Physics:Komar superpotential

Short description: Hilbert–Einstein Lagrangian

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

[math]\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}) \, , }[/math]

associated with a vector field [math]\displaystyle{ \xi=\xi^{\rho}\partial_{\rho} }[/math], and where [math]\displaystyle{ \nabla_{\sigma} }[/math] denotes covariant derivative with respect to the Levi-Civita connection.

The Komar two-form:

[math]\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} \, , }[/math]

where [math]\displaystyle{ \mathrm{d}x_{\alpha\beta}= \iota_{\partial{\alpha}}\mathrm{d}x_{\beta}= \iota_{\partial{\alpha}}\iota_{\partial{\beta}}\mathrm{d}^4x }[/math] denotes interior product, generalizes to an arbitrary vector field [math]\displaystyle{ \xi }[/math] 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