Geometric properties of the Bertotti–Kasner space-time

Abstract

Purpose

The purpose of this paper is to study the Bertotti–Kasner space-time and its geometric properties.

Design/methodology/approach

This paper is based on the features of λ-tensor and the technique of six-dimensional formalism introduced by Pirani and followed by W. Borgiel, Z. Ahsan et al. and H.M. Manjunatha et al. This technique helps to describe both the geometric properties and the nature of the gravitational field of the space-times in the Segre characteristic.

Findings

The Gaussian curvature quantities specify the curvature of Bertotti–Kasner space-time. They are expressed in terms of invariants of the curvature tensor. The Petrov canonical form and the Weyl invariants have also been obtained.

Originality/value

The findings are revealed to be both physically and geometrically interesting for the description of the gravitational field of the cylindrical universe of Bertotti–Kasner type as far as the literature is concerned. Given the technique of six-dimensional formalism, the authors have defined the Weyl conformal λW-tensor and discussed the canonical form of the Weyl tensor and the Petrov scalars. To the best of the literature survey, this idea is found to be modern. The results deliver new insight into the geometry of the nonstatic cylindrical vacuum solution of Einstein's field equations.

Citation

Manjunatha, H.M., Narasimhamurthy, S.K. and Nekouee, Z. (2022), "Geometric properties of the Bertotti–Kasner space-time", Arab Journal of Mathematical Sciences, Vol. 28 No. 1, pp. 77-86. https://doi.org/10.1108/AJMS-10-2020-0085

1. Introduction

Einstein's relativity theory is one of the successful theories of space-time and gravity. The space-time geometry describes one of the fundamental interactions in nature, namely gravity. Einstein's theory of relativity successfully reveals that space becomes curved in the presence of the gravitational field. The matter distribution determines the geometry of space-time. Albert Einstein introduced the field equations in 1915. Einstein's field equation (EFE) is a remarkable contribution in determining the motion of matter in a gravitational field as well as in determining the gravitational field from the distribution of matter. Among well-known exact solutions of EFE, the Schwarzschild solution is the most important. It preserves spherical symmetry. In 2011, Włodzimierz Borgiel [1] investigated the Schwarzschild space-time and its gravitational field. In [2], Musavvir Ali and Zafar Ahsan have studied the Kerr–Newman solution, which is the generalization of other well-known exact solutions of Einstein–Maxwell equations. The metric of Kerr–Newman space-time goes over into the Kerr metric, Reissner–Nordström metric, and Schwarzschild metric if the electric charge, the angular momentum per unit mass and both of them, respectively, are equal to zero. It reduces to Minkowski metric if the physical parameters such as mass, the angular momentum per unit mass and the electric charge vanish. They have studied the Schwarzschild soliton and its geometric properties in [3].

The Schwarzschild-de Sitter (SdS) solution is the generalization of the Schwarzschild solution. It is the spherically symmetric vacuum solution of EFE with a non-vanishing cosmological constant. SdS solution is not the only possible generalization of the Schwarzschild solution. Another possible generalization is the Bertotti–Kasner solution [4].

The Bertotti–Kasner solution is the non-static cylindrical vacuum solution of EFE. The Bertotti–Kasner space-time metric in Schwarzschild coordinates (t, r, θ, φ) with relativistic units (G=c=1) is as follows: (see [4, 5])

(1)ds2=dt2−e2Λtdr2−1Λ dΩ2,

where dΩ2=dθ2+sin2θdφ2 and Λ>0 denotes the cosmological constant.

According to Bertotti [6], Kasner [7] introduced the existence of this solution in 1925, but the explanation was not clear in the problem of the signature. In [6], it is found that the Bertotti–Kasner solution exists in the absence of the electromagnetic field. In the 1960s, many geometers and physicists have studied the spherically symmetric vacuum solutions. The Bertotti–Kasner solution characterizes the geometry of our universe as cylindrical. It is distinct from the solution due to the field around a spherical distribution of mass. So, Bonnor [8] neglected the Bertotti–Kasner solution. However, it has been drawn the attention of many geometers and physicists in recent days. One can see the discussion of geodesics on Bertotti–Kasner space, and hyper-spherical Bertotti–Kasner space in [9] with the famous Kruskal–Szekeres procedure. The Bertotti–Kasner solution is constructed in [10] from multiplets of scalar fields with a self-interacting potential in 3+1−dimensions. A discussion on Killing's equations, Killing vectors, and time-like Killing vectors on Bertotti–Kasner space-time is found in [11].

