the mass of the Sun, such an accumulation will fall inside its own Schwarzschild radius and thus it would be a supermassive black hole of 300,000 solar masses (Supermassive black holes up to a few billion solar masses are thought to exist). The supermassive black hole in the center of our galaxy (2.5 million solar masses) constitutes observationally the most convincing evidence for the existence of black holes in general. It is thought that large black holes like these don't form directly in one collapse of a cluster of stars. Instead they may start as a stellar-sized black hole and grow larger by the accretion of matter and other black holes. The larger the mass of a galaxy, the larger is the mass of the supermassive black hole in its center.

Stellar black hole

If one accumulates matter at nuclear density (the density of the nucleus of an atom, about 10¹⁸ kg/m³; neutron stars also reach this density), such an accumulation would fall within its own Schwarzschild radius at about 3 solar masses and thus would be a stellar black hole.

Primordial black hole

Conversely, a small mass has an extremely small Schwarzschild radius. A mass as big as Mount Everest has a Schwarzschild radius smaller than a nanometre. Its average density at that size would be so high that no known mechanism could form such extremely compact objects. Such black holes might possibly be formed in an early stage of the evolution of the universe, just after the Big Bang, when densities were extremely high. Therefore these hypothetical baby black holes are called primordial black holes.