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This is unfounded because that law has relativistic corrections. For example, the meaning of "r" is physical distance in that classical law, and merely a coordinate in General Relativity.] The Schwarzschild metric can also be derived using the known physics for a circular orbit and a temporarily stationary point mass. [1]
This often yields a result giving a family of solutions implicitly, but in many examples does yield the general solution in explicit form. In general relativity, to obtain timelike geodesics it is often simplest to start from the spacetime metric , after dividing by d s 2 {\displaystyle ds^{2}} to obtain the form
In an isotropic chart (on a static spherically symmetric spacetime), the metric (aka line element) takes the form = + (+ (+ ())), < <, < <, < <, < < Depending on context, it may be appropriate to regard , as undetermined functions of the radial coordinate (for example, in deriving an exact static spherically symmetric solution of the Einstein field equation).
The Schwarzschild solution, taken to be valid for all r > 0, is called a Schwarzschild black hole. It is a perfectly valid solution of the Einstein field equations, although (like other black holes) it has rather bizarre properties. For r < r s the Schwarzschild radial coordinate r becomes timelike and the time coordinate t becomes spacelike. [22]
where = + is the Riemann curvature tensor and is the Christoffel symbol. Because it is a sum of squares of tensor components, this is a quadratic invariant. Einstein summation convention with raised and lowered indices is used above and throughout the article.
The Levi-Civita connection (like any affine connection) also defines a derivative along curves, sometimes denoted by D. Given a smooth curve γ on (M, g) and a vector field V along γ its derivative is defined by = ˙ (). Formally, D is the pullback connection γ*∇ on the pullback bundle γ*TM.
The covariant derivative for a generic differential form of degree p is defined by = + (). Bianchi's second identity then becomes D R b a = 0. {\displaystyle DR_{\;\,b}^{a}=0.} The difference between a connection with torsion, and the unique torsionless connection is given by the contorsion tensor .
For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives. In general relativity , the connection plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor.