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This diagram gives the route to find the Schwarzschild solution by using the weak field approximation. The equality on the second row gives g 44 = −c 2 + 2GM/r, assuming the desired solution degenerates to Minkowski metric when the motion happens far away from the blackhole (r approaches to positive infinity).
To simplify the derivatives, the angles are given in radians (where d sin(x)/dx = cos(x), the degree values introduce an additional factor of 360 / 2 pi). At any location, the tangent directions are (up), (north) and (east) - you can also use indices 1,2,3.
1 Christoffel symbols, covariant derivative. 2 Curvature tensors. ... Springer-Verlag, Berlin, 1987. xii+510 pp. ISBN 3-540-15279-2 This page was last ...
Several sources [2] [12] [3] replace nσ λ with k λ r, where k λ is the absorption coefficient per unit density and r is the density of the gas. The absorption coefficient for spectral flux (a beam of radiation with a single wavelength, [W/m 2 /μm]) differs from the absorption coefficient for spectral intensity [W/sr/m 2 /μm] used in ...
Note that in the one-variable case, the Hessian condition simply gives the usual second derivative test. In the two variable case, (,) and (,) are the principal minors of the Hessian. The first two conditions listed above on the signs of these minors are the conditions for the positive or negative definiteness of the Hessian.
where (,) and (,) are two metric potentials dependent on Weyl's canonical coordinates {,}.The coordinate system {,,,} serves best for symmetries of Weyl's spacetime (with two Killing vector fields being = and =) and often acts like cylindrical coordinates, [2] but is incomplete when describing a black hole as {,} only cover the horizon and its exteriors.
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.
Using the above procedure, the Riemann tensor is defined as a type (1, 3) tensor and when fully written out explicitly contains the Christoffel symbols and their first partial derivatives. The Riemann tensor has 20 independent components.