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
Keywords
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
Publisher
:Emerald Publishing Limited
Copyright © 2021, H.M. Manjunatha, S.K. Narasimhamurthy and Zohreh Nekouee
License
Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
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
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
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,
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
2. Curvature of the Bertotti–Kasner space-time
We have considered that the matrix
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
The independent non-vanishing components of
Riemann and Christoffel introduced the tensor
The independent nonzero components of
The Ricci tensor
We found that
The scalar curvature of Bertotti–Kasner space-time is
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
Given the anti-symmetric property
The Weyl tensor is traceless, but it has the symmetric properties as Riemann tensor
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
Given
The bivector-tensor
Now we relabel the Riemann tensor
Given identification (6) of the six-dimensional formalism, we relabel the Weyl tensor
The λ-tensor is defined as
The curvature tensor has the following canonical form:
Also, we have
From the features of λ-tensor
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
The Weyl conformal
The determinant of the Weyl conformal
Now, let us consider that
The matrix
For the metric (11), the Riemann tensor has only one nonzero component, and is given by
Moreover, the Gaussian curvature at each point
We conclude that the Gaussian curvature of the hypersurface
Next, for the case
The matrix
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
Hence the curvature of three-dimensional space at each point
Since the Riemann tensor components
We pointed out that the curvature of Bertotti–Kasner space-time is determined by two quantities
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
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
The canonical forms of tensors
Figures
References
[1]Borgiel W. The gravitational field of the Schwarzschild spacetime. Diff Geom Appl. 2011; 29: S207-10. doi: 10.1016/j.difgeo.2011.04.042.
[2]Ali M, Ahsan Z. Geometry of charged rotating black hole. SUT J Math. 2013; 49(2): 129-43.
[3]Ali M, Ahsan Z. Gravitational field of Schwarzschild soliton. Arab J Math Sci. 2015; 21(1): 15-21. doi: 10.1016/j.ajmsc.2013.10.003.
[4]Rindler W. Birkhoff's theorem with
[5]Bernabéu J, Espinoza C, Mavromatos NE. Cosmological constant and local gravity. Phys Rev D. 2010; 81: 084002. doi: 10.1103/PhysRevD.81.084002.
[6]Bertotti B. Uniform electromagnetic field in the theory of general relativity. Phys Rev. 1959; 116(5): 1331-3. doi: 10.1103/PhysRev.116.1331.
[7]Kasner E, Solutions of the Einstein equations involving functions of only one variable, Trans Am Math Soc. 1925; 27(2): 155-62.
[8]Bonnor WB. in: Recent developments in general relativity. New York: Pergamon Press; 1962: 167.
[9]Lake K. Maximally extended, explicit and regular coverings of the Schwarzschild-de Sitter vacua in arbitrary dimension, Class. Quantum Grav. 2006; 23(20): 5883-5895. doi: 10.1088/0264-9381/23/20/010.
[10]Mazharimousavi SH, Halilsoy M. Black holes from multiplets of scalar fields in
[11]Pelavas N. Timelike Killing vectors and ergo surfaces in non-asymptotically flat spacetimes. Gen Relativ Gravit. 2005; 37(2): 313-26. doi: 10.1007/s10714-005-0021-3.
[12]Antoniou I, Perivolaropoulos L. Searching for a cosmological preferred axis: Union2 data analysis and comparison with other probes. J Cosmol Astropart Phys. 2010; 12: 012. doi: 10.1088/1475-7516/2010/12/012.
[13]Plank Collaboration. Planck 2013 results. XXIII. Isotropy and statistics of the CMB. Astron Astrophys. 2014; 571: A23. doi: 10.1051/0004-6361/201321534.
[14]Landau LD, Lifshitz EM. The classical theory of fields, Oxford: Butterworth-Heinemann; 1994.
[15]Pirani FAE. Invariant formulation of gravitational radiation theory. Phys Rev. 1957; 105(3): 1089-99. doi: 10.1103/PhysRev.105.1089.
[16]Manjunatha HM, Narasimhamurthy SK, Nekouee Z. The gravitational field of the SdS space-time. Int J Geom Methods Mod Phys. 2020; 17(5): 2050069. doi: 10.1142/S0219887820500693.
[17]Hall GS, Symmetries and curvature structure in general relativity, Singapore: World Scientific Publ.; 2004.
[18]Petrov AZ. The classification of spaces defining gravitational fields. Gen Rel Grav. 2000; 32(8): 1665-85. doi: 10.1023/A:1001910908054.
[19]Rindler W. Relativity: Special, general and cosmological, New York: Oxford University Press; 2006.
Acknowledgements
The author H.M. Manjunatha is very much grateful to Karnataka Science and Technology Promotion Society (KSTePS), Department of Science and Technology (DST), Govt. of Karnataka (Award Letter No. OTH-04: 2018-19), for awarding DST-PhD Fellowship.