On Spacetime Symmetries in General Relativity and Teleparallel Gravity

A Dissertation Submitted in Partial Fulﬁllment for the Requirement of the Degree of Doctor of Philosophy in Mathematics

By Tahir Hussain

Spervised by Prof. Dr. Gulzar Ali Khan Dr. Suhail Khan

DEPARTMENT OF MATHEMATICS UNIVERSITY OF PESHAWAR May, 2016 Author’s Declaration

I, Tahir Hussain S/O Hussain Ghulam, hereby declare that this dissertation is based on my Ph.D research carried out at the Department of Mathemat- ics, University of Peshawar, Pakistan. No part of this dissertation has been submitted elsewhere for the award of any other degree or qualiﬁcation. To the best of my knowledge, this dissertation does not contain any previously published or written material by any other person, unless referenced to the contrary in the text. Most of the work presented in this dissertation has been published in reputed journals.

Tahir Hussain Department of Mathematics, University of Peshawar, Pakistan

i Certiﬁcate

It is certiﬁed that the research work presented in this dissertation, entitled On Spacetime Symmetries in General Relativity and Teleparallel Gravity is the original work of Mr. Tahir Hussain and is carried out under the supervision of Prof. Dr. Gulzar Ali Khan and Dr. Suhail Khan at Department of Mathematics, University of Peshawar. This dissertation has been approved for the award of the degree of Doctor of Philosophy in Mathematics.

External Examiner : Dr. Siraj-ul-Haq

Supervisor : Prof. Dr. Gulzar Ali Khan

Co-Supervisor : Dr. Suhail Khan

Chairman : Dr. Imran Aziz

ii Dedicated To My Parents, Brothers and My Wife

iii Table of Contents

Abstract...... vii List of Publications...... viii Acknowledgements...... ix

1 Preliminaries1 1.1 Introduction...... 1 1.2 Fundamentals of General Relativity...... 5 1.2.1 Manifolds...... 5 1.2.2 Tangent Vectors...... 7 1.2.3 Tensors...... 9 1.2.4 Derivative Operators...... 11 1.2.5 Parallel Transport of Vectors...... 13 1.2.6 Geodesics...... 14 1.2.7 Curvature and some other important Tensors..... 15 1.2.8 Lie Derivative...... 17 1.2.9 Spacetime...... 18 1.2.10 Tetrad...... 19 1.3 Spacetime Symmetries in General Relativity...... 20

iv 1.4 Fundamentals of Teleparallel Gravity...... 26 1.4.1 Tetrad in Teleparallel Gravity...... 26 1.4.2 Weitzenb¨ock Connection and Torsion Tensor...... 26 1.5 Spacetime Symmetries in Teleparallel Gravity...... 28 1.6 Outlines of Work...... 31

2 Conformal Killing Vectors in LRS Bianchi Type V, Static and Non Static Plane Symmetric Spacetimes 33 2.1 Conformal Killing Vectors in LRS Bianchi Type V Spacetimes 34 2.2 Inheriting Conformal Killing Vectors in LRS Bianchi type V Spacetimes...... 47 2.3 Conformal Killing Vectors in Static Plane Symmetric Spacetimes 49 2.4 Inheriting Conformal Killing Vectors in Static Plane Symmet- ric Spacetimes...... 60 2.5 Conformal Killing Vectors in Non Static Plane Symmetric Spacetimes...... 62 2.6 Inheriting Conformal Killing Vectors in Non Static Plane Sym- metric Spacetimes...... 78 2.7 Summary...... 80

3 Teleparallel Conformal Killing Vectors in LRS Bianchi Type V, Static and Non Static Plane Symmetric Spacetimes 83 3.1 Teleparallel Conformal Killing Vectors in LRS Bianchi Type V Spacetimes...... 84 3.2 Teleparallel Conformal Killing Vectors in Static Plane Sym- metric Spacetimes...... 96

v 3.3 Teleparallel Conformal Killing Vectors in Non Static Plane Symmetric Spacetimes...... 108 3.4 Summary...... 124

4 Teleparallel Killing and Homothetic Vectors of Kantowski- Sachs and LTB Metrics 126 4.1 Proper Teleparallel Homothetic Vectors in Kantowski-Sachs Spacetimes...... 127 4.2 Teleparallel Killing Vectors in Kantowski-Sachs Spacetime.. 130 4.3 Proper Teleparallel Homothetic Vectors in LTB Metric.... 138 4.4 Teleparallel Killing Vectors in LTB Metric...... 140 4.5 Summary...... 147

5 Teleparallel Killing and Homothetic Vectors of 3-dimensional Static Circularly Symmetric Spacetimes 149 5.1 Teleparallel Killing Vectors of 3-dimensional Static Circularly Symmetric Spacetimes...... 150 5.2 Proper Teleparallel Homothetic Vectors for 3-dimensional Static Circularly Symmetric Spacetimes...... 161 5.3 Summary...... 166

6 Conclusion 168

References 177

vi Abstract

In this thesis we have investigated Killing, homothetic and conformal Killing vectors for some well known spacetimes. Conformal Killing vectors are investigated for locally rotationally symmetric (LRS) Bianchi type V, static and non static plane symmetric spacetimes in the context of general relativity as well as teleparallel gravity, while Killing and homothetic vectors are explored for Kantowski-Sachs, Lemaitre-Tolman-Bondi (LTB) and 3-dimensional static circularly symmetric spacetimes in the framework of teleparallel gravity. In general relativity, it is shown that Bianchi type V, static and non static plane symmetric spacetimes admit proper conformal Killing vectors for some speciﬁc values of the metric functions. In teleparallel gravity, it is observed that the LRS Bianchi type V spacetimes do not admit proper teleparallel conformal Killing vectors. Further, the number of proper teleparallel conformal Killing vectors for static and non static plane symmetric spacetimes turned out to be one or three for diﬀerent choices of the metric functions. Moreover, it is shown that the Kanstowski-Sachs and LTB metrics do not admit any proper teleparallel homothetic vector. The maximum number of teleparallel Killing vectors for Kantowski-Sachs spacetimes turned out to be seven, while for LTB metric, this maximum number is found to be six. Finally, our analysis shows that the 3-dimensional static circularly symmetric spacetimes admit a proper teleparallel homothetic vector in only one case, while the maximum number of teleparallel Killing vectors for these spacetimes is found to be six.

vii List of Publications

Out of the research work presented in this thesis, the following papers have been published in diﬀerent journals.

• S. Khan, T. Hussain, A. H. Bokhari and G. A. Khan, Conformal Killing vectors of plane symmetric four dimensional Lorentzian manifolds, European Physical Journal C, 75:523 (2015).

• S. Khan, T. Hussain and G. A. Khan, Conformal Killing symmetries of plane symmetric static spacetimes in teleparallel theory of gravitation, European Physical Journal Plus, 129:228 (2014).

• S. Khan, T. Hussain and G. A. Khan, A note on teleparallel Lie symmetries using non diagonal tetrad, Romanian Journal of Physics, 59, 488 (2014).

• S. Khan, T. Hussain, G. A. Khan and Amjad Ali, A note on teleparallel Killing symmetries in three-dimensional circularly symmetric static spacetime, International Journal of Theoretical Physics, 54, 2969 (2015).

• S. Khan, T. Hussain and G. A. Khan, A note on teleparallel conformal Killing vector ﬁelds in plane symmetric non static spacetimes, International Journal of Geometric Methods in Modern Physics, 13, 1650030 (2016).

• S. Khan, T. Hussain, A. H. Bokhari and G. A. Khan, Conformal Killing vectors in LRS Bianchi type V spacetimes, Communications in Theoretical Physics, 65, 315 (2016).

viii Acknowledgements

First and foremost, praises and gratitude to Almighty ALLAH whose showers of blessings made me able to complete my research successfully. I am forever thankful to Prof. Dr. Gulzar Ali Khan and Dr. Suhail Khan for having accepted to be my research supervisors and providing me indispensable guidance throughout this research. Their forbearance and efforts were vital for completing this research and to my formation as a future researcher. Their dynamism and motivation have deeply inspired me. I am extremely grateful to Prof. Dr. Ashfaque Hussain Bokhari, Depart- ment of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Saudi Arabia for his invaluable guidance regarding my research. I am also grateful to Dr. Abdul Samad, Chairman Department of Math- ematics, for providing me the opportunity to pursue my higher studies at the Department. I am extending my heartfelt thanks to all faculty members of Department of Mathematics, University of Peshawar for their sincerity and genuine support. I must acknowledge University of Peshawar for granting me ﬁnancial support and NOC to pursue my higher studies. My profound thanks to all friends, especially Dr. Muhammad Farooq, Rohul Amin, Imran Khan, Muhammad Asif and Fawad Khan for their nice company during my research work. Finally, I would like to thank my parents, brothers, sisters, nephews, nieces and wife for their love, prayers, caring and support throughout my academic trajectory that made me able to achieve my goals.

Tahir Hussain

ix Chapter 1

Preliminaries

1.1 Introduction

Albert Einstein is considered as the most inﬂuential thinker of the 20th cen- tury due to his accomplishments in science. In particular, he has truly risen to fame after publishing the famous scientiﬁc theory of relativity which stands out as one of the greatest achievements in science and became one of the important pillars of modern Physics. The importance of the theory of relativity is realized by the fact that it changed our thinking about the concept of space and time and a number of previously held basic results about space and time are proved to be wrong after the development of this theory. As a consequence of his theory of relativity, Einstein determined that space and time are interwoven into a single continuum, known as spacetime. Einstein’s theory of relativity was published in two versions, namely special relativity and general relativity. The theory of special relativity was published in 1905, in which Einstein proposed that laws of Physics are the same for all

1 observers in an inertial frame of reference and the speed of light in a vacuum is independent of the motion of all observers. This theory is called the special theory of relativity because it is limited to only the bodies which move in an inertial frame of reference. This theory became famous because of some perplexing properties of time and length such as time dilation and length contraction, which it proved to be true. These properties of time and length were proved through the theoretical and experimental study of light propa- gation observed by moving observers. Another important consequence of the theory of special relativity is the mass-energy equivalence relation described by the famous equation E = mc2, which states that in a physical system mass m and energy E both are present in a constant proportion and this equivalency of mass and energy is reliant on the speed of light c. After his stupendous efforts for about ten years, Einstein completed his jour- ney from special relativity towards a new geometric theory of space, time and gravitation in 1915, which is known as theory of general relativity. The aim of this theory was to work with Einstein’s ideas in the presence of gravitational forces. This theory generalized the theory of special relativity and Newton’s law of universal gravitation and it unified description of gravity as a geometric property of spacetime. The theory of general relativity was formulated in the language of differential geometry, the subject which studies curves is space and the space itself. This theory narrates gravitation as the curvature of a spacetime caused by the presence of matter and energy. The curvature of a spacetime is the amount by which a spacetime deviates from being flat and it is completely described by its metric. Unlike the case of special relativity, the spacetime metric need not to be flat in general relativ-

2 ity. The curvature of a spacetime is related to the energy-momentum tensor of the matter in a spacetime by the following system of partial differential equations, known as Einstein’s field equations [1]: R R − g + Λg = kT , (1.1.1) ab 2 ab ab ab where Rab, gab and Tab are the components of Ricci, metric and energy- momentum tensors respectively. Also R is the Ricci scalar, k is the gravitational constant and Λ represents the cosmological constant. The constant Λ in Einstein’s field equations is important in the cosmological context, otherwise it is usually taken as zero in all non-cosmological situations. Only a limited number of exact solutions of Einstein’s field equations are known because of their highly non-linear nature [1]. Some of the known exact solutions include spherically symmetric solutions of Schwarzchild, Kerr, Reissner and Nordstrm, Einstein Universe, de Sitter universe, Tolman and Friedmann Robertson Walker, axisymmetric static electromagnetic and vacuum solutions of Weyl and the plane wave metrics. All these exact solutions of Ein- stein’s field equations are found by imposing some symmetry restrictions on the metric, by taking some restrictions on the structure of Riemann tensor or under some initial and boundary conditions. The exact solutions of Einstein’s field equations have played an important role in the discussion of physical problems. For example Schwarzchild and Kerr solutions in black holes, the Friedmann solutions in cosmology and plane wave solutions played an important role in discovering the existence of gravitational radiations [1]. Beside their physical interpretation, exact solutions of Einstein’s field equations are also useful in checking the validity of the results obtained by approximation and numerical techniques used for the solution of Einstein’s field equations.

3 Although the theory of general relativity remained very successful in excellent agreement with observations and experiments at classical level, this theory faced some challenges at quantum level. Such challenges are often summarized as dark matter and dark energy problem. Dark matter is a hypothetical kind of matter which is an important ingredient for the dynamics of the entire universe and dark energy is a form of energy which is hypothesized to spread throughout the space and it is responsible for the accelerated expansion of the universe [2]. In order to find a reasonable explanation for the above mentioned problems, modifications were started in this theory by Einstein and his followers. Many of those early attempts were concerned to unify gravitation and electromag- netism. In 1918, Weyl [3] made one of such attempts whose proposal did not succeed but it led to the gauge transformations and gauge invariance. An- other attempt in the same direction was made by Albert Einstein [4], based on the mathematical structure of teleparallelism. This idea introduced the concept of tetrad, which is a field of orthonormal bases on tangent spaces at each point of a four dimensional spacetime. Like Weyl’s work, Einstein’s attempt of unification also did not succeed but it introduced certain concepts that remained important till now. After three decades, a new strength to the Einstein’s idea was given by Möller[5], not for the purpose of unification but in pursuing the gauge theory of gravitation. Following Möller’sidea, Pellegrini and Plebanski [6] found a Lagrangian formulation for teleparallel gravity. In 1967, Hayashi and Nakano [7] formulated gauge theory for translation groups. Hayashi [8] also presented a connection between gauge theory for translation groups and teleparallelism. Following this, Hayashi

4 and Shirafuji [9] made an attempt to unify these developments. The theory of general relativity was supplemented by teleparallel gravity in this way. Although general relativity and teleparallel gravity are equivalent theories, their interpretations are quite different. In the framework of general relativity, the gravitational potentials are contained in the metric of spacetime and the metric is responsible for the curvature of spacetime. In teleparallel approach, gravitational fields are represented by tetrad. Moreover general relativity is based on Riemannian geometry while teleparallel gravity is based on Weitzenböck geometry [10]. In the remaining portion of this chapter, we give a brief review of some basic concepts of general relativity and teleparallel gravity. Moreover, the literature review on spacetime symmetries in both theories is also presented here.

1.2 Fundamentals of General Relativity

In this section we brieﬂy discuss some important basic concepts of general relativity which are frequently used in this thesis. These basic concepts include manifolds, tangent vectors, tensors, covariant derivative, parallel transport of vectors, geodesics, curvature, Lie derivative and some other terms. All the materials of this section are taken from [11–15].

1.2.1 Manifolds

In the theory of general relativity, Manifolds play a central role in describing the complicated structures in terms of relatively easily understandable prop-

5 erties of the Euclidean space. They describe the interactions among matter, energy and gravitational forces. A spacetime in general relativity can be modeled by a 4-dimensional manifold carrying a certain type of geometric structure, which is known as a Lorentz metric and this metric satisfies the Einstein field equations. The curvature of the Lorentz metric then interprets the gravitational effects in spacetime. Generally speaking, manifolds resemble with curves and surfaces except that they might have higher dimension. The dimension of a manifold is the number of independent parameters required to locate a point on manifold. The

Euclidean space Rn is an example of an n-dimensional manifold. A one-dimensional manifold is called a curve, although it does not need to be curved in ordinary sense. Real line, circle and parabola are the trivial examples of one-dimensional manifolds. Manifolds of dimension 2 are known as surfaces. Examples of such manifolds are plane, sphere, cylinder, paraboloid and ellipsoid. Similarly a unit 3-sphere deﬁned by the set

4 2 2 2 2 {(x1, x2, x3, x4) ∈ R : x1 + x2 + x3 + x4 = 1} is an example of 3-dimensional manifold. Mathematically, a real smooth n-dimensional manifold M is a set along with a collection of subsets {Ok} which satisfy the following conditions [11]: i. The collection {Ok} forms a cover of M. ii. For each k, there exists a bijective mapping ψk : Ok → Uk, where Uk is some open subset of Rn. iii. If Ok ∩ Oj 6= φ, then we can consider the mapping

−1 n n ψj ◦ ψk : ψk[Ok ∩ Oj] ⊂ Uk ⊂ R → ψj[Ok ∩ Oj] ⊂ Uj ⊂ R

6 n such that the subsets Uk and Uj are open in R and this map is smooth, that is inﬁnitely continuously diﬀerentiable. Similarly a complex manifold can be

n n defined in the same way if we replace R by C . For each k, the mapping ψk in the above definition is known as a chart or a coordinate system. A topological manifold is a topological space which satisfies the properties of a manifold with all its charts ψk to be homeomorphisms. A mapping λ from a manifold M onto another manifold N is called a diffeomorphism if it is smooth, bijective and has a smooth inverse. In such a case we say that the manifolds M and N are diffeomorphic, that is they have the same manifold structures. A local diffeomorphism ϕ from a manifold M into another manifold N is a mapping ϕ : M → N such that every point p ∈ M has a neighborhood U with ϕ(U) open in N and ϕ|U : U → ϕ(U) as a diffeomorphism. For a manifold M, a one parameter group of diffeomorphisms ϕt is a smooth mapping R × M → M such that for any fixed t ∈ R, the mapping

ϕt : M → M is a diﬀeomorphism and ϕt ◦ ϕr = ϕt+r, for all t, r ∈ R.

