Covariant energy–momentum and an uncertainty principle for general relativity

• We present a totally invariant spacetime energy expression for general relativity incorporating the contribution from gravity.
• Demand for the general expression to reduce to the Tolman integral for stationary systems supports the Ricci integral as energy–momentum.
• Localized energy via the Ricci integral is consistent with the energy localization hypothesis.
• New localized energy supports the Bonnor claim that the Szekeres collapsing dust solutions are energy-conserving.
• Suggest the Heisenberg Uncertainty Principle be generalized in terms of spacetime energy–momentum in strong gravity extreme.

We introduce a naturally-defined totally invariant spacetime energy expression for general relativity incorporating the contribution from gravity. The extension links seamlessly to the action integral for the gravitational field. The demand that the general expression for arbitrary systems reduces to the Tolman integral in the case of stationary bounded distributions, leads to the matter-localized Ricci integral for energy–momentum in support of the energy localization hypothesis. The role of the observer is addressed and as an extension of the special relativistic case, the field of observers comoving with the matter is seen to compute the intrinsic global energy of a system. The new localized energy supports the Bonnor claim that the Szekeres collapsing dust solutions are energy-conserving. It is suggested that in the extreme of strong gravity, the Heisenberg Uncertainty Principle be generalized in terms of spacetime energy–momentum.