Komar superpotential

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


 * $$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 $$\xi=\xi^{\rho}\partial_{\rho}$$, and where $$\nabla_{\sigma}$$ denotes covariant derivative with respect to the Levi-Civita connection.

The Komar two-form:


 * $$\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 $$\mathrm{d}x_{\alpha\beta}= \iota_{\partial{\alpha}}\mathrm{d}x_{\beta}= \iota_{\partial{\alpha}}\iota_{\partial{\beta}}\mathrm{d}^4x$$ 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.