black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
physics, mathematical physics, philosophy of physics
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The theory called the theory of general relativity is the classical field theory that in physics describes the field of gravity.
In general relativity physical spacetime is modeled in terms of differential geometry as a Lorentzian manifold whose pseudo-Riemannian metric – or rather the Levi-Civita connection that corresponds to it – encodes the field of gravity.
The action functional describing the dynamics of this field is the Einstein-Hilbert action, in which the field of gravity enters in terms the integral of the scalar curvature of the Levi-Civita connection over spacetime.
As usual in classical field theory, the physically realized configurations – here: Levi-Civita connections – are those that extremize this functional. The Euler-Lagrange equations characterizing these extrema are the Einstein equations.
Historically the theory of general relativity was developed by Albert Einstein based on the theory known as special relativity. Given the conceptual simplicity of the Einstein-Hilbert action, there are several variations of his original version of the theory that are immediately obtained by adding certain terms to the action functional. One of these generalization is supergravity, which extends the theory from ordinary differential geometry to supergeometry.
Matthias Blau, Lecture notes on general relativity (web)
Emil T. Akhmedov, Lectures on General Theory of Relativity (arXiv:1601.04996)
Pietro Menotti, Lectures on gravitation (arXiv:1703.05155)
For some introductory exposition, see