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ADM Mass

The ADM mass is an asymptotic invariant obtained from the energy and momentum fluxes of an asymptotically flat initial data set. In spatial dimension n>=3, let omega_(n-1) be the area of the unit sphere and S_R a large Euclidean coordinate sphere. One normalization is

E=1/(2(n-1)omega_(n-1))lim_(R->infty)int_(S_R)(partial_jg_(ij)-partial_ig_(jj))nu_delta^idA_delta,
(1)
P_i=1/((n-1)omega_(n-1))lim_(R->infty)int_(S_R)(K_(ij)-(tr_gK)g_(ij))nu_delta^jdA_delta,
(2)

where repeated coordinate indices are summed. If E>||P||_delta, the Lorentz-invariant mass is

m=sqrt(E^2-||P||_delta^2).
(3)

The name ADM mass also occurs for the energy E, so the convention matters when P!=0. The invariant-mass formulation is used in the spacetime Penrose inequality.

See also

Asymptotically Flat Manifold, Initial Data Set, Lorentz Transformation, Penrose Inequality

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References

Bray, H. L. and Lee, D. A. "On the Riemannian Penrose Inequality in Dimensions Less than Eight." Duke Math. J. 148, 81-106, 2009. https://doi.org/10.1215/00127094-2009-020.Dain, S. and Gabach-Clement, M. E. "Geometrical Inequalities Bounding Angular Momentum and Charges in General Relativity." Living Rev. Relativ. 21, Article 5, 2018. https://doi.org/10.1007/s41114-018-0014-7.Mars, M. "Present Status of the Penrose Inequality." Class. Quantum Grav. 26, 193001, 2009. https://doi.org/10.1088/0264-9381/26/19/193001.

Cite this as:

Weisstein, Eric W. "ADM Mass." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ADMMass.html

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