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

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 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 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 , ∇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 . 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 . (1 − 3gz)5/3 − (1 − 3gz) 1/3 ∂t2 ∂x2 We have not obtained a general analytical solution for 2 2 the above equation but in the following sections we present − ∂ φ ∂ φ − (1 − 3gz) 1/3 − (1 − 3gz) solutions for two particular situations. The first one is the ∂y2 ∂z2 case of a massless scalar field propagating in the +z direc- ∂φ + 3g + m2(1 − 3gz)φ = 0. (8) tion, while the second one is a solution for small z¯, i.e., when ∂z the field is propagating in a region near the massive plate Notice that in obtaining the above equation, we have used and far away from the singular plane. the fact that R = 0forthemetric(4) since it was derived for an empty space, i.e., T μν = 0. Obviously, due to the pres- ence of the plane, the energy-momentum tensor could not 3 Analytic Solution for Zero Mass and vanish at z = 0 (for a detailed examination of the solution Perpendicular Propagation of Einstein’s equations at the plane, we refer the reader to the Ref. [13]). Considering the (13) for the particular case of zero mass It is important to notice that (8) is linear for the field φ, m = 0 and choosing the field propagation in the perpendic- so any linear combination of independent solutions for φ is ular direction to the plate, i.e., kxy = 0 with the functions X also a solution for this equation. and Y constants, we have Supposing the field as a separable function 2 d Z 1 dZ  = −iωt − + ω 2(1 −¯z)2/3Z = 0, (14) φ(t,x) X(x)Y(y)Z(z)e , (9) dz¯2 (1 −¯z) dz¯ we have which has the following general solution in terms of the zero 2 2 order Bessel and Neumann functions 1 d X + 1 d Y X dx2 Y dy2 Z(u) = AJ0(u) + BN0(u), (15) 1 d2Z 1 dZ + (1 − 3gz)4/3 − (1 − 3gz)1/33g with u = 3 ω(z¯ − 1)4/3. The function J (u) hasalwaysa Z dz2 Z dz 4 0 finite value, for every ω, in the region between the massive − 2 − 4/3 + − 2 2 = m (1 3gz) (1 3gz) ω 0, (10) plate and the singular plane. However, the Neumann func- where ω is the angular frequency of the field measured at tion N0(u) decreases abruptly near the singular plane and z = 0. goes to −∞ as z¯ → 1. Notice that the singularity appear- For the two first terms, we have ing in the function N0 does not mean that we should discard 1 d2X 1 d2Y this solution, but this is an expected behaviour of the scalar =−k2, =−k2, (11) field at z¯ = 1 since the spacetime is singular at that plane as X dx2 x Y dy2 y it is evidenced by the evaluation of the Kretschmann scalar and we are left with the following ordinary homogeneous as mentioned previously. Therefore, the propagation of the differential equation for Z(z), scalar field turns out to be a useful tool to identify some 2 properties of the spacetime like the presence of singularities. d Z − 3g dZ 2 − A natural question the reader can raise relies on the fact dz (1 3gz) dz  2 that a singular solution for φ could induce non-negligible kxy + (1 − 3gz)2/3ω2 − − m2 Z = 0, modifications in the metric and, so, produce backreaction − 4/3 (1 3gz) effects. This problem can be solved in two alternative ways. (12) The first one is simply to discard the divergent solutions Braz J Phys (2016) 46:784–792 787 in (15), by setting B = 0. The second approach is more of the Bessel functions of the first and second kind for large subtle and consists in adjusting the initial conditions