Schwarzschild radius

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The above diagram illustrates five interrelated properties of mass together with the proportionality constants that relate these properties. Every sample of mass is believed to exhibit all five properties, however, due to extremely large proportionality constants, it is generally impossible to verify more than two or three properties for a specific sample of mass.
  • The Schwarzschild radius (rs) represents the ability of mass to cause curvature in space and time.
  • The standard gravitational parameter (μ) represents the ability of a massive body to exert Newtonian gravitational forces on other bodies.
  • Inertial mass (m) represents the Newtonian response of mass to forces.
  • Rest energy (E0) represents the ability of mass to be converted into other forms of energy.
  • The Compton wavelength (λ) represents the response of mass to local geometry.

The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a characteristic radius associated with every mass. It is the radius for a given mass where, if that mass could be compressed to fit within that radius, no known force or degeneracy pressure could stop it from continuing to collapse into a gravitational singularity. 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[1][2] 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[3], 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.

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[edit] 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

rs 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 / c2, 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.

This can be extended to show that an object of any density can be large enough to fall within its own Schwarzschild radius,

V_s \propto \rho^{-3/2},

where

Vs is the volume of the object, and
ρ is its density.

[edit] Classification by Schwarzschild radius

[edit] 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[4].) 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.

[edit] 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.

[edit] 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.

[edit] 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".[5]

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.[6] 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.[7]

[edit] Other uses for the Schwarzschild radius

The Schwarzschild radius is primarily used to characterize black holes. However, every mass has a well defined Schwarzschild radius, and certain properties of massive bodies can be determined using the body’s Schwarzschild radius.

[edit] 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:

  • tr 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),
  • rs is the Schwarzschild radius.

The Pound, Rebka experiment in 1959 proved general relativity by measuring the time dilation of the earth’s gravity. By measuring the earth’s time dilation, this experiment indirectly measured the earth’s Schwarzschild radius.

[edit] 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:

This equality can be generalized to elliptic orbits as follows:

\frac{a}{r_s} \left( \frac{2 \pi a}{c T} \right)^2 = \frac{1}{2}

where:

[edit] 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.

[edit] See also

Classification of black holes by type:

A classification of black holes by mass:

[edit] References

  1. ^ K. Schwarzschild, "Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie", Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin, Klasse fur Mathematik, Physik, und Technik (1916) pp 189.
  2. ^ K. Schwarzschild, "Uber das Gravitationsfeld einer Kugel aus inkompressibler Flussigkeit nach der Einsteinschen Theorie", Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin, Klasse fur Mathematik, Physik, und Technik (1916) pp 424.
  3. ^ Encyclopedia Britannica [1]
  4. ^ Bryner, Jenna (January 9, 2008). "Colossal Black Hole Shatters the Scales". SPACE.COM. http://www.space.com/scienceastronomy/080109-aas-massive-black-holes.html. Retrieved on 2008-04-02. 
  5. ^ Hamed Moradi, "An Early History of Black Holes", (2004) Monash University
  6. ^ A. Einstein, "On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses", Annals of Mathematics, (1939)
  7. ^ J.R. Oppenheimer, H. Snyder, "On Continued Gravitational Contraction", Physical Review 56 (1939) p455.
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