Schwarzschild radius

Template:Properties of mass The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a characteristic radius associated with every quantity of mass. It is the radius (and therefore the four (4) dimensional volume (e.g., including the dimension of time)) which when it contains a specified quantity of matter (mass), and thereby is of a specified density (where density is defined to be mass/volume), no known force or degeneracy pressure could stop it from continuing to collapse, that is, to decrease in volume (and therefore become in effect infinitely dense, and therefore a gravitational singularity, a point of infinite density. The term is used in physics and astronomy, especially in the theory of gravitation, and general relativity.

In 1916, Karl Schwarzschild obtained an exact solution to Einstein's field equations for the gravitational field outside a non-rotating, spherically symmetric body (see Schwarzschild metric). The solution contained a term of the form $1/(2M-r)$; the value of $r$ making this term singular has come to be known as the Schwarzschild radius. The physical significance of this singularity, and whether this singularity could ever occur in nature, was debated for many decades; a general acceptance of the possibility of a black hole did not occur until the second half of the 20th century.

The Schwarzschild radius of an object is proportional to the mass. Accordingly, the Sun has a Schwarzschild radius of approximately 3 km, while the Earth's is only about 9 mm, the size of a peanut.

An object smaller than its Schwarzschild radius is called a black hole. The surface at the Schwarzschild radius acts as an event horizon in a non-rotating body. (A rotating black hole operates slightly differently.) Neither light nor particles can escape through this surface from the region inside, hence the name "black hole". The Schwarzschild radius of the (currently hypothesized) Supermassive black hole at our Galactic Center would be approximately 7.8 million km.

History

The significance of the singularity at $r=2M$ (in natural units) was first raised by Jacques Hadamard, who, during a conference in Paris in 1922, asked what might happen if a physical system could ever obtain this singularity. Albert Einstein insisted that it could not, pointing out the dire consequences for the universe, and jokingly referred to the singularity as the "Hadamard disaster".

Schwarzschild's original model of a star assumed an incompressible fluid; Einstein pointed out that this was an unreasonable assumption, as sound waves would propagate at infinite speed. In his own work, Einstein reconsidered a model of a star where the components of the star were orbiting masses, and showed that the orbital velocities would exceed the speed of light at the Schwarzschild radius. In 1939, he used this to argue that no such thing can happen, and so the singularity could not occur in nature. The same year, Robert Oppenheimer and Hartland Snyder considered a model of a dust cloud, where the dust particles of the cloud were moving radially, towards a single point, and showed that the dust particles could reach the singularity in finite proper time. After passing the limit, Oppenheimer and Snyder noted that light cones were directed inwards, and that no signal could escape outside.

Formula for the Schwarzschild radius

The Schwarzschild radius is proportional to the mass with a proportionality constant involving the gravitational constant and the speed of light:

$r_{s}={\frac {2Gm}{c^{2}}},$

where

$r_{s}$ is the Schwarzschild radius,

$G$ is the gravitational constant,

$m$ is the mass of the gravitating object, and

$c$ is the speed of light.

The proportionality constant, $2G/c^{2}$, is approximately 1.48×10^-27 m/kg, or 2.95 km/solar mass.

The formula for the Schwarzschild radius can be found by setting the escape velocity to the speed of light. Note that although the result is correct, general relativity must be used to properly derive the Schwarzschild radius. It is only a coincidence that Newtonian physics produces the same result.

An object of any density can be large enough to fall within its own Schwarzschild radius,

$V_{s}\propto \rho ^{-3/2},$

where

$V_{s}$ is the volume of the object, and

$\rho$ is its density.

Classification by Schwarzschild radius

Supermassive black hole

If one accumulates matter at normal density (1000 kg/m³, for example, the density of water) up to about 150,000,000 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 150,000,000 solar masses. (Supermassive black holes up to 18 billion solar masses have been observed.) The supermassive black hole in the center of our galaxy (3.7 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 similar to 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.

Other uses for the Schwarzschild radius

The Schwarzschild radius in gravitational time dilation

Gravitational time dilation near a large, slowly rotating, nearly spherical body, such as the earth or sun can be reasonably approximated using the Schwarzschild radius as follows:

${\frac {t_{r}}{t}}={\sqrt {1-{\frac {r_{s}}{r}}}}$

where:

• $t_{r}$ is the elapsed time for an observer at radial coordinate "r" within the gravitational field,
• $t$ is the elapsed time for an observer distant from the massive object (and therefore outside of the gravitational field),
• $r$ is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object),
• $r_{s}$ is the Schwarzschild radius.

The results of the Pound, Rebka experiment in 1959 were found to be consistent with predictions made by general relativity. By measuring Earth’s gravitational time dilation, this experiment indirectly measured Earth’s Schwarzschild radius.

The Schwarzschild radius in Newtonian gravitational fields

The Newtonian gravitational field near a large, slowly rotating, nearly spherical body can be reasonably approximated using the Schwarzschild radius as follows:

${\frac {g}{r_{s}}}\left({\frac {r}{c}}\right)^{2}={\frac {1}{2}}$

where:

• $g$ is the gravitational acceleration at radial coordinate "r",
• $r_{s}$ is the Schwarzschild radius of the gravitating central body,
• $r$ is the radial coordinate, and
• $c$ is the speed of light.

On the surface of the earth:

${\frac {9.80665m/s^{2}}{8.870056mm}}\left({\frac {6375416m}{299792458m/s}}\right)^{2}=\left(1105.59s^{-2}\right)\left(0.0212661s\right)^{2}={\frac {1}{2}}$

The Schwarzschild radius in Keplerian orbits

For all circular orbits around a given central body:

${\frac {r}{r_{s}}}\left({\frac {v}{c}}\right)^{2}={\frac {1}{2}}$

where:

• $r$ is the orbit radius,
• $r_{s}$ is the Schwarzschild radius of the gravitating central body,
• $v$ is the orbital speed, and
• $c$ is the speed of light.

This equality can be generalized to elliptic orbits as follows:

${\frac {a}{r_{s}}}\left({\frac {2\pi a}{cT}}\right)^{2}={\frac {1}{2}}$

where:

• $a\$ is the semi-major axis, and
• $T\$ is the orbital period.

Relativistic circular orbits and the photon sphere

The Keplerian equation for circular orbits can be generalized to the relativistic equation for circular orbits by accounting for time dilation in the velocity term:

${\frac {r}{r_{s}}}\left({\frac {v}{c}}{\sqrt {1-{\frac {r_{s}}{r}}}}\right)^{2}={\frac {1}{2}}$

${\frac {r}{r_{s}}}\left({\frac {v}{c}}\right)^{2}\left(1-{\frac {r_{s}}{r}}\right)={\frac {1}{2}}$

$\left({\frac {v}{c}}\right)^{2}\left({\frac {r}{r_{s}}}-1\right)={\frac {1}{2}}$

This final equation indicates that an object orbiting at the speed of light would have an orbital radius of 1.5 times the Schwarzschild radius. This is a special orbit known as the photon sphere.

See also

• black hole, a general survey
• Chandrasekhar limit, a second requirement for black hole formation

Classification of black holes by type:

• Schwarzschild or static black hole
• rotating or Kerr black hole
• charged black hole or Newman black hole and Kerr-Newman black hole

A classification of black holes by mass:

• micro black hole and extra-dimensional black hole
• primordial black hole, a hypothetical leftover of the Big Bang
• stellar black hole, which could either be a static black hole or a rotating black hole
• supermassive black hole, which could also either be a static black hole or a rotating black hole
• visible universe, if its density is the critical density