Komar superpotential

In general relativity, the Komar superpotential,^[1] 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.

Notes

↑ Arthur Komar (1959). "Covariant Conservation Laws in General Relativity". Phys. Rev. 113 (3): 934. Bibcode:1959PhRv..113..934K. doi:10.1103/PhysRev.113.934.

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[1] Arthur Komar (1959). "Covariant Conservation Laws in General Relativity". Phys. Rev. 113 (3): 934. Bibcode:1959PhRv..113..934K. doi:10.1103/PhysRev.113.934.

Komar superpotential

See also

Notes