Deriving The Schwarzschild Solution - Calculating The Christoffel Symbols

Calculating The Christoffel Symbols

Using the metric above, we find the Christoffel symbols, where the indices are . The sign denotes a total derivative of a function.

\Gamma^0_{ik} = \begin{bmatrix} A'/\left( 2A \right) & 0 & 0 & 0\\ 0 & -r/A & 0 & 0\\ 0 & 0 & -r \sin^2 \theta /A & 0\\ 0 & 0 & 0 & -B'/\left( 2A \right) \end{bmatrix}
\Gamma^1_{ik} = \begin{bmatrix} 0 & 1/r & 0 & 0\\ 1/r & 0 & 0 & 0\\ 0 & 0 & -\sin\theta\cos\theta & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}
\Gamma^2_{ik} = \begin{bmatrix} 0 & 0 & 1/r & 0\\ 0 & 0 & \cot\theta & 0\\ 1/r & \cot\theta & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}
\Gamma^3_{ik} = \begin{bmatrix} 0 & 0 & 0 & B'/\left( 2B \right)\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \\ B'/\left( 2B \right) & 0 & 0 & 0\end{bmatrix}

Read more about this topic:  Deriving The Schwarzschild Solution

Famous quotes containing the words calculating the, calculating and/or symbols:

    “What our children have to fear is not the cars on the highways of tomorrow but our own pleasure in calculating the most elegant parameters of their deaths.”
    —J.G. (James Graham)

    “I know that the right kind of leader for the Labour Party is a kind of desiccated calculating machine.”
    —Aneurin Bevan (1897–1960)

    “Children became an obsessive theme in Victorian culture at the same time that they were being exploited as never before. As the horrors of life multiplied for some children, the image of childhood was increasingly exalted. Children became the last symbols of purity in a world which was seen as increasingly ugly.”
    —C. John Sommerville (20th century)