On the viability of f(Q) gravity models

Abstract

In general relativity, the contracted Bianchi identity makes the field equation compatible with the energy conservation, likewise in $f(R)$ theories of gravity. We show that this classical phenomenon is not guaranteed in the symmetric teleparallel theory, and rather generally $f(Q)$ model specific. We further prove that the energy conservation criterion is equivalent to the affine connection's field equation of $f(Q)$ theory, and except the $f(Q)=\alpha Q+\beta$ model, the non-linear $f(Q)$ models do not satisfy the energy conservation or, equivalently the second field equation in every spacetime geometry; unless $Q$ itself is a constant. So the problem is deep-rooted in the theory, several physically motivated examples are provided in the support.

Non-metricity with boundary terms: 𝖿(𝖰,𝖢) gravity and cosmology

“…We close this section with a discussion on the energy conservation in f (Q, C) theories of gravity. In general relativity, as well as in f ( R) theories of gravity, the field equations are compatible with the classical energy conservation due to the second Bianchi identity, which is not the case in simple f (Q) gravity [52]. On the other hand, in the covariant formulation of f (Q, C) gravity, equation (3.12) leads to…”

“…where ζ is an arbitrary scalar field. Firstly, the left-hand-side of (B.1) can be expanded as As shown in [52]…”

“…Alternatively, one can start from the torsional formulation of gravity and obtain f (T) gravity [11,12], f (T, T G ) gravity [13,14], etc. Finally, there is an alternative way to build classes of modified gravities, namely to use non-metricity [15,16] resulting to f (Q) gravity [51][52][53][54][55][56].…”

Cosmology of f(Q) gravity in non-flat Universe

“…where γ (t) is a non-zero function of time. The corresponding non-metricity scalar Q can be calculated from (12) as [43,[84][85][86][87][88]]…”

Reconstructing Modified and Alternative Theories of Gravity

“…Nonetheless, the isotropic and homogeneous Robertson-Walker metric (19) under present consideration also admits three non-trivial connections [14,48], apart from the coincidence gauge. The study of the three other connections have been initiated recently [49,50]. More recently, it has been shown that all these three connections can also explain the cosmic evolutionary history, for a linear form of F(Q) = f 0 Q, without further modification [51].…”