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

The Bel-Robinson tensor $B_{αβμν}$ gives a positive definite gravitational energy in the quasilocal small sphere limit approximation. However, there is an alternative tensor $V_{αβμν}$ that was proposed recently that offers the same positivity as $B_{αβμν}$ does. We have found that $V_{αβμν}$ is the unique alternative tensor with $B_{αβμν}$ 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 $B_{αβμν}$ and $V_{αβμν}$ are on equal footing at the same limit.

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