G\"odel-type spacetimes in $f(Q)$ gravity

TL;DR

This paper explores G"odel-type solutions within $f(Q)$ gravity, finding that causality-violating solutions exist for perfect fluids, while causal solutions are possible with scalar fields, highlighting the theory's diverse cosmological implications.

Contribution

It demonstrates that $f(Q)$ gravity admits G"odel-type solutions, including both causal and noncausal cases, and clarifies conditions under which causality is preserved or violated.

Findings

G"odel-type metrics are permitted for any $f(Q)$ in this theory.

Only G"odel's original model is allowed with perfect fluids, implying causality violation.

Causal solutions exist with scalar fields and cosmological constant.

Abstract

There is ongoing interest in the nonmetricity formulation of gravity. The nonlinear extension of the theory, called $f(Q)$ gravity, has recently been proposed and offers a promising avenue for addressing some of the long-standing challenges in cosmology and fundamental physics. A number of solutions have already been found that have confirmed the usefulness of the theory for astrophysics and cosmology. Motivated by earlier work on G\"odel-type spacetimes we investigate whether $f(Q)$ gravity admits such cosmological solutions, which entail causality violation. In the coincident gauge, which is shown to be a legitimate gauge because the connection field equations are satisfied, we find that G\"odel-type metrics are allowed by this modified gravity theory for any function $f(Q)$. Noncausal and causal solutions exist depending on the matter content. For a perfect fluid, out of all G\"odel-type metrics only G\"odel's original model is allowed in $f(Q)$ gravity, implying violation of causality. The requirement that energy conditions be obeyed by the fluid leads to mild restrictions on $f(Q)$. For a massless scalar field in the presence of a cosmological constant, there are causal solutions for any reasonable function $f(Q)$.