The connection between Noether's Theorems and symmetries in physics is well established. However, the role they play in gravitation, especially with the conservation of energy, is often overlooked. In this article, we will review Noether's Theorems and their application in General Relativity. This will include what is usually referred to as the Third Theorem. We will present Noether's Theorems in their original form. Then, they will be restated as they are usually applied to physics. Some basic equations of Special Relativity will be reviewed. Several applications in flat spaces are examined. This is so they can be contrasted with the equations in General Relativity. The equations of General Relativity are then developed, and applications are contrasted with those of Special Relativity. We will be most interested with the role of Noether's Theorems in conservation laws. A formulation of Noether's Theorems in curved space is presented. Of particular interest is the conservation of energy in General Relativity. This involves Noether's First Theorem. Here, there is no general form for a conserved tensor, as there is in the flat-space case. Pseudotensor formulations can be found. The Landau and Lifshitz and the Dirac pseudotensors are given as examples. In special cases, one can formulate a tensor equation for the conservation of energy. The standard cosmology is an example. The formulation of Noether's Second and Third Theorems are also presented. These formulations are related to things like the Bianchi identity and the response equation.
| Published in | American Journal of Modern Physics (Volume 15, Issue 5) |
| DOI | 10.11648/j.ajmp.20261505.13 |
| Page(s) | 162-180 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2026. Published by Science Publishing Group |
Noether's Theorems, General Relativity, Conservation of Energy, Variational Methods
| [1] | Travel, M. A. (1971). "Noether's theorem," Transport Theory and Statistical Physics, 1: 183 - 207. Emmy Noether (1918). "Invariante variationsprobleme," Nachr. d. Konig. Gesellsch. d. Wiss. zu Gottingen, Math-phys. Klasse, 235 - 257. |
| [2] | Katherine A. Brading (2002). "Which symmetry? Noether, Weyl, and conservation of electric charge," Studies in History and Philosophy of Modern Physics, 33: 3 - 22. |
| [3] | Katherine Brading (2005). "A Note on General Relativity, energy conservation, and Noether's theorems," in The University of General Relativity, Einstein Studies, vol. II, Jean Eisenstaedt and Anne J. Knox, Birkh"auser, Eds. Boston, pp. 125 - 135. |
| [4] | Katherine Brading and Harvey R. Brown (2000). "Noether's theorems and gauge symmetries," arXiv:hep-th/0009058v1, unpublished. |
| [5] | Brown, H. R. and Brading,K. A. (2002). "General covariance from the perspective of Noether's theorems," Di'alogas, 70: 59 - 86. |
| [6] | Katherine Brading and Harvey R. Brown (2003). "Symmetries and Noether's Theorems," in Symmetries in Physics: Philosophical Reflections, Katherine Brading and E. Castellani, Eds. Cambridge University Press , pp. 89-109. |
| [7] | Herbert Goldstein (1965). Classical Mechanics. Addison-Wesley. |
| [8] | Landau L. D. and Lifshitz E. M. (1965). Mechanics (3rd ed.). Pergamon Press. |
| [9] | Berestetskiui V. B., Lifshitz E. M. and Pitaevskiui L. P. (1971). Relativistic Quantum Theory: Part I. Pergamon Press. |
| [10] | Landau L. D. and Lifshitz E. M. (1975). The Classical Theory of Fields (4th ed.). Pergamon Press. |
| [11] | Steven Weinberg (1972). Gravitation and Cosmology: Principles and applications of the General Theory of Relativity. Wiley. |
| [12] | Dirac P. A. M. (1975). General Theory of Relativity. John Wiley and Sons. |
| [13] | Schutz, Jr.,Bernard F. (1970). "Perfect Fluids in General Relativity: Velocity Potentials and a Variational Principle," Phys. Rev. D, 2: 2762. |
| [14] | Einstein A. "The foundation of the general theory of relativity," in Lorentz, H. A., Einstein, A, Minkowski H. and Weyl, H. (1952). The Principle of Relativity, Dover Pub. Inc. pp. 109 - 164. |
| [15] | Hilbert, D. (1916). "Die Grundlagen der physik", in K"onigliche Gesellschaft der Wissenschaften zu G"ottingen," Mathematisch-physikalische Klasse. Nachrichten, pp. 395 - 407. |
| [16] | Klein, F. (1917). "Zu Hilberts erster note über die grundlagen der physik," in K"onigliche Gesellschaft der Wissenschaften zu G"ottingen. Mathematisch-physikalische Klasse. Nachrichten, pp. 469 - 82. |
| [17] | Klein, F. (1918). " "Uber die differentialgesetze für die erhaltung von impuls und energie in der einsteinschen gravitationstheorie," in K"onigliche Gesellschaft der Wissenschaften zu G"ottingen. Mathematisch-physikalische Klasse. Nachrichten, pp. 171 - 89. |
| [18] | Richard C. Tolman (1950). Relativity, Thermodynamics and Cosmology. Oxford University Press. |
| [19] | Andrzej Trautmann (1962). " Conservation Laws in General Relativity," in Gravitation: An Introduction to Current Research, editor, Louis Witten. John Wiley and Sons, pp. 319-336. |
