Thermal Casimir eﬀect in Gödel-type universes A. F. Santos 1and Faqir C. Khanna

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Thermal Casimir eﬀect in Gödel-type universes

A. F. Santos 1, ∗and Faqir C. Khanna b2, ‡

1Instituto de Física, Universidade Federal de Mato Grosso,

78060-900, Cuiabá, Mato Grosso, Brazil

2Department of Physics and Astronomy, University of Victoria,

3800 Finnerty Road, Victoria BC V8P 5C2, Canada

In this paper, a massless scalar ﬁeld coupled to gravity is considered. Then the Casimir

eﬀect at ﬁnite temperature is calculated. Such development is carried out in the Thermo

Field Dynamics formalism. This approach presents a topological structure that allows for

investigating the eﬀects of temperature and the size eﬀect in a similar way. These eﬀects

are calculated considering Gödel-type solutions as a gravitational background. The Stefan-

Boltzmann law and its consistency are analyzed for both causal and non-causal Gödel-type

regions. In this space-time and for any region, the Casimir eﬀect at zero temperature is

always attractive. However, at ﬁnite temperature, a repulsive Casimir eﬀect can emerge

from a critical temperature.

I. INTRODUCTION

The Casimir eﬀect is a quantum remarkable phenomenon with numerous applications. This eﬀect

was ﬁrst proposed by H. Casimir, in 1948 [1]. It describes an attractive force that arises between

two parallel conducting plates placed in the vacuum of a quantum ﬁeld. About ten years after

the theoretical proposal, experimental conﬁrmation was carried out [2]. Recently, the experimental

accuracy has increased signiﬁcantly [3–7]. The original idea was developed using the electromagnetic

ﬁeld. However, nowadays this phenomenon appears in any quantum ﬁeld. The Casimir eﬀect

emerges when boundary conditions or topological eﬀects are imposed on a quantum ﬁeld. As a

consequence, the vacuum energy of the ﬁeld is modiﬁed [8–10]. In this paper, the quantum ﬁeld

to be considered is the massless scalar ﬁeld coupled to gravity. In this context, thermal eﬀects are

investigated having as a gravitational background the Gödel-type solutions.

Temperature changes the properties and behavior of any system. Furthermore, phenomena

at zero temperature generally do not occur in nature. To be more realistic let us introduce the

bProfessor Emeritus - Physics Department, Theoretical Physics Institute, University of Alberta

Edmonton, Alberta, Canada

∗alesandroferreira@ﬁsica.ufmt.br

‡fkhanna@ualberta.ca; khannaf@uvic.ca

arXiv:2210.02869v1 [hep-th] 6 Oct 2022

temperature eﬀect in our theory. There are diﬀerent approaches in the literature to introduce

temperature eﬀects into a quantum ﬁeld theory [11,12]. Here the Thermo Field Dynamics (TFD)

formalism [13–18] is considered. TFD is a thermal quantum ﬁeld theory that exhibits a topological

structure. This topology is deﬁned as Γd

D= (S1)d×RD−d, where Dare the space-time dimensions

and dis the number of compactiﬁed dimensions. This implies that any set of dimensions of the

manifold can be compactiﬁed in a circumference S1. Due to this characteristic, diﬀerent eﬀects

such as the Stefan-Boltzmann law and Casimir eﬀect at zero and non-zero temperatures can be

calculated in this formalism in the same way, just considering diﬀerent compactiﬁcations along the

space-time dimensions. In this work, the TFD formalism is used to calculate the Stefan-Boltzmann

law and Casimir eﬀect on a gravitational background described by the Gödel-type universe.

In 1949, Kurt Gödel proposed a cosmological model that is a solution to Einstein’s equations

[19]. It is a rotating cosmological model with a non-vanishing cosmological constant and a dust-like

matter source. The main feature of this solution is the possibility of the existence of Closed Time-

like Curves (CTCs) that lead to the breakdown of causality. Violation of causality is not a unique

characteristic of the Gödel solution, there are other solutions from the general theory of relativity

that allow such CTCs [20,21]. This metric has been generalized to the so-called Gödel-type metric

[22]. Various properties of this metric have been investigated [23–25]. In the Gödel-type solution

there is a speciﬁc relationship between two parameters that allow the investigation of three classes

of solutions that lead to three diﬀerent regions: (i) non-causal; (ii) causal and (iii) there is an inﬁnite

sequence of alternating causal and non-causal regions. Here, the Gödel-type universe is considered

to calculate the Stefan-Boltzmann law and Casimir eﬀect at ﬁnite temperature. The Casimir eﬀect

in the Gödel space-time has been investigated [26,27]. However, these works do not analyze what

happens in the energy density and in the Casimir eﬀect in the causal and non-causal Gödel-type

regions. Therefore, here it is investigated whether there is an inﬂuence of these regions on these

phenomena.

