Komar superpotential

In general relativity, the Komar superpotential, named after Arthur Komar who wrote about it in 1952,¹ 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