Math 2.0

Relativity and gravitational theory Definition

Komar mass

Builds on Lorentzian manifoldKilling vector fieldHodge star operatorStress–energy tensor

Definition

Let (M,g)(M, g) be an oriented 4-dimensional Lorentzian manifold of signature (−,+,+,+)(-,+,+,+) with volume form ϵabcd\epsilon_{abcd}. Let ξ\xi be a Killing vector field that is timelike near infinity and normalised by g(ξ,ξ)→−1g(\xi, \xi) \to -1 at infinity. Let SS be a closed spacelike 2-surface in MM. In units with G=1G = 1, the Komar mass of SS is MK(S)=−18π∫Sϵabcd∇cξd,M_K(S) = -\frac{1}{8\pi} \int_S \epsilon_{abcd} \nabla^c \xi^d, where SS is oriented as the boundary of a spacelike region that it encloses. In the notation of forms, MK(S)=−18π∫S⋆dξ♭M_K(S) = -\frac{1}{8\pi} \int_S \star d\xi^\flat up to the factor fixed by the convention for the Hodge star.

Remarks

Assume that SS bounds a spacelike hypersurface Σ\Sigma with future unit normal nn, and that the Einstein equations hold. Then Stokes' theorem gives MK(S)=2∫Σ(Tab−12Tgab)naξb dV,M_K(S) = 2 \int_\Sigma \Bigl( T_{ab} - \tfrac12 T g_{ab} \Bigr) n^a \xi^b \, dV, where TabT_{ab} is the stress–energy tensor and TT its trace. Hence MK(S)M_K(S) depends only on the matter enclosed, and it is the same for all surfaces SS that are homologous in a vacuum region. Example: for the Schwarzschild metric with ξ=∂t\xi = \partial_t, one has ξ♭=−(1−2M/r) dt\xi^\flat = -(1 - 2M/r) \, dt and dξ♭=(2M/r2) dt∧drd\xi^\flat = (2M/r^2) \, dt \wedge dr. Integrating its dual over the sphere of radius rr, of area 4πr24\pi r^2, gives ±8πM\pm 8\pi M, and with the orientation above MK=MM_K = M. Non-example: if ξ\xi is not a Killing field, the integral generally depends on the surface SS, so it does not define a mass.

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Differential form, concept, no evidence levelDifferential formconceptExterior algebra, concept, no evidence levelExterior algebraconceptInner product, concept, no evidence levelInner productconceptLevi-Civita connection, concept, no evidence levelLevi-Civita connectionconceptLie derivative, concept, no evidence levelLie derivativeconceptOrientation of a real vector space, concept, no evidence levelOrientation of a real…conceptPseudo-Riemannian manifold, concept, no evidence levelPseudo-Riemannian man…conceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptSignature of a quadratic form, concept, no evidence levelSignature of a quadra…conceptStress–energy tensor, concept, no evidence levelStress–energy tensorconceptTangent bundle, concept, no evidence levelTangent bundleconceptHodge star operator, definition, no evidence levelHodge star operatordefinitionKilling vector field, definition, no evidence levelKilling vector fielddefinitionLorentzian manifold, definition, no evidence levelLorentzian manifolddefinitionKomar mass, definition, no evidence levelKomar massdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Differential form concept
  2. Exterior algebra concept
  3. Inner product concept
  4. Levi-Civita connection concept
  5. Lie derivative concept
  6. Orientation of a real vector space concept
  7. Pseudo-Riemannian manifold concept
  8. Riemannian manifold concept
  9. Signature of a quadratic form concept
  10. Stress–energy tensor concept
  11. Tangent bundle concept
  12. Hodge star operator definition
  13. Killing vector field definition
  14. Lorentzian manifold definition
  15. Komar mass definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Komar_mass
About this page
TagZ6MZG1FMD5
URIm2://Z6MZG1FMD5
Kinddefinition
WritingW0 stub
MSC83C40, 83C57
DefinesKomar mass
UsesLorentzian manifold, Killing vector field, Hodge star operator, Stress–energy tensor
Changed3 November 2026

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