The interesting feature of Bertotti–Kasner space-time metric is its mathematical simplicity and is purely geometric. It leads to the impression that our universe expands more in one particular direction. Some recent experimental evidence shows that our universe may have a particular direction, and in that direction the expansion velocity of the universe is maximum. In a galactic coordinate system, the experimental data [12] of Union2 type Ia supernova has given the evidence for the preferred direction, (l, b)=(309∘, 18∘) of the universe. The experimental report of the Planck Collaboration [13] has confirmed the deviations from the isotropy with a significance level ∼3σ and hence given the evidence for the preferred direction.

The article is organized as follows. In Section 2, we discuss the canonical form, and the curvature invariants based on the technique of six-dimensional formalism. Hence, we analyze the curvature of Bertotti–Kasner space-time. The description of gravitational field is given by the features of λ-tensor. A glimpse of Weyl conformal λW-tensor is also given in Section 2. The paper ends with Section 3, where we have mentioned some important conclusions.

2. Curvature of the Bertotti–Kasner space-time

We have considered that the matrix (gμν) has the signature (+, −, −, −). In the Schwarzschild coordinates x=(t, r, θ, φ), the matrix (gμν) is as follows:

(2)gμν(x)=(10−e2Λt−1Λ0−sin2θΛ),

where μ, ν=0, 1, 2, 3. The determinant of a matrix (gμν) is equal to −e2Λtsin2θΛ2, which is less than zero. Hence it is a real space-time [14]. The Riemannian metric tensor gμν determines the nature of gravitational field potential, and for the Bertotti–Kasner space-time metric, it is given by

Φ≈12(g00−1)=0.

We deduce that the gravitational field potential of Bertotti–Kasner space-time metric is approximately equal to zero. Christoffel symbols are the functions constructed by certain combinations of partial differential coefficients of the metric tensor gμν. Let Γβγα denote the Christoffel symbols of second kind defined by

(3)Γβγα=12 gαδ(∂γgβδ+∂βgδγ−∂δgγβ).

The independent non-vanishing components of Γβγα are as follows:

Γ011=Γ101=Λ, Γ110=Λe2Λt,Γ233=Γ323=cos θsin θ, Γ332=−cos θ sin θ.

Riemann and Christoffel introduced the tensor Rβγδα of type (1,3). It is formed by metric tensor gμν and its partial derivatives up to second order. The curvature tensor (or Riemann tensor) of type (0, 4) can be expressed as

(4)Rαβγδ=12(∂βγ2gαδ+∂αδ2gβγ−∂βδ2gαγ−∂αγ2gβδ)+gμν(ΓβγμΓαδν−ΓβδμΓαγv).

The independent nonzero components of Rαβγδ are

R1010(x)=Λe2Λt,R3232(x)=− sin2θΛ.

The Ricci tensor Rαβ is a covariant tensor of order 2 and is given by

Rαβ=Rαβγγ.

We found that Rαβ=Λgαβ. Let R denote the scalar curvature. It is a tensor of order zero given by

R=gαβRαβ.

The scalar curvature of Bertotti–Kasner space-time is 4Λ. Therefore, it has a constant scalar curvature. Since components of Ricci tensor are proportional to metric tensor components, scalar curvature is directly proportional to the cosmological constant, and hence Bertotti–Kasner space-time is an Einstein space.

The Kretschmann scalar is found to be

Rαβγδ Rαβγδ=8Λ2.

For Bertotti–Kasner space-time, Kretschmann scalar is also constant and is directly proportional to the square of the cosmological constant. The tensor Gαβ of type (0, 2) is defined by

Gαβ=Rαβ−12 gαβR

is called the Einstein tensor. It plays a supreme role in Einstein's relativity theory. We found that Gαβ=−Λgαβ. Hence EFE is given by

Gαβ+Λgαβ=0.

Given the anti-symmetric property

A[αβ]=12(Aαβ−Aβα),

the Weyl curvature tensor can be evaluated as follows:

(5)Cαβγδ=Rαβγδ+gγ[αRβ]δ−gδ[αRβ]γ−13 gγ[αgβ]δR.

The Weyl tensor is traceless, but it has the symmetric properties as Riemann tensor Rαβγδ. If we contract on the couple of indices αδ or βγ, the obtained tensor vanishes. The independent non-vanishing components of the Weyl tensor Cαβγδ are as follows:

C1010(x)=2Λe2Λt3, C2020(x)=−13, C3030(x)=−sin2θ3,C2121(x)=e2Λt3, C3131(x)=sin2θe2Λt3, C3232(x)=−2sin2θ3Λ.

We observed that at a point of Bertotti–Kasner space-time, some components of the Weyl tensor are non-vanishing. Hence Bertotti–Kasner space-time is not conformally flat.

