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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]
1 Christoffel symbols, covariant derivative. 2 Curvature tensors. Toggle Curvature tensors subsection. 2.1 Definitions. 2.1.1 (3,1) Riemann curvature tensor.
If the derivative does not lie on the tangent space, the right expression is the projection of the derivative over the tangent space (see covariant derivative below). Symbols of the second kind decompose the change with respect to the basis, while symbols of the first kind decompose it with respect to the dual basis.
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]
The Christoffel symbols of this connection are given in terms of partial derivatives of the metric in local coordinates by the formula = (+) = (, +,,) (where commas indicate partial derivatives). The curvature of spacetime is then given by the Riemann curvature tensor which is defined in terms of the Levi-Civita connection ∇.
where the comma indicates a partial derivative with respect to the coordinates: g a b , c = ∂ g a b ∂ x c {\displaystyle g_{ab,c}={\frac {\partial {g_{ab}}}{\partial {x^{c}}}}} As the manifold has dimension n {\displaystyle n} , the geodesic equations are a system of n {\displaystyle n} ordinary differential equations for the n ...
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection.
This fact has a widely application, such as to analytically distort a Schwarzschild black hole. We employed the axisymmetric Laplace and gradient operators to write Eqs(7.a-7.e) and Eqs(8.a-8.d) in a compact way, which is very useful in the derivation of the characteristic relation Eq(7.f).