1.2.2 Tangent Vectors

In special relativity, a spacetime is considered as a 4-dimensional vector space. However, this structure of spacetime as a vector space is lost in general relativity. For example there is no notion of adding two points on a sphere to get another point on the same sphere. This structure of vector space can be recovered by the concept of tangent vectors.

For a manifold M which is embedded in Rn (like a sphere), a tangent vector at a point p ∈ M is a vector which lies in the tangent plane to M at p. But every manifold need not to be embedded in Rn, a tangent vector for such a

7 manifold is deﬁned as follows:

Let τ be the set of all smooth mappings from a manifold M to R and p ∈ M. A tangent vector v at p is a mapping v : τ → R, which satisﬁes the following two properties [11]: i. v(c1f1 + c2f2) = c1 v(f1) + c2 v(f2), for all f1, f2 ∈ τ and c1, c2 ∈ R. ii. v(f1f2) = f1(p) v(f2) + f2(p) v(f1), for all f1, f2 ∈ τ.

The above two properties of a tangent vector are respectively referred as Lin- earity and Leibnitz rule. If we denote the collection of all tangent vectors at a point p on a manifold M by Vp, then it can be checked easily that Vp con- stitutes a vector space with respect to the addition and scalar multiplication defined by (v1 + v2)(f) = v1(f) + v2(f) and (av)(f) = av(f) respectively. This vector space is called a tangent space to M at the point p. It is well known that for a manifold of dimension n, dim Vp = n. Sometimes the tangent space Vp is also denoted by TpM. The union of all tangent spaces at all points of a manifold M is denoted by TM and it is called a tangent bundle to M. A vector field X on a manifold M is a mapping X : M → TM, which assigns a tangent vector vp to each point p ∈ M. The vector field X is called global or local if it is defined on the whole manifold M or on some open subset of M respectively. If X and Y are two vector fields on M, then their commutator [X,Y ] is another vector field which assigns, to each point p of

M, a tangent vector [X,Y ]p such that [X,Y ]p(f) = Xp Y (f) − Yp X(f).

8 1.2.3 Tensors

In Physics, particularly in general relativity, tensors are of vital importance as they provide a comprehensive mathematical framework for solving physical problems. The concept of tensor gave a new way to formulate diﬀerential geometry of a manifold in terms of Riemann curvature tensor. A tensor T of type or valence (k, l) over a ﬁnite dimensional vector space V

∗ ∗ ∗ is a multilinear map T : V × V × ....V × V × V × ..... × V → R with | {z } | {z } k−times l−times k independent variables form V ∗ and l independent variables from V , where V ∗ denotes the vector space of all real valued linear functions on V , called the dual space of V [12]. The numbers k and l are referred as the degrees of tensor T with k as the contravariant degree and l the covariant degree. The sum of contravariant and covariant degrees of a tensor is known as rank or order of that tensor. Tensors of type (1, 0), (0, 1), (k, 0), (0, l) and (k, l) are called contravariant vector, covariant vector, contravariant tensor, covariant tensor and mixed tensor respectively. A tensor of type (0, 0) is deﬁned to be a scalar. If τ(k, l) denotes the collection of all tensors of type (k, l) over an n- dimensional vector space, then τ(k, l) forms a vector space of dimension nk+l with respect to the usual operations of addition and scalar multiplication of mappings. A mapping which assigns a tensor to each point of a manifold is called a tensor ﬁeld. In tensor analysis, some basic algebraic techniques are used to produce new tensors. Being the elements of a vector space, two tensors of the same type can be added to get a new tensor of the same type. Similarly a tensor can be multiplied by a scalar. If T1 and T2 are tensors of type (p, q) and (r, s) respectively over a vector space V , then their tensor product or outer product

9 ∗ p+r q+s T1 ⊗ T2 is a tensor of type (p + r, q + s) deﬁned on the set (V ) × V such that [13]:

1∗ p+r∗ 1 q+s T1 ⊗ T2(v , ..., v , u , ..., u )

1∗ p∗ 1 q p+1∗ p+r∗ q+1 q+s = T1(v , ..., v , u , ..., u ) T2(v , ..., v , u , ..., u ).

Contraction is another operation which is used to obtain a tensor of type (k − 1, l − 1) from a tensor T of type (k, l), provided that one contravariant and one covariant indices of T are equal. Beside these operations, the inner product of tensors is defined as the process of outer product followed by contraction. A tensor T is said to be symmetric in pth and qth variables of the same type if it gives the same values, as a multilinear map, after interchanging these variables. A tensor is called contravariant symmetric if it is symmetric in any pair of its contravariant indices. Similarly if a tensor is symmetric in any pair of its covariant indices, it is known as covariant symmetric. If a tensor is both contravariant and covariant symmetric, then it is called a symmetric tensor. The contravariant skew symmetric, covariant skew symmetric and skew symmetric tensors can be defined in a similar way except that the sign of the tensor is changed in skew symmetry instead of leaving it unchanged. As the Einstein’s field equations are tensor equations, therefore tensors play an important role in the theory of general relativity. One of the important tensors used in general relativity is the metric tensor which tells us the infinitesimal squared distance associated with an infinitesimal displacement.

It is a tensor of type (0, 2), that is a bilinear map g : TpM × TpM → R, which is symmetric and non degenerate. The metric tensor g is sometimes

2 also denoted by ds and it can be expressed in terms of its components gab

10 as [11]:

2 X a b ds = gab dx dx (1.2.3.1) a,b For every metric g, there exists an orthonormal basis of the tangent space at each point p of M, say v1, v2, ..., vn, which satisfy g(vi, vj) = 0 for i 6= j and g(vi, vi) = ±1 [11]. Here the number of + and − signs is called signature of the metric g. In particular, a metric of the signature (+, +, ..., +) is called positive deﬁnite or Riemannian metric and a metric of the signature (−, +, ..., +) is known as a Lorentzian metric. The metric of a spacetime is Lorentzian.

1.2.4 Derivative Operators

In Physics, the most important laws are usually expressed as differential equations. Moreover these differential equations have the form of tensor equations in order to ensure that the physical laws represented by these equations are independent of the coordinate system. Thus it is important to see whether tensors can be differentiated to produce new tensors, and if so, how it can be done. The usual partial derivative of a tensor do not produce another tensor. To find the derivative of a tensor to produce another tensor in curved spaces or in a spacetime, we need to define covariant derivative. A derivative operator or covariant derivative, denoted by ∇, on a manifold

a1...ak M diﬀerentiates a tensor T b1...bl of type (k, l) and produces another

a1...ak tensor ∇cT b1...bl of type (k, l + 1) such that the following properties are satisﬁed [11]: i. ∇ is linear, which means that for any two tensors T a1...ak and T a1...ak 1 b1...bl 2 b1...bl

11 of type (k, l) and c1, c2 ∈ R, we have ∇ (c T a1...ak +c T a1...ak ) = c ∇ T a1...ak +c ∇ T a1...ak c 1 1 b1...bl 2 2 b1...bl 1 c 1 b1...bl 2 c 2 b1...bl ii. ∇ satisﬁes Leibnitz rule, which means that for a tensor T a1...ak of type 1 b1...bl 0 c1...ck 0 0 (k, l) and a tensor T 0 of type (k , l ), we have 2 d1...dl

0 0 a1...ak c1...ck a1...ak c1...ck ∇e[T T 0 ] = [∇eT ]T 0 1 b1...bl 2 d1...dl 1 b1...bl 2 d1...dl 0 a1...ak c1...ck + T [∇eT 0 ] 1 b1...bl 2 d1...dl

a1...ak iii. ∇ commutes with contraction, that is for any tensor T b1...bl of type

a1..c..ak a1..c..ak (k, l), we have ∇e(T b1..c..bl ) = ∇eT b1..c..bl . iv. For any smooth real valued function f on M and any tangent vector

a a t ∈ TpM, t(f) = t ∇af, which shows the consistency of ∇ with the notion of tangent vectors as directional derivatives on scalar ﬁelds. v. ∇ is torsion free, that is for any smooth real valued function f on M,

∇a∇bf = ∇b∇af.

The above defined derivative operator is also known as absolute or covariant derivative and sometimes it is denoted by a semicolon (; ). It is worth noting here that in some gravitational theories, like teleparallel gravity, the condition (v) in the above definition is dropped because of the presence of a torsion tensor. However in general relativity, this condition must be satisfied. In fact, if we drop condition (v) from the above definition,

c then one can show the existence of a tensor Tab, known as torsion tensor, such c c c that Tab = −Tba and ∇a∇bf − ∇b∇af = −Tab∇cf [11]. In general relativity, torsion tensor vanishes and hence condition (v) of the above deﬁnition holds.

12 Following are some important results which can be derived using the above ﬁve conditions of covariant derivative. We state these results without proof.

(a) If va and wb are two vector ﬁelds, then their commutator can be ex-

b a b a b pressed as: [v, w] = v w ;a − w v ;a.

˜ (b) For given two derivative operators ∇a and ∇a and a dual vector field c wb, we can find a tensor field Cab, symmetric in its lower indices, such ˜ c that ∇awb = ∇awb − Cabwc.

a (c) If t is any vector ﬁeld, then for given two derivative operators ∇a and ˜ b ˜ b b c ∇a, we have ∇at = ∇at + Cact .

Similarly, a general formula can be derived for the action of ∇a on any tensor c ˜ in terms of Cab and ∇a. In a particular case, when the derivative operator ˜ c ∇a is the ordinary derivative operator ∂a, the tensor field Cab is denoted by c c Γab and it is known as Christoffel symbol. The Christoffel symbol Γab can be expressed in terms of partial derivatives of the components of metric tensor as [11,14]: 1 Γc = gcd{g + g + g } (1.2.4.1) ab 2 bd,a ad,b ab,d

1.2.5 Parallel Transport of Vectors

Generally, a vector v is called parallelly transported along a curve C if the following two conditions hold [14]: i. The transported vector is parallel to v. ii. The transported vector have the same length as that of v.

13 As an example, we can consider a circle C in a plane and a vector v can be transported parallelly along C such that it comes back to its original position with the same direction and having the same length. In general relativity, we deﬁne parallel transport of a vector along a curve C with the help of derivative operator. If ∇a is some derivative operator, then we say that the vector va is parallelly transported at each point along C if

a b a b a b c a we have t ∇av = 0 or equivalently t ∂av + t Γacv = 0, where t denotes

b1...bk the tangent vector to the curve C at that point. Similarly if T c1...cl is a tensor of type (k, l), then it is called parallelly transported along C if

a b1...bk t ∇aT c1...cl = 0 [11].

1.2.6 Geodesics

In general relativity, the idea of geodesics is of particular interest. The free falling particles follow the paths which are deﬁned by geodesics and the path of a planet revolving around a star is the projection of a geodesic of the four dimensional spacetime geometry around the star onto three dimensional space. Roughly speaking, the straightest possible lines which one can draw in curved geometry are called geodesics. For example, the geodesics in Euclidean geometry are straight lines. Similarly, the geodesics on a sphere are the shortest arcs of the great circles from one point to another point on the sphere. In terms of derivative operator, we deﬁne the notion of geodesics as follows:

If ∇a is some given derivative operator, then a geodesic is deﬁned to be a curve whose tangent vector at any point is parallelly transported along itself.

a b b Such curves satisfy the equation t ∇at = 0, where t is the tangent vector

14 to the curve [11].

1.2.7 Curvature and some other important Tensors

The curvature of a geometric object is the quantitative measure of how it bends from being flat. The word flat used here may have different meanings depending upon the situation. If we are considering curves, the word flat will mean straight line, while in case of surfaces it might mean the Euclidean plane. There are two notions of curvature, namely extrinsic and intrinsic curvature. The extrinsic notion of curvature is limited to those objects which are embedded in another higher dimensional surfaces. The intrinsic curvature is defined for any manifold without the reference of its embedding in a higher dimensional space. As our spacetime manifold is not embedded in any higher dimensional space, the intrinsic notion of curvature is preferred in the theory of general relativity. An example of extrinsic curvature is that of a circle which is equal to the reciprocal of its radius. Circles with greater radii have smaller curvature and vice versa. The intrinsic curvature arises in different cases. For example if a vector fails to return to its starting position after parallelly transported along a small closed path, it gives rise to curvature. Similarly if we have some parallel geodesics and they fail to be parallel after transportation, then it mean there is a curvature. Also, if the successive operations of differentiations fail to commute when applied to a dual vector, it shows that there is a curvature. The curvature of a manifold is expressed by a tensor, known as Riemann curvature tensor. The Riemann curvature tensor is actually a tensor field which assigns a tensor to each point of the manifold. In terms of christoffel

15 symbols, it is deﬁned as [14]:

a a a a e a e Rbcd = Γbd,c − Γbc,d + ΓecΓbd − ΓedΓbc. (1.2.7.1)

n2(n2−1) For an n-dimensional manifold, the Riemann curvature tensor has 12 non zero independent components. Thus the number of such components for a spacetime manifold is 20. A spacetime is ﬂat if all the components of the Riemann curvature tensor vanish. The Riemann curvature tensor can be written in completely covariant form by lowering the ﬁrst index with metric, that is [14]:

e e e Rabcd = gaeRbcd = Γbda,c − Γbca,d + ΓadeΓbc − ΓaceΓbd. (1.2.7.2)

We calculate another type of tensor, known as Ricci tensor, form Riemann curvature tensor by contracting ﬁrst and third indices:

c Rab = Racb (1.2.7.3)

Contracting the Ricci tensor with metric, we get a scalar which is called Ricci scalar:

ab g Rab = R (1.2.7.4)

With the help of these known quantities, we deﬁne the Einstein tensor Gab as [14]: 1 G = R − Rg . (1.2.7.5) ab ab 2 ab Finally another important quantity, known as Weyl tensor or conformal tensor, is deﬁned as [14]: 1 1 C = R + {g R + g R − g R − g R } + R{g g − g g }. abcd abcd 2 ad bc bc ad ac bd bd ac 6 ac bd ad bc (1.2.7.6)

16 A spacetime metric gab is called conformally flat if there exists a differentiable function f(x) such that gab is conformally related to the Minkowski metric 2 ηab, that is gab = f (x) ηab. Equivalently, a spacetime metric is conformally flat if all the components of Weyl tensor vanish.

1.2.8 Lie Derivative

In geometry, Lie derivative is used to compare the value of some geometric object at a point along a curve relative to its value after dragging the object to some other point along the curve. For us, the Lie derivative of vector fields and that of tensor fields are of particular interest. The Lie derivative of a vector field X on any differentiable manifold determines the rate of change of the vector field X along the flow of another vector field ξ, denoted by LξX. The Lie derivative of a tensor field can be defined in a similar way. In general relativity, especially important is to see the behavior of a metric tensor when it is dragged along curves on a manifold. The study of Killing, homothetic and conformal Killing vectors in a spacetime metric is carried out in this way, which will be given with details in next section of this thesis. Before giving a proper definition of Lie derivative, we recall the definitions of derived and pullback maps. Consider a smooth map φ : M → M 0 between smooth manifolds M and M 0. Let p ∈ M and set φ(p) = p0 ∈ M 0. The

0 mapping φ∗p : TpM → Tp0 M deﬁned by φ∗pv(f) = v(fo φ), for v ∈ TpM is called the derived map of φ at p. If φ : M → M 0 is a smooth map and f : M 0 → R is some smooth function on M 0, then the composition map fo φ is known as pullback of f under φ and it is denoted by φ∗f [11]. Let M be a smooth manifold and T,X be global smooth tensor ﬁeld and

17 vector field on M respectively. Let φt be the local diffeomorphisms associated with X, then the Lie derivative of T at a point p ∈ M along X is a global smooth tensor field on M and it is defined as [11,14]:

∗ (φt T )(p) − T (p) LX T (p) = lim . (1.2.8.1) t→0 t

Following are some important properties of Lie derivative operator.

(a) LX (c1T1 + c2T2) = c1 LX T1 + c2 LX T2, that is L is linear

(b) LX (T1 ⊗ T2) = T1 ⊗ LX T2 + LX T1 ⊗ T2

(d) Lc1X+c2Y T = c1 LX T + c2 LY T

(e) LX f = X(f) where X,Y denote global smooth vector ﬁelds, T,T1,T2 are global smooth tensor ﬁelds and f is a smooth real valued function on M.

1.2.9 Spacetime

Before the development of the theory of relativity, time was believed to be universal, progressing with a constant rate in all frame of references and independent of the motion of an observer. The theory of relativity proved that time cannot be separated from three dimensions of space because the rate of passaging of time for an object is dependent on the object’s velocity and on the gravitational ﬁelds. It was proved experimentally that time slows at higher speed of the frame of reference relative to another frame of reference.

18 This phenomenon is called time dilation. Mathematically, a spacetime (M, g) is defined as a four dimensional, smooth, connected and Hausdorff manifold M with Lorentz metric g [11]. The elements of a spacetime M are known as events. The Lorentz metric g of a spacetime M determines the geometry of M. The spacetime is assumed to be curved in general relativity because of the presence of energy and the curvature of spacetime is given by Riemann curvature tensor. On the other hand, the Riemann curvature tensor vanishes in special relativity and hence the spacetime is called flat or Minkowski space. The metric tensor of Minkowski space is called Minkowski metric, denoted by η.

1.2.10 Tetrad

Consider the Minkowski space M. The Minkowski inner product for u, v ∈ M is denoted by u · v or η(u, v), which has the following properties [15]: i. η(cu + v, w) = c η(u, w) + η(v, w), for all u, v ∈ M and c ∈ R. ii. η(u, v) = η(v, u), for all u, v ∈ M. iii. If for all v ∈ M, η(u, v) = 0, then u = 0.

