Schwarzschild metric

The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun.

According to Birkhoff's theorem, the Schwarzschild metric is the most general spherically symmetric vacuum solution of the Einstein field equations.

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 (within the theory of general relativity) a Schwarzschild black hole of any mass could exist if conditions became sufficiently favorable to allow for its formation.

The Schwarzschild metric is a solution of Einstein's field equations in empty space, meaning that it is valid only outside the gravitating body.

The ratio becomes large only in close proximity to black holes and other ultra-dense objects such as neutron stars.

[7] The Schwarzschild solution is analogous to a classical Newtonian theory of gravity that corresponds to the gravitational field around a point particle.

Schwarzschild died shortly after his paper was published, as a result of a disease (thought to be pemphigus) he developed while serving in the German army during World War I.

[10] Johannes Droste in 1916[11] independently produced the same solution as Schwarzschild, using a simpler, more direct derivation.

[12] 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 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.[14] In 1950, John Synge produced a paper[15] 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.

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 the singularity at the Schwarzschild radius was physical.

[19] Real progress was made in the 1960s when the mathematically rigorous formulation cast in terms 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, i.e., a hypersurface in spacetime that can be crossed in only one direction.

[14] The Schwarzschild solution appears to have singularities at r = 0 and r = rs; some of the metric components "blow up" (entail division by zero or multiplication by infinity) at these radii.

Since the Schwarzschild metric is expected to be valid only for those radii larger than the radius R of the gravitating body, there is no problem as long as R > rs.

The exterior Schwarzschild solution with r > rs is the one that is related to the gravitational fields of stars and planets.

The Schwarzschild coordinates therefore give no physical connection between the two patches, which may be viewed as separate solutions.

One such important quantity is the Kretschmann invariant, which is given by At r = 0 the curvature becomes infinite, indicating the presence of a singularity.

However, a greater understanding of general relativity led to the realization that such singularities were a generic feature of the theory and not just an exotic special case.

It is a perfectly valid solution of the Einstein field equations, although (like other black holes) it has rather bizarre properties.

[22] A curve at constant r is no longer a possible worldline of a particle or observer, not even if a force is exerted to try to keep it there; this occurs because spacetime has been curved so much that the direction of cause and effect (the particle's future light cone) points into the singularity.

[citation needed] The surface r = rs demarcates what is called the event horizon of the black hole.

The spatial curvature of the Schwarzschild solution for r > rs can be visualized as the graphic shows.

Then replace the (r, φ) plane with a surface dimpled in the w direction according to the equation (Flamm's paraboloid) This surface has the property that distances measured within it match distances in the Schwarzschild metric, because with the definition of w above, Thus, Flamm's paraboloid is useful for visualizing the spatial curvature of the Schwarzschild metric.

Noncircular orbits, such as Mercury's, dwell longer at small radii than would be expected in Newtonian gravity.

This can be seen as a less extreme version of the more dramatic case in which a particle passes through the event horizon and dwells inside it forever.

Intermediate between the case of Mercury and the case of an object falling past the event horizon, there are exotic possibilities such as knife-edge orbits, in which the satellite can be made to execute an arbitrarily large number of nearly circular orbits, after which it flies back outward.

This is thus the subgroup of the ten-dimensional Poincaré group which takes the time axis (trajectory of the star) to itself.

To understand the physical meaning of these quantities, it is useful to express the curvature tensor in an orthonormal basis.