- #1

ergospherical

- 889

- 1,222

I'd appreciate some clarification of this passage in the paper Spinning test particles in general relativity by Papapetrou,

The definition is easy enough to understand, but what's the motivation? ##X^{\alpha}## are the coordinates of points on the worldline whilst ##x^{\alpha}## are presumably arbitrary spacetime coordinates (of points near the worldline).

n.b. ##\mathfrak{T}^{\mu \nu} = \sqrt{-g} T^{\mu \nu}## and\begin{align*}

\nabla_{\nu} T^{\mu \nu} = \partial_{\nu} T^{\mu \nu} + \Gamma^{\nu}_{\sigma \nu} T^{\mu \sigma} + \Gamma^{\mu}_{\sigma \nu} T^{\sigma \nu} &= 0 \\ \\

\implies \dfrac{1}{\sqrt{-g}} \partial_{\nu} \left( \sqrt{-g} T^{\mu \nu} \right) + \Gamma^{\mu}_{\sigma \nu} T^{\sigma \nu} &= 0\\

\partial_{\nu} \left( \sqrt{-g} T^{\mu \nu} \right) + \Gamma^{\mu}_{\sigma \nu} \sqrt{-g} T^{\sigma \nu} &= 0 \\

\partial_{\nu} \mathfrak{T}^{\mu \nu} + \Gamma^{\mu}_{\sigma \nu}\mathfrak{T}^{\sigma \nu} &= 0

\end{align*}

The definition is easy enough to understand, but what's the motivation? ##X^{\alpha}## are the coordinates of points on the worldline whilst ##x^{\alpha}## are presumably arbitrary spacetime coordinates (of points near the worldline).

n.b. ##\mathfrak{T}^{\mu \nu} = \sqrt{-g} T^{\mu \nu}## and\begin{align*}

\nabla_{\nu} T^{\mu \nu} = \partial_{\nu} T^{\mu \nu} + \Gamma^{\nu}_{\sigma \nu} T^{\mu \sigma} + \Gamma^{\mu}_{\sigma \nu} T^{\sigma \nu} &= 0 \\ \\

\implies \dfrac{1}{\sqrt{-g}} \partial_{\nu} \left( \sqrt{-g} T^{\mu \nu} \right) + \Gamma^{\mu}_{\sigma \nu} T^{\sigma \nu} &= 0\\

\partial_{\nu} \left( \sqrt{-g} T^{\mu \nu} \right) + \Gamma^{\mu}_{\sigma \nu} \sqrt{-g} T^{\sigma \nu} &= 0 \\

\partial_{\nu} \mathfrak{T}^{\mu \nu} + \Gamma^{\mu}_{\sigma \nu}\mathfrak{T}^{\sigma \nu} &= 0

\end{align*}

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