These are the 16 Riemann invariants for Schwarzschild solution, using the formulas by Carminati and McLenaghan The related Weyl scalars in the context of the Newman-Penrose formalism the definition is in terms of the Weyl tensor and the tetrad of tensors of the Newman-Penrose formalism These are the matrix components of the Christoffel symbols of the second kind (that describe, in. I am calculating the Riemann tensor for the Schwarzschild solution. For a specialized case orbits in a Schwarzschild- (anti-)de Sitter space-time have been presented in. The Christoffel symbol becomes: Γ l ij = g l ∂ 2 r ∂x i ∂x j That shows the symmetry with respect to the two indexes i an j: Γ l ij = Γ l ji Γ l ij = Γ l ji Using the metric g ij to lower an index, we can write: g kl Γ l ij = Γ kij Γ kij = g kl Γ l ij (3.2) The term Γ kij is called the Christoffel symbol of first kind.

In the metric, terms like d t 2 are shorthand for the tensor.

A consistent derivation of (2.23) is only possible with Einstein's equations. To determine Christoffel symbols, one uses the following relatiorns (See (2.3) VA - gi) I 1 sm C)g,mj + ag.k _ gjkX Received by the editors JUne 1, 1964, and in re vised form July 26, 1966. Therefore a process such as a supernova explosion, which. For the basic tensorial properties, we have corresponding functions, but you need to take care with the arguments. (2.2) It is symmetric with respect to two lower indices.

Christoffel symbols for flat frw metric