If u, v ∈ M such that η(u, v) = 0, then u and v are called orthogonal. A vector v ∈ M is called a unit vector if η(v, v) = ±1. A basis {v1, v2, v3, v4} of M is called orthonormal basis if η(vi, vj) = 0, for i 6= j and η(vi, vi) = ±1. An orthonormal basis of M is also called an orthonormal tetrad. A vector v 6= 0 ∈ M is called spacelike, timelike or null if η(v, v) > 0, η(v, v) < 0 or η(v, v) = 0 respectively. Out of four elements of an orthonormal

19 tetrad, one is always timelike while the other three elements are spacelike.

Another type of tetrad, known as real null tetrad, is a basis {f1, f2, f3, f4} of the Minkowski space such that η(f1, f2) = η(f3, f3) = η(f4, f4) = 1 and all other inner products vanish. In real null tetrad, f1 and f2 are null, while f3 and f4 are unit spacelike vectors.

1.3 Spacetime Symmetries in General Rela- tivity

As mentioned in the introduction section, the theory of general relativity is governed by highly non-linear Einstein’s field equations. Due to the highly non-linearity of Einstein’s field equations, it is quite difficult to find their exact solutions. The symmetry restrictions on the metric of a spacetime assist in finding new exact solutions and in the classification of the known solutions of Einstein’s field equations. Also the existence of symmetries in spacetimes give rise to the conservation laws [16], which are equally important in general relativity and all physical systems which are expressed in terms of the invariance property of physical systems under a continuous symmetry. Roughly speaking, a symmetry of a spacetime M is a smooth vector field on M whose associated local diffeomorphisms preserve some geometric feature of M [17]. The geometric feature preserved by the associated local diffeomorphisms of a symmetry vector field may be geodesics, metric, Riemann curvature tensor or any other fundamental part of spacetime geometry. De- pending upon these geometric features preserved by the symmetry vector fields, there are various types of spacetime symmetries but we will focus only

20 on three important symmetries, namely Killing, homothetic and conformal Killing symmetries. Conformal Killing symmetry is represented by a vector field whose associated local diffeomorphisms preserve the metric up to a conformal factor [17]. If the conformal factor is a constant, the corresponding symmetry is known as homothetic symmetry and if the conformal factor vanishes, the symmetry is called Killing symmetry. The Lie derivative operator is very helpful in the investigation of the above mentioned types of symmetries in a spacetime. Lie derivative operator is used to compare the value of a geometric object at some point along a curve on the spacetime manifold relative to its value after dragging the object along the curve to the same point. For us, the most important is to see the behavior of the metric tensor when it is dragged along curves on manifolds. A conformal Killing symmetry or a conformal Killing vector X is a vector field on a spacetime manifold M such that when the metric of M is dragged along the curves generated by X, its Lie derivative satisfies the relation [17]:

LX gab = 2αgab, (1.3.1) where α is some real-valued function on M, called conformal factor. If in equation (1.3.1) we have α = 0, then the the vector ﬁeld X reduces to a Killing vector and in such a case the metric of spacetime is left invariant when it is dragged along the curves on M. In such a case equation (1.3.1) is called Killing equation. Moreover if α in equation (1.3.1) becomes a constant, the vector ﬁeld X becomes a homothetic vector and in this case the metric is being changed by a constant factor while dragging along the curves on M. A conformal Killing vector which is not homothetic is known as proper conformal Killing vector and a homothetic vector which is not Killing is

21 called proper homothetic vector. An important class of conformal Killing vectors contain the special conformal killing vectors whose conformal factor satisﬁes the relation αa;b = 0. Note that equation (1.3.1) can also be written in explicit form as [17]:

c c c gab,cX + gbcX,a + gacX,b = 2αgab, (1.3.2)

Some other alternate forms of equation (1.3.1) can be seen in [17]. The sets consisting all Killing, homothetic, conformal Killing and special conformal Killing vectors on a spacetime manifold M are respectively denoted as K(M),H(M),C(M) and SC(M) and each of these sets forms a Lie algebra with resect to the Lie bracket operation such that dimK(M) ≤ 10, dimH(M) ≤ 11 and dimC(M) ≤ 15. It is well know that if M is a conformally flat spacetime, then dimC(M) = 15 and conversely if dimC(M) = 15 (in fact when dimC(M) > 7), then the spacetime M is conformally flat [17]. Moreover K(M),H(M) and SC(M) all are subalgebras of C(M) and the sets K(M) and H(M) attain their maximum dimensions if and only if M is a flat spacetime. The notion of a particular type of conformal Killing vectors in fluid spacetimes, known as inheriting conformal Killing vectors, was introduced by Co- ley and Tupper [18]. A conformal Killing vector X is said to be inherited by the physical fluid source represented by the energy-momentum tensor if the fluid flow lines are mapped conformally into the fluid flow lines and such type of conformal Killing vectors satisfy [18]:

b b LX ua = ua,bX + ubX,a = αua, (1.3.3)

22 where ua signifies the four velocity of the fluid. Spacetime symmetries have a wide range of applications in general relativity and other branches of Physics. As mentioned before, spacetime symmetries give rise to conservation laws. Noether’s theorem guarantees the existence of conservation laws corresponding to every continuous symmetry [19]. These conservation laws in a spacetime are usually provided by Killing vectors. However sometimes it will be not possible to find the conservation laws with Killing vectors. In such a case, conformal transformations are employed to find the conservation laws which are not given by Killing vectors. For example in Friedman metrics, there exist no translational invariance which give law of conservation of energy but it admits a conformal time translational invariance which provides a conformal analogue of conservation of energy. Besides this, conformal Killing vectors play an important role at kinemat- ics and dynamics level. Kinematic variables such as rotation, expansion and shear can be studied under the assumption that a spacetime admit conformal Killing vectors [20–23]. Apart from general relativity, conformal symmetries have valuable applications in other branches of Physics, like astrophysics and cosmology [24–27]. In the recent literature of general relativity, much attention is given to study different types of spacetime symmetries. Petrov [16] was the first to explore Killing vectors in four dimensional spaces by solving the Killing’s equation. Bokhari and Qadir [28,29] classified static spherically symmetric spacetimes according to their Killing vectors and concluded that the minimum number of linearly independent Killing vectors admitted by these spacetimes is 4. Qadir and Ziad [30, 31] were able to achieved the classification of static

23 cylindrically symmetric and non static spherically symmetric spacetimes according to their Killing vectors. They found that the static cylindrically symmetric spacetimes admit 3, 4, 5, 6, 7 or 10 Killing vectors. Feroze et. al. [32] solved the Killing’s equation and achieve a complete classification of plane symmetric Lorentzian manifolds. They obtained some plane symmetric metrics possessing 3, 4, 5, 6, 7 or 10 Killing vectors. Bokhari and his collaborators [33] investigated all possible Killing vectors possessed by a three-dimensional circularly symmetric static metric and concluded that this metric admits 2, 3, 4 or 6 Killing vectors. Killing vectors in Bianchi type VIo and VIIo spacetimes have been explored by Ali and his collaborators [34]. Recently Khan et. al. [35] found Killing vectors for Lemaitre-Tolman-Bondi metric. They found that this metric admits only three or four Killing vectors. Similarly homothetic vectors are also investigated for different spacetimes. Hall and Steele [36] proved that the maximum number of homothetic vectors admitted by a four dimensional spacetime is 11. Ahmad and Ziad [37] gave a complete classification of spherically symmetric spacetimes according to their homothetic vectors and showed that these spacetimes possess 4, 5, 6, 8 or 11 homothetic vectors. A study of proper homothetic vectors in Bianchi type I spacetimes was carried out by Shabbir and Amur [38] who concluded that Bianchi type I spacetimes admit 4, 5, 7 or 11 homothetic vectors for some special choices of the metric functions. Proper homothetic vectors in Bianchi type IV and V spacetimes have been found by Ali and his collaborators [39, 40]. In Bianchi type IV spacetimes, the dimension of homothetic algebra turned out to be 4 or 5, while Bianchi type V spacetimes admit 7

24 homothetic vectors only in one case. Homothetic vectors in static cylindrically symmetric spacetimes are explored by Shabbir and Ramzan [41] and it is proved that the dimension of homothetic algebra for these spacetimes is 4, 5, 7 or 11. The details of classification of plane symmetric spacetimes according to their homothetic vectors can be seen in [42]. Parallel to Killing and homothetic symmetries, conformal Killing symmetry has been discussed for some well known spacetimes. Conformal Killing vectors in Robertson-Walker spacetimes are studied by Maartens and Ma- haraj [43]. Hall and Steele proved [44] that the dimension of conformal algebra for a conformally flat spacetime is 15, while for a non-conformally flat spacetime the maximum number of conformal Killing vectors is 7. Moopa- nar and Maharaj [45] studied the complete conformal geometry of spherically symmetric spacetimes with vanishing shear. For static spherically symmetric spacetimes, the conformal Killing’s equation is solved by Maartens et. al. [46]. The authors found that static spherically symmetric spacetimes admit at most two proper conformal Killing vectors. Hall and Capocci [47] published a remarkable paper about the maximum dimension of conformal algebra for three dimensional spacetimes in which they proved that for a non conformally flat three dimensional spacetime M, dim C(M) ≤ 4. For details of conformal symmetry in some other well known spacetimes, we re- fer [48–51]. As far as the inheriting conformal Killing vectors are concerned, they are investigated for spherically symmetric, spherically symmetric anisotropic fluid and synchronous perfect fluid spacetimes [18,52,53].

25 1.4 Fundamentals of Teleparallel Gravity

In this section we give a brief review of some basics of teleparallel gravity. Throughout this section, the Greek alphabets (µ, ν, ρ, ... = 0, 1, 2, 3) and the Latin alphabets (a, b, c, ... = 0, 1, 2, 3) are used to represent the indices related to spacetime and tangent space respectively.

1.4.1 Tetrad in Teleparallel Gravity

The tangent space TpM, at each point p of a four dimensional diﬀerentiable spacetime manifold M, is a Minkowski space whose Lorentz metric have the form:

ηab = diag (+1, −1, −1, −1) (1.4.1.1)

A relation between a spacetime manifold and its tangent space is established by a tetrad ﬁeld. If g and η signify the metrics of a spacetime and its

Minkowski tangent space with components gµν and ηab respectively, then a µ tetrad ﬁeld ha = ha ∂µ relates g with η by the relation [54]:

µ µ ηab = g(ha, hb) = gµνha hb (1.4.1.2)

µ A tetrad ﬁeld is also called vierbein. The inverse ﬁeld of the tetrad ha is a denoted by hµ and the following relation holds [55]:

a ν ν a µ a hµ ha = δµ , hµ hb = δb (1.4.1.3)

1.4.2 Weitzenb¨ock Connection and Torsion Tensor

In general relativity, the only geometric object which is considered as the basic entity is the metric tensor of a spacetime. The Riemann curvature

26 tensor is expressed in terms of Levi-Civita connections. The Einstein tensor is then expressed in terms of metric and Riemann curvature tensors and in this way the Einstein’s field equations are formed. However, gravitation can be described by torsion instead of curvature on a globally flat spacetime. Weitzenböck noted that a specific choice of the connections will ensure that the spacetime is globally flat. This idea gave rise to Weitzenböck connections, which can be defined with the help of tetrad field as follows. a The covariant derivative of a tetrad field hµ is given by [55]:

a a θ a ∇νhµ = ∂νhµ − Γ µνhθ , (1.4.2.1)

θ where Γ µν denotes the Weitzenböck connection which introduces the distant parallelism in a spacetime manifold. As mentioned in the introduction section of this thesis, teleprallel theory of gravitation depends on Weitzenböck geometry. The tetrads in Weitzenböck spacetimes are parallelly transported, a so that ∇νhµ = 0 which gives the following form of the Weitzenböck connection in terms of tetrad.

θ θ a Γ µν = ha ∂νhµ (1.4.2.2)

The torsion tensor can be expressed in terms of Weitzenb¨ock connections as [55]:

θ θ θ T µν = Γ νµ − Γ µν (1.4.2.3)

θ θ The torsion tensor is antisymmetric in its lower indices, that is T µν = −T νµ. If 0Γ denotes the Levi-Civita connection, then it is related to the Weitzenb¨ock connection Γ as follows:

0 θ θ θ Γµν = Γµν − Kµ ν, (1.4.2.4)

27 where K is the contorsion tensor and it can be written in terms of torsion tensor as: 1 K θ = {T θ − T θ + T θ } (1.4.2.5) µ ν 2 µν νµ µ ν

We can obtain the torsion vector Tµ by contracting the torsion tensor as:

θ Tµ = T θµ (1.4.2.6)

1.5 Spacetime Symmetries in Teleparallel Grav- ity

In section (1.3), we have already discussed the importance and applications of spacetime symmetries in general relativity. It was also mentioned in the same section that Lie derivative operator is very helpful to find the spacetime symmetries because it gives the explicit form of the Killing’s equation and by solving it we can find Killing vectors admitted by a spacetime metric. Similar remarks hold in case of finding homothetic and conformal Killing vectors. In 2008, Sharif and Amir [56] defined teleparallel Lie derivative of a tensor of rank p + q in the context of teleparallel gravity. In particular, they noted that the teleparallel version of Lie derivative of the metric tensor is different from that of general relativity because of introducing the torsion tensor terms. This new version of teleparallel Lie derivative of a metric tensor is given by [56]:

T c c c c d d LX gab = gab,cX + gbcX,a + gacX,b + X [gadTbc + gbdTac] (1.5.1)

28 The authors in [56] deﬁned teleparallel Killing vectors to be those vectors which satisfy the following condition.

T c c c c d d LX gab = gab,cX + gbcX,a + gacX,b + X [gadTbc + gbdTac] = 0 (1.5.2)

Similarly teleparallel homothetic and teleparallel conformal Killing vectors can be deﬁned as the vectors which satisfy:

T c c c c d d LX gab = gab,cX + gbcX,a + gacX,b + X [gadTbc + gbdTac] = 2αgab, (1.5.3) where α is a constant or it depends on spacetime coordinates when X is a homothetic or a conformal Killing vector. As a pioneer, Sharif and Amir [56] used equation (1.5.2) to investigate teleparallel Killing vectors in Einstein universe. Although the teleparallel Killing vectors obtained for Einstein universe in this way were coincident with those of general relativity, this idea opened a new way of exploring the spacetime symmetries in the context of teleparallel gravity. Sharif and Majeed [57] followed this way to find teleparallel Killing vectors for static spherically symmetric spacetimes. They observed that for static spherically symmetric spacetimes, the teleparallel Killing vectors are not the same as the Killing vectors of general relativity, however the results of both theories may coincide under certain conditions. Shabir et. al. [58–63] solved equation (1.5.2) for finding teleparallel Killing vectors in Bianchi type I, II, III, Kantowski-Sachs, static cylindrically symmetric, non static cylindrically symmetric and spatially homogeneous rotating spacetimes. In Bianchi type I spacetimes, teleparallel Killing vectors turned out to be the same in number as those in general relativity, however teleparallel Killing vectors arise as multiples of some specific functions of t, which are actually the components of the inverse tetrad field [58]. For Bianchi type

29 II, non static cylindrically symmetric and spatially homogeneous rotating spacetimes, it was observed that the teleparallel Killing vectors are more than the Killing vectors of general relativity [59, 62, 63], which in turn give more conservation laws and this increase in the conservation laws arises because of the absence of curvature and presence of torsion in the spacetimes. The teleparallel Killing vectors in Bianchi type III, Kantowski-Sachs and static cylindrically symmetric spacetimes are either same or multiples of the corresponding Killing vectors of general relativity in different cases [60, 61]. Beside these spacetimes, recently teleparallel Killing vectors are also explored in Bianchi type V and FRW spacetimes by Khan et. al. [64,65]. Teleparallel homothetic vectors are also investigated for some well known spacetimes. Shabir and Khan [66–68] classified static cylindrically symmetric, non static plane symmetric and Bianchi type I spacetimes according o their teleparallel homothetic vectors. The authors concluded that in these spacetimes, a proper teleparallel homothetic vector exists for specific values of the metric functions and one can recover the homothetic vectors of general relativity from the obtained teleparallel homothetic vectors by vanishing torsion tensor. As far as teleparallel conformal Killing vectors are concerned, they are found for only few spacetimes. Before us, teleparallel conformal Killing vectors were explored only for Bianchi type I and static cylindrically symmetric spacetimes [69,70]. Although spacetime symmetries are widely discussed in the context of general relativity and teleparallel gravity, the field is still open for further research. A number of spacetimes are still left to be considered for spacetime symmetries in both the theories, especially the conformal Killing vectors are investigated

30 for a less number of spacetimes as compared to Killing and homothetic vectors.

1.6 Outlines of Work

This thesis has been designed as follows:

1. In chapter 1, we have brieﬂy introduced some basics of general relativity and teleparallel gravity. These basics will be helpful to understand this thesis.

2. In chapter 2, conformal Killing symmetries for LRS Bianchi type V, static plane symmetric and non static plane symmetric spacetimes are found in the context of general relativity. Out of this work, two research papers [71,72] have been published.

3. Chapter 3 provides a study of teleparallel conformal Killing symmetries for LRS Bianchi type V, static plane symmetric and non static plane symmetric spacetimes in the framework of teleparallel gravity. On the basis of this work, two research papers [73,74] have been published.

4. Chapter 4 deals with teleparallel Killing and homothetic symmetries of Kantowski-Sachs and Lemaitre-Tolman-Bondi metrics. One research paper [75] on this work has been published.

