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Propagation of Scalar Fields in a Plane Symmetric Spacetime

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Propagation of Scalar Fields in a Plane Symmetric Spacetime Braz J Phys (2016) 46:784–792 DOI 10.1007/s13538-016-0458-8 PARTICLES AND FIELDS Propagation of Scalar Fields in a Plane Symmetric Spacetime Juliana Celestino1 · Marcio´ E. S. Alves2 · F. A. Barone 3 Received: 6 June 2016 / Published online: 3 October 2016 © Sociedade Brasileira de F´ısica 2016 Abstract The present article deals with solutions for a min- for the scalar field in terms of the Kummer and Tricomi imally coupled scalar field propagating in a static plane confluent hypergeometric functions. symmetric spacetime. The considered metric describes the curvature outside a massive infinity plate and exhibits an Keywords Gravitational waves · Taub metric · Scalar field intrinsic naked singularity (a singular plane) that makes the accessible universe finite in extension. This solution can be interpreted as describing the spacetime of static domain 1 Introduction walls. In this context, a first solution is given in terms of zero order Bessel functions of the first and second kind and A variety of theories of fundamental physics predict the presents a stationary pattern which is interpreted as a result existence of scalar fields [1]. In cosmology, scalar fields are of the reflection of the scalar waves at the singular plane. central to the implementation of the inflationary models [2, This is an evidence, at least for the massless scalar field, of 3], and also in the development of alternatives to solve the an old interpretation given by Amundsen and Grøn regard- puzzle of the present accelerated cosmic expansion [4, 5]. ing the behaviour of test particles near the singularity. A All these subjects together with the recent discovery of the second solution is obtained in the limit of a weak gravita- Higgs boson have been motivated a better understanding of tional field which is valid only far from the singularity. In the dynamical effects due to the coupling of scalar fields this limit, it was possible to find out an analytic solution with gravity. One of the most simple gravitational fields that we could conceive in Newtonian grounds is the homogeneous gravi- tational field produced by an infinite plate with superficial uniform mass density σ. The corresponding Newtonian Juliana Celestino potential is [email protected] ϕ(z) = 2πGσ|z|, (1) Marcio´ E.S. Alves [email protected] where we have supposed that the plate is located at the plane F.A. Barone z = 0. [email protected] The General Relativity version of this problem is the plane-symmetric solutions of Einstein’s field equations. In 1 Departmnt of Physics, Rio de Janeiro State University, R. Sao˜ Francisco Xavier, 524 - Maracana,˜ Rio de Janeiro, RJ, 1951, Taub found the general vacuum solution [6]. A gen- 20550-900, Brazil eralization to a spacetime with cosmological constant was found by Novotny´ and Horsky[´ 7], and the solution for 2 Instituto de Cienciaˆ e Tecnologia, UNESP - Univ Estadual a plane-symmetric scalar field was found by Singh [8]. Paulista, Sao˜ Jose´ dos Campos, SP, 12247-004, Brazil Finally, a generalization of the Singh’s solution with the 3 IFQ, University Federal of Itajuba,´ Av. BPS 1303, presence of a massless plane symmetric scalar field and cep: 37500-903, Itajuba,´ Brazil a cosmological constant was recently found by Vuille. [9] Braz J Phys (2016) 46:784–792 785 μν One particular solution which leads to the above poten- where = g ∇μ∇ν is the generalized D’Alembertian tial in the Newtonian limit is the Rohrlich[10] solution that operator in curved spacetimes, ∇ν is the covariant deri- actually describes flat spacetime in a special coordinate sys- vative, R is the curvature scalar, and ξ is the coupling tem. On the other hand, in this article, we will consider the constant which specifies the “strength” of the interaction metric first found by Horsky[´ 11] which represents a truly between φ and the gravitational field. There are two val- curved spacetime. It is a particular solution of Einstein’s ues of ξ that are of particular interest, namely, 0 and 1/6. If equations of General Relativity with vanishing cosmolog- ξ = 0, we say that the field φ is minimally coupled to grav- ical constant and vanishing surface charge density. For a ity since the interaction occurs only through the modified detailed revision of general plane symmetric solutions of D’Alembertian. If ξ = 1/6, the scalar field is conformally Einstein-Maxwell equations including discussions about the coupled to gravity, i.e., the (2) is invariant under a conformal particular results, we refer the interested reader to the Ref. transformation of the metric. [12]. These solutions can be interpreted as describing the It will be useful to consider the following identity spacetime of domain walls. 