Abstract
The issue of energy and its potential localizability in general relativity has challenged physicists for more than a century. Many non-invariant measures were proposed over the years but an invariant measure was never found. We discovered the invariant localized energy measure by expanding the domain of investigation from space to spacetime. We note from relativity that the finiteness of the velocity of propagation of interactions necessarily induces indefiniteness in measurements. This is because the elements of actual physical systems being measured as well as their detectors are characterized by entire four-velocity fields, which necessarily leads to information from a measured system being processed by the detector in a spread of time. General relativity adds additional indefiniteness because of the variation in proper time between elements. The uncertainty is encapsulated in a generalized uncertainty principle, in parallel with that of Heisenberg, which incorporates the localized contribution of gravity to energy. This naturally leads to a generalized uncertainty principle for momentum as well. These generalized forms and the gravitational contribution to localized energy would be expected to be of particular importance in the regimes of ultra-strong gravitational fields. We contrast our invariant spacetime energy measure with the standard 3-space energy measure which is familiar from special relativity, appreciating why general relativity demands a measure in spacetime as opposed to 3-space. We illustrate the misconceptions by certain authors of our approach.
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Notes
Some have argued that the equivalence principle negates any possibility that gravitational energy could be localized, reading into it that gravity is eliminated by free-fall. However, this is incorrect in that this principle is only an approximation. Real gravity is characterized by spacetime curvature and this cannot be obliterated by a coordinate transformation. Synge has forcefully promoted this point.
We use relativist’s preferred units where c = G = 1 which renders all physical quantities as powers of L.
This sidesteps the issue of how definitely the energy and momentum can be ascertained by an observer, a very basic issue that we discuss in what follows.
In what follows, we will see that this expansion blends quite naturally into considerations regarding uncertainty.
To be noted as well is that the dimensions of spacetime energy \(E^*\), being conventional energy times time, lends itself quite naturally to a formal uncertainty relationship in the form of an inequality.
To contemplate pure stationarity in this regard is an idealization. In practice, since the act of measurement is an inherently dynamic operation, pure stationarity is incompatible with the pursuit of a measurement.
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Cooperstock, F.I., Dupre, M.J. Energy and Uncertainty in General Relativity. Found Phys 48, 387–394 (2018). https://doi.org/10.1007/s10701-018-0137-4
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DOI: https://doi.org/10.1007/s10701-018-0137-4