5. In chapter 5, we have explored teleparallel Killing and homothetic symmetries in three dimensional circularly symmetric static metric. Out of this work, one research paper [76] is published.

31 6. In last chapter, a brief summary, our ﬁndings, comparison of the obtained results in general relativity and teleparallel gravity and some open problems are mentioned.

32 Chapter 2

Conformal Killing Vectors in LRS Bianchi Type V, Static and Non Static Plane Symmetric Spacetimes

In this chapter, we investigate conformal Killing vectors in locally rotationally symmetric (LRS) Bianchi type V, static and non static plane symmetric spacetimes in the context of general relativity. For all the three mentioned spacetimes, the components of conformal Killing vectors are obtained up to some unknown functions of the variables t and x, subject to some integrability conditions. To find these unknown functions, the integrability conditions are solved for some specific choices of the metric functions in each case, giving the final form of conformal Killing vectors. Inheriting conformal Killing vectors are also explored for each of the three mentioned spacetimes. Lie algebra of

33 conformal and inheriting conformal Killing vectors is calculated in each case. It is important to mention here that conformal Killing vectors for static plane symmetric spacetimes are already discussed by Saifullah and Yazdan [77]. They concluded with the remarks that these spacetimes do not admit any proper conformal Killing vector, unless they become conformally flat. However, we found some proper conformal killing vectors in non conformally flat static plane symmetric spacetimes by solving the integrability conditions using a different approach to that of Saifullah and Yazdan.

2.1 Conformal Killing Vectors in LRS Bianchi Type V Spacetimes

The Bianchi type I-IX spacetime metrics are cosmological models which admit 3-dimensional isometry group G3, acting on the hypersurfaces of homo- geneity. Out of these models, the Bianchi type V universe is the general- ization of FRW model, which is homogeneous and isotropic. In Cartesian coordinates, the LRS Bianchi type V spacetimes metric is of the form [1]:

ds2 = −dt2 + A2 dx2 + e2mxB2 dy2 + dz2 , (2.1.1) where m is a constant and the metric functions A and B are dependent on t only. The above metric reduces to LRS Bianchi type I spacetime metric if m = 0 and conformal symmetries in Bianchi type I spacetimes have been recently discussed by Tsamparlis et. al. [20]. Thus throughout our analysis, we will take m as a non zero constant. The metric given in (2.1.1) admits

34 the following four linearly independent spatial Killing vectors [1]:

X1 = ∂y,X2 = ∂z,X3 = ∂x − my∂y − mz∂z,X4 = y ∂z − z ∂y. (2.1.2)

As mentioned in chapter 1, a conformal Killing vector X satisﬁes [17]:

c c c LX gab = gab,c X + gbc X,a + gac X,b = 2αgab, (2.1.3) where α, for the Bianchi type V spacetimes, is a function of the variables t, x, y and z. For the metric (2.1.1), Eq. (2.1.3) give rise to the following system of ten coupled partial diﬀerential equations:

0 X,0 = α, (2.1.4)

2 1 0 A X,0 − X,1 = 0, (2.1.5)

2mx 2 2 0 e B X,0 − X,2 = 0, (2.1.6)

2mx 2 3 0 e B X,0 − X,3 = 0, (2.1.7) ˙ 0 1 AX + AX,1 = αA, (2.1.8)

2mx 2 2 2 1 e B X,1 + A X,2 = 0, (2.1.9)

2mx 2 3 2 1 e B X,1 + A X,3 = 0, (2.1.10) ˙ 0 1 2 BX + mBX + BX,2 = αB, (2.1.11)

2 3 X,3 + X,2 = 0, (2.1.12) ˙ 0 1 3 BX + mBX + BX,3 = αB. (2.1.13)

In the above set of equations, a dot on a metric function is used to denote its derivative with respect to t and the commas in subscript are used to represent partial derivatives with respect to the spacetime coordinates. The simultaneous solution of this system will give conformal Killing vectors in

35 LRS Bianchi type V spacetimes. The procedure of solving the above system is explained below: Diﬀerentiating Eqs. (2.1.9), (2.1.10) and (2.1.12) with respect to z, y and x respectively, we have:

2mx 2 2 2 1 e B X,13 + A X,23 = 0, (2.1.14)

2mx 2 3 2 1 e B X,12 + A X,23 = 0, (2.1.15)

2 3 X,13 + X,12 = 0. (2.1.16)

Subtracting Eq. (2.1.15) from Eq. (2.1.14) and then adding the resulting

2 3 equation with Eq. (2.1.16), we get X,13 = 0 and X,12 = 0. Putting back 1 these values in Eq. (2.1.14), we obtain X,23 = 0. Thus we have:

1 2 3 X,23 = X,13 = X,12 = 0. (2.1.17)

Similarly if we diﬀerentiate Eqs. (2.1.6), (2.1.7) and (2.1.12) with respect to z, y and t respectively, we ﬁnd that:

0 2 3 X,23 = X,03 = X,02 = 0. (2.1.18)

Solving Eqs. (2.1.17) and (2.1.18) simultaneously, we obtain the following system:

X0 = f 1(t, x, y) + f 2(t, x, z),

X1 = f 3(t, x, y) + f 4(t, x, z),

2 5 1 2 X = fy (t, x, y) + Fy (t, y) + F (y, z),

3 6 3 4 X = fz (t, x, z) + Fz (t, z) + Fz (y, z), (2.1.19) where f i and F j are unknown functions of integration which arise during the integration process, for i = 1, ..., 6 and j = 1, ..., 4. Using the system (2.1.19)

36 in Eqs. (2.1.6), (2.1.7), (2.1.9), (2.1.10) and (2.1.12) and doing some simple algebraic calculations, we get: 0 2mx 2 7 8 5 X = e B ft (t, x, y) + ft (t, x, z) + F (t, x), B2 X1 = − e2mx f 7(t, x, y) + f 8(t, x, z) + F 6(t, x), A2 x x 2 7 7 X = fy (t, x, y) − Fy (y, z),

3 8 7 X = fz (t, x, z) + Fz (y, z), (2.1.20) where f 7(t, x, y) = f 5(t, x, y) + F 1(t, y), f 8(t, x, z) = f 6(t, x, z) + F 3(t, z) and F k denote functions of integration, for k = 5, 6, 7. Subtracting Eq. (2.1.13)

2 3 form Eq. (2.1.11), we get X2 = X3 , which by using the above system gives:

7 7 8 7 fyy(t, x, y) − Fyy(y, z) = fzz(t, x, z) + Fzz(y, z). (2.1.21)

Diﬀerentiating Eq. (2.1.21) with respect to t, we have:

7 8 ftyy(t, x, y) = ftzz(t, x, z). (2.1.22)

7 Diﬀerentiating Eq. (2.1.22) with respect to y, we obtain ftyyy(t, x, y) = 0 and integrating this equation repeatedly with respect to y and t, we get:

1 f 7(t, x, y) = y2F 8(t, x) + yF 9(t, x) + F 10(t, x) + F 11(x, y). (2.1.23) 2

Using this value of f 7(t, x, y) in Eq. (2.1.22) and integrating the resulting equation, we ﬁnd that:

1 f 8(t, x, z) = z2F 8(t, x) + zF 12(t, x) + F 13(t, x) + F 14(x, z). (2.1.24) 2

The functions F n, for n = 8, ..., 14, appearing in the last two equations are functions of integration. Using the above values of f 7(t, x, y) and f 8(t, x, z)

37 in Eq. (2.1.21) and diﬀerentiating the resulting equation with respect to x, we obtain:

11 14 Fxyy(x, y) = Fxzz(x, z). (2.1.25)

11 If we diﬀerentiate Eq. (2.1.25) with respect to y, it gives Fxyyy(x, y) = 0, which in turn yields:

1 F 11(x, y) = y2 G1(x) + y G2(x) + G3(x) + G4(y). (2.1.26) 2

Putting back this value in Eq. (2.1.25), we have:

1 F 14(x, z) = z2 G1(x) + z G5(x) + G6(x) + G7(z), (2.1.27) 2 where Gl, for l = 1, ..., 7 are functions of integration. Substituting all these values in the system (2.1.20), we get:

1 X0 = e2mxB2 (y2 + z2)F 8(t, x) + yF 9(t, x) + zF 12(t, x) + F 10(t, x) 2 t t t t 13 5 + Ft (t, x) + F (t, x), B2 1 X1 = − e2mx (y2 + z2)F 8(t, x) + yF 9(t, x) + zF 12(t, x) + F 10(t, x) A2 2 x x x x 1 + F 13(t, x) + (y2 + z2)G1(x) + yG2(x) + zG5(x) + G3(x) + G6(x) x 2 x x x x x + F 6(t, x),

2 8 9 1 2 4 7 X = yF (t, x) + F (t, x) + yG (x) + G (x) + Gy(y) − Fy (y, z),

3 8 12 1 5 7 7 X = zF (t, x) + F (t, x) + zG (x) + G (x) + Gz(z) + Fz (y, z). (2.1.28)

Once we get the ﬁnal form of X0, the conformal factor α can be easily found from Eq. (2.1.4). In the above system , some functions are redundant and

38 can be removed if we set F 10(t, x) + F 13(t, x) + G3(x) + G6(x) = F 15(t, x), F 8(t, x) + G1(x) = F 16(t, x), F 9(t, x) + G2(x) = F 17(t, x) and F 12(t, x) + G5(x) = F 18(t, x) . With these substitutions, the system (2.1.28) gets the form:

1 X0 = e2mxB2 (y2 + z2)F 16(t, x) + yF 17(t, x) + zF 18(t, x) + F 15(t, x) 2 t t t t + F 5(t, x), B2 1 X1 = − e2mx (y2 + z2)F 16(t, x) + yF 17(t, x) + zF 18(t, x) + F 15(t, x) A2 2 x x x x + F 6(t, x),

2 16 17 4 7 X = yF (t, x) + F (t, x) + Gy(y) − Fy (y, z),

3 16 18 7 7 X = zF (t, x) + F (t, x) + Gz(z) + Fz (y, z). (2.1.29)

2mx 2 15 B2 2mx 15 Moreover, we can merge the terms e B Ft (t, x) and − A2 e Fx (t, x) in to the functions F 5(t, x) and F 6(t, x) respectively, which allow us to rewrite the system (2.1.29) as follows:

1 X0 = e2mxB2 (y2 + z2)F 16(t, x) + yF 17(t, x) + zF 18(t, x) + F 5(t, x), 2 t t t B2 1 X1 = − e2mx (y2 + z2)F 16(t, x) + yF 17(t, x) + zF 18(t, x) + F 6(t, x), A2 2 x x x 2 16 17 4 7 X = yF (t, x) + F (t, x) + Gy(y) − Fy (y, z),

3 16 18 7 7 X = zF (t, x) + F (t, x) + Gz(z) + Fz (y, z). (2.1.30)

Using this system in Eq. (2.1.4), we can ﬁnd the conformal factor as:

1 α = e2mxB2 (y2 + z2)F 16(t, x) + yF 17(t, x) + zF 18(t, x) 2 tt tt tt 1 + 2e2mxBB˙ (y2 + z2)F 16(t, x) + yF 17(t, x) + zF 18(t, x) + F 5(t, x). 2 t t t t (2.1.31)

39 Diﬀerentiating Eq. (2.1.11) with respect to y and z respectively, we get

2 X,223 = 0 and using the system (2.1.30) in this identity, we have: 1 F 7 (y, z) = 0 ⇒ F 7(y, z) = G8(y)+ y2G9(z)+yG10(z)+G11(z), (2.1.32) yyyz 2

Gk, for k = 8, ..., 11, being functions of integration. The above value of F 7(y, z) can be rewritten in the following form if we diﬀerentiate Eq. (2.1.13) with respect to y and z:

1 nc o nc o F 7(y, z) = G8(y) + y2 1 z2 + c z + c + y 4 z2 + c z + c + G11(z). 2 2 2 3 2 5 6 (2.1.33) Subtracting Eq. (2.1.13) from Eq. (2.1.11) and using the system (2.1.30) and the above value of F 7(y, z) in the resulting equation, we obtain:

c c G4(y) − G8(y) = 1 y3 + 4 y2 + c y + c , (2.1.34) y y 6 2 7 8 c c G7(z) + G11(z) = − 1 z3 − 2 z2 + c z − c z + c . (2.1.35) z z 6 2 7 3 9

Let us denote c7 − c3 = c10 and c8 − c6 = c11. Using the values from Eqs. (2.1.33)-(2.1.35) in the system (2.1.30), we get the following form of the components of conformal Killing vectors, up to some unknown functions of t and x.

1 X0 = e2mxB2 (y2 + z2)F 16(t, x) + yF 17(t, x) + zF 18(t, x) + F 5(t, x), 2 t t t B2 1 X1 = − e2mx (y2 + z2)F 16(t, x) + yF 17(t, x) + zF 18(t, x) + F 6(t, x), A2 2 x x x c X2 = yF 16(t, x) + F 17(t, x) + 4 y2 − z2 − c yz − c z + c y + c , 2 2 5 10 11 c X3 = zF 16(t, x) + F 18(t, x) + 2 y2 − z2 + c yz + c y + c z + c . 2 4 5 10 9 (2.1.36)

40 The conformal factor is same as given in Eq. (2.1.31). Using the system (2.1.36) and the conformal factor in Eqs. (2.1.5), (2.1.8), (2.1.11) and equating the like terms of the resulting equations, we obtain the following set of integrability conditions:

2 16 2 ˙ 16 16 A BFtt + A BFt + mBFx = 0, (2.1.37) 2 2 17 2 ˙ 17 2 17 2 −2mx A B Ftt + A BBFt + mB Fx − c4A e = 0, (2.1.38) 2 2 18 2 ˙ 18 2 18 2 −2mx A B Ftt + A BBFt + mB Fx + c2A e = 0, (2.1.39) 2 16 ˙ ˙ 16 16 16 A BFtt + A(2AB − AB)Ft + BFxx + 2mBFx = 0, (2.1.40) 2 17 ˙ ˙ 17 17 17 A BFtt + A(2AB − AB)Ft + BFxx + 2mBFx = 0, (2.1.41) 2 18 ˙ ˙ 18 18 18 A BFtt + A(2AB − AB)Ft + BFxx + 2mBFx = 0, (2.1.42) B2 ˙ 2B2F 16 + 2mB2F 16 + A2 F 16 = 0, (2.1.43) tx t A2 x B2 ˙ 2B2F 17 + 2mB2F 17 + A2 F 17 = 0, (2.1.44) tx t A2 x B2 ˙ 2B2F 18 + 2mB2F 18 + A2 F 18 = 0, (2.1.45) tx t A2 x 2 6 5 A Ft − Fx = 0, (2.1.46) ˙ 5 6 5 AF + AFx − AFt = 0, (2.1.47) ˙ 5 6 16 5 BF + mBF + BF − BFt = 0. (2.1.48)

y2+z2 At this stage, we introduce the new variables βi = (β1, β2, β3) = 2 , z, y , P i = (P 1,P 2,P 3) = (F 16,F 18,F 17), P 0 = F 5 and P 4 = F 6. With these variables, the conformal Killing vectors in (2.1.36) and the conformal factor in (2.1.31) can be rewritten in a more compact form as:

0 2mx 2 i 0 X = e B βiPt + P , B2 X1 = − e2mxβ P i + P 4, A2 i x 41 c X2 = β P i + 4 y2 − z2 − c yz − c z + c y + c , i,2 2 2 5 10 11 c X3 = β P i + 2 y2 − z2 + c yz + c y + c z + c , i,3 2 4 5 10 9 2mx 2 i 2mx ˙ i 0 α = e B βiPtt + 2e BBβiPt + Pt . (2.1.49)

Accordingly, the integrability conditions reduce to the following equations:

2 2 i 2 ˙ i 2 i 2 −2mx A B Ptt + A BBPt + mB Px − kiA e = 0, (2.1.50) B2˙ 2B2P i + 2mB2P i + A2 P i = 0, (2.1.51) tx t A2 x 2 i ˙ ˙ i i i A BPtt + A 2AB − AB Pt + BPxx + 2mBPx = 0, (2.1.52)

2 4 0 A Pt − Px = 0, (2.1.53) ˙ 0 4 0 AP + APx − APt = 0, (2.1.54) ˙ 0 4 1 0 BP + mBP + BP − BPt = 0, (2.1.55) where ki = 0, −c2, c4, for i = 1, 2, 3 respectively. To find the final form of conformal Killing vectors, we need to solve the above integrability conditions. It can be noticed that the above equations are highly non linear and cannot be solved directly as they stand. We solve the above system by assuming that P k(t, x) = Rk(t) + Hk(x), for k = 1, ..., 4. First let us find P 1 by setting P 1(t, x) = R1(t) + H1(x). Taking i = 1 in Eq. (2.1.50) and differentiating

1 1 it with respect to x, we obtain Hxx(x) = 0 ⇒ H (x) = c12x + c13, where c12, c13 ∈ R. Multiplying Eq. (2.1.50) by 2, Eq. (2.1.52) by B and then 1 ˙ 1 subtracting the resulting equations, we get ARtt(t) + ARt (t) = 0, so that 1 R −1 1 R −1 R (t) = c14 A dt + c15. Hence P = c12x + c14 A dt + c16, where 1 c16 = c13 + c15. Putting this value of P in Eqs. (2.1.50)-(2.1.52), for i = 1, we have the following two equations.

