Vacuum polarization in the field of a multidimensional global monopole
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, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Vacuum Polarization in the Field of a Multidimensional Global Monopole Yu. V. Grats* and P. A. Spirin Faculty of Physics, Moscow State University, Moscow, 119991 Russia *e-mail: [email protected] Received May 15, 2016
Abstract—An approximate expression for the Euclidean Green function of a massless scalar field in the spacetime of a multidimensional global monopole has been derived. Expressions for the vacuum expectation values 〈φ2〉ren and 〈T00〉ren have been derived by the dimensional regularization method. Comparison with the results obtained by alternative regularization methods is made. DOI: 10.1134/S1063776116110091
1. INTRODUCTION In recent years, multidimensional generalizations of known solutions of Einstein’s gravity theory have become the object of intensive research in connection with the theories under active development in a spacetime with more than four dimensions. Although the existence of extra dimensions has not been experimentally confirmed to date (for the results of the latest experiments, see [1]), modern theories have stimulated the investigation of general relativity (GR) in spacetimes with dimensionality d > 4. One of the goals of such an investigation is to ascertain which of the GR predictions are unique to the four-dimensional (4D) case and which are universal and extend to higher dimensions. In this paper we consider the vacuum polarization effect in a spacetime that is a multidimensional generalization of the solution of the 4D theory known as the spacetime of a point global monopole. A global monopole is one of the types of topological defects that, having formed during the phase transitions in the early Universe, could “survive” until now [2, 3]. The spacetime of a point monopole is an ultrastatic space whose metric is generally written as
ds 2 = −dt 2 + d 2 + β 2 2(d θ 2 + sin 2 θ d ϕ 2 ),
(1)
where 0 < β < 1. This metric corresponds to a spherically symmetric space with a deficit solid angle δΩ = 4π(1 – β2). Any surface passing through the coordinate origin and dividing the spacetime (1) into two symmetric parts is a cone with a deficit angle Δ = 2π(1 – β). This makes the monopole metric similar to the metric of a gauge cosmic string. A significant dis-
tinction is that, in contrast to the string spacetime, the spacetime (1) is not locally flat. The simplest model that predicts the existence of global monopoles is a triplet of scalar fields with the Lagrangian
+ = 1 (∂ μφ a )(∂ μφ a ) − λ (φ aφ a − η2 )2, a = 1,3, 2 4 whose global O(3) symmetry is spontaneously broken to U(1). Strictly speaking, the metric of the self-consistent spherically symmetric solution of Einstein’s equations and the equations of motion of the scalar triplet contains the mass term [4, 5]. However, as has been shown, the mass term is too small to play a prominent role in astrophysical-scale phenomena, and it is neglected by writing the metric in form (1). In this case, the gravitational field of the monopole is entirely determined by the deficit solid a
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