Maxwell extension of fR gravity

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arXiv:2210.09454v2 [hep-th] 31 Jan 2023

Maxwell extension of f(R)gravity

Oktay Cebecio˘glu1,∗Ahmet Saban1,†and Salih Kibaro˘glu2,3‡

1Department of Physics, Kocaeli University, 41380 Kocaeli, Turkey,

2Department of Basic Sciences, Faculty of Engineering and Natural Sciences,

Maltepe University, 34857, Istanbul, Turkey and

3Institute of Space Sciences (CSIC-IEEC) C. Can Magrans s/n, 08193 Cerdanyola (Barcelona) Spain

(Dated: February 1, 2023)

Inspired by the Maxwell symmetry generalization of general relativity (Maxwell gravity), we have

constructed the Maxwell extension of f(R) gravity. We found that the semi-simple extension of

the Poincare symmetry allows us to introduce geometrically a cosmological constant term in four-

dimensional f(R) gravity. This symmetry also allows the introduction of a non-vanishing torsion to

the Maxwell f(R) theory. It is found that the antisymmetric gauge ﬁeld Bab associated with Maxwell

extension is considered as a source of the torsion. It is also found that the gravitational equation of

motion acquires a new term in the form of an energy-momentum tensor for the background ﬁeld.

The importance of these new equations is brieﬂy discussed.

PACS numbers: 04.50.Kd; 11.15.-q; 02.20.Sv

Keywords: Cosmological constant, f(R) gravity, Gauge theory of gravity, Maxwell symmetry.

I. INTRODUCTION

Although general relativity (GR) is widely accepted as a fundamental theory to describe the gravitational phe-

nomena on an astrophysical scale, it does not explain for the rotational curves of galaxies that have been measured

do not ﬁt the predictions of GR with baryonic matter and predict the accelerated expansion of the universe that

was observed at the end of the last century [1]. The explanation in the case of rotational curves is to introduce a

new type of non-baryonic matter (dark matter) [2,3]. The accelerated expansion of the universe is usually explained

by invoking a mysterious substance called dark energy. The simplest candidate for dark energy is the cosmological

constant [4]. Needless to say, the cosmological constant problem is one of the major challenges in theoretical physics.

Introducing mysterious substances to match experimentally observed values with the theoretical predictions of GR is

one of the approaches to overcome the problem. In this approach, one modiﬁes the matter part of the Einstein ﬁeld

equations. Another approach is to modify the left-hand side (geometric part) of the Einstein ﬁeld equation, called

as a modiﬁed gravitational theory, in which the standard Einstein-Hilbert (E-H) action is replaced by an arbitrary

function of the Ricci scalar R. Such a modiﬁcation ﬁrst was put forward by Buchdahl in 1970 [5]. This theory is

called today f(R) gravity and became an established ﬁeld of theoretical gravity and cosmology after the inﬂuential

work by Starobinsky [6]. The current acceleration of the universe can be explained by f(R) gravity [7–13]. Viable

models of dark energy satisfying the Solar system and cosmological observational data based on f(R) gravity where

f(R) is ﬁnite at R= 0 were ﬁrst independently constructed in [11–13] and previous models where f(R) diverges at

R= 0 were shown to be not viable in [14]. For more information as well as recent developments and their applications

to the physically relevant models of f(R) theories, see one of the excellent reviews [15–23] and references therein.

There exists another interesting class of modiﬁed gravity theory which may easily produce the cosmological constant

by gauging the Maxwell algebra, so-called Maxwell-gravity [24]. Starting with the work of Bacry et al. [25,26], the

idea of Maxwell symmetry has been systematically studied by Schrader [27]. Such a symmetry describes a charged

particle moving in a four-dimensional Minkowski background in the presence of a constant electromagnetic ﬁeld.

The Maxwell algebra is an extension of the Poincare algebra by six additional tensorial abelian symmetry generators

that make the four-momenta non-commutative [Pa, Pb] = iλZab [28]. In 2012, the semi-simple tensor extension of the

Poincare group was given by Soroka with a new non-abelian tensorial generator [29]. In this study, another alternative

approach to the cosmological term problem is proposed. After the work of Azcarraga and Soroka, there has been a

renewed interest in the cosmological constant problem due to Maxwell symmetry. Various studies on the gauge theory

of the (super) Maxwell symmetry algebras carried out and diﬀerent aspects has been studied in [30–41].

