On the equivalence between Saez-Ballester theory and Einstein-scalar field system

Here, we discuss a topic that comes up more often than expected: A same theory or theoretical model arises in two different presentations which are assumed to be actually different theories so that these are independently developed. Sometimes this leads to an unwanted doubling of the results. In this paper, we illustrate this issue with the example of two apparently different gravitational theories: (i) the (minimally coupled) Einstein-massless-scalar (EMS) system and (ii) the Saez-Ballester theory (SBT). We demonstrate that the latter is not a scalar-tensor theory of gravity, as widely acknowledged. Moreover, SBT is identified with the EMS theory. As illustrations of this identification we show that several known solutions of SBT are also solutions of the EMS system and vice versa. Cosmological arguments are also considered. In particular, a dynamical systems-based demonstration of the dynamical equivalence between these theories is given. The study of the asymptotic dynamics of the Saez-Ballester-based cosmological model shows that there are no equilibrium points which could be associated with accelerated expansion, unless one includes a cosmological constant term or a self-interacting scalar field. This is a well-known result for cosmological models which are based on the Einstein-self-interacting-scalar theory, also known as quintessence.

Automated summary

In this paper, we discuss the issue of duplicate results generated by the independent development of apparently different theories that share the same theoretical model. Using the examples of the Einstein-massless-scalar (EMS) system and the Saez-Ballester theory (SBT), we demonstrate that SBT is in fact equivalent to the EMS theory, contradicting the widely held belief that SBT is a scalar-tensor theory of gravity. We also analyze cosmological arguments and show that the Saez-Ballester-based cosmological model does not exhibit accelerated expansion unless additional terms, such as a cosmological constant or a self-interacting scalar field, are included.