This paper is organized as follows. In section II, the theory is presented. The vacuum expectation

value of the energy-momentum tensor associated with the massless scalar ﬁeld coupled to gravity

is calculated. In section III, a brief introduction to the TFD formalism is given. In order to obtain

physical quantities, the energy-momentum tensor is rewritten, where a renormalization procedure is

performed. In section IV, some characteristics of the Gödel-type universe are explored. In section V,

thermal applications on a Gödel-type background are investigated. Diﬀerent topologies are chosen,

then thermal eﬀects and the size eﬀects are calculated in this cosmological model. In section VI,

some concluding remarks are made.

II. THE THEORY: SCALAR FIELD COUPLED TO GRAVITY

Here the theoretical model describes the gravitational ﬁeld coupled with a massless scalar ﬁeld.

Its Lagrangian is given as [28]

L=1

2gµν ∂µφ(x)∂νφ(x)−ξRφ(x)2,(1)

where gµν is the metric tensor, φ(x)is the massless scalar ﬁeld, Ris the Ricci scalar and ξis the

coupling constant. The main objective of this work is to investigate this model in the Gödel-type

universe at ﬁnite temperature. To develop such a study, the energy-momentum tensor associated

with Eq. (1) must be calculated. Using the deﬁnition,

Tµν =−2

√−g

δL

δgµν ,(2)

the energy-momentum tensor is given as

Tµν (x) = 1

2gµν ∂ρφ(x)∂ρφ(x)−∂µφ(x)∂νφ(x) + ξRµν −1

2gµν R+gµν 2−∂µ∂νφ(x)2,(3)

with Rµν being the Ricci tensor and 2=gµν ∂µ∂νis the d’Alembertian operator.

It is important to observe that, due to the product of two ﬁelds at the same space-time point, the

energy-momentum tensor becomes a divergent quantity. In order to obtain a ﬁnite quantity, this

tensor is written at diﬀerent points in space-time. This is a well-known technique used in quantum

ﬁeld theory, for examples see [29–31]. Then

Tµν (x) = lim

x0→xn∆µν τφ(x)φ(x0)−Σµν δ(x−x0)o,(4)

where τis the time ordering operator and

∆µν =1

2gµν ∂ρ∂0

ρ−∂µ∂0

ν+ξRµν −1

2gµν R+gµν 2−∂µ∂0

ν(5)

Σµν =−i

2gµν nρ

0n0ρ+in0µn0ν.(6)

Here nµ

0= (1,0,0,0) is a time-like vector and the canonical quantization for the scalar ﬁeld, i.e.

[φ(x), ∂0µφ(x0)] = inµ

0δ(~x −~

x0), has been used.

To make the proposed applications, the vacuum expectation value of the energy-momentum

tensor is calculated, i.e.

hTµν (x)i= lim

x0→xni∆µν G0(x−x0)−Σµν δ(x−x0)o,(7)

where the deﬁnition of the massless scalar ﬁeld propagator

0

τ[φ(x)φ(x0)]

0=iG0(x−x0)(8)

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ThermalCasimireectinGödel-typeuniversesA.F.Santos1,andFaqirC.Khannab2,z1InstitutodeFísica,UniversidadeFederaldeMatoGrosso,78060-900,Cuiabá,MatoGrosso,Brazil2DepartmentofPhysicsandAstronomy,UniversityofVictoria,3800FinnertyRoad,VictoriaBCV8P5C2,CanadaInthispaper,amasslessscalareldcoupledtogravityi...

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Thermal Casimir eﬀect in Gödel-type universes A. F. Santos 1and Faqir C. Khanna

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