1 √ gμν∇ ∇ φ = √ ∂ ( −g gμν∂ φ), (3) Following a different approach, instead of searching for μ ν −g μ ν exact solutions of the Einstein’s field equations, in this arti- = cle, we consider the solutions of the equations of motion of where g det(gμν) is the determinant of the metric gμν. “weak” scalar fields propagating in a plane symmetric back- In this article we consider the following metric [11]: − ground spacetime that describes the curvature outside an ds2 = (1 − 3gz) 2/3dt2 infinity massive plate located at z = 0. We shall restrict to −(1 − 3gz)4/3(dx2 + dy2) − dz2, (4) situations where the scalar field is said “weak” in the sense that the associated energy density and pressure of the field where g is a constant. This metric describes a plane symmet- are weak enough to produce significant changes in the back- ric spacetime for z ≥ 0 out of an infinity plate with uniform ground curvature. As pointed out by Amundsen and Grøn, mass density σ located at z = 0. [12] this spacetime has the peculiarity of having an intrinsic The temporal component of the metric is related to the naked singularity (a singular plane) in a finite distance from potential (1) in the Newtonian limit by the massive plate that makes the accessible universe finite in 2ϕ g 1 + . (5) extension. If this is true then a massless scalar field would 00 c2 be reflected or destroyed by the singular plane, i.e., it is not Therefore, the connection between the constant g present possible to escape from the gravity generated by the plate. in the metric (4)andσ can be found by Taylor expanding In this context, the first solution we find is a particular solu- the 00 component of the metric (4) to first order in gz tion of the equations of motion of a massless scalar field propagating perpendicular to the plate. The behaviour of the g00 1 + 2gz, (6) field in the vicinity of the massive plate and near the singu- and comparing (6)and(5) together with (1)iswhatleadsto lar plane are then analysed. In a second approach, we aim 2πG to work out the solution of the equations of motion without g = σ, (7) restricting the direction of propagation of the scalar field. It c2 is shown that this can be analytically achieved by focusing where we identify gc2 = 2πGσ as the Newtonian acceler- in the limiting case of weak gravitational field in the vicin- ation of gravity in the vicinity of the plate. ity of the massive plate. In the last section, we present our The fact that the spacetime described by the above metric conclusions and final remarks. is truly curved, can be seen by calculating the Kretschmann Along the paper, we use a method commonly employed curvature invariant (see, e.g., Ref. [12]) which is not null. If in Quantum Field theory and in Optics. In this approach, it were null, the metric would not represent a curvature, but non-linear equations are linearized for weak propagating just the Minkowski spacetime described in a particular coor- fields. dinate system. Furthermore, another important information can be obtained from this invariant; since the Kretschmann scalar diverges for z+ = 1/3g, we are led to conclude that 2 General Equations for the Propagation of Scalar there is a singular plane at z+ that, according to the physical Fields in a Planar Symmetric Spacetime interpretation given by Ref. [12], makes the universe finite in extension. In other words, a particle that resides above the The equation for a scalar field of mass m coupled to gravity massive plate is confined in the region z ∈[0,z+]. Notice is also that in the plate at z = 0,themetric(4) reduces to the Minkowski metric and, on the other hand, the curvature ( + m2 + ξR)φ = 0, (2) becomes more prominent as we approach z+. 786 Braz J Phys (2016) 46:784–792 At this point, it is important to have in mind the regime of = 2 + 2 where we interpret kxy kx ky as the wave vector validity of the calculations in this article. When we consider component which is parallel to the plate. the metric (4) as representing the spacetime where the scalar It is convenient to introduce a new variable z¯ = 3gz, field propagates, we are assuming that the resulting com- where z¯ ∈[0, 1], such that z¯ = 1 locates the singular plane. ponents of the energy-momentum tensor associated with φ For this variable, the (12)forZ becomes are small enough to affect the spacetime curvature. In other 2 words, we are assuming that “backreaction effects” of the d Z − 1 dZ ¯2 −¯ ¯ field φ in the curvature are negligible in comparison with the dz (1 z) dz 2 2 effects produced by the massive plane in all over the space. kxy m + ω 2 − − (1 −¯z)2/3Z = 0, Now, let us substitute the metric (4) and its determinant (1 −¯z)2 (1 −¯z)2/3 g =−(1 − 3gz)2 into the (2) in order to write the following second order partial differential equation (13) k 2 2 = ω = xy = m ∂ φ − ∂ φ where ω 3g , kxy 3g and m 3g .
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