Hilbert–Einstein Lagrangian
In general relativity, the Komar superpotential,[1] corresponding to the invariance of the Hilbert–Einstein Lagrangian
, 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 })\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7676b9d91a09b902e17f72f40e7c26d7a2226b)
associated with a vector field
, and where
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 }\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36aab0af71c19e25e3b838c954833897c96315cb)
where
denotes interior product, generalizes to an arbitrary vector field
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
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.
- ^ 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