Now, we switch onto the six-dimensional formalism to examine the bivector-tensors, the Riemann tensor and the Weyl tensor in a pseudo-Euclidean space ℝ6 [15]. Let us consider the following identification to adopt the six-dimensional formalism [16]:

(6)αβ:23 31 12 10 20 30

U: 1  2  3  4  5  6.

Given gαβ are the metric tensor components at a point of Bertotti–Kasner space-time, we define the bivector-tensor as follows:

gUV=gαβγδ=gαγgβδ−gαδgβγ.

The bivector-tensor gUV is non-singular, and has the signature (+, +, +, −, −, −). It satisfies the symmetric property, that is, gUV=gVU. The suffix pairs αβ, γδ are skew-symmetric. The bivector-tensor gUV has the following non-vanishing components:

g11(x)=sin2θΛ2, g22(x)=e2Λtsin2θΛ, g33(x)=e2ΛtΛ,g44(x)=−e2Λt, g55(x)=−1Λ, g66(x)=−sin2θΛ.

Now we relabel the Riemann tensor Rαβγδ as RUV given the scheme of six-dimensional formalism. Because of the property Rαβγδ=Rγδαβ, the tensor RUV satisfies the symmetric property, that is, RUV=RVU. The tensor RUV has the following nonvanishing components:

R11(x)=−sin2θΛ, R44(x)=Λe2Λt,R22(x)=R33(x)=R55(x)=R66(x)=0.

Given identification (6) of the six-dimensional formalism, we relabel the Weyl tensor Cαβγδ as CUV. Because of the property Cαβγδ=Cγδαβ, the tensor CUV is symmetric, that is, CUV=CVU. The tensor CUV has the following nonvanishing components:

C11(x)=−2sin2θ3Λ, C22(x)=sin2θ e2Λt3, C33(x)=e2Λt3,C44(x)=2Λe2Λt3, C55(x)=−13,    C66(x)=−sin2θ3.

The λ-tensor is defined as RUV−λgUV. The curvature invariants are roots of the characteristic equation |RUV(x)−λgUV(x)|=0, and for Bertotti–Kasner space-time, they are obtained as follows:

(7)λ1(r)=λ4(r)=−Λ,

(8)λ2(r)=λ3(r)=λ5(r)=λ6(r)=0.

The curvature tensor has the following canonical form:

RU′V′=(−Λ000Λ000).

Also, we have

gU′V′=(1011−1−10−1).

From the features of λ-tensor RUV−λgUV, we analyze the description of the gravitational field in the Segre symbols (see Ref. [17]), and we found that it is of the type G1[(1111)(11)].

Under the algebraic structure of the Riemann tensor, we conclude that the Bertotti–Kasner space-time (1) belongs to Type I in the Petrov's classification (see Ref. [18]). It is important to notice that the geometry of Bertotti–Kasner space-time is both flat and curved. It reduces smoothly into Minkowski space-time and hence will become flat as Λ→0. This is shown in Figure 1.

The Weyl conformal λW-tensor CUV−λWgUV is constructed from the symmetric tensors CUV and gUV. The roots of the characteristic equation |CUV(x)−λWgUV(x)|=0 are called Weyl invariants, Petrov invariants or Petrov scalars. For Bertotti–Kasner space-time, Weyl invariants are as follows:

(9)λW1(r)=λW4(r)=−2Λ3,

(10)λW2(r)=λW3(r)=λW5(r)=λW6(r)=Λ3.

The determinant of the Weyl conformal λW-matrix CUV(x)−λWgUV(x) is zero for any of the above Weyl invariants. The canonical form of the Weyl tensor or Petrov canonical form is given by

CU′V′=(−2Λ30Λ3Λ32Λ3−Λ30−Λ3).

Now, let us consider that θ=0 or θ=π. This implies that dθ=0. Then the Bertotti–Kasner space-time metric (1) reduces to the following form:

(11)ds′2=dt2−e2Λtdr2.

The matrix (gμν′) of metric tensor components in coordinates x′=(t, r) is given by

(12)gμν′(x′)=(100−e2Λt),

where μ, ν=0, 1. The matrix (gμν′) has the determinant −e2Λt. The hypersurface H0′ or Hπ′ degenerates to the two-dimensional surface. Let Γβγ′α denote the Christoffel symbols of second kind. The non-vanishing components of Γβγ′α are

Γ01′0=Γ10′0=Λ, Γ11′0=Λe2Λt.

For the metric (11), the Riemann tensor has only one nonzero component, and is given by

R1010′(x′)=Λe2Λt.

Moreover, the Gaussian curvature at each point x′=(t, r) of the hypersurface H0′ or Hπ′ is as follows:

(13)K′(x′)=R1010′(x′)|100−e2Λt|=−Λ.

