Komar superpotential

In general relativity, the Komar superpotential, corresponding to the invariance of the Hilbert-Einstein Lagrangian ${\mathcal {L}}_{\mathrm {G} }={1 \over 2\kappa }R{\sqrt {-g}}\,\mathrm {d} ^{4}x$, is the tensor density:

$U^{\alpha \beta }({{\mathcal {L}}_{\mathrm {G} }},\xi )={{\sqrt {-g}} \over {\kappa }}\nabla ^{[\beta }\xi ^{\alpha ]}={{\sqrt {-g}} \over {2\kappa }}(g^{\beta \sigma }\nabla _{\sigma }\xi ^{\alpha }-g^{\alpha \sigma }\nabla _{\sigma }\xi ^{\beta })\,,$

associated with a vector field $\xi =\xi ^{\rho }\partial _{\rho }$, and where $\nabla _{\sigma }$ denotes covariant derivative with respect to the Levi-Civita connection.

The Komar two-form:

${\mathcal {U}}({{\mathcal {L}}_{\mathrm {G} }},\xi )={1 \over 2}U^{\alpha \beta }({{\mathcal {L}}_{\mathrm {G} }},\xi )\mathrm {d} x_{\alpha \beta }={1 \over {2\kappa }}\nabla ^{[\beta }\xi ^{\alpha ]}{\sqrt {-g}}\,\mathrm {d} x_{\alpha \beta }\,,$

where $\mathrm {d} x_{\alpha \beta }=\iota _{\partial {\alpha }}\mathrm {d} x_{\beta }=\iota _{\partial {\alpha }}\iota _{\partial {\beta }}\mathrm {d} ^{4}x$ denotes interior product, generalizes to an arbitrary vector field $\xi$ the so-called above Komar superpotential, which was originally derived for timelike Killing vector fields.

See also

• Einstein–Hilbert action
• Komar mass
• Tensor calculus
• Christoffel symbols
• Riemann curvature tensor