The Geometry of Energy in General Relativity

How does one measure energy in general relativity? This seemingly simple question leads to some remarkable connections between geometry and physics. We will begin by reviewing the celebrated positive energy theorem of Schoen and Yau, emphasizing its connection to scalar curvature. We will then introduce the Brown–York quasi-local energy, which provides a way to measure the energy contained within a bounded region of space using the mean curvature of its boundary. Finally, motivated by the expansion of the universe, we will discuss an analogue of the Brown–York energy adapted to an expanding de Sitter background, developed in joint work with Rodrigo Avalos and Annachiara Piubello. We will describe some positivity and rigidity results, as well as surprising limitations of these properties in the expanding setting.