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For some background, the stress-energy tensor has been described as the density of energy and momentum, in for instance MTW. So if one says that the represention of a volume element is a vector, one can basically say that a rank 2 tensor (the stress energy tensor) maps the volume element (a rank 1 tensor) into another rank 1 tensor (the energy-momentum 4-vector), where the energy-momentum 4-vector representing the total energy and momentum contained in the specified volume.

But MTW, for instance, does not, as far as I've seen, ever explicitly say that a volume element can be represented by a vector. Instead, they say that one multiples the stress-energy tensor by the 4-velocity of an observer.