˙ ˙ c14AB − c14AB + c12mB = 0 , (2.1.56)

42 ˙ ˙ c12AB − c12AB + c14mB = 0 . (2.1.57)

One can easily solve the above two equations to get c12 = c14. Putting back c12 = c14 in Eqs. (2.1.56) and (2.1.57), we obtain:

AB˙ − AB˙ + mB = 0. (2.1.58)

1 1 R −1 Also, the value of the function P becomes P = c12x + c12 A dt + c16. This value of P 1 satisfy the integrability conditions (2.1.50)-(2.1.52) provided that the condition given in Eq. (2.1.58) holds. Now since P 4 = R4(t) + H4(x), so rearranging Eq. (2.1.54) and integrating

0 4 R −1 12 it with respect to t, we can write P = AHx(x) A dt + AG (x), where G12(x) is a function of integration. Putting this value of P 0 in Eq. (2.1.53) and then diﬀerentiating it with respect to t, we obtain:

n o 4 ˙ 4 4 A ARtt(t) + ARt (t) = Hxx(x) = c17,

where c17 is a separation constant. Integrating the above equation, we have

4 R −1 R −1 R −1 4 c17 2 R (t) = c17 A A dt dt+c18 A dt+c19, H (x) = 2 x +c20x+c21. 12 Putting back these values in Eq. (2.1.53), we get G (x) = c18x + c22. Hence we have: Z 0 −1 P = (c17x + c20) A A dt + (c18x + c22) A, Z Z Z c P 4 = c A−1 A−1dt dt + c A−1dt + 17 x2 + c x + c , 17 18 2 20 23 (2.1.59)

where c23 = c19 +c21. Diﬀerentiating Eq. (2.1.55) twice with respect to x and doing some simple algebraic manipulation, we obtain c12 = c17 = 0, c18 = c20

43 c20 c16 0 4 and c23 = c22 + m − m . Thus the values of P and P in (2.1.59) are reduced to: Z 0 −1 P = c18 A A dt + c18xA + c22A, Z c c P 4 = c A−1dt + c x + c + 18 − 16 . (2.1.60) 18 18 22 m m

1 1 Also by putting c12 = 0, the function P becomes P = c16. To ﬁnd the function P 2, we assume that P 2(t, x) = R2(t) + H2(x). Diﬀeren-

2 tiating Eq. (2.1.51) for i = 2, we get Pxx = 0. Diﬀerentiating Eq. (2.1.50) 2 with respect to x and using Pxx = 0, we obtain ki = 0 ⇒ c2 = c4 = 0. The remaining procedure for ﬁnding P 2 is same as that of P 1, consequently

2 3 we have P = c24. Similarly we can show that P = c25. Putting back all these values in (2.1.49), we have the following ﬁnal form of conformal Killing vectors: Z 0 −1 X = c18 A A dt + c18xA + c22A, Z c c X1 = c A−1dt + c x + c + 18 − 16 , 18 18 22 m m 2 X = −c5z + c16y + c11,

3 X = c5y + c16z + c9, Z ˙ −1 ˙ ˙ α = c18 + c18 A A dt + c18xA + c22A, (2.1.61)

where we have merged the constants c10, c24 and c25 in c16, c9 and c11 respectively. The above conformal Killing vectors are given subject to the condition given in Eq. (2.1.58). From (2.1.61), it is clear that LRS Bianchi type V spacetimes admit six conformal Killing vectors, provided that the metric functions A and B satisfy the condition given in Eq. (2.1.58). In order to

44 obtain the conformal Killing vectors in some particular cases, we consider the following additional cases:

Case 1: If we take A = A(t) and B = m1, where m1 is a non zero constant, then Eq. (2.1.58) can be easily solved to get A = mt + m2, where m2 ∈ R. Thus the metric given by Eq. (2.1.1), after a suitable rescaling, can be written as:

ds2 = −dt2 + m2t2 dx2 + e2mx dy2 + dz2 . (2.1.62)

For this metric, the conformal Killing vectors and conformal factor in (2.1.61) reduce to:

nc o X0 = mt 18 ln mt + c x + c , m 18 22 c c c X1 = 18 ln mt + c x + c + 18 − 16 , m 18 22 m m 2 X = −c5z + c16y + c11,

3 X = c5y + c16z + c9,

α = c18 + c18 ln mt + c18mx + c22m. (2.1.63)

Thus the metric given in Eq. (2.1.62) admits six conformal Killing vectors with one proper conformal Killing vector given by:

1 1 X = (t ln mt + mtx) ∂ + ln mt + x + ∂ . 6 t m m x

The dimension of homothetic algebra is ﬁve with one proper homothetic vector, which can be written as X5 = mt ∂t + ∂x. The dimension of group of isometries in this case is four with the four Killing vectors same as mentioned in (2.1.2). The non zero commutators of Lie algebra of the above six

45 conformal Killing vectors are given by:

[X1,X3] = −m X1, [X1,X4] = −X2, [X2,X3] = −m X2,

[X2,X4] = X1, [X3,X6] = X5, [X5,X6] = 2X5.

Case 2: Here we take A = m1, where m1 6= 0 ∈ R and B = B(t). With mt − m these values, Eq. (2.1.58) yields B = m2 e 1 . Assuming m1 = m2 = 1, the LRS Bianchi type V spacetimes metric (2.1.1) takes the form:

ds2 = −dt2 + dx2 + e2m(x−t) dy2 + dz2 . (2.1.64)

For this metric, the conformal Killing vectors in (2.1.61) reduce to:

0 X = c18t + c18x + c22, c c X1 = c t + c x + c + 18 − 16 , 18 18 22 m m 2 X = −c5z + c16y + c11,

3 X = c5y + c16z + c9,

α = c18, (2.1.65) which shows that the metric given in (2.1.64) admit no proper conformal Killing vector and the conformal Killing vectors in this case are homothetic vectors with one proper homothetic vector, given by: 1 X = (t + x) ∂ + t + x + ∂ . 6 t m x Also the dimension of isometry group is ﬁve. Four Killing vectors are same as mentioned in (2.1.2) and the ﬁfth Killing vector is expressible in the form

X5 = ∂t + ∂x. In this case the Lie algebra of the above homothetic vectors has the following non zero commutators:

[X1,X3] = −m X1, [X1,X4] = −X2, [X2,X3] = −m X2,

[X2,X4] = X1, [X3,X6] = X5, [X5,X6] = 2X5.

46 2.2 Inheriting Conformal Killing Vectors in LRS Bianchi type V Spacetimes

The conformal Killing vectors, up to some unknown functions of t and x, for LRS Bianchi type V spacetimes are given in (2.1.49) along with some integrability conditions (2.1.50)-(2.1.55). In this section, we take X as an inheriting conformal Killing vector. As mentioned in chapter 1, such type of conformal Killing vectors satisfy the following additional condition [18]:

b b LX ua = ua,b X + ub X,a = α ua, (2.2.1)

a a where ua signiﬁes four velocity of the ﬂuid. We choose u = δ0 , then the relation (2.2.1) produces the following equations:

0 X,0 = α, (2.2.2)

0 X,i = 0, i = 1, 2, 3 (2.2.3) which suggests that in (2.1.49) we must have X0 = P 0(t) and P i = P i(x), for i = 1, 2, 3. Also Eq. (2.1.53) implies that P 4 = P 4(x). The remaining integrability conditions reduce to:

2 i 2 −2mx mB Px(x) − kiA e = 0 , (2.2.4) B2 ˙ P i(x) = 0 , (2.2.5) A2 x i i Pxx(x) + 2mPx(x) = 0 , (2.2.6) ˙ 0 4 0 AP (t) + APx (x) − APt (t) = 0 , (2.2.7) ˙ 0 4 1 0 BP (t) + mBP (x) + BP (x) − BPt (t) = 0 . (2.2.8)

From Eq. (2.2.5), two possible cases arise, namely, B 6= cA and B = cA, where c is a non zero constant. Taking B 6= cA, one can easily solve the

47 system of Eqs. (2.2.4)-(2.2.8) to get the following inheriting conformal Killing vectors:

0 X = c12A, c c X1 = 12 − 13 , m m 2 X = c13y − c5z + c14,

3 X = c13z + c5y + c15, ˙ α = c12A, (2.2.9)

A ˙ subject to the condition B B =1. This result reveals that in this case the LRS Bianchi type V metric admits ﬁve inheriting conformal Killing vectors with only one proper inheriting conformal Killing vector, which is given by

1 X5 = A ∂t + m ∂x. The dimension of homothetic algebra turned out to be 4, with no proper homothetic vector and the dimension of isometry group is also four with four Killing vectors as given in (2.1.2). The Lie algebra of above inheriting conformal Killing vectors has the following non-vanishing commutators:

[X1,X3] = −m X1, [X1,X4] = −X2, [X2,X3] = −m X2, [X2,X4] = X1.

For the second case, that is when B = cA, the metric given in (2.1.2) reduces to: ds2 = −dt2 + A2dx2 + A2e2mx dy2 + dz2 . (2.2.10)

Solving Eqs. (2.2.4)-(2.2.8)) for the above metric, we obtain the following inheriting conformal Killing vectors:

0 X = c12 A, c z c y c X1 = 2 − 4 − 10 , m m m 48 c c X2 = − 4 e−2mx + 4 y2 − z2 − c yz − c z + c y + c , 2c2m2 2 2 5 10 11 c c X3 = 2 e−2mx + 2 y2 − z2 + c yz + c y + c z + c , 2c2m2 2 4 5 10 9 ˙ α = c12 A, (2.2.11) which shows that the metric given in (2.2.10) admits seven inheriting conformal Killing vectors, one being proper inheriting conformal Killing vector which is expressible as X7 = A ∂t. In this case no proper homothetic vector exist and the dimension of isometry group is six. Four Killing vectors are same as given in (2.1.2)) and the other two Killing vectors can

z 1 1 −2mx 2 2 be written as X5 = m ∂x − yz∂y + 2 c2m2 e + y − z ∂z and X6 = −y 1 −1 −2mx 2 2 m ∂x + yz∂z + 2 c2m2 e + y − z ∂y. The Lie algebra of above inheriting conformal Killing vectors has the following non-vanishing commutators: 1 [X ,X ] = −m X , [X ,X ] = −X , [X ,X ] = X , [X ,X ] = − X , 1 3 1 1 4 2 1 5 4 1 6 m 3 1 [X ,X ] = −m X , [X ,X ] = X , [X ,X ] = − X , [X ,X ] = −X , 2 3 2 2 4 1 2 5 m 3 2 6 4

[X3,X5] = −m X5, [X3,X6] = −m X6, [X4,X5] = −X6, [X4,X6] = X5.

2.3 Conformal Killing Vectors in Static Plane Symmetric Spacetimes

Plane symmetric spacetimes are Lorentzian manifolds admitting 3-dimensional isometry group G3 in such a way that the group orbits are spacelike surfaces of constant curvature. In particular, static plane symmetric spacetimes admit an additional timelike Killing vector. The most general form of static plane symmetric spacetimes metric is [1]:

ds2 = −e2A(x)dt2 + dx2 + e2B(x) dy2 + dz2 , (2.3.1)

49 where the metric functions A and B are dependent on x only. The above metric admits the following four linearly independent Killing vectors [1]:

X1 = ∂t,X2 = ∂y,X3 = ∂z,X4 = z∂y − y∂z. (2.3.2)

Following are the conformal Killing’s equations produced by Eq. (1.3.1) for the metric (2.3.1).

0 1 0 A X + X,0 = α, (2.3.3)

1 2A 0 X,0 − e X,1 = 0, (2.3.4)

2B 2 2A 0 e X,0 − e X,2 = 0, (2.3.5)

2B 3 2A 0 e X,0 − e X,3 = 0, (2.3.6)

1 X,1 = α, (2.3.7)

2B 2 1 e X,1 + X,2 = 0, (2.3.8)

2B 3 1 e X,1 + X,3 = 0, (2.3.9)

0 1 2 B X + X,2 = α, (2.3.10)

2 3 X,3 + X,2 = 0, (2.3.11)

0 1 3 B X + BX,3 = α, (2.3.12) where a prime on a metric function is used to denote its derivative with respect to x and the commas in subscript are used to represent partial derivatives with respect to the spacetime coordinates. The conformal factor α is a function of t, x, y and z. Using the same procedure which we have used for the solution of conformal Killing’s equations of Bianchi type V spacetimes, Saifullah and Yazdan [77] solved the above system to obtain the following conformal Killing vectors,

50 up to some unknown functions of t and x: 1 X0 = e2(B−A) (y2 + z2)P 1(t, x) + zP 2(t, x) + yP 3(t, x) + P 0(t, x), 2 t t t 1 X1 = −e2B (y2 + z2)P 1(t, x) + zP 2(t, x) + yP 3(t, x) + P 4(t, x), 2 x x x c X2 = yP 1(t, x) + P 3(t, x) + 3 z2 − y2 + c yz + c z, 2 2 4 c X3 = zP 1(t, x) + P 2(t, x) + 2 y2 − z2 − c yz − c y, 2 3 4 1 α = −e2B (y2 + z2)P 1 (t, x) + zP 2 (t, x) + yP 3 (t, x) 2 xx xx xx 1 − 2B0e2B (y2 + z2)P 1(t, x) + zP 2(t, x) + yP 3(t, x) + P 4(t, x), 2 x x x x (2.3.13) subject to the following integrability conditions:

1 0 0 1 Ptx(t, x) + (B − A ) Pt (t, x) = 0, (2.3.14)

2 0 0 2 Ptx(t, x) + (B − A ) Pt (t, x) = 0, (2.3.15)

3 0 0 3 Ptx(t, x) + (B − A ) Pt (t, x) = 0, (2.3.16)

0 1 1 B Px (t, x) + Pxx(t, x) = 0, (2.3.17)

0 2 2 −2B B Px (t, x) + Pxx(t, x) + c2e = 0, (2.3.18)

0 3 3 −2B B Px (t, x) + Pxx(t, x) − c3e = 0, (2.3.19)

0 0 1 −2A 1 1 (2B − A ) Px (t, x) + e Ptt(t, x) + Pxx(t, x) = 0, (2.3.20)

0 0 2 −2A 2 2 (2B − A ) Px (t, x) + e Ptt(t, x) + Pxx(t, x) = 0, (2.3.21)

0 0 3 −2A 3 3 (2B − A ) Px (t, x) + e Ptt(t, x) + Pxx(t, x) = 0, (2.3.22)

2A 0 4 e Px (t, x) − Pt (t, x) = 0, (2.3.23)

0 4 0 4 A P (t, x) + Pt (t, x) − Px (t, x) = 0, (2.3.24)

0 4 1 4 B P (t, x) + P (t, x) − Px (t, x) = 0. (2.3.25)

51 The authors in [77] solved the above integrability conditions for the cases when (i) A0 = 0, B0 6= 0 (ii) A0 6= 0, B0 = 0 and (iii) A0 = 0, B0 = 0. They observed that in all these cases, either the spacetimes under consideration become conformally flat or the conformal Killing vectors reduce to Killing or homothetic vectors. Thus non conformally flat static plane symmetric spacetimes do not admit any proper conformal Killing vector [77]. How- ever, we believe that this study of conformal Killing vectors in static plane symmetric spacetimes is incomplete. In fact one can check that each Weyl tensor component for the metric in (2.3.1) is a multiple of the expression A0 (B0 − A0) + B00 − A00. For the chosen values of the metric functions A and B in [77], this expression vanishes and consequently static plane symmetric spacetimes become coformally flat. In our analysis, we restrict ourselves to choosing the metric functions A and B such that:

A0 (B0 − A0) + B00 − A00 6= 0. (2.3.26)

The conformal Killing vectors given in (2.3.13) can be written in the following y2+z2 form by introducing the new variables βi = (β1, β2, β3) = 2 , z, y :

0 2(B−A) i 0 X = e βiPt + P ,

1 2B i 4 X = −e βiPx + P , c X2 = β P i + 3 (z2 − y2) + c yz + c z, i,2 2 2 4 c X3 = β P i + 2 (z2 − y2) − c yz − c y, i,3 2 3 4 2B i 0 2B i 4 α = −e βiPxx − 2B e βiPx + Px . (2.3.27)

52 Also, the integrability conditions (2.3.14)-(2.3.25) reduce to:

i 0 0 i Ptx(t, x) + (B − A ) Pt (t, x) = 0, (2.3.28)

0 i i −2B B Px(t, x) + Pxx(t, x) = kie , (2.3.29)

0 0 i −2A i i (2B − A ) Px(t, x) + e Ptt(t, x) + Pxx(t, x) = 0, (2.3.30)

2A 0 4 e Px (t, x) − Pt (t, x) = 0, (2.3.31)

0 4 0 4 A P (t, x) + Pt (t, x) − Px (t, x) = 0, (2.3.32)

0 4 1 4 B P (t, x) + P (t, x) − Px (t, x) = 0. (2.3.33)

where ki = 0, −c2, c3 for i = 1, 2, 3 respectively. In the following two cases, we solve the above equations either by assuming the separability of the unknown functions P i as sum of two functions in their arguments or by directly choosing some speciﬁc metric functions A and B, satisfying the relation (2.3.26).

