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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]
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).
Use of LaTeX for separately displayed formulas and more complicated inline formulas; Use of LaTeX for formulas involving symbols that are not regularly rendered in Unicode (see MOS:BBB) Avoid formulas in section headings, and when this is necessary, use raw HTML (see Finite field for an example)
The paler hyperbolas represent contours of the Schwarzschild r coordinate, and the straight lines through the origin represent contours of the Schwarzschild t coordinate. In general relativity , Kruskal–Szekeres coordinates , named after Martin Kruskal and George Szekeres , are a coordinate system for the Schwarzschild geometry for a black hole .
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres.In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coordinate chart on a static and spherically symmetric spacetime, which is adapted to these nested round spheres.
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.
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
The covariant derivative is a generalization of the directional derivative from vector calculus.As with the directional derivative, the covariant derivative is a rule, , which takes as its inputs: (1) a vector, u, defined at a point P, and (2) a vector field v defined in a neighborhood of P. [7]