∗ocebecioglu@kocaeli.edu.tr

†ahmetsaban55@gmail.com

‡salihkibaroglu@maltepe.edu.tr

As is well known, the simplest candidate for describing dark energy is the cosmological constant. Then, it becomes

interesting to study the ways in which cosmological constant terms can be introduced in the f(R) theories. In

particular, the f(R) theory of gravity with a cosmological term can be developed in a geometric formulation, where

the theory is constructed from the curvatures of the semi-simple extended Poincare algebra. The main purpose of

the present paper is to generalize the metric f(R) theory to a situation with extra degrees coming from Maxwell

symmetry extension. In this way, we give an alternative way of introducing the cosmological term to the f(R) theory.

The paper is organized as follows: In Section II, we brieﬂy recall the construction of f(R) theory and summarize its

basic equations in both metric and Palatini formalism. In Section III, we present the construction of a four-dimensional

gravity model containing a cosmological constant term only from the (linear in) curvatures of semi-simple Poincare

algebra, the action we reached corresponds to an Einstein-Hilbert like action. In Section IV, we propose our simple

model of the Maxwell generalized f(R) gravity action involving a cosmological term and comment on the obtaining

of the equations of motion. Finally, in Section V, we discuss the obtained results and possible future developments.

II. f(R)THEORY IN TERMS OF DIFFERENTIAL FORMS

There are three versions of f(R) gravity: Metric-f(R) theory (second order formalism) is fully described by the

metric ﬁeld alone, Palatini-f(R) theory (ﬁrst order formalism) in which metric and connection are handled as inde-

pendent ﬁelds, similarly the "metric-aﬃne f(R) theory"in which matter Lagrangian also includes connection [15]. In

this section, we very brieﬂy summarize the main ingredients of f(R) gravity in both metric and Palatini approaches.

The starting point for the metric-f(R) gravity is the Einstein-Hilbert action,

SEH =1

2κˆd4x√−gR, (1)

in which κ= 8πG/c4with Gbeing Newton’s gravitational constant and the Ricci scalar Ris constructed from the

Riemann curvature tensor. One of the simplest modiﬁcations to general relativity is the f(R) gravity. It generalizes

the Lagrangian density of the E-H action. Speciﬁcally, it replaces the Ricci scalar Rin action Eq.(1), with some

function f(R) of the scalar curvature:

Sf(R)=1

2κˆd4x√−gf (R).(2)

The source-free vacuum ﬁeld equations that are obtained taking the variations of action with respect to the metric

gµν are

f′Rµν −1

2fgµν + (gµν ∆− ∇µ∇ν)f′= 0,(3)

where ∆ = gµν ∇µ∇νis the d’Alembertian operator. Note that the cosmological constant term does not appear in

this ﬁeld equation. These equations can be re-arranged in the Einstein-like form

Gµν =Rµν −1

2gµν R=Teff

µν ,(4)

where

Teff

µν =1

f′∇µ∇νf′−gµν ∆f′+1

2gµν f−f′R,(5)

is an eﬀective energy-momentum tensor which can be interpreted as an extra gravitational energy-momentum tensor

due to higher order curvature eﬀects. Including the function f(R) gives extra freedom in deﬁning the behavior of

gravity. The detailed structure of f(R) gravity theories arising from the action Eq.(2), in 4Dspace-time has been

discussed in ref. [15,16,18–20,22].

In the context of the ﬁrst order (Palatini) formalism, the entities basis 1-forms eaand the connection 1-forms ωab are

independent from each other. That is, the connection is not previously ﬁxed to be given by Christoﬀel’s symbols, but

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摘要：

arXiv:2210.09454v2[hep-th]31Jan2023Maxwellextensionoff(R)gravityOktayCebecio˘glu1,∗AhmetSaban1,†andSalihKibaro˘glu2,3‡1DepartmentofPhysics,KocaeliUniversity,41380Kocaeli,Turkey,2DepartmentofBasicSciences,FacultyofEngineeringandNaturalSciences,MaltepeUniversity,34857,Istanbul,Turkeyand3InstituteofSpa...

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Maxwell extension of fR gravity

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