Case 1: In this case we solve the system of equations (2.3.28)-(2.3.33) by taking P k(t, x) = Rk(t) + Hk(x), for k = 1, ..., 4. Using this assumption in

0 0 i i Eq. (2.3.28), we get (B − A ) Pt (t, x) = 0 ⇒ Pt (t, x) = 0, because for B0 = A0 the spacetimes become conformally ﬂat, which is not the case of our interest. Subtracting Eq. (2.3.29) from Eq. (2.3.30) and integrating the re-

i R e−2B sulting equation with respect to x, we obtain P = −ki B0−A0 dx + ri, where ri denote three diﬀerent constants of integration for i = 1, 2, 3. Putting back i 0 0 0 00 00 this value of P in Eq. (2.3.29) , we have ki {A (B − A ) + B − A } = 0, i so that ki = 0 ⇒ c2 = c3 = 0. Hence P = ri, for i = 1, 2, 3. For consistency, 1 2 3 let us denote P = r1 = c5,P = r2 = c6 and P = r3 = c7. The integrability conditions (2.3.28)-(2.3.30) are now identically satisﬁed by these values of

53 P i. The conformal Killing vectors given in (2.3.27) are now reduced to:

X0 = P 0,

X1 = P 4,

2 X = c4z + c5y + c7,

3 X = −c4y + c5z + c6,

4 α = Px . (2.3.34)

The values of P 0 and P 4 are to be determined from the remaining three integrability conditions (2.3.31)-(2.3.33). Diﬀerentiating Eq. (2.3.33) with

0 4 0 respect to t, we get B Pt (t, x) = 0. Now if B = 0, then Eq. (2.3.33) yields 4 Px (t, x) = c5. It can be seen from (2.3.34) that the conformal factor becomes constant. Thus there is no proper conformal Killing vector in this case. So

0 4 4 4 we must choose B 6= 0, so that Pt (t, x) = 0 ⇒ P = P (x). Putting this 4 0 0 0 value of P in Eq. (2.3.31), we obtain Px (t, x) = 0 ⇒ P = P (t). We rewrite the remaining two equation (2.3.32) and (2.3.33) as:

0 4 0 4 A P (x) + Pt (t) − Px (x) = 0, (2.3.35)

0 4 4 B P (x) + c5 − Px (x) = 0. (2.3.36)

0 Diﬀerentiating Eq. (2.3.35) with respect to t, we ﬁnd that P (t) = c8t + c9. Substituting back this value of P 0 in Eq. (2.3.35) and then subtracting the

4 c5−c8 resulting equation from Eq. (2.3.36), we obtain P (x) = A0−B0 . Simplifying Eq. (2.3.35) or Eq. (2.3.36) with the help of these obtained values of P 0 and P 4, we have the following relation:

0 0 0 00 00 0 0 0 00 00 c8 {B (B − A ) + B − A } − c5 {A (B − A ) + B − A } = 0. (2.3.37)

54 The conformal Killing vectors in (2.3.34) will get their ﬁnal form if the metric functions A and B satisfy the above equation. We see that Eq. (2.3.37) is highly non linear and cannot be solved directly, however we can choose some values of the metric functions such that it holds true. As mentioned earlier, the metric function B must not be a constant. Also for proper conformal

0 4 Killing vectors, we must take A 6= 0, otherwise (2.3.35) gives Px (x) = c8

⇒ α = c8. Moreover it is important to note that c8 6= 0, as for c8 = 0 Eq.

(2.3.37) yields c5 = 0 and hence the conformal factor α in (2.3.34) vanishes, which is not the case of our interest. Following are some trivial examples of non conformally ﬂat static plane symmetric metrics which admit proper conformal Killing vectors.

Example 2.3.1. If we take c5 = 0 and A(x) = x, then solving Eq. (2.3.37) x we get B(x) = ln(m1e + m2), where m1 6= 0, m2 ∈ R. For these values of A and B, the static plane symmetric metric given in (2.3.1) gets the form:

2 2x 2 2 x 2 2 2 ds = −e dt + dx + (m1e + m2) dy + dz , (2.3.38) which is clearly non conformally ﬂat. Assuming m1 = m2 = 1, the conformal Killing vectors given in (2.3.34) become:

0 X = c8t + c9,

1 x X = −c8 (e + 1) ,

2 X = c4z + c7,

3 X = −c4y + c6,

x α = −c8e . (2.3.39)

This indicates the existence of a proper conformal Killing vector, in addition to the four basic Killing vectors of static plane symmetric metric mentioned

55 in (2.3.2). There is no proper homothetic vector in this case. The proper

x conformal Killing vector can be expressed as X5 = t∂t − (e + 1) ∂x. The Lie algebra of the above conformal Killing vectors has the following non zero commutators:

[X1,X5] = X1, [X2,X4] = −X3, [X3,X4] = X2.

Example 2.3.2. If we choose c5 = 0 and A(x) = ln x, then Eq. (2.3.37) gives B(x) = ln(m1x − m2x ln x), where m1, m2 ∈ R. With these values of the metric functions, the static plane symmetric spacetimes metric given in (2.3.1) can be rewritten as:

2 2 2 2 2 2 2 ds = −x dt + dx + (m1x − m2x ln x) dy + dz , (2.3.40)

which is a non conformally ﬂat metric. Assuming m1 = m2 = 1, the conformal Killing vectors given in (2.3.34) get their ﬁnal form as:

0 X = c8t + c9,

1 X = c8x (ln x − 1),

2 X = c4z + c7,

3 X = −c4y + c6,

α = c8 ln x. (2.3.41)

It shows that the metric given in (2.3.40) admits ﬁve conformal Killing vectors with one proper conformal Killing vector X5 = t ∂t + x (ln x − 1) ∂x. The dimension of homothetic and Killing algebra turned out to be 4, with no proper homothetic vector and four Killing vectors which are same as mentioned in (2.3.2). The non zero commutators of Lie algebra for the above

56 conformal Killing vectors are:

[X1,X5] = X1, [X2,X4] = −X3, [X3,X4] = X2.

Example 2.3.3. Let c5 = 0 and B(x) = x. Solving Eq. (2.3.37), we obtain −x A(x) = −m1e + x + m2, where m1, m2 ∈ R. The metric given in (2.3.1) becomes:

2 −x 2 2 2 2 ds = − exp 2(−m1e + x + m2) dt + dx + exp(2x) dy + dz , (2.3.42) which is non conformally ﬂat. Assuming m1 = 1, the conformal Killing vectors in (2.3.34) become:

0 X = c8t + c9,

1 x X = −c8 e ,

2 X = c4z + c7,

3 X = −c4y + c6,

x α = −c8 e . (2.3.43)

In this case the static plane symmetric metric (2.3.42) admits ﬁve conformal Killing vectors in which four are the basic Killing vectors mentioned in (2.3.2)

x and one is a proper conformal Killing vector, which is X5 = t ∂t − e ∂x. No proper homothetic vector exist in this case. The non vanishing commutators of Lie algebra of the above conformal Killing vectors are:

[X1,X5] = X1, [X2,X4] = −X3, [X3,X4] = X2.

Note that in each of the above three examples, static plane symmetric metric admits a proper conformal Killing vector, in addition to the four

57 Killing vectors of static plane symmetric spacetimes. Apart from these examples, one can ﬁnd a number of metric functions A and B satisfying Eq. (2.3.37) to get the proper conformal Killing vectors in non conformally ﬂat static plane symmetric spacetimes.

Case 2: In this case we solve the integrability conditions (2.3.28)-(2.3.33) for some more choices of the metric functions A and B satisfying the relation (2.3.26), without imposing the condition of separability of the functions P i(t, x) in their arguments. In each of the following two examples, a non conformally ﬂat static plane symmetric metric is found admitting a proper conformal Killing vector as well as a proper homothetic vector, in addition to the four basic Killing vectors of static plane symmetric spacetimes.

Example 2.3.4. Here we consider A = m1 and B = ln x, where m1 6= 0 ∈ R. The metric given in (2.3.1), after a suitable rescaling, gets the form:

ds2 = −dt2 + dx2 + x2 dy2 + dz2 . (2.3.44)

Rearranging Eq. (2.3.29) and then integrating it twice with respect to x, we

i ki 2 i i i i obtain P (t, x) = 2 (ln x) + ln x R (t) + H (t), where R (t) and H (t) are functions of integration. Simplifying Eq. (2.3.28) with the help of this value

i i i i i of P (t, x), we have Rt(t) = Ht (t) = 0. Thus R (t) = pi and H (t) = ri, where pi, ri ∈ R. Substituting all these values in Eq. (2.3.30), we get ki = pi = 0. i Hence P (t, x) = ri, for i = 1, 2, 3. For consistency of constant, we denote 1 2 3 P (t, x) = r1 = c5,P (t, x) = r2 = c6, and P (t, x) = r3 = c7. Integrating 4 4 4 Eq. (2.3.33) with respect to x, we get P (t, x) = c5x ln x+xR (t), where R (t) is a function of integration. We use this value of P 4(t, x) in Eq. (2.3.31) to

58 0 x2 4 5 5 get P (t, x) = 2 Rt (t)+R (t), where R (t) is a function of integration. Using all these values in Eq. (2.3.32) and diﬀerentiating the resulting equation with

4 5 c8 2 respect to x, we obtain c5 = 0, R (t) = c8t + c9 and R (t) = 2 t + c9t + c10. Substituting back all these values in (2.3.27), we get the following ﬁnal form of conformal Killing vectors: c X0 = 8 x2 + t2 + c t + c , 2 9 10 1 X = c8tx + c9x,

2 X = c4z + c7,

3 X = −c4y + c6,

α = c8t + c9. (2.3.45)

Here we see that the non conformally ﬂat metric (2.3.44) admits six conformal

Killing vectors, with one proper homothetic vector X5 = t ∂t + x ∂x and one 1 2 2 proper conformal Killing vector X6 = 2 (x + t ) ∂t + xt ∂x. The remaining four conformal Killing vectors are actually the basic Killing vectors of static plane symmetric spacetimes mentioned in (2.3.2). The Lie algebra of these conformal Killing vectors has the following non vanishing commutators:

[X1,X5] = X1, [X1,X6] = X5, [X2,X4] = −X3, [X3,X4] = X2, [X5,X6] = X6.

Example 2.3.5. Here we take A = ln x2 and B = ln x. For these values of metric functions, the relation (2.3.26) holds and we get the following non conformally ﬂat static plane symmetric metric.

ds2 = −x4dt2 + dx2 + x2 dy2 + dz2 . (2.3.46)

Following the same steps as that of the previous example, we can easily solve the integrability conditions (2.3.28)-(2.3.33) to get the following ﬁnal form

59 of conformal Killing vectors:

c X0 = − 8 t2 + x−2 − c t + c , 2 9 10 1 X = c8tx + c9x,

2 X = c4z + c7,

3 X = −c4y + c6,

α = c8t + c9, (2.3.47) which shows that the metric (2.3.46) admit six conformal Killing vectors in which four are the basic Killing vectors mentioned in (2.3.2), one is proper homothetic vector X5 = −t ∂t + x ∂x and one is proper conformal Killing 1 2 −2 vector which can be written as X6 = − 2 (t + x ) ∂t +xt ∂x. The Lie algebra of these six conformal Killing vectors has the following non zero commutators:

[X1,X5] = −X1, [X1,X6] = X5, [X2,X4] = −X3, [X3,X4] = X2, [X5,X6] = X6.

2.4 Inheriting Conformal Killing Vectors in Static Plane Symmetric Spacetimes

The conformal Killing vectors, up to some unknown functions of t and x, for static plane symmetric spacetimes are given in (2.3.27), subject to the differential constraints (2.3.28)-(2.3.33). In this section we solve these differential constraints to find the final form of conformal Killing vectors, by considering X as an inheriting conformal Killing vector. In such a case, X satisfies the relation:

b b LX ua = ua,b X + ub X,a = αua, (2.4.1)

60 a a −A a where u is the four velocity vector of the ﬂuid. Here we take u = e δ0 , so that the relation (2.4.1) generates the following equations:

0 1 0 A X + X,0 = α, (2.4.2)

0 X,i = 0, i = 1, 2, 3. (2.4.3)

Thus in the system (2.3.27), we must have X0 = P 0(t) and P i = P i(x), for i = 1, 2, 3. Also Eq. (2.3.31) gives P 4 = P 4(x). With these values, two integrability conditions (2.3.28) and (2.3.31) are identically satisﬁed. The remaining four integrability conditions reduce to:

0 i i −2B B Px(x) + Pxx(x) = kie , (2.4.4)

0 0 i i (2B − A ) Px(x) + Pxx(x) = 0, (2.4.5)

0 4 0 4 A P (x) + Pt (t) − Px (x) = 0, (2.4.6)

0 4 1 4 B P (x) + P (x) − Px (x) = 0. (2.4.7)

Subtracting Eq. (2.4.4) from Eq. (2.4.5) and integrating the resulting equa-

i R e−2B tion with respect to x, we have P (x) = −ki B0−A0 dx+ri, where ri represent three diﬀerent constants of integration for i = 1, 2, 3. Putting this value of

i 0 0 0 00 00 P (x) in Eq. (2.4.4), we get ki {A (B − A ) + B − A } = 0 ⇒ ki = 0, oth- i erwise the spacetime becomes conformally ﬂat. Hence P = ri, for i = 1, 2, 3. 1 2 3 For consistency, we denote P = r1 = c5,P = r2 = c6 and P = r3 = c7. 0 If we diﬀerentiate Eq. (2.4.6) with respect to t, it gives P (t) = c8t + c9.

4 c5−c8 Subtracting eq. (2.4.6) from (2.4.7), we ﬁnd that P (x) = A0−B0 and putting back this value in Eq. (2.4.6) or Eq. (2.4.7), we obtain the following relation:

0 0 0 00 00 0 0 0 00 00 c8 {B (B − A ) + B − A } − c5 {A (B − A ) + B − A } = 0. (2.4.8)

61 Thus the solution of the system of equations (2.4.4)-(2.4.7) is now reduced to the solution of Eq. (2.4.8). The conformal Killing vectors given in (2.3.27) will be now referred as inheriting conformal Killing vectors and now they reduce to:

0 X = c8t + c9, c − c X1 = 5 8 , A0 − B0 2 X = c5y + c4z + c7,

3 X = c5z − c4y + c6, A00 − B00 α = (c − c ) , (2.4.9) 8 5 (A0 − B0)2 which are same as the conformal Killing vectors given in (2.3.34). Rest of the analysis is same as we have done in Case 1 of section (2.3). Any metric functions satisfying Eq. (2.4.8) will give the ﬁnal form of the above inheriting conformal Killing vectors. Hence the inheriting conformal Killing vectors in this case are same as the conformal Killing vectors.

2.5 Conformal Killing Vectors in Non Static Plane Symmetric Spacetimes

The most general form of non static plane symmetric spacetimes metric is [1]:

ds2 = −e2A(t,x)dt2 + e2B(t,x)dx2 + e2C(t,x) dy2 + dz2 , (2.5.1) where the metric functions A, B and C are dependent on t and x only. This metric admits the following three spatial Killing vectors [1]:

X1 = ∂y,X2 = ∂z,X3 = z∂y − y∂z, (2.5.2)

62 in which X1,X2 represent conservation of linear momentum along y and z directions and X3 represents conservation of angular momentum. Using the metric (2.5.1) in Eq. (1.3.1), we obtain the following conformal Killing’s equations:

˙ 0 0 1 0 AX + A X + X,0 = α, (2.5.3)

2B 1 2A 0 e X,0 − e X,1 = 0, (2.5.4)

2C 2 2A 0 e X,0 − e X,2 = 0, (2.5.5)

2C 3 2A 0 e X,0 − e X,3 = 0, (2.5.6) ˙ 0 0 1 1 BX + B X + X,1 = α, (2.5.7)

2C 2 2B 1 e X,1 + e X,2 = 0, (2.5.8)

2C 3 2B 1 e X,1 + e X,3 = 0, (2.5.9) ˙ 0 0 1 2 CX + C X + X,2 = α, (2.5.10)

2 3 X,3 + X,2 = 0, (2.5.11) ˙ 0 0 1 3 CX + C X + X,3 = α, (2.5.12) where a prime and a dot on metric functions are used to denote their partial derivative with respect to x and t respectively. The commas in subscript represent partial derivatives with respect to the spacetime coordinates and the conformal factor α is a function of t, x, y and z. We solve the above system of equations to get conformal Killing vectors for the metric (2.5.1). Diﬀerentiating Eq. (2.5.5), (2.5.6) and (2.5.11) with respect to z, y and t respectively, we have:

2C 2 2A 0 e X,03 − e X,23 = 0, (2.5.13)

2C 3 2A 0 e X,02 − e X,23 = 0, (2.5.14)

63 2 3 X,03 + X,02 = 0. (2.5.15)

Subtracting Eq. (2.5.14) from Eq. (2.5.13) and then adding the resulting equation with Eq. (2.5.15), we get:

0 2 3 X,23 = X,03 = X,02 = 0. (2.5.16)

Similarly diﬀerentiating Eqs. (2.5.8), (2.5.9) and (2.5.11) with respect to z, y and x respectively, we ﬁnd that:

1 2 3 X,23 = X,13 = X,12 = 0. (2.5.17)

2 Keeping (2.5.16) in mind, the following value of X,3 can be found if we subtract Eq. (2.5.10) from Eq. (2.5.7) and then diﬀerentiate the resulting equation with respect to y and z respectively:

2 1 2 X,3 = yF (x, z) + F (x, z), (2.5.18) where F 1(x, z) and F 2(x, z) are functions of integration. Substituting this

2 value of X,3 in Eq. (2.5.11), we ﬁnd that: 1 X3 = − y2F 1(x, z) − yF 2(x, z) + f 1(t, x, z), (2.5.19) 2 where f 1(t, x, z) is a function of integration. The following relation can be found by subtracting Eq. (2.5.12) from Eq. (2.5.10):

2 3 X,2 = X,3. (2.5.20)

Diﬀerentiating Eq. (2.5.20) with respect to z and using Eqs. (2.5.18) and (2.5.19), we ﬁnd that F 1(x, z) = z G1(x)+G2(x), F 2(x, z) = z G3(x)+G4(x)

