01 January 1970 25 1K Report

Free spacetime contains no energy-momentum*, so when objects m are travelling at constant velocity in it, they do not exchange energy-momentum. Non-free spacetime contains energy-momentum. The Einstein field equation of general relativity,

Rµν - (1/2)gµνR = G*Tµν,

expresses the relationship between the energy-momentum (mass) and the structure of spacetime ( metric) at a point (region) in spacetime**. Usually we think that "Gravity couples universally to all forms of energy" [1]. Then, we need to ask three basic questions:

1) What is the best way to express the energy-momentum of the gravitational field? or how are the "long-standing problems about energy-momentum localisation in GR" [2][3][4] addressed? The energy-momentum of the gravitational field is the energy-momentum of the spacetime field, which must be localizable. The energy-momentum of the spacetime field must involve only the spacetime parameter xi(i=0,1,2,3), because the independent spacetime field has no other parameter (or it has some other hidden parameter that does not play an explicit role). But it cannot be expressed directly in terms of spatio-temporal coordinates (t,x,y,z) because they must be background independent, nor can it be expressed in terms of time lengths T and space lengths L because we have no way of determining the measurement boundaries. So what are the remaining covariates? The rates of measure change, curvature, and deflection, etc.. which are the most appropriate? Even if we consider space-time as a "medium", what are the properties of the medium? Density, elasticity? What density? What elasticity?

2) By what means are gravitational fields and other forms of energy-momentum exchanged with each other? Obviously it must be through a common covariate, and then the only option available is the spacetime covariate. Does this qualify that all other forms of energy-momentum must contain spacetime covariates? Includes energy-momentum of dark matter (no dark energy involved). And more critically, the form of these spacetime Attributes and the form in which the spacetime energy-momentum is expressed should be the same, i.e., if the energy-momentum of spacetime is expressed in terms of a change of metric, the other forms of energy-momentum must be related to a change in the spacetime metric; and if it is expressed in terms of a curvature, the other forms must be related to a change in the curvature.

3) Is the energy-momentum of the gravitational field conserved[5]? If the energy-momentum of the gravitational field is not conserved, what will become of the gravitation dominated evolution of galaxies?

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Notes

* We need to distinguish between the concepts of space-time and vacuum.“Are Vacuum and Space Two Separate Things?”,https://www.researchgate.net/post/NO34How_the_View_of_Space-Time_is_Unified_6-Are_Vacuum_and_Space_Two_Separate_Things;

** The concept of a strict "point" interaction does not really exist in physics.

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References

[1] Kiefer, C. (2006). Quantum gravity: general introduction and recent developments. Annalen der Physik, 518(1-2), 129-148.

[2] Einstein Ann. d. Phys. 49, 769 (1916).

[3] Hestenes, D. (2021). Energy-Momentum Complex in General Relativity and Gauge Theory. Advances in Applied Clifford Algebras, 31(3), 51.

[4] Møller, C. (1958). On the localization of the energy of a physical system in the general theory of relativity. Annals of Physics, 4(4), 347-371.

[5] Szabados, L. B. (2009). Quasi-local energy-momentum and angular momentum in general relativity. Living Reviews in Relativity, 12(1), 1-163.

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