Shing-Tung Yau's work on the notion of mass in general relativity

Document Type : Research Article


Department of Mathematics‎, ‎Columbia University‎, ‎2990 Broadway‎, ‎New York‎, ‎NY 10027.


The notion of mass or energy has been one of the most challenging problems in general relativity since Einstein's time. As is well known from the equivalence principle, there is no well-defined concept of energy density for gravitation. On the other hand, when there is asymptotic symmetry, concepts of total energy and momentum can be defined. This is the ADM energy-momentum and the Bondi energy-momentum when the system is viewed from spatial infinity and null infinity, respectively. These concepts are fundamental in general relativity but there are limitations to such definitions if the physical system is not isolated and cannot quite be viewed from infinity where asymptotic symmetry exists.

The positive energy conjecture states that the total energy of a nontrivial isolated physical system must be positive. This conjecture lies in the foundation of general relativity upon which stability of the system rests.This long standing conjecture had attracted many physicists and mathematician,
but only very special cases were verified up until the seventies.


R. Schoen and S.-T. Yau, Positivity of the total mass of a general space-time, Phys. Rev. Lett. 43 (1979), no. 20, 1457--1459.
R. Schoen and S.-T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), no. 1, 45--76.
R. Schoen and S.-T. Yau, Proof of the positive mass theorem II, Comm. Math. Phys. 79 (1981), no. 2, 231--260.
R. Schoen and S.-T. Yau, Proof that the Bondi mass is positive, Phys. Rev. Lett. 48 (1982), no. 6, 369--371.
R. Schoen and S.-T. Yau, The existence of a black hole due to condensation of matter, Comm. Math. Phys. 90 (1983), no. 4, 575--579
G. Huisken and S.-T. Yau, Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature, Invent. Math. 124 (1996), no. 1-3, 281--311.
S.-T. Yau, Geometry of three manifolds and existence of black hole due to boundary effect, Adv. Theor. Math. Phys. 5 (2001), no. 4, 755--767.
M.-T. Wang and S.-T. Yau, Quasilocal mass in general relativity, Phys. Rev. Lett. 102 (2009), no. 2, no. 021101, 4 pages.
M.-T. Wang and S.-T. Yau, Isometric embeddings into the Minkowski space and new quasi-local mass, Comm. Math. Phys. 288 (2009), no. 3, 919--942.