Abstract
In this paper, it is shown that the super-potential defined by Einstein, Landau–Lifshitz, and Bergmann–Thompson in general relativity theory and in teleparallel gravity theory are equivalent for a general diagonal space–time metric. Therefore, the energy–momentum tensor defined by Einstein, Landau–Lifshitz, and Bergmann–Thompson is equivalent in general relativity theory and in teleparallel gravity theory for the general diagonal space–times. Two different examples are also given to ensure the validity of the established statements. However, this fact is not true in general for the non-diagonal space–times. Non-diagonal Phantom black hole metric and non-diagonal stationary axi-symmetric space–time are introduced as a counter-examples to prove the non-equivalence of the super-potential and energy–momentum tensor defined in general relativity theory and in teleparallel gravity theory.
•The problem of localization of energy is unresolved and controversial, although much attention has been given by different scientists to resolve it.•We have discussed the problem of localization of energy-momentum in the two different frameworks of general relativity (GR) and teleparallel gravity (TG) by using different energy-momentum complexes.•This paper continues the investigation of comparing various distributions presented in the literature in the framework of GR and TG.•We compare the results obtained in the framework of GR and TG.•It is worth mentioning that GR theory and TG theory gave the same results under a certain choice of the metric functions.•In the other word, Einstein, Landau–Lifshitz and Bergmann–Thompson super-potentials components and energy-momentum densities components in GR and TG yield the same results for some space-times and different results in the case of others.•We can see that for a given space-time many quasi-local mass definitions do not give agreed results.•Energy momentum complexes are not unique. Thus, it makes it difficult to decide that which one is to be used to compute the energy momentum distribution and causes doubt as they would give different energy-momentum distributions for a given space-time.•We conclude that the use of the energy-momentum complexes may not be sufficient to find the energy-momentum distribution of the physical systems.•It is pointed out that TG is an alternate description of gravitation that corresponds to a gauge theory for the translation groups. The energy localization problem was reconsidered in the framework of this theory by many authors.