Conservation Law of Energy

Although, according to Noether’s theorem, the time translation symmetry of the Lagrangian is directly related to the conservation of energy of the system, the crucial point of the importance of the energy conservation is what is the range that the law specifically covers? Does it cover the entire universe or only the local system?

To elaborate the point of the argument, here is a suggested Gedanken experiment. For an example, suppose there is a test mass in a thermally isolated container. One decided to influence the energy of the test mass inside the container from outside and moved a heavy mass back and forth to change the gravitational force on the test mass. Following the external influence, the test mass initially set at rest started to oscillate and gained a definite kinetic energy delta E.

Of course, one will argue that the applicable range of the total energy of the system has to be extended beyond the closed container where the test mass is located. But suppose that the heavy mass outside of the container is located so far away from the test mass that the existence of the heavy mass’ external influence is not verifiable. Now one can see that the boundary of the local conservation of the energy law has to be stretched to the infinity which is the practical range of the force of gravity.

Therefore, in general, it can be stated that the local conservation of the energy law can not be respected for the long range forces. One can always try to reformulate and enforce the law but when the boundary is infinite, what is the purpose of such an enforcement? There are always possibilities that one can devise a system that the local energy is practically gained at the expense of the energy loss from somewhere else in the universe.

The mentioned influence of the external force can be initiated by manipulating the specific local configuration of the test mass, for example, by the rotation of the longitudinally asymmetric object which produces the dipole gavity effect.

After all, general relativity and Maxwell’s equation are the theories describing how the matters are interacting with each other within their own principles of interactions in the universe, and they are not about acting like a watchdog for the local conservation of the energy. The inherent nature of the long rangedness (infinity) of those forces makes the local conservation of the energy within these principles meaningless.

However, the strong force which is extermely short ranged and holds the nucleus together and also subsequently the atoms and molecules will definitely conserve the local energy as proven to be correct abundantly in the kinetic theory of gases.