Abstract
Apart from the familiar structure firmly-rooted in the general relativistic field equations where the energy–momentum tensor has a null divergence i.e., it conserves, there exists a considerable number of extended theories of gravity allowing departures from the usual conservative framework. Many of these theories became popular in the last few years, aiming to describe the phenomenology behind dark matter and dark energy. However, within these scenarios, it is common to see attempts to preserve the conservative property of the energy–momentum tensor. Most of the time, it is done by means of some additional constraint that ensures the validity of the standard conservation law, as long as this option is available in the theory. However, if no such extra constraint is available, the theory will inevitably carry a non-trivial conservation law as part of its structure. In this work, we review some of such proposals discussing the theoretical construction leading to the non-conservation of the energy–momentum tensor.
Highlights
The principle of matter–energy conservation is one of the main pillars of GeneralRelativity (GR)
The presence of curvature spoils the interpretation of such a equation as a conservation law, since in this case there is no well-defined notion of parallel vectors at different points, which harms the introduction of a global family of inertial observers able to measure the energy of a distant particle
If one evades the diffeomorphism invariance of the matter action, the physical meaning of the above constraint can be translated to the notion of a non-conservation, i.e., the mere application of the Bianchi identities to the field Equation (96) yields to σ
Summary
The principle of matter–energy conservation is one of the main pillars of General. Relativity (GR). Considering a family of models whose action carries both non-minimal coupled terms and arbitrary functions of the scalar curvature, he finds expressions for the modified conservation law for both of the variational formalisms. He shows that it is possible to generalize the Bianchi identity so that the usual conservation law is ensured. In [16], the authors obtain the energy–momentum conservation for an even wider class of theories of gravity, where the geometric dependence of the non-minimal coupling function is not restricted to scalar curvature, as it may depend on the square of the Ricci and Riemann tensors. At the end of this work, we revisit the notion of energy conditions in modified gravity theories (Section 11) and present our final considerations
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