of the real arguments [14] scalar field in such a way that its contribution to the metric   is negligible in comparison with the ones produced by the 2 π 2 π J0(u) ∼ cos(u − ), N0(u) ∼ sin(u − ), massive plane. Let us start by the second approach. πu 4 πu 4 The constants A and B can be found by imposing initial (21) conditions for the function φ and its first derivative at the we find massive plate, located at z = 0. Hence, let us consider for  + instance, φ(t,z) φ0(1 4gz)cos ωz cos ωt. (22)   The position of the nodes for this asymptotic limit ∂φ φ(z = 0,t = 0) = φ0,  = 0. (16) depends on ω andaregivenby ∂z z=0,t=0 π(1 + 2n) z = ,n= 0, 1, 2, ... (23) By using conditions (16) as well as the Wronskian for n 2ω Bessel functions of (15), we obtain By taking z¯ → 0, we note that the above expression is  3πω  a solution of the usual wave equation in the flat space; this A =−φ N (3ω /4) (17) ω 0 8 1 is in accordance with the fact that the metric we are using reduces to the Minkowski metric at the massive plane. Also, and for the same limit, it is possible to define the z component  3πω  of the wave vector by the following eigenvalue equation Bω = φ0 J1(3ω /4). (18)   8 2 ∂ φ(t,z) =− 2 lim kz φ(t,z), (24) The final real solution for the scalar field is z→0 ∂z2  = 3πω  from which we identify kz ω and, by restoring physical φ(t,z) = φ cos ωt[J (3ω /4)N (u) = 0 8 1 0 units, we find the phase velocity at the plate ω/kz c.  How do we physically interpret the obtained solution? −N1(3ω /4)J0(u)]. (19) Our stationary solution suggests that we have a combination The constant φ0 must be small enough in such a way of two waves, one travelling to the right and the other travel- that any possible metric effects induced by the scalar field ling to the left after being reflected by the singular plane at are negligible in comparison with the ones produced by the z+. For the asymptotic solution (22), we can write explicitly massive plane. In this way, we shall not have backreaction φ(t,z) = φ−(z − t) + φ+(z + t), (25) effects. A similar result can be found for the case ω = 0and where kxy = 0, for which the solution is given in terms of modified φ0 π φ−(z − t) = (1 + 4gz) sin[ω(z − t) + ], (26) Bessel functions, as follows, 2 2 and φ(t,z) = cos ωt[C1I0(v) + C2K0(v)], (20) φ0 π φ+(z + t) = (1 + 4gz) sin[ω(z + t) + ]. (27) =  ¯ − 1/3 2 where we defined v 3kxy(z 1) and C1 and C2 are 2 constants. Also, for this case, there is a singularity in the This is a proof of the interpretation given in Ref. [12]by plane z¯ = 1. Amundsen and Grøn, which comes from the analysis of the In the Fig. 1, we show the spatial behaviour of the scalar geodesic equation of particles in the spacetime described by field φ of the solution (19)att = 0 for three values of the metric (4). According to those authors, only massless frequency. Note that the solution (19) represents a stationary particles, like photons, can reach the singular plane at z+ pattern for the scalar field. A stationary wave solution can and then they fall back. With the geodesic equation it is pos-  be particularly noticed for high frequency waves ω 1in sible to show that photons reach z+ but it is not possible to the vicinity of the massive plate z¯ 1 (it is worth to notice show that they are reflected, this was a matter of interpre-  that ω 1 implies ω c/z+,wherez+ = 1/3g;that tation at that time. The same happens for massive particles, means that the period of the wave as measured by a clock but in this case there is a maximum height zM for which located at z = 0 is much shorter than the time it takes light zM

