The positive energy theorem in general relativity

Abstract

The self-consistency of general relativity requires that the spacetime total energy must be positive for isolated gravitational systems with positive mass density. It is called the positive energy conjecture. It plays a fundamental role in general relativity, moreover, the total mass is well-defined if it holds true. In the case of zero cosmological constant, the positive energy conjecture was first proved by Schoen and Yau in the late 1970s, and soon later by Witten using different method. In this paper, we provide a review on its recent development for 4-dimensional physical spacetimes, which includes the positive energy theorem, Kerr constraint, the positive energy theorem near null infinity and the positivity of the Bondi energy, as well as the positive energy theorem for positive cosmological constant and for negative cosmological constant.