Energy conservation law.

Ruslan A. Sharipov.

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The energy conservation law is one of the fundamental laws that permeates all sections of physics. In classical field physics it has two forms: integral form and differential form. In integral form, it is written as follows:

The first integral in the formula expresses the total energy contained in the volume Ω. The second integral in the formula is the energy flow through the boundary of the volume Ω. Verbally the energy conservation law is expressed as follows: the change in the amount of total energy in the volume Ω occurs only due to the energy flowing through the boundary of this volume.

H in the formula of the energy conservation law denotes the total energy density of all fields, including the gravitational field. Jin this formula denotes the vector of the total energy flux density of all fields, including the gravitational field. The components of this vector are numbered with the superscript i. These same quantities are included in the differential form of the energy conservation law:

Upon transition to relativistic physics, the situation with the energy conservation law changes dramatically. The integral form of the energy conservation law in general relativity is completely absent, while the differential form of this law is written as

T with indices i and k denotes the components of the energy-momentum tensor. This tensor includes contributions from all matter fields, but does not contain a direct contribution from the gravitational field. The gravitational field in general relativity is as if it was isolated from other fields and does not contribute to the energy conservation law. Soviet and Russian academician L. D. Faddeev characterizes this situation in general relativity with the following words: "The energy of the gravitational field cannot be localized, i.e. there is no uniquely determined energy density" (see the journal Soviet Physics Uspekhi. 1982, vol 25, issue 3, pages 130-142).

In the new theory of gravity, which is called "3D-brane universe model", the situation with the energy conservation law is corrected. The integral form of the law returns and has the same form as in classical field physics:

The energy density H and the energy flux density J in this formula again contain contributions from all fields, including the gravitational field. The differential form of the energy conservation law differs slightly from the classical one:

The additional term in this formula is called the Hubble term. It arises because in the new theory, as in general relativity, there is a possibility of expansion or contraction of the universe and of deformation of space near massive gravitating bodies.