Komar superpotential explained

l{L} _{G} = { 1 \over 2 \kappa} R \sqrt{ - g} d ^{4x} , is the tensor density:U ^{\alpha\beta} ({ l{L} _{G} },\xi) =\nabla^=(g^ \nabla_\xi^ - g^ \nabla_\xi^)\,, \xi = \xi ^{\rho} \partial _{\rho} , and where \nabla _{\sigma} denotes covariant derivative with respect to the Levi-Civita connection.

The Komar two-form:

l{U}({ l{L} _{G} },\xi) =U^(\xi)\mathrmx_=\nabla^\sqrt\,\mathrmx_\,,

where

d x _{\alpha\beta} = \iota _{\partial{\alpha} }\mathrmx_= \iota_\iota_\mathrm^4x 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.

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 and References

1. Covariant Conservation Laws in General Relativity. Arthur Komar. Phys. Rev.. 3. 113. 1959. 934. 10.1103/PhysRev.113.934. Arthur Komar. 1959PhRv..113..934K.
2. A note on Komar's anomalous factor. J. Katz. Class. Quantum Gravity. 3. 2 . 1985. 423. 10.1088/0264-9381/2/3/018. 250898281 .