Pure and Applied Mathematics Quarterly

Volume 15 (2019)

Number 3

Special Issue: In Honor of Robert Bartnik (Part 2 of 2)

Guest Editors: Piotr T. Chruściel, Greg Galloway, Jim Isenberg, Pengzi Miao, Mu-Tao Wang, and Shing-Tung Yau

Quasi-local mass at null infinity in Bondi–Sachs coordinates

Pages: 875 – 895

DOI: https://dx.doi.org/10.4310/PAMQ.2019.v15.n3.a5


Po-Ning Chen (Department of Mathematics, University of California at Riverside)

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, Taiwan)

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


There are two chief statements regarding the Bondi–Trautman mass [3, 29, 37, 33, 34] at null infinity: one is the positivity [30, 20], and the other is the mass loss formula [3], which are both global in nature. In this note, we compute the limit of the Wang–Yau quasi-local mass on unit spheres at null infinity of an asymptotically flat spacetime in the Bondi–Sachs coordinates. The quasi-local mass leads to a local description of radiation that is purely gravitational at null infinity. In particular, the quasi-local mass is evaluated in terms of the news function of the Bondi–Sachs coordinates.

P.-N. Chen is supported by NSF grant DMS-1308164 and Simons Foundation collaboration grant #584785, M.-T. Wang is supported by NSF grant DMS-1405152 and DMS-1810856, Y.-K.Wang is supported by MOST Taiwan grant 105-2115-M-006-016-MY2, 107-2115-M-006-001-MY2, and S.-T. Yau is supported by NSF grants PHY-0714648 and DMS-1308244.

The authors would like to thank the National Center for Theoretical Sciences at National Taiwan University where part of this research was carried out.

Received 9 April 2019

Received revised 7 September 2019

Accepted 18 September 2019

Published 2 January 2020