Geometric properties of the Bertotti–Kasner space-time

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 sixdimensional formalism introduced by Pirani and followed byW. 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.


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 spacetime 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-Nordstr€ om 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 ðt; r; θ; fÞ with relativistic units ðG ¼ c ¼ 1Þ is as follows: (see [4,5]) where dΩ 2 ¼ dθ 2 þ sin 2 θdf 2 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 þ 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, ðl; bÞ ¼ ð309 + ; 18 + Þ 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 λ W -tensor is also given in Section 2. The paper ends with Section 3, where we have mentioned some important conclusions.

Curvature of the Bertotti-Kasner space-time
We have considered that the matrix ðg μν Þ has the signature ðþ; − ; − ; − Þ. In the Schwarzschild coordinates x ¼ ðt; r; θ; fÞ, the matrix ðg μν Þ is as follows: , which is less than zero. Hence it is a real space-time [14]. The Riemannian metric tensor g μν determines the nature of gravitational field potential, and for the Bertotti-Kasner space-time metric, it is given by 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 g μν . Let Γ α βγ denote the Christoffel symbols of second kind defined by The independent non-vanishing components of Γ α βγ are as follows: Riemann and Christoffel introduced the tensor R α βγδ of type ð1; 3Þ. It is formed by metric tensor g μν and its partial derivatives up to second order. The curvature tensor (or Riemann tensor) of type ð0; 4Þ can be expressed as The independent nonzero components of R αβγδ are The Ricci tensor R αβ is a covariant tensor of order 2 and is given by We found that R αβ ¼ Λg αβ . Let R denote the scalar curvature. It is a tensor of order zero given by 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 For Bertotti-Kasner space-time, Kretschmann scalar is also constant and is directly proportional to the square of the cosmological constant. The tensor G αβ of type ð0; 2Þ is defined by is called the Einstein tensor. It plays a supreme role in Einstein's relativity theory. We found that G αβ ¼ −Λg αβ . Hence EFE is given by G αβ þ Λg αβ ¼ 0: Given the anti-symmetric property the Weyl curvature tensor can be evaluated as follows: The Weyl tensor is traceless, but it has the symmetric properties as Riemann tensor R αβγδ . If we contract on the couple of indices αδ or βγ, the obtained tensor vanishes. The independent non-vanishing components of the Weyl tensor C αβγδ are as follows: 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 ℝ 6 [15]. Let us consider the following identification to adopt the six-dimensional formalism [16]: Given g αβ are the metric tensor components at a point of Bertotti-Kasner space-time, we define the bivector-tensor as follows: g UV ¼ g αβγδ ¼ g αγ g βδ À g αδ g βγ : The bivector-tensor g UV is non-singular, and has the signature ðþ; þ; þ; −; −; − Þ. It satisfies the symmetric property, that is, g UV ¼ g VU . The suffix pairs αβ, γδ are skewsymmetric. The bivector-tensor g UV has the following non-vanishing components: Now we relabel the Riemann tensor R αβγδ as R UV given the scheme of six-dimensional formalism. Because of the property R αβγδ ¼ R γδαβ , the tensor R UV satisfies the symmetric property, that is, R UV ¼ R VU . The tensor R UV has the following nonvanishing components: Given identification (6) of the six-dimensional formalism, we relabel the Weyl tensor C αβγδ as C UV . Because of the property C αβγδ ¼ C γδαβ , the tensor C UV is symmetric, that is, The tensor C UV has the following nonvanishing components: The λ-tensor is defined as R UV − λg UV . The curvature invariants are roots of the characteristic equation jR UV ðxÞ − λg UV ðxÞj ¼ 0, and for Bertotti-Kasner space-time, they are obtained as follows: λ 2 ðrÞ ¼ λ 3 ðrÞ ¼ λ 5 ðrÞ ¼ λ 6 ðrÞ ¼ 0: The curvature tensor has the following canonical form: Also, we have From the features of λ-tensor R UV − λg UV , we analyze the description of the gravitational field in the Segre symbols (see Ref. [17]), and we found that it is of the type G 1 ½ð1111Þð11Þ.
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 λ W -tensor C UV − λ W g UV is constructed from the symmetric tensors C UV and g UV . The roots of the characteristic equation jC UV ðxÞ − λ W g UV ðxÞj ¼ 0 are called Weyl invariants, Petrov invariants or Petrov scalars. For Bertotti-Kasner space-time, Weyl invariants are as follows: The determinant of the Weyl conformal λ W -matrix C UV ðxÞ − λ W g UV ðxÞ is zero for any of the above Weyl invariants. The canonical form of the Weyl tensor or Petrov canonical form is given by

AJMS
Now, let us consider that θ ¼ 0 or θ ¼ π. This implies that dθ ¼ 0. Then the Bertotti-Kasner space-time metric (1) reduces to the following form: The matrix ðg 0 μν Þ of metric tensor components in coordinates x 0 ¼ ðt; rÞ is given by For the metric (11), the Riemann tensor has only one nonzero component, and is given by Moreover, the Gaussian curvature at each point x 0 ¼ ðt; rÞ of the hypersurface H 0 0 or H 0 π is as follows: We conclude that the Gaussian curvature of the hypersurface H 0 0 or H 0 π is constant. Further, every point is isotropic in the two-dimensional surface.
Next, for the case θ ∈ ð0; πÞ and f ¼ 0, the Bertotti-Kasner space-time metric (1) takes the form ds 00 2 ¼ dt 2 À e 2 ffiffi ffi Λ p t dr 2 À 1 Λ dθ 2 : The matrix ðg 00 μν Þ of metric tensor components in coordinates x 00 ¼ ðt; r; θÞ has the form g 00 μν ðx 00 Þ ¼ 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