Non-Abelian Gravitoelectromagnetism and Applications at Finite Temperature

Hindawi Advances in High Energy Physics Volume 2020, Article ID 5193692, 8 pages https://doi.org/10.1155/2020/5193692 Research Article Non-Abelian Gravitoelectromagnetism and Applications at Finite Temperature A. F. Santos ,1 J. Ramos,2 and Faqir C. Khanna3 1Instituto de Física, Universidade Federal de Mato Grosso, 78060-900, Cuiabá, Mato Grosso, Brazil 2Faculty of Science, Burman University, Lacombe, Alberta, Canada T4L 2E5 3Department of Physics and Astronomy, University of Victoria, BC, Canada V8P 5C2 Correspondence should be addressed to A. F. Santos; alesandroferreira@fisica.ufmt.br Received 22 January 2020; Revised 9 March 2020; Accepted 18 March 2020; Published 3 April 2020 Academic Editor: Michele Arzano Copyright © 2020 A. F. Santos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3. Studies about a formal analogy between the gravitational and the electromagnetic fields lead to the notion of Gravitoelectromagnetism (GEM) to describe gravitation. In fact, the GEM equations correspond to the weak-field approximation of the gravitation field. Here, a non-abelian extension of the GEM theory is considered. Using the Thermo Field Dynamics (TFD) formalism to introduce temperature effects, some interesting physical phenomena are investigated. The non-abelian GEM Stefan- Boltzmann law and the Casimir effect at zero and finite temperatures for this non-abelian field are calculated. 1. Introduction between equations for the Newton and Coulomb laws and the interest has increased with the discovery of the The Standard Model (SM) is a non-abelian gauge theory with Lense-Thirring effect, where a rotating mass generates a symmetry group Uð1Þ × SUð2Þ × SUð3Þ. SM describes the- gravitomagnetic field [15–17]. Some experiments that oretically and experimentally three of the four fundamental study this effect have been developed, such as LAGEOS forces of nature, i.e., the electromagnetic, weak, and strong (Laser Geodynamics Satellites) and LAGEOS 2 [18], the forces. The electromagnetism is a Uð1Þ abelian gauge theory Gravity Probe B [19], and the mission LARES (Laser Relativity which has been tested to a high precision. The generalization Satellite) [20, 21]. ff of an abelian gauge theory to the non-abelian gauge theory The GEM theory may be analyzed by three di erent was proposed by Yang and Mills [1]. The last one approaches: (i) using the similarity between the linearized describes the electroweak unification and quantum chro- Einstein and Maxwell equations [22], (ii) a theory based on modynamics. The electroweak interaction is described by an an approach using tidal tensors [23], and (iii) the decompo- ℬ ∈ kl SUð2Þ × Uð1Þ group and while the SUð3Þ group satisfies sition of the Weyl tensor (Cijkl) into ij = 1/2 iklC0j and ℰ − the quantum chromodynamics [2–4]. ij = C0i0j, the gravitomagnetic and gravitoelectric compo- Gravity is not a part of SM. This implies that the SM is nents, respectively [24]. In this paper, the Weyl tensor not a fundamental theory that describes all fundamental approach is used. A Lagrangian formulation for GEM is interactions of nature. In this paper, an extension of non- developed [25], and a gauge transformation in GEM is stud- abelian gravity is discussed. Some applications of such a the- ied [26]. Here, an extension to non-abelian GEM fields is ory are developed. The gravitational theory studied here is introduced. Applications of the non-abelian GEM at finite the Gravitoelectromagnetism (GEM). GEM is an approach temperature are investigated. The temperature effects are based on describing gravity in a way analogous to the electro- introduced using Thermo Field Dynamics (TFD) formalism. magnetism [5–7]. Several studies about the GEM theory have There are two ways to introduce the temperature effect: been developed [8–14]. These ideas arise from the analogy- (i) using the imaginary time formalism [27] and (ii) using 2 Advances in High Energy Physics the real-time formalism [28–36]. In this paper, TFD formal- ℬ = curl A~ , ism is chosen. It is a real-time finite temperature formalism. 