Einstein and Rastall theories of gravitation in comparison

Abstract

We profit by a recent paper of Visser claiming that Rastall gravity is equivalent to Einstein gravity to compare the two gravitational theories in a general way. Our conclusions are different from Visser's ones. We indeed argue that these two theories are not equivalent. In fact, Rastall theory of gravity is an "open" theory when compared to Einstein general theory of relativity. Thus, it is ready to accept the challenges of observational cosmology and quantum gravity.The framework of extended theories of gravity [1][2][3][4][5] is today considered an useful and popular approach to attempt to solve the important problems of the standard model of cosmology like Dark Energy [6,7] and Dark Matter [8,9]. A key point is that all of the potential alternatives to Einstein's general theory of relativity (GTR) must be viable. This implies that alternative theories must be in agreement with the Einstein's equivalence principle, which has today a strong, unchallengeable empirical evidence [2]. The fundamental consequence is that the alternatives to Einstein gravity must be metric theories [2]. Another important point is that such alternatives must pass the solar system tests. Hence, deviations from the standard GTR must be weak [1,2].Within the framework of extended gravity, one finds the theory proposed by Rastall in 1972 [10], which recently obtained a renewed interest in the literature [11][12][13][14]. This interest is due to some good behavior of the Rastall theory. In fact, on one hand it seems in good agreement with observational data on the Universe age and on the Hubble parameter [15]. On the other hand it may, in principle, provide an alternative description for the matter dominated era with respect to the GTR [16]. Observational data from the helium nucleosynthesis seem also in agreement with Rastall gravity [17]. Based on these observational evidences, there have been some recently studies of the various cosmic eras in the framework of Rastall gravity [18][19][20][21][22]. Other interesting a e-mail: cordac.galilei@gmail.com issues are that Rastall gravity seems to do not suffer from the entropy and age problems of standard cosmology [23] and is consistent with the gravitational lensing phenomena [24,25]. Further interesting works on Rastall gravity are given by [26][27][28][29][30] and references within.Differently from Einstein gravity, Rastall gravity considers a non-divergence-free energy-momentum . A different, but similar theory is the so called Curvature-matter theory of gravity [31][32][33][34][35]. In this theory the matter and geometry are coupled to each other in a non-minimal way [31][32][33][34][35]. Then, the standard energy-momentum conservation law does not work in this case too [31][32][33][34][35].In a recent paper [36], Visser claimed that Rastall gravity is equivalent to Einstein gravity. In this letter, we argue that these two theories are not equivalent instead. Let us see the key points of our interpretation.Rastall did not define a new energy-momentum tensor (EMT) in his origina...

Rastall gravity is equivalent to Einstein gravity

“…This is a very strong constraint on their assumed split between the matter and interaction terms; one that they do not even attempt to justify or discuss in the rest of their article. The rest of reference [25] does not actually address any of the points I have raised in this article; those authors appear to have significantly misinterpreted my actual comments.…”

“…For another example, consider reference [25]. In that article the authors first split the total action S into an Einstein-Hilbert term S EH plus the rest -S not−EH .…”

Impact of the Rastall parameter on perfect fluid spheres

“…In addition Visser does not offer any experimental method to determine which energy momentum tensor is the physically correct one. The matter has been adequately dealt with by Darabi et al [37] who have shown that the incorrect energy momentum tensor was alluded to by Visser. In the present article we consider, as a simple counter-example, the famous Schwarzschild interior metric and show that while this metric yields an incompressible fluid sphere (constant density) in general relativity, the model in Rastall theory using the exact same metric does not display constant density but is a variable quantity dependent on the radial value.…”

Equilibrium stellar configurations in Rastall theory and astrophysical implications

“…which is independent of the Rastall parameter, α. Hence, we are left with a system of three differential equations, (11), (12) and (13), which are not enough to solve for four variables λ, ν, ρ, and p. It is required that one of the variables must be specified at the start and the remaining three determined through integrating the field equations. Alternatively, a functional dependence of one variable on another may be postulated motivated on physical grounds.…”

“…An important advance was made by dos Santos and Nogale [9] who generated a Lagrangian density for Rastall gravity that seemed to indicate that Rastall gravity is a special case of f (R, T ) gravity theory. Visser [10] recently erred in claiming that Rastall theory was equivalent to Einstein gravity [11,12].…”