Local gravitational energy in higher dimensions

Abstract

In general relativity, the local gravitational energy is best characterised by the quasilocal mass. The small sphere limit of quasilocal mass provides us the most local notion of gravitational energy. In four dimensions, the limits were shown be the stress tensor in non-vacuum and the Bel-Robinson tensor in vacuum. We study the local gravitational energy in higher dimensions through the lens of the small sphere limits of various quasilocal mass proposals which can be appropriately generalised beyond four dimensions, and report a new quantity W which potentially characterises the local gravitational energy content in vacuum. We find that the limits at presence of matter yield the stress tensor as expected, but the vacuum limits are not proportional to the Bel-Robinson superenergy Q in dimensions n > 4. The result defies the role of the Bel-Robinson superenergy as characterising the gravitational energy in higher dimensions, albeit the fact that it uniquely generalises. More surprisingly, W replace the Bel-Robinson superenergy Q in all three quasilocal mass proposals that we study. However, W cannot be directly interpreted as a local gravitational energy because of its non-positivity. The physical meaning of W awaits more evidence from investigating other quasilocal masses in higher dimensions.Introduction.-The gravitational field itself carries energy, but it is tricky to locally describe it in general relativity. It is well know that the equivalence principle forbids a covariant stress tensor characterising the energy content of the gravitational field [1]. Nevertheless, there is no obstruction in giving nonlocal prescriptions and the quasilocal mass (QLM) is such an attempt. Over the past half-century, QLM is an ongoing research subject studied by both physicists and mathematicians [2][3][4][5][6][7][8][9][10][11][12]. Nevertheless, QLM is rarely studied in dimensions beyond four.