In Einstein's theory of general relativity, the Schwarzschild metric 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 a
Schwarzschild black hole of any mass could exist if conditions became
sufficiently favorable to allow for its formation.
The Schwarzschild metric In Schwarzschild coordinates, the line
element for the Schwarzschild metric has the form
where is the proper time,
c is the speed of light, t is the time coordinate,
is the radial coordinate, θ is the colatitude,
φ is the longitude, and is the Schwarzschild radius of the
massive body, a scale factor which is related to its mass M by rs = 2GM/c2,
where G is the gravitational constant. 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, 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 Paul Painlevé and in 1922 Allvar
Gullstrand independently produced a metric, a spherically symmetric solution
of Einstein's equations, which we now know is coordinate transformation of the
Schwarzschild metric, Gullstrand–Painlevé coordinates, in
which there was no singularity at r = rs. They, however, did not recognize
that their solutions were just coordinate transforms, and in fact used
their solution to argue that Einstein's theory was wrong. In 1924 Arthur
Eddington produced the first coordinate transformation that showed that the
singularity at r = rs was a coordinate artifact, although he also seems to have
been unaware of the significance of this discovery. Later, in 1932, Georges
Lemaître gave a different coordinate transformation 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, John Synge produced a paper that showed the maximal analytic
extension of the Schwarzschild metric, again showing that the singularity at r
= rs was a coordinate artifact and that it represented two horizons. A similar
result was later rediscovered by George Szekeres, and independently Martin
Kruskal. The new coordinates nowadays known as Kruskal-Szekeres coordinates
were much simpler than Synge's but both provided a single set of coordinates
that covered the entire spacetime. However, perhaps due to the obscurity of
the journals in which the papers of Lemaître and Synge were published their
conclusions went unnoticed, with many of the major players in the field including
Einstein believing that singularity at the Schwarzschild radius was physical.
Progress was only made in the 1960s when the more exact tools of differential
geometry entered the field of general relativity, allowing more exact
definitions of what it means for a Lorentzian manifold to be singular. This
led to definitive identification of the r = rs singularity in the Schwarzschild
metric as an event horizon. Singularities and black holes
The Schwarzschild solution appears to have singularities at r = 0 and r = rs;
some of the metric components "blow up" at these radii. Since the Schwarzschild
metric is only expected to be valid for radii larger than the radius R of the
gravitating body, there is no problem as long as R > rs. For ordinary stars and
planets this is always the case. For example, the radius of the Sun is
approximately 700,000 km, while its Schwarzschild radius is only 3 km.
The singularity at r = rs divides the Schwarzschild coordinates in two
disconnected patches. The exterior Schwarzschild solution with r > rs is
the one that is related to the gravitational fields of stars and
planets. The interior Schwarzschild solution with 0 ≤ r 0, is called a
Schwarzschild black hole. It is a perfectly valid solution of the Einstein
field equations, although it has some rather bizarre properties. For r