Energy Conservation in Expanding Universe

The problem of energy and linear momentum conservation in a gravitational field is considered one of the problems for which physicists have not yet reached an agreed-upon solution. This problem becomes more severe and more complex in cases of an expanding universe, where there is almost a consensus among scientists that no law of energy conservation or momentum conservation exists. It is also well known that all the mathematical treatments that have been proposed to solve this problem suffer from many shortcomings, such as lack of localization, lack of covariance, and the multiplicity of formulations. Here, we aim to propose a new and different solution to this problem, a solution that is free of all these issues. In this approach, we present a local and covariant formulation of conserved energy and momentum. This formulation is based on expanding the concept of energy and momentum, and also on slightly modifying the concept of conservation. In the solution presented here, the energy and momentum of a body are conserved in a gravitational field, and also in cases of instability of the gravitational field, such as the expansion of the universe. Instead of defining the energy and momentum of a body by a four-vector, the energy and momentum of the body are defined by an extended object, as shown in the figure, which includes energy, momentum, and other complementary quantities of the same type. The concept of conservation also changes slightly: conservation of energy and momentum does not mean conserving each individual component that represents energy or momentum, but rather conserving the sum of these components. In the case of a static gravitational field, the energy remains constant and the sum of the momentum components remains constant. When the body moves in the field, it neither gains nor loses energy or momentum; rather, momentum is merely transferred from one component to other momentum components. When the gravitational field changes, such as in cases of cosmic expansion, the body also does not lose energy or momentum; instead, part of its energy is transformed into momentum, or vice versa, while the sum of the energy and the components of momentum remains constant. This idea is not a hypothesis; rather, it can be derived mathematically from the principles of general relativity and the general principles of physics, and this is summarized in a publicly available paper on this subject.