Homogeneous Gödel-type solutions in hybrid metric-Palatini gravity

J. Santos; M. J. Rebouças; A. F. F. Teixeira

doi:10.1140/epjc/s10052-018-6025-4

2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2018) 78: 567
https://doi.org/10.1140/epjc/s10052-018-6025-4

Regular Article - Theoretical Physics

Homogeneous Gödel-type solutions in hybrid metric-Palatini gravity

J. Santos¹, M. J. Rebouças²^* and A. F. F. Teixeira²

¹ Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal, RN, 59072-970, Brazil
² Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Rio de Janeiro, RJ, 22290-180, Brazil

^* e-mail: reboucas@cbpf.br

Received: 10 May 2018
Accepted: 25 June 2018
Published online: 10 July 2018

Abstract

The hybrid metric-Palatini $f({\mathscr {R}})$ gravity is a recently devised approach to modified gravity in which it is added to the metric Ricci scalar R, in the Einstein–Hilbert Lagrangian, a function $f({\mathscr {R}})$ of Palatini curvature scalar ${\mathscr {R}}$ , which is constructed from an independent connection. These hybrid metric-Palatini gravity theories provide an alternative way to explain the current accelerating expansion without a dark energy matter component. If gravitation is to be described by a hybrid metric-Palatini $f({\mathscr {R}})$ gravity theory there are a number of issues that ought to be examined in its context, including the question as to whether its equations allow homogeneous Gödel-type solutions, which necessarily leads to violation of causality. Here, to look further into the potentialities and difficulties of $f({\mathscr {R}})$ theories, we examine whether they admit Gödel-type solutions for physically well-motivated matter source. We first show that under certain conditions on the matter sources the problem of finding out space-time homogeneous (ST-homogeneous) solutions in $f({\mathscr {R}})$ theories reduces to the problem of determining solutions of Einstein’s field equations with a cosmological constant. Employing this far-reaching result, we determine a general ST-homogeneous Gödel-type solution whose matter source is a combination of a scalar with an electromagnetic fields plus a perfect fluid. This general Gödel-type solution contains special solutions in which the essential parameter $$m^2$$ can be $m^{2} > 0$ hyperbolic family, $$m=0$$ linear class, and $m^{2} < 0$ trigonometric family, covering thus all classes of homogeneous Gödel-type spacetimes. This general solution also contains all previously known solutions as special cases. The bare existence of these Gödel-type solutions makes apparent that hybrid metric-Palatini $f({\mathscr {R}})$ gravity does not remedy causal anomaly in the form of closed timelike curves that are permitted in general relativity.