1 1 3 1 1 2 2 3 4 and f (t, x, z) = 6 z G (x) + 2 z G (x) + z F (t, x) + F (t, x). Here the

64 functions Gi(x) and F k(t, x) are functions of integration, for i = 1, ..., 4 and k = 3, 4. Using these values in Eqs. (2.5.18) and (2.5.19), we obtain:

1 1 X2 = y z2 G1(x) + z G2(x) + z2 G3(x) + z G4(x) + f 2(t, x, y), 2 2 1 1 X3 = − y2 z G1(x) + G2(x) − y z G3(x) + G4(x) + z3 G1(x) 2 6 1 + z2 G2(x) + z F 3(t, x) + F 4(t, x). (2.5.21) 2

The function f 2(t, x, y) appearing in the above system arises in the integration process. The functions Gi(x), for i = 1, ..., 4 become constant by the

1 2 3 relation (2.5.17). Let us denote G (x) = c1,G (x) = c2,G (x) = c3 and 4 G (x) = c4. Using the system (2.5.21) in Eq. (2.5.20) and integrating the resulting equation with resect to y, we get:

c c f 2(t, x, y) = − 1 y3 − 3 y2 + y F 3(t, x) + F 5(t, x), (2.5.22) 6 2 where F 5(t, x) is a function of integration. Also, using the system (2.5.21) in Eq. (2.5.6) and integrating the resulting equation with resect to z, we have:

1 X0 = e2(C−A) z2F 3(t, x) + zF 4(t, x) + f 3(t, x, y), (2.5.23) 2 t t where f 3(t, x, y) is a function of integration. Similarly using the system (2.5.21) in Eq. (2.5.9) and integrating the resulting equation with resect to z, we obtain:

1 X1 = −e2(C−B) z2F 3(t, x) + zF 4(t, x) + f 4(t, x, y), (2.5.24) 2 x x f 4(t, x, y) being a function of integration. The following values of f 3(t, x, y) and f 4(t, x, y) are found by using Eqs. (2.5.21)-(2.5.24) in Eqs. (2.5.5) and

65 (2.5.8):

1 f 3(t, x, y) = e2(C−A) y2F 3(t, x) + yF 5(t, x) + F 6(t, x), 2 t t 1 f 4(t, x, y) = −e2(C−B) y2F 3(t, x) + yF 5(t, x) + F 7(t, x). (2.5.25) 2 x x

Finally Eqs. (2.5.21), (2.5.23) and (2.5.24) give:

1 X0 = e2(C−A) (z2 + y2)F 3(t, x) + zF 4(t, x) + yF 5(t, x) + F 6(t, x), 2 t t t 1 X1 = −e2(C−B) (z2 + y2)F 3(t, x) + zF 4(t, x) + yF 5(t, x) + F 7(t, x), 2 x x x c c c c X2 = y 1 z2 + c z + 3 z2 + c z − 1 y3 − 3 y2 + y F 3(t, x) + F 5(t, x), 2 2 2 4 6 2 1 c c X3 = − y2 (c z + c ) − y (c z + c ) + 1 z3 + 2 z2 + z F 3(t, x) + F 4(t, x), 2 1 2 3 4 6 2 (2.5.26) which determine the conformal Killing vectors, up to some unknown functions of t and x. As far as the conformal factor is concerned, it gets the following form if we use the above system in Eq. (2.5.7).

1 α = Be˙ 2(C−A) (z2 + y2)F 3(t, x) + zF 4(t, x) + yF 5(t, x) 2 t t t 1 − e2(C−B) (z2 + y2)F 3 (t, x) + zF 4 (t, x) + yF 5 (t, x) 2 xx xx xx 1 + (B0 − 2C0) e2(C−B) (z2 + y2)F 3(t, x) + zF 4(t, x) + yF 5(t, x) 2 x x x ˙ 6 0 7 7 + BF (t, x) + B F (t, x) + Fx (t, x). (2.5.27)

Using the system (2.5.26) and the above conformal factor in Eqs. (2.5.3), (2.5.4), (2.5.10) and equating the like terms of the resulting equations, we

66 get c1 = 0 and the following integrability conditions are generated.

3 0 0 3 ˙ ˙ 3 Ftx + (C − A ) Ft + (C − B)Fx = 0, (2.5.28) 4 0 0 4 ˙ ˙ 4 Ftx + (C − A ) Ft + (C − B)Fx = 0, (2.5.29) 5 0 0 5 ˙ ˙ 5 Ftx + (C − A ) Ft + (C − B)Fx = 0, (2.5.30) 0 0 −2B 3 −2B 3 ˙ ˙ −2A 3 (C − B ) e Fx + e Fxx + (C − B)e Ft = 0, (2.5.31) 0 0 −2B 4 −2B 4 ˙ ˙ −2A 4 −2C (C − B ) e Fx + e Fxx + (C − B)e Ft + c2e = 0, (2.5.32) 0 0 −2B 5 −2B 5 ˙ ˙ −2A 5 −2C (C − B ) e Fx + e Fxx + (C − B)e Ft − c3e = 0, (2.5.33) n o 0 0 0 3 3 2(B−A) ˙ ˙ ˙ 3 3 (2C − A − B ) Fx + Fxx + e (2C − A − B)Ft + Ftt = 0, (2.5.34) n o 0 0 0 4 4 2(B−A) ˙ ˙ ˙ 4 4 (2C − A − B ) Fx + Fxx + e (2C − A − B)Ft + Ftt = 0, (2.5.35) n o 0 0 0 5 5 2(B−A) ˙ ˙ ˙ 5 5 (2C − A − B ) Fx + Fxx + e (2C − A − B)Ft + Ftt = 0, (2.5.36)

2B 7 2A 6 e Ft − e Fx = 0, (2.5.37) ˙ ˙ 6 0 0 7 6 7 (A − B)F + (A − B ) F + Ft − Fx = 0, (2.5.38) ˙ ˙ 6 0 0 7 3 7 (C − B)F + (C − B ) F + F − Fx = 0, (2.5.39) where F k are functions of t and x, for k = 3, ..., 7. Introducing the new

y2+z2 i 1 2 3 3 4 5 variables βi = (β1, β2, β3) = 2 , z, y , P = (P ,P ,P ) = (F ,F ,F ), P 0 = F 6 and P 4 = F 7, we can write the conformal Killing vectors in (2.5.26) and the conformal factor in (2.5.27) as follows:

0 2(C−A) i 0 X = e βiPt + P ,

1 2(C−B) i 4 X = −e βiPx + P , c X2 = β P i + 3 (z2 − y2) + c yz + c z, i,2 2 2 4 c X3 = β P i + 2 (z2 − y2) − c yz − c y, i,3 2 3 4

67 ˙ 2(C−A) i 2(C−B) i 0 0 2(C−B) i α = Be βiPt − e Pxx + (B − 2C )e βiPx ˙ 0 0 4 4 + BP + B P + Px . (2.5.40)

The integrability conditions (2.5.28)-(2.5.39) now reduce to:

i 0 0 i ˙ ˙ i Ptx + (C − A ) Pt + (C − B)Px = 0, (2.5.41) 0 0 −2B i −2B i ˙ ˙ −2A i −2C (C − B ) e Px + e Pxx + (C − B)e Pt + kie = 0, (2.5.42) n o 0 0 0 i i 2(B−A) ˙ ˙ ˙ i i (2C − A − B ) Px + Pxx + e (2C − A − B)Pt + Ptt = 0, (2.5.43)

2B 4 2A 0 e Pt − e Px = 0, (2.5.44) ˙ ˙ 0 0 0 4 0 4 (A − B)P + (A − B ) P + Pt − Px = 0, (2.5.45) ˙ ˙ 0 0 0 4 1 4 (C − B)P + (C − B ) P + P − Px = 0, (2.5.46)

where ki = 0, c2, −c3 for i = 1, 2, 3 respectively. The problem of ﬁnding the ﬁnal form of conformal Killing vectors, given in (2.5.40), is now reduced to the solution of equations (2.5.41)-(2.5.46). One can see that these equations are highly non linear and cannot be solved easily in the present form. Even the assumption of separability of the functions P i as the sum of two functions in their arguments, which we have used in case of Bianchi type V and static plane symmetric spacetimes, does not work well here. However, in the following cases, we solve these equations by imposing some condition either on the form of conformal Killing vectors or on metric functions. Case 1: In this case we assume that the conformal Killing vector X has only the temporal component, that is X = (X0, 0, 0, 0). Thus in (2.5.40), we

i 4 0 0 4 must have P = P = c2 = c3 = c4 = 0 and X = P . Putting P = 0 in Eq. (2.5.44), we ﬁnd that P 0 = P 0(t). Also substituting P 1 = P 4 = 0 in Eq. (2.5.46), we get C = γB, where γ is a constant. Finally Eq. (2.5.45) implies

68 ˙ ˙ 0 0 0 B−A (A − B)P + Pt = 0, which on integration yields P ∝ e . This shows the existence of a timelike conformal Kiling vector parallel to timelike vector

a −A a u = e δ0 .

Case 2: Here we assume that A = 0, while B and C are dependent on t only. In such a case the metric given in (2.5.1) reduces to:

ds2 = −dt2 + e2B(t)dx2 + e2C(t) dy2 + dz2 . (2.5.47)

Also the integrability conditions (2.5.41)-(2.5.46) reduce to:

i ˙ ˙ i Ptx + (C − B)Px = 0, (2.5.48) −2B i ˙ ˙ i −2C e Pxx + (C − B)Pt + kie = 0, (2.5.49) −2B i ˙ ˙ i i e Pxx + (2C − B)Pt + Ptt = 0, (2.5.50)

2B 4 0 e Pt − Px = 0, (2.5.51) ˙ 0 0 4 −BP + Pt − Px = 0, (2.5.52) ˙ ˙ 0 1 4 (C − B)P + P − Px = 0. (2.5.53)

The non zero components of Ricci tensor for the above metric are:

¨ ¨ ˙ 2 ˙ 2 R00 = −2C − B − 2C − B , n o 2B ˙ 2 ˙ ˙ ¨ R11 = e B + 2BC + B , n o 2C ˙ 2 ˙ ˙ ¨ R22 = R33 = e 2C + BC + C . (2.5.54)

To obtain a vacuum solution, we put R00 = R11 = R22 = R33 = 0. Solving 2 these equations with the help of (2.5.54), we obtain B(t) = 3 ln (c5t + c6) 1 and C(t) = − 3 ln (c5t + c6), where c5, c6 ∈ R. Subtracting Eq. (2.5.49) from

69 Eq. (2.5.50) and then integrating the resulting equation twice with respect to t, we have:

9ki 4 3 1 i 3 3 i i P (t, x) = 2 (c5t + c6) + (c5t + c6) H (x) + R (x), 4c5 c5 where Hi(x) and Ri(x) are functions of integration. Putting this value in

i i Eq. (2.5.48), we ﬁnd that H (x) = pi and R (x) = ri, where pi, ri ∈ R. Using i all these values in Eq. (2.5.49), we get ki = pi = 0. Hence P (t, x) = ri, for 1 2 i = 1, 2, 3. For consistency of constants, we denote P = r1 = c7,P = r2 = c8 3 and P = r3 = c9. We subtract Eq. (2.5.52) from Eq. (2.5.53) and integrate the resulting equation with respect to t to get:

c7 2 0 3 4 P (t, x) = 3 (c5t + c6) + (c5t + c6) Hx(x), c5

4 where Hx(x) is a function of integration. Substituting back this value in Eq. (2.5.52) and integrating it with respect to x, we obtain:

4 − 1 4 5 P (t, x) = 3c7x + c5(c5t + c6) 3 H (x) + H (t), where H5(t) is a function of integration. Simplifying Eq. (2.5.51) by using

4 5 all these values, we ﬁnd that H (x) = 0 and H (t) = c10. Hence we have

0 c7 4 P (t, x) = 3 (c5t+c6) and P (t, x) = 3c7x+c10. Thus the conformal Killing c5 vectors given in (2.5.40) get their ﬁnal form as:

0 c7 X = 3 (c5t + c6), c5 1 X = 3c7x + c10,

2 X = c7y + c4z + c9,

3 X = c7z − c4y + c8,

α = 3c7. (2.5.55)

70 This result reveals that in this case there exist no proper conformal Killing vector and the conformal Killing vectors reduce to homothetic vectors. The dimension of homothetic algebra turned out to be 5 with only one proper homothetic vector and four Killing vectors. Out of four Killing vectors, three are the same as given in (2.5.2) and the fourth Killing vector is X4 = ∂x. The proper homothetic vector can be expressed as X5 = 3t∂t + 3x∂x + y∂y + z∂z.

Case 3: In this case we solve the integrability conditions (2.5.41)-(2.5.46) for some known classes of non conformally flat plane symmetric metrics. In each of the following examples, the considered metric is taken from the literature which is obtained either by solving the Einstein’s field equations or by imposing some symmetry restrictions on the spacetime metric. Since the calculations involved in all examples are basic and similar, we give the details in only the first example. In the rest of the examples, we omit the basic calculations; only the final results are presented.

Example 2.5.1. Consider the following non conformally ﬂat plane symmetric metric [78]:

√ ds2 = −x2+2 2dt2 + dx2 + x2 dy2 + dz2 , (2.5.56) which is static and admits at least four Killing vectors, which are mentioned in (2.3.2). Comparing the above metric with the most general metric of plane symmetric non static spacetimes, given in (2.5.1), we see that the metric √ functions get the values A = ln x1+ 2,B = 0 and C = ln x. We solve Eqs. (2.5.41)-(2.5.46) for these values of the metric functions to get the ﬁnal form of conformal Killing vectors.

71 After putting the above values of A, B and C in Eq. (2.5.42), we have:

2 i i x Pxx + xPx + ki = 0.

Rearranging and integrating this equation twice with respect to x, we obtain

i ki 2 i i i i P = − 2 (ln x) + ln x R (t) + H (t), where R (t) and H (t) are functions of integration, for i = 1, 2, 3. Substituting this value of P i in Eq. (2.5.41)

i i and differentiating it with respect to x, we get R (t) = ri and H (t) = hi, where ri, hi ∈ R. Differentiating Eq. (2.5.43) with respect to x, we obtain i ki = ri = 0. Hence we have P = hi, for i = 1, 2, 3. For consistency of the 1 2 3 constant, we denote P = h1 = c5,P = h2 = c6 and P = h3 = c7. If 1 we put P = c5 in Eq. (2.5.46) and integrate it with respect to x, we get 4 4 4 P = c5x ln x + xR (t), where R (t) is a function of integration. Integrating Eq. (2.5.44) with respect to x, after using the obtained value of P 4, we have √ P 0 = − √1 x−2 2 R4(t) + R5(t), R5(t) being a function of integration. Using 2 2 t the values of P 0 and P 4 in Eq.(2.5.45) and differentiating it with respect to √ 4 5 c8 2 x, we get c5 = 0, R (t) = c8t+c9 and R (t) = − 2 2 t + c9t +c10. Finally √ √ we have P 0 = − c√8 x−2 2 − 2 c8 t2 + c t + c and P 4 = x (c t + c ) . 2 2 2 9 10 8 9 Hence the conformal Killing vectors, given in (2.5.40), get the final form as: √ √ 0 c8 −2 2 c8 2 X = − √ x − 2 t + c9t + c10, 2 2 2 1 X = x (c8t + c9) ,

2 X = c4z + c7,

3 X = −c4y + c6,

α = c8t + c9, (2.5.57) which shows that the plane symmetric metric (2.5.56) admits six conformal Killing vectors with one proper conformal Killing vector, which can

72 √ be expressed as X = − √1 x−2 2 + √1 t2 ∂ + tx ∂ . The dimension of 6 2 2 2 t x homothetic algebra turned out to be 5 with one proper homothetic vector √ X5 = − 2t ∂t. The constants c4, c6, c7 and c10 in (2.5.57) correspond to the four basic Killing vectors of static plane symmetric spacetimes, which are same as mentioned in (2.3.2). The Lie algebra of the above six conformal Killing vectors has the following non zero commutators: √ [X1,X5] = − 2 X1, [X1,X6] = X5, [X2,X4] = −X3, √ [X3,X4] = X2, [X5,X6] = − 2 X6.

Example 2.5.2. Here we consider the following non conformally ﬂat plane symmetric metric, which admit self similarity of second kind [78]:

ds2 = −dt2 + dx2 + 2t dy2 + dz2 . (2.5.58)

Solving the integrability conditions (2.5.41)-(2.5.46) for the above metric by using the same procedure as that of the previous example, we get:

0 X = 2c5t,

1 X = 2c5x + c6,

2 X = c4z + c5y + c7,

3 X = −c4y + +c5z + c8,

α = 2c5, (2.5.59) which shows that the metric given in (2.5.58) admit no proper conformal Killing vector. The dimension of homothetic algebra turned out to be 5 with

y z one proper homothetic vector, which is X5 = t ∂t + x ∂x + 2 ∂y + 2 ∂z. Also the dimension of isometry group is 4 with three Killing vectors same as

73 mentioned in (2.5.2) and the fourth Killing vector can be written as X4 = ∂x. The non zero commutators of these ﬁve homothetic vectors are listed below: 1 [X ,X ] = −X , [X ,X ] = X , [X ,X ] = X , 1 3 2 1 5 2 1 2 3 1 1 [X ,X ] = X , [X ,X ] = X . 2 5 2 2 4 5 4 Example 2.5.3. Here we take the following plane symmetric metric admitting a ﬁve dimensional isometry group [79]:

2 2x 2 2 2x 2 2 ds = −e a dt + dx + e b dy + dz , a 6= b 6= 0. (2.5.60)

Solving the integrability conditions (2.5.41)-(2.5.46) for the above metric, we see that the conformal factor α vanishes, which means that the conformal Killing vectors for this metric are just the Killing vectors, which are given below: b X0 = c t + c , a 5 6 1 X = −c5b,

2 X = c4z + c5y + c7,

3 X = −c4y + +c5z + c8. (2.5.61)

Out of the ﬁve Killing vectors admitted by the metric given in (2.5.60), four are the basic Killing vectors of static plane symmetric spacetimes and the

b ﬁfth Killing vector can be written as X5 = a t∂t − b ∂x + y ∂y + z ∂z. The non zero commutators of these ﬁve Killing vectors can be seen in [79].