| [20] | Erwin Schr"odinger (1950). Space-Time Structure. Cambridge University Press. |
| [21] | Misner C. W., Thorne K. S. and Wheeler J. A. (1973). Gravitation. W. H. Freeman. |
| [22] | Gelfand I. M. and Fomin S. V. (1963). Calculus of Variations. W. H. Freeman. |
| [23] | Herbert Goldstein (1980). Classical Mechanics (2nd ed.). Addison-Wesley. |
APA Style
McLeod, R. J. (2026). A Brief Review Noether's Theorems and Their Application to General Relativity. American Journal of Modern Physics, 15(5), 162-180. https://doi.org/10.11648/j.ajmp.20261505.13
ACS Style
McLeod, R. J. A Brief Review Noether's Theorems and Their Application to General Relativity. Am. J. Mod. Phys. 2026, 15(5), 162-180. doi: 10.11648/j.ajmp.20261505.13
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Download@article{10.11648/j.ajmp.20261505.13,
author = {Robert John McLeod},
title = {A Brief Review Noether's Theorems and Their Application to General Relativity},
journal = {American Journal of Modern Physics},
volume = {15},
number = {5},
pages = {162-180},
doi = {10.11648/j.ajmp.20261505.13},
url = {https://doi.org/10.11648/j.ajmp.20261505.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20261505.13},
abstract = {The connection between Noether's Theorems and symmetries in physics is well established. However, the role they play in gravitation, especially with the conservation of energy, is often overlooked. In this article, we will review Noether's Theorems and their application in General Relativity. This will include what is usually referred to as the Third Theorem. We will present Noether's Theorems in their original form. Then, they will be restated as they are usually applied to physics. Some basic equations of Special Relativity will be reviewed. Several applications in flat spaces are examined. This is so they can be contrasted with the equations in General Relativity. The equations of General Relativity are then developed, and applications are contrasted with those of Special Relativity. We will be most interested with the role of Noether's Theorems in conservation laws. A formulation of Noether's Theorems in curved space is presented. Of particular interest is the conservation of energy in General Relativity. This involves Noether's First Theorem. Here, there is no general form for a conserved tensor, as there is in the flat-space case. Pseudotensor formulations can be found. The Landau and Lifshitz and the Dirac pseudotensors are given as examples. In special cases, one can formulate a tensor equation for the conservation of energy. The standard cosmology is an example. The formulation of Noether's Second and Third Theorems are also presented. These formulations are related to things like the Bianchi identity and the response equation.},
year = {2026}
}
TY - JOUR T1 - A Brief Review Noether's Theorems and Their Application to General Relativity AU - Robert John McLeod Y1 - 2026/10/20 PY - 2026 N1 - https://doi.org/10.11648/j.ajmp.20261505.13 DO - 10.11648/j.ajmp.20261505.13 T2 - American Journal of Modern Physics JF - American Journal of Modern Physics JO - American Journal of Modern Physics SP - 162 EP - 180 PB - Science Publishing Group SN - 2326-8891 UR - https://doi.org/10.11648/j.ajmp.20261505.13 AB - The connection between Noether's Theorems and symmetries in physics is well established. However, the role they play in gravitation, especially with the conservation of energy, is often overlooked. In this article, we will review Noether's Theorems and their application in General Relativity. This will include what is usually referred to as the Third Theorem. We will present Noether's Theorems in their original form. Then, they will be restated as they are usually applied to physics. Some basic equations of Special Relativity will be reviewed. Several applications in flat spaces are examined. This is so they can be contrasted with the equations in General Relativity. The equations of General Relativity are then developed, and applications are contrasted with those of Special Relativity. We will be most interested with the role of Noether's Theorems in conservation laws. A formulation of Noether's Theorems in curved space is presented. Of particular interest is the conservation of energy in General Relativity. This involves Noether's First Theorem. Here, there is no general form for a conserved tensor, as there is in the flat-space case. Pseudotensor formulations can be found. The Landau and Lifshitz and the Dirac pseudotensors are given as examples. In special cases, one can formulate a tensor equation for the conservation of energy. The standard cosmology is an example. The formulation of Noether's Second and Third Theorems are also presented. These formulations are related to things like the Bianchi identity and the response equation. VL - 15 IS - 5 ER -
School of Chemical and Physical Sciences, Flinders University Bedford Park, Adelaide, Australia
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