An unique alternative non-negative gravitational energy tensor to the Bel-Robinson tensor in the quasilocal small sphere limit

Abstract

The Bel-Robinson tensor $B_{\alpha\beta\mu\nu}$ gives a positive definite gravitational energy in the quasilocal small sphere limit approximation. However, there is an alternative tensor $V_{\alpha\beta\mu\nu}$ that was proposed recently that offers the same positivity as $B_{\alpha\beta\mu\nu}$ does. We have found that $V_{\alpha\beta\mu\nu}$ is the unique alternative tensor with $B_{\alpha\beta\mu\nu}$ which implies that these two tensors are a basis for expressions that have the desirable non-negative gravitational energy in the small sphere limit. In other words, the `energy-momentum' density according to $B_{\alpha\beta\mu\nu}$ and $V_{\alpha\beta\mu\nu}$ are on equal footing at the same limit.