Math 2.0

Relativity and gravitational theory Definition

Mass in general relativity

Builds on Asymptotically flat spacetimeADM energyBondi massKomar mass

Definition

Let (M,g)(M, g) be a solution of the Einstein equations that is asymptotically flat. The mass of (M,g)(M, g) is a quantity defined from the asymptotic behaviour of gg, by one of the following definitions.

  1. The ADM mass, defined by a surface integral at spatial infinity on an asymptotically flat initial data set.
  2. The Bondi mass, a function of retarded time defined by an integral over a cut of future null infinity.
  3. The Komar mass, defined when gg is stationary, by a surface integral of the derivative of the timelike Killing field.

Remarks

The ADM mass is constant in time, while the Bondi mass does not increase, and it decreases when gravitational waves carry energy away. For a stationary asymptotically flat vacuum spacetime outside a compact region, the Komar mass and the ADM mass agree. Example: for the Schwarzschild metric g=−(1−2M/r)dt2+(1−2M/r)−1dr2+r2dΩ2g = -(1 - 2M/r) dt^2 + (1 - 2M/r)^{-1} dr^2 + r^2 d\Omega^2, all three definitions give MM. Non-example: de Sitter space is not asymptotically flat, so none of these definitions applies to it. Several other definitions are quasi-local: they assign a mass to a closed 2-surface, such as the Hawking mass and the Brown–York mass.

Reading path

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

Show the path as a graph
ADM energy, concept, no evidence levelADM energyconceptAsymptotically flat spacetime, concept, no evidence levelAsymptotically flat s…conceptBondi mass, concept, no evidence levelBondi massconceptKomar mass, concept, no evidence levelKomar massconceptMass in general relativity, definition, no evidence levelMass in general relat…definition
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. Asymptotically flat spacetime concept
  3. Bondi mass concept
  4. Komar mass concept
  5. Mass in general relativity definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Mass_in_general_relativity
About this page
TagES3P447MKT
URIm2://ES3P447MKT
Kinddefinition
WritingW0 stub
MSC83C40, 83C30, 83C05
DefinesMass in general relativity
UsesAsymptotically flat spacetime, ADM energy, Bondi mass, Komar mass
Changed3 November 2026

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