Fig. 1 Behaviour of the scalar field φ at the time t = 0 between themassiveplateatz¯ = 0and the singular plane at z¯ = 1for three values of frequency

Also, in analysing the spacetime structure our solution rein- the metric field induced by the scalar field are negligible in forces the interpretation that the physical universe outside comparison with the contribution produced by the massive the massive plate has a finite extension in the z coordinate. plane. A quantity of particular interest is the scalar field energy In Fig. 2 we can see the spatial distribution of the scalar distribution in the region between z¯ = 0 and 1. This quantity field energy density between the massive plate and the can be evaluated from the 00 component of the energy- naked planar singularity. Near z¯ = 0, the function ρ(z)¯ momentum tensor of the scalar field which, in its general has a smooth increase, but it increases very fast when it form for curved spacetimes, is [15] approaches the singularity and goes to infinity for z¯ = 1. Remember that we have assumed that the energy density of the scalar field was small enough to affect the space- T = (1 − 2ξ)∇ φ∇ φ + (2ξ − 1/2)g ∇ φ∇λφ μν μ ν μν λ time geometry. As we can see, although that could be true 1 −2ξφ∇ ∇ φ + 2ξφφg + ξG φ2 − g m2φ2. (28) near the massive plate, the energy density gained by the μ ν μν μν 2 μν scalar field from the prominent curvature near z¯ = 1 makes ρ(z)¯ arbitrarily large in such a way that T could not be This expression is greatly simplified if we remember that μν disregarded in order to determine the spacetime metric. A for the case we are studying G = 0, m = 0 and therefore μν new metric that results from the solution of the Einstein’s φ = 0. Also, for simplicity, we will perform the calcula- field equations with the T givenby(29) as a source tions only for the minimally coupled case ξ = 0 (note that μν term was first obtained by Singh [8] and a generalization although the coupling constant ξ does not appear in the field with cosmological constant was recently obtained by Vuille equations for φ since R = 0, in general it affects the com- [9]. It is interesting to notice that the Singh solution con- ponents of the energy-momentum tensor). With all these tains the same singularity that appears in the metric we are considerations, the energy-momentum tensor reduces to considering; therefore, there are no qualitative changes in 1 λ the spacetime structure when a non-null energy-momentum Tμν =∇μφ∇νφ − gμν∇λφ∇ φ. (29) 2 tensor for the scalar field is assumed. Let us define the averaged energy density As mentioned previously, one could discard the divergent solution by setting B = 0in(15). In this case, the solution ρ = T  , (30) 00 for (14) reads where the brackets denote time average over one period. Z(u) = AJ (u) . (32) After performing the calculations we obtain 0 2 The constant A is determined by the initial conditions. ω 2 ρω (z) = [Aω J0(u) + Bω N0(u)] As an example, let us use the ones of (16). In this case, we 4 have that A = φ0 and the possible values for the frequency +[  +  ]2 Aω J1(u) Bω N1(u) . (31) ω become discrete, given by

It is important o enphasize that the constants Aω and Bω  4 ω = α , (33) must be small enough in such a way that the contribution to 3 1,n Braz J Phys (2016) 46:784–792 789

Fig. 2 Distribution of the energy density ρ(z)¯ in the region between the massive plate and the singular plane

where α1,n stands for the roots of the Bessel function J1, where C1 and C2 are constants determined by the initial namely, J1(α1,n) = 0. conditions and we defined the constant The corresponding averaged energy density (30)is 2  m 1      ω 2 a = − 16k 4 + (3 + 16ω 2)k 2 + 4ω 4 + 15ω 2 ,  = 2 2 + 2 xy xy ρω (z) Aω (J0(u)) (J1(u)) . (34) 2 18 4 (37) ∼ In Fig. 3, we have a plot for ρ given in (34) with α1,2 = ∼ ∼ 7.01559, α1,5 = 16.4706 and α1,10 = 32.1897. We can see and the function that on the singular plane, z¯ = 1, the energy density (3)is ¯ + 2 + 2 + 2 always non-divergent. For higher zeros α1,n,wehavemore (3z 4ω 8kxy 3) R(z)¯ = . (38) oscillations for the energy density, which exhibits its higher 18 values in the vicinity of the singular plane. Equation (35) is valid only for small values of z¯,sowe must consider the solution (36)forz<<¯ 1. For this task, 4 Solution in the Vicinity of the Massive Plate we use the fact that in the first order in z¯,wehave   In this section, we consider the scalar field propagation in M(a,1/2; R(z))¯ =∼ M a,1/2,α2/18 the vicinity of the massive plane and far away from the sin-   2 gular plane. In this situation, we must have z<<¯ 1, so we + aα M a + 1, 3/2,α2/18 z,¯ (39) 3 can linearize (13)inz¯ as follows,

