The energy-momentum problem for the gravitational field in general relativity has been a stumbling block for theoreticians for more than six decades. During this time different authors have proposed in the scientific literature numerous approaches to the problem, but none of them has led to sucoess. Extensive analysis of the energy-momentum problem in general relativity shows [8, 9, 11–17] that in Einstein's theory there is in principle no solution to this problem, since it does not contain conservation laws for the matter and gravitational field taken together.

The energy determined in general relativity on the basis of the traditional Hamiltonian approach does not have physical meaning

Misconceptions about energy conservation abound due to the gap between physics and secondary school chemistry. This paper surveys this difference and its relevance to the 1690s-2010s Leibnizian argument that mind-body interaction is impossible due to conservation laws. Justifications for energy conservation are partly empirical, such as Joule's paddle wheel experiment, and partly theoretical, such as Lagrange's statement in 1811 that energy is conserved if the potential energy does not depend on time. In 1918 Noether
generalized results like Lagrange's and proved a converse: symmetries imply conservation laws and vice versa. Conservation holds if and only if nature is uniform.

The rise of field physics during the 1860s-1920s implied that energy is located in particular places and conservation is primordially local: energy cannot disappear in Cambridge and reappear in Lincoln instantaneously or later; neither can it simply disappear in Cambridge or simply appear in Lincoln. A global conservation law can be inferred in some circumstances.

Einstein's General Relativity, which stimulated Noether's work, is another source of difficulty for conservation laws. As is too rarely realized, the theory admits conserved
quantities due to symmetries of the Lagrangian, like other theories. Indeed General Relativity has _more_ symmetries and hence (at least formally) _more_ conserved energies. An argument akin to Leibniz's finally gets some force.
While the mathematics is too advanced for secondary school, the ideas that conservation is tied to uniformities of nature and that energy is in particular places, are
accessible. Improved science teaching would serve the truth and enhance the social credibility of science.

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Conservation of Energy: Missing Features in Its Nature and Justification and Why They Matter

The conservation of energy and momentum have been viewed as undermining Cartesian mental causation since the 1690s. Modern discussions of the topic tend to use mid-nineteenth century physics, neglecting both locality and Noether’s theorem and its converse. The relevance of General Relativity (GR) has rarely been considered. But a few authors have proposed that the non-localizability of gravitational energy and consequent lack of physically meaningful local conservation laws answers the conservation objection to mental causation: conservation already fails in GR, so there is nothing for minds to violate. This paper is motivated by two ideas. First, one might take seriously the fact that GR formally has an infinity of rigid symmetries of the action and hence, by Noether’s first theorem, an infinity of conserved energies-momenta (thus answering Schrodinger’s 1918 false-negative objection). Second, Sean Carroll has asked (rhetorically) how one should modify the Dirac–Maxwell–Einstein equations to describe mental causation. This paper uses the generalized Bianchi identities to show that General Relativity tends to exclude, not facilitate, such Cartesian mental causation. In the simplest case, Cartesian mental influence must be spatio-temporally constant, and hence 0. The difficulty may diminish for more complicated models. Its persuasiveness is also affected by larger world-view considerations. The new general relativistic objection provides some support for realism about gravitational energy-momentum in GR (taking pseudotensor laws seriously). Such realism also might help to answer an objection to theories of causation involving conserved quantities, because energies-momenta would be conserved even in GR.

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General Relativity, Mental Causation, and Energy Conservation

For the relativistic uniform system with an invariant mass density the exact expressions are determined for the potentials and strengths of the gravitational field, the energy of particles and fields. It is shown that, as in the classical case for bodies with a constant mass density, in the system with a zero vector potential of the gravitational field, the energy of the particles, associated with the scalar field potential, is twice as large in the absolute value as the energy defined by the tensor invariant of the gravitational field. The problem of inaccuracy of the use of the field’s stress-energy tensors for calculating the system’s mass and energy is considered. The found expressions for the gravitational field strengths inside and outside the system allow us to explain the occurrence of the large-scale structure of the observable Universe, and also to relate the energy density of gravitons in the vacuum field with the limiting mass density inside the proton. Both the Universe and the proton turn out to be relativistic uniform systems with the maximum possible parameters. The described approach allows us to calculate the maximum possible Lorentz factor of the matter particles at the center of the neutron star and at the center of the proton, and also to estimate the radius of action of the strong and ordinary gravitation in cosmological space.

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The Gravitational Field in the Relativistic Uniform Model within the Framework of the Covariant Theory of Gravitation

The conservation of energy and momentum have been viewed as undermining Cartesian mental causation since the 1690s. Modern discussions of the topic tend to use mid-19th century physics, neglecting both locality and Noether's theorem and its converse. The relevance of General Relativity (GR) has rarely been considered. But a few authors have proposed that the non-localizability of gravitational energy and consequent lack of physically meaningful local conservation laws answers the conservation objection to mental causation: conservation already fails in GR, so there is nothing for minds to violate. 
This paper is motivated by two ideas. First, one might take seriously the fact that GR formally has an infinity of rigid symmetries of the action and hence, by Noether's first theorem, an infinity of conserved energies-momenta (thus answering Schrodinger's 1918 false-negative objection). Second, Sean Carroll has asked (rhetorically) how one should modify the Dirac-Maxwell-Einstein equations to describe mental causation. This paper uses the generalized Bianchi identities to show that General Relativity tends to exclude, not facilitate, such Cartesian mental causation. In the simplest case, Cartesian mental influence must be spatio-temporally constant, and hence 0. The difficulty may diminish for more complicated models. Its persuasiveness is also affected by larger world-view considerations. 
The new general relativistic objection provides some support for realism about gravitational energy-momentum in GR (taking pseudotensor laws seriously). Such realism also answers an objection to theories of causation involving conserved quantities, because energies-momenta would be conserved even in GR.

V. I. Denisov

Papers

The inertial mass defined in the general theory of relativity has no physical meaning