In general relativity, the Komar superpotential, named after Arthur Komar who wrote about it in 1952, corresponding to the invariance of the Hilbert–Einstein Lagrangian L G = 1 2 κ R − g d 4 x {\displaystyle {\mathcal {L}}_{\mathrm {G} }={1 \over 2\kappa }R{\sqrt {-g}}\,\mathrm {d} ^{4}x} , is the tensor density:
U α β ( L G , ξ ) = − g κ ∇ [ β ξ α ] = − g 2 κ ( g β σ ∇ σ ξ α − g α σ ∇ σ ξ β ) , {\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 ξ = ξ ρ ∂ ρ {\displaystyle \xi =\xi ^{\rho }\partial _{\rho }} , and where ∇ σ {\displaystyle \nabla _{\sigma }} denotes covariant derivative with respect to the Levi-Civita connection. The Komar two-form:
U ( L G , ξ ) = 1 2 U α β ( L G , ξ ) d x α β = 1 2 κ ∇ [ β ξ α ] − g d x α β , {\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 d x α β = ι ∂ α d x β = ι ∂ α ι ∂ β d 4 x {\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 ξ {\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.
See also Superpotential Einstein–Hilbert action Komar mass Tensor calculus Christoffel symbols Riemann curvature tensor
Notes
References Misner, Charles W.; Thorne, Kip. S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 978-0-7167-0344-0