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# Schwarzschild metric

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 Title: Schwarzschild metric Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Schwarzschild metric

In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild vacuum or Schwarzschild solution) is the solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. The solution is named after Karl Schwarzschild, who first published the solution in 1916.

According to Birkhoff's theorem, the Schwarzschild metric is the most general spherically symmetric, vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a black hole that has no charge or angular momentum. A Schwarzschild black hole is described by the Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass.

The Schwarzschild black hole is characterized by a surrounding spherical surface, called the event horizon, which is situated at the Schwarzschild radius, often called the radius of a black hole. Any non-rotating and non-charged mass that is smaller than its Schwarzschild radius forms a black hole. The solution of the Einstein field equations is valid for any mass M, so in principle (according to general relativity theory) a Schwarzschild black hole of any mass could exist if conditions became sufficiently favorable to allow for its formation.

## Contents

• The Schwarzschild metric 1
• History 2
• Singularities and black holes 3
• Alternative coordinates 4
• Flamm's paraboloid 5
• Orbital motion 6
• Symmetries 7
• Quotes 8
• Notes 10
• References 11

## The Schwarzschild metric

In Schwarzschild coordinates, the line element for the Schwarzschild metric has the form

c^2 {d \tau}^{2} = \left(1 - \frac{r_s}{r} \right) c^2 dt^2 - \left(1-\frac{r_s}{r}\right)^{-1} dr^2 - r^2 \left(d\theta^2 + \sin^2\theta \, d\varphi^2\right),

where

The analogue of this solution in classical Newtonian theory of gravity corresponds to the gravitational field around a point particle.

In practice, the ratio rs/r is almost always extremely small. For example, the Schwarzschild radius rs of the Earth is roughly 8.9 mm, while the Sun, which is 3.3×105 times as massive has a Schwarzschild radius of approximately 3.0 km. Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The ratio only becomes large close to black holes and other ultra-dense objects such as neutron stars.

The Schwarzschild metric is a solution of Einstein's field equations in empty space, meaning that it is valid only outside the gravitating body. That is, for a spherical body of radius R the solution is valid for r > R. To describe the gravitational field both inside and outside the gravitating body the Schwarzschild solution must be matched with some suitable interior solution at r = R.

## History

The Schwarzschild solution is named in honor of Karl Schwarzschild, who found the exact solution in 1915 and published it in 1916, a little more than a month after the publication of Einstein's theory of general relativity. It was the first exact solution of the Einstein field equations other than the trivial flat space solution. Schwarzschild died shortly after his paper was published, as a result of a disease he contracted while serving in the German army during World War I.

Johannes Droste in 1916 independently produced the same solution as Schwarzschild, using a simpler, more direct derivation.

In the early years of general relativity there was a lot of confusion about the nature of the singularities found in the Schwarzschild and other solutions of the Einstein field equations. In Schwarzschild's original paper, he put what we now call the event horizon at the origin of his coordinate system. In this paper he also introduced what is now known as the Schwarzschild radial coordinate (r in the equations above), as an auxiliary variable. In his equations, Schwarzschild was using a different radial coordinate that was zero at the Schwarzschild radius.

A more complete analysis of the singularity structure was given by David Hilbert in the following year, identifying the singularities both at r = 0 and r = rs. Although there was general consent that the singularity at r = 0 was a 'genuine' physical singularity, the nature of the singularity at r = rs remained unclear.

In 1921 Lemaître coordinates) to the same effect and was the first to recognize that this implied that the singularity at r = rs was not physical. In 1939 Howard Robertson showed that a free falling observer descending in the Schwarzschild metric would cross the r = rs singularity in a finite amount of proper time even though this would take an infinite amount of time in terms of coordinate time t.

In 1950, Schwarzschild, K. (1916). "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie".

• Schwarzschild, K. (1916). "Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit".
• Flamm, L. (1916). "Beiträge zur Einstein'schen Gravitationstheorie".
• Adler, R.; Bazin, M.; Schiffer, M. (1975). Introduction to General Relativity (2nd ed.).
• Landau, L. D.; Lifshitz, E. M. (1951). The Classical Theory of Fields.
• Misner, C. W.; Thorne, K. S.; Wheeler, J. A. (1970). Gravitation.
• Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity.
• Taylor, E. F.; Wheeler, J. A. (2000). Exploring Black Holes: Introduction to General Relativity.
• Heinzle, J. M.; Steinbauer, R. (2002). "Remarks on the distributional Schwarzschild geometry".
• Foukzon, J. (2008). "Distributional Schwarzschild Geometry from nonsmooth regularization via Horizon".
• Text of the original paper, in Wikisource
• Translation: Antoci, S.; Loinger, A. (1999). "On the gravitational field of a mass point according to Einstein's theory".
• A commentary on the paper, giving a simpler derivation: Bel, L. (2007). "Über das Gravitationsfeld eines Massenpunktesnach der Einsteinschen Theorie".