# The origin of the energy–momentum conservation law

## Abstract

The interplay between the action–reaction principle and the energy–momentum conservation law is revealed by the examples of the Maxwell–Lorentz and Yang–Mills–Wong theories, and general relativity. These two statements are shown to be equivalent in the sense that both hold or fail together. Their mutual agreement is demonstrated most clearly in the self-interaction problem by taking account of the rearrangement of degrees of freedom appearing in the action of the Maxwell–Lorentz and Yang–Mills–Wong theories. The failure of energy–momentum conservation in general relativity is attributed to the fact that this theory allows solutions having nontrivial topologies. The total energy and momentum of a system with nontrivial topological content prove to be ambiguous, coordinatization-dependent quantities. For example, the energy of a Schwarzschild black hole may take any positive value greater than, or equal to, the mass of the body whose collapse is responsible for forming this black hole. We draw the analogy to the paradoxial Banach–Tarski theorem; the measure becomes a poorly defined concept if initial three-dimensional bounded sets are rearranged in topologically nontrivial ways through the action of free non-Abelian isometry groups. - Highlights: • The action–reaction principle and energy–momentum conservation hold or fail together. • Nontrivial topology is the reason formore »

- Authors:

- Escuela de Física, Universidad Autónoma de Zacatecas, Apartado Postal C-580 Zacatecas 98068, Zacatecas (Mexico)
- Russian Federal Nuclear Center, Sarov, 607189 Nizhniĭ Novgorod Region (Russian Federation)
- (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22701529

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics

- Additional Journal Information:
- Journal Volume: 384; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; BLACK HOLES; DEGREES OF FREEDOM; GENERAL RELATIVITY THEORY; MATHEMATICAL SOLUTIONS; REST MASS; THREE-DIMENSIONAL CALCULATIONS; YANG-MILLS THEORY

### Citation Formats

```
Chubykalo, Andrew E., E-mail: achubykalo@yahoo.com.mx, Espinoza, Augusto, Kosyakov, B.P., E-mail: kosyakov.boris@gmail.com, and Moscow Institute of Physics & Technology, Dolgoprudniĭ, 141700 Moscow Region.
```*The origin of the energy–momentum conservation law*. United States: N. p., 2017.
Web. doi:10.1016/J.AOP.2017.06.018.

```
Chubykalo, Andrew E., E-mail: achubykalo@yahoo.com.mx, Espinoza, Augusto, Kosyakov, B.P., E-mail: kosyakov.boris@gmail.com, & Moscow Institute of Physics & Technology, Dolgoprudniĭ, 141700 Moscow Region.
```*The origin of the energy–momentum conservation law*. United States. doi:10.1016/J.AOP.2017.06.018.

```
Chubykalo, Andrew E., E-mail: achubykalo@yahoo.com.mx, Espinoza, Augusto, Kosyakov, B.P., E-mail: kosyakov.boris@gmail.com, and Moscow Institute of Physics & Technology, Dolgoprudniĭ, 141700 Moscow Region. Fri .
"The origin of the energy–momentum conservation law". United States. doi:10.1016/J.AOP.2017.06.018.
```

```
@article{osti_22701529,
```

title = {The origin of the energy–momentum conservation law},

author = {Chubykalo, Andrew E., E-mail: achubykalo@yahoo.com.mx and Espinoza, Augusto and Kosyakov, B.P., E-mail: kosyakov.boris@gmail.com and Moscow Institute of Physics & Technology, Dolgoprudniĭ, 141700 Moscow Region},

abstractNote = {The interplay between the action–reaction principle and the energy–momentum conservation law is revealed by the examples of the Maxwell–Lorentz and Yang–Mills–Wong theories, and general relativity. These two statements are shown to be equivalent in the sense that both hold or fail together. Their mutual agreement is demonstrated most clearly in the self-interaction problem by taking account of the rearrangement of degrees of freedom appearing in the action of the Maxwell–Lorentz and Yang–Mills–Wong theories. The failure of energy–momentum conservation in general relativity is attributed to the fact that this theory allows solutions having nontrivial topologies. The total energy and momentum of a system with nontrivial topological content prove to be ambiguous, coordinatization-dependent quantities. For example, the energy of a Schwarzschild black hole may take any positive value greater than, or equal to, the mass of the body whose collapse is responsible for forming this black hole. We draw the analogy to the paradoxial Banach–Tarski theorem; the measure becomes a poorly defined concept if initial three-dimensional bounded sets are rearranged in topologically nontrivial ways through the action of free non-Abelian isometry groups. - Highlights: • The action–reaction principle and energy–momentum conservation hold or fail together. • Nontrivial topology is the reason for energy–momentum nonconservation in gravity. • The total energy of a black hole is an ambiguous coordinatization-dependent quantity. • This affair closely parallels that in the paradoxical Banach–Tarski theorem.},

doi = {10.1016/J.AOP.2017.06.018},

journal = {Annals of Physics},

issn = {0003-4916},

number = ,

volume = 384,

place = {United States},

year = {2017},

month = {9}

}