1 ∂A~ ð2Þ In this formalism, a thermal state is developed where the ℰ + = − grad φ, main objective is to interpret the statistical average of an c ∂t arbitrary operator as an expectation value in a thermal vacuum. Two elements are necessary to construct this thermal where φ is the GEM counterpart of the electromagnetic state: (i) doubling of the original Hilbert space and (ii) the (EM) scalar potential ϕ. μνα use of Bogoliubov transformations. These are two Hilbert Defining ℱ as the gravitoelectromagnetic tensor, the fi spaces, the original space S and the tilde space ~S, which are GEM eld equations become related by a mapping, called the tilde conjugation rules, π while the Bogoliubov transformation consists in a rotation μνα 4 G να ∂μℱ = J , involving these two spaces that ultimately introduce the c ð3Þ temperature effects. μναhi ∂μG =0, The Stefan-Boltzmann law and the Casimir effect for the fi fi non-abelian GEM eld at nite temperature are calculated. να The Stefan-Boltzmann law describes the power radiated from where J depends on quantities ρi and Jij that are the mass a black body in terms of its temperature. The Casimir effect, and the current density, respectively. In addition, the gravi- proposed by H. Casimir [37], is a quantum phenomenon that toelectromagnetic tensor is defined as appears due to vacuum fluctuations of any quantum field. μνα μ να ν μα The results in this case may be at zero or finite temperatures. ℱ = ∂ A − ∂ A , ð4Þ This paper is organised as follows. In section II, a brief introduction to the abelian GEM Lagrangian formalism is and the dual GEM tensor is defined as presented. In section III, an extension to non-abelian GEM fi eld is developed. The energy-momentum tensor associated μνα 1 μνγσ αρ fi G = ∈ η − ℱ γσρ: ð5Þ to the non-abelian gauge eld is calculated. In section IV, 2 the TFD formalism is introduced. In section V, some applica- fi fi tions considering the non-abelian GEM eld at nite temper- Using these definitions, the GEM Lagrangian density is ature are analysed. (i) The Stefan-Boltzmann law is calculated. given as [25]. (ii) The Casimir effect at zero temperature is obtained, and (iii) the Casimir effect at finite temperature is calculated. In 1 μνα G να section VI, some concluding remarks are presented. ℒ = − ℱ μναℱ − J A να: ð6Þ G 16π c 2. Lagrangian Formulation of Abelian Gem This Lagrangian allows considering several gravitational In this section, an introduction to the Lagrangian formula- applications involving the graviton, such as interactions with tion of abelian GEM is presented. The GEM field equations, other fundamental particles. This makes it possible to study Maxwell-like equations, are several related topics. In this way, the GEM theory is described by two fields ℰij and ℬij, which are symmetric and traceless tensors of the ∂iℰij = −4πGρj, second order. These fields can be expressed in terms of the symmetric gravitoelectromagnetic potential Aμν [25, 26], ∂iℬij =0, analogous to that of electromagnetism Aμ. Thus, Aμν is the ∂ℰij π k 1 4 G ð1Þ fi ∈hikl∂ ℬlji + = − Jij, fundamental eld in GEM and naturally, it has two indices c ∂t c [25, 26]. 1 ∂ℬij It is important to note that GEM equations correspond to ∈hikl∂kℰlji + =0, the weak-field approximation of General Relativity. They do c ∂t not describe strong fields and, therefore, do not include the full Einstein equations. To be more specific, the abelian where G is the gravitational constant, εikl is the Levi-Civita GEM corresponds to the linear part of Einstein equations symbol, ρj is the vector mass density, Jij is the mass current and the non-abelian GEM corresponds up to the second order in the weak-field approach. density, and c is the speed of light. The quantities ℰij, ℬij, ij fi fi and J are the gravitoelectric eld, the gravitomagnetic eld, 3. Non-Abelian Gem and the mass current density, respectively. The symbol h⋯i fi denotes symmetrization of the rst and last indices, i.e., i Let us consider an extension of the GEM field to include the and j. non-abelian gauge transformations [38]. Then, in this sec- The fields ℰij and ℬij are expressed in terms of a tion, the Lagrangian for the non-abelian GEM field is pre- symmetric rank-2 tensor field, A~ , with components A ij, sented and the energy-momentum tensor associated to the such that non-abelian field is calculated. Advances in High Energy Physics 3 In order to obtain the non-abelian gauge transformation oneform pα is introduced [26]. The one form makes the for the GEM field, let us investigate the Dirac Lagrangian phase function to split into phase factors each associated with under global and local gauge transformations. The free Dirac one of the four directions in spacetime. μ Lagrangian is given as Using these results and replacing the derivative ∂ by the covariant derivative Dμ, the Dirac Lagrangian is gauge ℒ − ψ γ ∂μψ ψ ψ D = i ðÞx μ ðÞx + m ðÞx ðÞx , ð7Þ invariant, i.e., ℒ − ψ γ μψ ψ ψ : where ψðxÞ is a two-component column vector. This D = i ðÞx μD ðÞx + m ðÞx ðÞx ð15Þ Lagrangian is invariant under global SUð2Þ gauge transformation given as In this formulation, three new gauge tensor fields are introduced. To write a full Lagrangian invariant under ψ′ðÞx = UψðÞx , ð8Þ local gauge transformation, a kinetic term of AμαðxÞ must be constructed. To do that, an analogue of the electromag- iH with U being a 2×2unitary matrix that is written as U = e , netic tensor Fμν is constructed.

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