Komar superpotential

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

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

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

The Komar two-form:

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

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

Komar superpotential is affected by the anomalous factor problem: In fact, when computed, for example, on the Kerr–Newman solution, produces the correct angular momentum, but just one-half of the expected mass.[2]

See also

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

Notes

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.
2. J. Katz (1985). "A note on Komar's anomalous factor". Class. Quantum Gravity. 2 (3): 423. doi:10.1088/0264-9381/2/3/018. S2CID 250898281.

References

• Misner, Charles W.; Thorne, Kip. S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 978-0-7167-0344-0