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In
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...
, the Komar superpotential, corresponding to the invariance of the Hilbert–Einstein Lagrangian \mathcal_\mathrm = R \sqrt \, \mathrm^4x, is the
tensor density In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see tensor field), except that it is ...
: : 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 In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
. The Komar two-form: : \mathcal(,\xi) =U^(,\xi) \mathrmx_= \nabla^\xi^\sqrt\,\mathrmx_ \, , where \mathrmx_= \iota_\mathrmx_= \iota_\iota_\mathrm^4x denotes
interior product In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of ...
, generalizes to an arbitrary vector field \xi the so-called above Komar superpotential, which was originally derived for timelike
Killing vector field In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal gen ...
s. 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

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Superpotential In theoretical physics, the superpotential is a function in supersymmetric quantum mechanics. Given a superpotential, two "partner potentials" are derived that can each serve as a potential in the Schrödinger equation. The partner potentials hav ...
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Einstein–Hilbert action The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the act ...
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Komar mass The Komar mass (named after Arthur Komar) of a system is one of several formal concepts of mass that are used in general relativity. The Komar mass can be defined in any stationary spacetime, which is a spacetime in which all the metric compon ...
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Tensor calculus In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Lev ...
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Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing dis ...
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Riemann curvature tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds ...


Notes


References

* Equations of physics Tensors General relativity {{relativity-stub Potentials