Advances in Theoretical and Mathematical Physics

Volume 25 (2021)

Number 3

Supertranslation invariance of angular momentum

Pages: 777 – 789



Po-Ning Chen (Department of Mathematics, University of California, Riverside, Calif., U.S.A.)

Mu-Tao Wang (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Ye-Kai Wang (Department of Mathematics, National Cheng Kung University, Tainan City, Taiwan)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


LIGO’s successful detection of gravitational waves has revitalized the theoretical understanding of the angular momentum carried away by gravitational radiation. An infinite dimensional supertranslation ambiguity has presented an essential difficulty for decades of study. Recent advances were made to address and quantify the supertranslation ambiguity in the context of compact binary coalescence. Here we present the first definition of angular momentum in general relativity that is completely free from supertranslation ambiguity. The new definition was derived from the limit of the quasilocal angular momentum defined previously by the authors. A new definition of center of mass integral at null infinity is also proposed and shown to be supertranslation invariant. Together with the classical Bondi–Sachs energy-momentum, they form a complete set of conserved quantities at null infinity that transform according to basic physical laws.

P.-N. Chen is supported by Simons Foundation collaboration grant #584785, M.-T. Wang is supported by NSF grant DMS-1810856.

Y.-K. Wang is supported by Taiwan MOST grant 109-2628-M-006-001-MY3.

S.-T. Yau is supported by by the John Templeton Foundation (award number 61497).

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1810856.

Published 21 March 2022