The values for the gravitational energy-momentum density, given by the famous classical pseudotensors: Einstein, Papapetrou, Landau-Lifshitz, Bergmann-Thompson, Goldberg, Moller and Weinberg, in the small region limit are found to the lowest non-vanishing order in normal coordinates. All except Moller's have the zeroth-order material limit required by the equivalence principle. However for small vacuum regions, we find that none of these classical holonomic pseudotensors satisfies the criterion of being proportional to the Bel-Robinson tensor. Generalizing an earlier work which had identified one case, we found another independent linear combination satisfying this requirement-and hence a one-parameter set of linear combinations of the classical pseudotensors with this desirable property.
