Physics:Komar superpotential

Short description: Hilbert–Einstein Lagrangian

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 ${{ℒ}}_{\mathrm{G}}=\frac{1}{2\kappa }R\sqrt{-g}{\mathrm{d}}^{4}x$, is the tensor density:

$U^{\alpha \beta }({{ℒ}}_{\mathrm{G}},\xi )=\frac{\sqrt{-g}}{\kappa }{\mathrm{\nabla }}^{[\beta }{\xi }^{\alpha ]}=\frac{\sqrt{-g}}{2\kappa }(g^{\beta \sigma }{\mathrm{\nabla }}_{\sigma }{\xi }^{\alpha }-g^{\alpha \sigma }{\mathrm{\nabla }}_{\sigma }{\xi }^{\beta }),$

associated with a vector field $\xi ={\xi }^{\rho }{\partial }_{\rho }$, and where ${\mathrm{\nabla }}_{\sigma }$ denotes covariant derivative with respect to the Levi-Civita connection.

The Komar two-form:

${𝒰}({{ℒ}}_{\mathrm{G}},\xi )=\frac{1}{2}{U}^{\alpha \beta }({{ℒ}}_{\mathrm{G}},\xi )\mathrm{d}{x}_{\alpha \beta }=\frac{1}{2\kappa }{\mathrm{\nabla }}^{[\beta }{\xi }^{\alpha ]}\sqrt{-g}\mathrm{d}{x}_{\alpha \beta },$

where $\mathrm{d}{x}_{\alpha \beta }={\iota }_{\partial \alpha }\mathrm{d}{x}_{\beta }={\iota }_{\partial \alpha }{\iota }_{\partial \beta }{\mathrm{d}}^{4}x$ 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.

See also

• Superpotential
• Einstein–Hilbert action
• Komar mass
• Tensor calculus
• Christoffel symbols
• Riemann curvature tensor

References

• Misner, Charles W.; Thorne, Kip. S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 978-0-7167-0344-0