We conclude that the Gaussian curvature of the hypersurface H0′ or Hπ′ is constant. Further, every point is isotropic in the two-dimensional surface.

Next, for the case θ∈(0, π) and φ=0, the Bertotti–Kasner space-time metric (1) takes the form

(14)ds″2=dt2−e2Λtdr2−1Λ dθ2.

The matrix (gμν″) of metric tensor components in coordinates x″=(t, r, θ) has the form

(15)gμν″(x″)=(10−e2Λt0−1Λ),

where μ, ν=0, 1, 2. The matrix (gμν″) has the determinant e2ΛtΛ. Let Γβγ″α denote the Christoffel symbols of the second kind. The non-vanishing components of Γβγ″α for the metric (14) are as follows:

Γ01″1=Γ10″1=Λ, Γ11″0=Λe2Λt.

Similar to the case of a two-dimensional surface, we have only one non-vanishing component of the Riemann tensor for the metric (14) and is given by

R1010″(x″)=Λe2Λt.

Hence the curvature of three-dimensional space at each point x″ is determined by one Gaussian curvature quantity Kθ″(x″). Further, we have

(16)Kθ″(x″)=R1010″(x″)|100−e2Λt|=−Λ.

Since the Riemann tensor components R2020″(x″) and R1212″(x″) are equal to zero, the Gaussian curvature quantities Kr″(x″) and Kt″(x″) vanish at each point. Further, we have observed that the quantity Kθ″(x″) (Eqn (16)) is identical with λ1(r), λ4(r) and the quantities Kr″(x″) and Kt″(x″) are identical with λ2(r), λ3(r), λ5(r), λ6(r).

We pointed out that the curvature of Bertotti–Kasner space-time is determined by two quantities Kt,r(x) and Kθ,φ(x) of Gaussian curvature at each point x. The four quantities Kt,θ(x), Kt,φ(x), Kr,θ(x), and Kr,φ(x) of Gaussian curvature vanish at each point. The curvature index [19] of Kt,r(x) and Kθ,φ(x) is −1, and that of Kt,θ(x), Kt,φ(x), Kr,θ(x) and Kr,φ(x) is 0. The six quantities of Gaussian curvature are shown in Eqn (17):

(17)Kθ,φ(x)=Kθ″(x″)=K′(x′)=λ4(r)=−Λ,Kt,φ(x)=Kt″(x″)=λ2(r)=0,Kr,φ(x)=Kr″(x″)=λ3(r)=0,Kt,r(x)=λ1(r)=−Λ,Kt,θ(x)=λ5(r)=0,Kr,θ(x)=λ6(r)=0.}

From Eqn (17) we have observed that all the six Gaussian curvature quantities are expressed in terms of curvature invariants. The radial coordinate r is plotted versus Kt,r(x) and Kθ,φ(x) in Figure 1. Here, we have considered different values of the cosmological constant like Λ=0.01, Λ=0.005, Λ=0.001, and Λ=0.0001. We deduce from Figure 1 that the quantities Kt,r(x) and Kθ,φ(x) of Gaussian curvature tend to zero as Λ decreases. So that as Λ→0, all six quantities of Gaussian curvature in Eqn (17) approach to zero, and hence Bertotti–Kasner space-time smoothly becomes flat. In other words, it reduces into Minkowski space-time.

3. Conclusions

We have studied the geometric properties of the Bertotti–Kasner space-time. It is found that in Bertotti–Kasner space-time, every point is isotropic. In two orientations K=−Λ and the other four orientations K=0. Further, we have analyzed the following:

Rαβγδ=K(gαγgβδ−gαδgβγ),Rαβγδ′=K(gαγ′gβδ′−gαδ′gβγ′),Rαβγδ″=K(gαγ″gβδ″−gαδ″gβγ″),

where K denotes the Gaussian curvature equal to −Λ, hence we conclude that Bertotti–Kasner space-time is completely isotropic.

The canonical forms of tensors RUV and CUV are achieved in the pseudo-Euclidean space ℝ6 concerning the orthonormal basis p1, p2, p3, p4, p5, p6 by six-dimensional formalism. Both curvature invariants and canonical form of the Riemann tensor RUV describe the space-time curvature, and hence they lead to the analysis of nature of the gravitational field. In four orthonormal directions pi(i=1, …, 4), the curvature invariants are equal to zero. However, in the remaining two orthonormal directions, they are nonzero and equal to −Λ each. Therefore, we may deduce that the Bertotti–Kasner space-time has a constant gravitational field.

Figures

Figure 1

The quantities Kt,r(x) and Kθ,φ(x) of Gaussian curvature for different values of Λ