Komar Superpotential

In general relativity, the Komar superpotential, corresponding to the invariance of the Hilbert–Einstein Lagrangian $\mathcal_\mathrm = R \sqrt \, \mathrm^4x$, is the tensor density:

: $U^(,\xi) =\nabla^\xi^ = (g^ \nabla_\xi^ - g^ \nabla_\xi^) \, ,$

associated with a vector field $\xi=\xi^\partial_$, and where $\nabla_$ denotes covariant derivative with respect to the Levi-Civita connection.

The Komar two-form:

: $\mathcal(,\xi) =U^(,\xi) \mathrmx_= \nabla^\xi^\sqrt\,\mathrmx_ \, ,$

where $\mathrmx_= \iota_\mathrmx_= \iota_\iota_\mathrm^4x$ 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.

See also

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

Notes

References

*

Equations of physics
Tensors
General relativity

{{relativity-stub
Potentials