2 d Z dZ 2 2 2 − (1 +¯z) + ω − k − m ∼ 2 ¯2 ¯ xy U(a,1/2; R(z))¯ = U(a,1/2,α /18) dz  dz     √ + 2 − 2 2 + 4 2 ¯ = 2 aαM a 1, 3/2,α /18 ω kxy z Z 0. (35) +¯z π 3 3 3 (a + 1/2) √   It is important to mention that (35) is valid just in the 2 + 2 − 2 α (1 2a) + 3 5 α vicinity of the massive plane (far away from the singular M a , , 27 (a) 2 2 18 plane), and so, their solutions must be considered only in √   2 this region. − M a+1/2, 3/2,α2/18 ,(40) The solution for (35) is given in terms of the so-called (a) Kummer functions of first and second types, M(a,b; j)and U(a,b; j), respectively, as follows, where stands for the Euler Gamma function, and we

− defined the quantity 2 zω¯ 2− 4 zk¯ 2 Z = e 3 3 xy [C1M (a,1/2; R(z)¯ ) = 2 + 2 + +C2U (a,1/2; R(z)¯ )], (36) α 4ω 8kxy 3 . (41) 790 Braz J Phys (2016) 46:784–792

Fig. 3 Distribution of the energy density ρ(z)¯ in the region between the massive plate and the singular plane for non-singular solutions. We have ∼ ∼ ∼ used α1,2 = 7.01559 (dashed line), α1,5 = 16.4706 (grey line)andα1,10 = 32.1897 (black line)

As we said, the constants C1 and C2 are determined by and the initial conditions of the propagating field φ(x).Since   2 2 we are interested only in the behaviour of the field, we shall 1(ω, kxy) = C1 aα M a + 1, 3/2,α /18 not impose specific boundary conditions for the scalar fields 3    and let the solution in terms of these constants √ 2 aαM a + 1, 3/2,α2/18 +C π Using (39)and(40)in(36), we can write expression (36) 2 3 (a + 1/2) in first order in z¯ as follows, √   2 α2(1 + 2a) − M a + 3/2, 5/2,α2/18 27 (a) −2 ¯ 2− 4 ¯ 2    ∼ zω zkxy √   Z(z) = e 3 3 0(ω, kxy) + 1(ω, kxy)z¯ 1 − 2 M a + 1/2, 3/2,α2/18 , (a) ∼ − 2 + 2 ¯ [ (44) = 1 (2ω /3 4kxy/3)z 0(ω, kxy) + (ω, k )z¯] 1 xy which do not depend on z¯. With the (43), one can write the propagating scalar field ∼ = 0(ω, kxy) +[1(ω, kxy) − 2 + 2 ]¯ (2ω /3 4kxy/3)0(ω, kxy) z (42)  −i(ωt+kx x+ky y) φ = Re e 0(ω, kxy)   + − 2 + 2 ¯ where we expanded the exponential and defined the func- 1(ω, kxy) (2ω /3 4kxy/3)0(ω, kxy) z .(45) tions Substituting the solution (45) in the 00 component of the   2 2 tensor (29)wehave 0(ω, kxy) = C1M a,1/2,α /18 +C2U(a,1/2,α /18),

(43) T00(t,x,y,z)¯ = [F(t,x,y) + G(t,x,y)z¯] , (46) Braz J Phys (2016) 46:784–792 791 where we defined the functions find either an analytic or a numerical solution for the mas-

1 sive scalar field propagation. For this case, it is expected that F(t,x,y) = (ω2 + k2 + k2)2(ω, k ) 2 x y 0 xy the scalar field would not reach the singular plane z+,butit 2 2 would be reflected in a maximum position zM

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