Math 2.0

Relativity and gravitational theory Mass, energy and conserved quantities Conjecture

Penrose inequality

Builds on ADM energyTrapped surfaceEnergy conditionsInitial value formulation of general relativity

Statement

Let (Σ,h,K)(\Sigma, h, K) be an asymptotically flat initial data set for the Einstein equations whose matter satisfies the dominant energy condition. Let mm be its ADM mass. Let S⊆ΣS \subseteq \Sigma be an outermost apparent horizon, a closed marginally trapped surface. Let AA be the least area of a closed surface in Σ\Sigma that encloses SS. Conjecture: then m≥A16π,m \ge \sqrt{\frac{A}{16\pi}}, with equality only for data in the Schwarzschild spacetime.

Remarks

Penrose derived the inequality by a heuristic argument that combines weak cosmic censorship with the Hawking area theorem. A counterexample would therefore be evidence against cosmic censorship. For time-symmetric data, K=0K = 0, the dominant energy condition becomes nonnegative scalar curvature and marginally trapped surfaces become minimal surfaces. In this case the conjecture is the Riemannian Penrose inequality, proved in 2001 by Huisken and Ilmanen for a connected horizon and by Bray in general. The general case is open, and the precise choice of the surface SS and of the area AA in the general formulation is itself discussed in the literature. For the Schwarzschild spacetime of mass mm, the horizon has area 16πm216\pi m^2, so equality holds.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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ADM energy, concept, no evidence levelADM energyconceptEnergy conditions, concept, no evidence levelEnergy conditionsconceptInitial value formulation of general relativity, concept, no evidence levelInitial value formula…conceptTrapped surface, concept, no evidence levelTrapped surfaceconceptPenrose inequality, conjecture, no evidence levelPenrose inequalityconjecture
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.
  • no evidence level
  1. ADM energy concept
  2. Energy conditions concept
  3. Initial value formulation of general relativity concept
  4. Trapped surface concept
  5. Penrose inequality conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Riemannian_Penrose_inequality
About this page
TagNBBBDBETP1
URIm2://NBBBDBETP1
Kindconjecture
WritingW0 stub
MSC83C57, 53C21, 83C40
StatusThis copy of the site has no build data. Run m2 status NBBBDBETP1.
AboutADM energy, Trapped surface
UsesADM energy, Trapped surface, Energy conditions, Initial value formulation of general relativity
Changed3 November 2026

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