Example 2.5.4. Here we solve the integrability conditions (2.5.41)-(2.5.46) for the following non static plane symmetric metric admitting six Killing vectors [79]: 2 2 2t 2 2 2 ds = −dt + e a dx + dy + dz , a 6= 0. (2.5.62)

74 It can be checked easily that the solution of Eqs. (2.5.41)-(2.5.46) yields α = 0, showing that the conformal Killing vectors for the metric (2.5.62) are just Killing vectors which, along with the corresponding Lie algebra, are already found in [79].

Besides the above four examples, we have also solved the integrability conditions (2.5.41)-(2.5.46) for the following plane symmetric metrics:

t ds2 = −dt2 + cos2 dx2 + dy2 + dz2, a 6= 0 (2.5.63) a x ds2 = − cosh2 dt2 + dx2 + dy2 + dz2, a 6= 0 (2.5.64) a x ds2 = − cos2 dt2 + dx2 + dy2 + dz2, a 6= 0 (2.5.65) a

For each of the above three metrics, the conformal factor α vanishes and hence no proper homothetic or conformal Killing vector is admitted by any of these three metrics. As far as the Killing vectors are concerned, they are already presented in [79] along with their corresponding Lie algebra for each of the above three metrics.

Case 4: As mentioned in chapter 1, every conformally flat spacetime metric admits fifteen conformal Killing vectors. In this case we solve the integrability conditions (2.5.41)-(2.5.46) for some known conformally flat plane symmetric metrics to obtain the explicit form of fifteen conformal Killing vectors. In each of the following examples, a conformally flat plane symmetric metric is taken from [79] or it is obtained by imposing certain conditions on the metric functions of plane symmetric spacetimes.

Example 2.5.5. Here we consider the following plane symmetric metric

75 admitting seven independent Killing vectors [79]:

2 2 2 2x 2 2 ds = −dt + dx + e a dy + dz , a 6= 0. (2.5.66)

It is straightforward to solve Eqs. (2.5.41)-(2.5.46) for the above metric and get the following ﬁfteen conformal Killing vectors:

0 1 2 2 x − t t z x − t t X = y + z e a −c e a + c e a + e a −c e a + c e a 2a 7 8 a 12 13 y x − t t a − x − t t + e a −c e a + c e a − e a c e a − c e a a 15 16 2 7 8 x − t t − e a c10e a − c14e a + c9,

1 1 2 2 x − t t z x − t t X = y + z e a c e a + c e a + e a c e a + c e a 2a 7 8 a 12 13 y x − t t a − x − t t + e a c e a + c e a − e a c e a + c e a a 15 16 2 7 8 x − t t + e a c10e a + c14e a − c2a z + c3a y − c6a,

2 c3 2 2 2 − 2x − x − t t − x − t t X = z − y + a e a + ye a c e a + c e a + e a c e a + c e a 2 7 8 15 16

+ c2yz + c4z + c6y + c5,

3 c2 2 2 2 − 2x − x − t t − x − t t X = z − y − a e a + ze a c e a + c e a + e a c e a + c e a 2 7 8 12 13

− c3yz − c4y + c6z + c11,

1 x 2 2 2 − 2x − t t z x − t t α = e a y + z + a e a c e a + c e a + e a c e a + c e a 2a2 7 8 a2 12 13 y x − t t 1 x − t t + e a c e a + c e a + e a c e a + c e a . a2 15 16 a 10 14

From above it is clear that the metric given in (2.5.66) admits ﬁfteen conformal Killing vectors, out of which seven are Killing vectors that are same as given in [79] and the remaining eight are proper conformal Killing vectors. There exist no proper homothetic vector in this case.

Example 2.5.6. Here we consider another conformally ﬂat plane symmetric

76 metric admitting seven independent Killing vectors, that is [79]:

2 2 2 2t 2 2 ds = −dt + dx + e a dy + dz , a 6= 0. (2.5.67)

Solving Eqs. (2.5.41)-(2.5.46) for the above metric, we see that it admits fifteen conformal Killing vectors which are same as in the above example with the only difference that the variables t and x are interchanged. Out of fifteen conformal Killing vectors, seven are Killing vectors which are same as given in [79] and the remaining eight are proper conformal Killing vectors with no proper homothetic vector.

Example 2.5.7. Here we impose the conditions on the metric functions as A = C = A(x) and B = 0. Under these assumptions, the metric in (2.5.1) reduces to: ds2 = dx2 + e2A(x) −dt2 + dy2 + dz2 , (2.5.68) which is conformally ﬂat. Solving the integrability conditions (2.5.41)-(2.5.46) for the above metric, we obtain:

c X0 = 7 t2 + y2 + z2 + c tz − c ty + c z + c y + c t 2 2 3 12 15 9 Z Z Z −A −A −A + (c5t + c8) e dx + c7 e e dx dx + c13, c X1 = − 5 eA y2 + z2 − 2t2 + (c t − c z − c y + c ) eA 2 8 10 16 11 Z Z Z A −A A −A −A + (c2z − c5y + c7t + c9) e e dx + c5 e e e dx dx, c X2 = 3 z2 − y2 − t2 + c yz + c ty + c z + c y + c t 2 2 7 4 9 15 Z Z Z −A −A −A + (c5y + c16) e dx + c3 e e dx dx + c6,

77 c X3 = − 2 y2 − z2 − t2 − c yz + c tz − c y + c z + c t 2 3 7 4 9 12 Z Z Z −A −A −A + (c7z + c10) e dx − c2 e e dx dx + c14, c α = − 5 A0eA y2 + z2 − 2t2 + (c t − c z − c y + c ) A0eA + c z 2 8 10 16 11 2 Z Z 0 A −A −A − c5y + c7t + c9 + (c2z − c5y + c7t + c9) A e e dx + c5 e dx Z Z 0 A −A −A + c5 A e e e dx dx, which shows that the conformally ﬂat metric given in (2.5.68) admits ﬁfteen conformal Killing vectors out of which eight are proper conformal Killing vectors and the remaining seven are homothetic vectors. From the conformal factor α, it can be seen that the metric under consideration will admit a proper homothetic vector if all constants except c9 appearing in α vanish and A0eA R e−Adx = const. In such a case the proper homothetic vector admitted by the metric (2.5.68) can be written as t ∂t + x ∂x + y ∂y + z ∂z.

2.6 Inheriting Conformal Killing Vectors in Non Static Plane Symmetric Spacetimes

In previous section we have found conformal Killing vectors for non static plane symmetric spacetimes up to some unknown functions of t and x, which are given in (2.5.40) along with some integrability conditions (2.5.41)- (2.5.46). In this section we assume that X is an inheriting conformal Killing vector, then X satisﬁes the additional condition:

b b LX ua = ua,b X + ub X,a = α ua, (2.6.1)

78 where ua is the four velocity vector of the ﬂuid source of spacetime. Here we

a −A a take u = e δ0 , so that the above relation generates the following equations:

˙ 0 0 1 0 AX + A X + X,0 = α, (2.6.2)

0 X,j = 0, j = 1, 2, 3. (2.6.3)

Thus in the system (2.5.40), we must have X0 = P 0(t) and P i = P i(x), for i = 1, 2, 3. Also Eq. (2.5.44) gives P 4 = P 4(x). With these values, the integrability condition (2.5.44) is identically satisﬁed. The remaining integrability conditions reduce to:

˙ ˙ i (C − B)Px = 0, (2.6.4)

0 0 −2B i −2B i −2C (C − B ) e Px + e Pxx + kie = 0, (2.6.5)

0 0 0 i i (2C − A − B ) Px + Pxx = 0, (2.6.6) ˙ ˙ 0 0 0 4 0 4 (A − B)P + (A − B ) P + Pt − Px = 0, (2.6.7) ˙ ˙ 0 0 0 4 1 4 (C − B)P + (C − B ) P + P − Px = 0. (2.6.8)

i If we assume that B 6= C, then Eq. (2.6.4) yields Px(x) = 0 and hence i P (x) = ri, where ri denote constants of integration, for i = 1, 2, 3. For 1 2 3 consistency we denote P = r1 = c5,P = r2 = c6 and P = r3 = c7. Putting i Px(x) = 0 in Eq. (2.6.5), we get ki = 0 ⇒ c2 = c3 = 0. Subtracting Eq. (2.6.7) from Eq. (2.6.8), we have:

˙ ˙ 0 0 0 4 1 0 (C − A)P + (C − A ) P + P − Pt = 0. (2.6.9)

Thus the solution of Eqs. (2.6.4)-(2.6.8) is now reduced to the solution of Eq. (2.6.9). Also the system (2.5.40) becomes:

X0 = P 0,

X1 = P 4,

79 2 X = c5y + c4z + c7,

3 X = c5z − c4y + c6, ˙ 0 0 4 4 α = BP + B P + Px , (2.6.10) which are the desired inheriting conformal Killing vectors for non static plane symmetric spacetimes, subject to the solution of Eq. (2.6.9).

2.7 Summary

In this chapter we have explored conformal and inheriting conformal Killing vectors for LRS Bianchi type V, static plane symmetric and non static plane symmetric spacetimes in the context of general relativity. For Bianchi type V spacetimes, it is shown that either they admit one proper conformal Killing vector, along with one proper homothetic vector and four Killing vectors or the conformal Killing vectors reduce to homothetic vectors which are six in number with one proper homothetic vector and five Killing vectors. The number of inheriting conformal Killing vectors turned out to be five or seven, with one proper inheriting conformal Killing vector In case of non conformally flat static plane symmetric spacetimes, it is observed that a proper conformal Killing vector exists for these spacetimes if the relation (2.3.37) holds. Five non conformally flat static plane symmetric spacetime metrics are presented, each admitting a proper conformal Killing vector. The inheriting conformal Killing vectors for static plane symmetric spacetimes turned out to be same as conformal Killing vectors. Finally we have discussed conformal Killing vectors for non static plane symmetric spacetimes. Our analysis revealed that these spacetimes admit time-

80 a −A a like conformal Killing vector parallel to the timelike vector u = e δ0 . Fur- ther, we have observed that the conformal Killing vectors for vacuum plane symmetric spacetimes reduce to homothetic vectors. Apart from this, the explicit form of 15-dimensional algebra of conformal Killing vectors for some known conformally ﬂat plane symmetric spacetimes is presented. In case of non conformally ﬂat plane symmetric metrics, we have either obtained one proper conformal Killing vector or the conformal Killing vectors are reduced to homothetic or Killing vectors. The inheriting conformal Killing vectors for non static plane symmetric spacetimes are found which are given in (2.6.10), subject to the solution of Eq. (2.6.9). It is worth noticing that the conformal Killing’s equation is covariant with respect to conformal transformation. Thus if we transform the spacetime conformally, we get the same number of solutions of conformal Killing equation. For the study of conformal Killing vectors we may perform a conformal transformation and choose arbitrary coordinates. As an example, if we multiply the line element of non static plane symmetric spacetimes given in (2.5.1) with the conformal factor e−2C , we have the equivalent line element:

2 2(A−C) 2 2(B−C) 2 2 2 dse = −e dt + e dx + dy + dz = −e2Aedt2 + e2Bedx2 + dy2 + dz2

If we drop the tildes, then since A and B are dependent only on dx and dt, we may treat xt subspace independently. Since any 2-dimensional spacetime is conformally ﬂat, we may rewrite the line element as:

−e2Aedt2 + e2Bedx2 = e2φ(X,T ) −dT 2 + dX2 ,

81 for some coordinates X and T and function φ(X,T ). Therefore, for the purpose of ﬁnding conformal symmetries, any of the three metrics considered in this chapter may be written as:

2 2φ(x,t) 2 2 2 2 dse = e −dt + dx + dy + dz

This would substantially shorten some calculations and will give the same number of conformal symmetries for all the three metrics.

82 Chapter 3

Teleparallel Conformal Killing Vectors in LRS Bianchi Type V, Static and Non Static Plane Symmetric Spacetimes

In this chapter, we explore teleparallel conformal Killing vectors for LRS Bianchi type V, static and non static plane symmetric spacetimes in the context of teleparallel gravity. For each of the three mentioned spacetimes, teleparallel conformal Killing vectors are obtained in terms of some unknown functions of t and x by solving Eq. (1.5.3). These teleparallel conformal Killing vectors are found subject to some integrability conditions in each case. For some speciﬁc choices of metric functions of the concerned spacetime, these integrability conditions are solved to get the explicit form of teleparallel conformal Killing vectors.

83 3.1 Teleparallel Conformal Killing Vectors in LRS Bianchi Type V Spacetimes

The LRS Bianchi type V spacetimes metric is given by [1]:

ds2 = −dt2 + A2dx2 + e2mxB2 dy2 + dz2 , (3.1.1) where m ∈ R and the metric functions A and B are dependent on t only. If m = 0, then the above metric reduces to LRS Bianchi type I spacetimes metric. However, we will focus on the case when m 6= 0, because teleparallel conformal Killing vectors in Bianchi type I spacetimes are already explored [69]. As mentioned in chapter 1, a teleparallel conformal Killing vector X satisﬁes the relation:

T c c c c d d LX gab = gab,c X +gbc X,a +gac X,b +X gad Tbc + gbd Tac = 2αgab, (3.1.2)

T d d where LX signiﬁes teleparallel Lie derivative operator along X, Tbc and Tac represent components of torsion tensor and the function α on right hand side depends on spacetime coordinates t, x, y and z. Using the relation (1.4.1.2), µ a the tetrad ha and its inverse hµ for the metric (3.1.1) become: 1 e−mx e−mx h µ = diag (1, A, emxB, emxB) , h a = diag 1, , , . (3.1.3) a µ A B B Using (3.1.3) in Eqs. (1.4.2.2) and (1.4.2.3), we get the following non zero components of torsion tensor: A˙ B˙ T 1 = ,T 2 = T 3 = ,T 2 = T 3 = m, (3.1.4) 01 A 02 03 B 12 13 where a dot on a metric function denotes its derivative with respect to t. The following set of partial diﬀerential equations is obtained by using the above

84 torsion tensor components and the metric functions in Eq. (3.1.2):

0 X,0 = α, (3.1.5) 2 1 0 ˙ 1 A X,0 − X,1 + AAX = 0, (3.1.6) 2mx 2 2 0 ˙ 2mx 2 e B X,0 − X,2 + BB e X = 0, (3.1.7) 2mx 2 3 0 ˙ 2mx 3 e B X,0 − X,3 + BB e X = 0, (3.1.8)

1 X,1 = α, (3.1.9)

2mx 2 2 2 1 2mx 2 2 e B X,1 + A X,2 + m e B X = 0, (3.1.10)

2mx 2 3 2 1 2mx 2 3 e B X,1 + A X,3 + m e B X = 0, (3.1.11)

2 X,2 = α, (3.1.12)

2 3 X,3 + X,2 = 0, (3.1.13)

3 X,3 = α. (3.1.14)

The simultaneous solution of the above system of equations will give explicit form of teleparallel conformal Killing vectors in LRS Bianchi type V spacetimes. The detailed procedure of solving the above system is given below: Diﬀerentiating Eqs. (3.1.10), (3.1.11) and (3.1.13) with respect to z, y and x respectively, we have:

2mx 2 2 2 1 2mx 2 2 e B X,13 + A X,23 + m e B X,3 = 0, (3.1.15)

2mx 2 3 2 1 2mx 2 3 e B X,12 + A X,23 + m e B X,2 = 0, (3.1.16)

2 3 X,13 + X,12 = 0. (3.1.17)

Using Eqs. (3.1.13) and (3.1.17) in the sum of Eqs. (3.1.15) and (3.1.16), we get:

1 X,23 = 0. (3.1.18)

85 1 2 Subtracting Eq. (3.1.12) from Eq. (3.1.9), we have X,1 = X,2. If we diﬀer- 2 entiate this equation with respect to y and z, it gives X,223 = 0. Integrating this equation twice with respect to y, we obtain:

2 1 2 X,3 = y f (t, x, z) + f (t, x, z), (3.1.19) where f 1(t, x, z) and f 2(t, x, z) are functions which arise during the process of integration. Using the above value in Eq. (3.1.13), we have:

y2 X3 = − f 1(t, x, z) − y f 2(t, x, z) + f 3(t, x, z), (3.1.20) 2 where f 3(t, x, z) is a function of integration. Comparing Eqs. (3.1.12) and (3.1.14), we get:

2 3 X,2 = X,3. (3.1.21)

The following values can be obtained if we diﬀerentiate the above equation three times with respect to y and use Eqs. (3.1.19) and (3.1.20) in the resulting equation:

f 1(t, x, z) = z F 1(t, x) + F 2(t, x),

f 2(t, x, z) = z F 3(t, x) + F 4(t, x), z3 z2 f 3(t, x, z) = F 1(t, x) + F 2(t, x) + z F 5(t, x) + F 6(t, x). (3.1.22) 6 2

Here F i(t, x) denote functions of integration, for i = 1, ..., 6. Integrating Eq. (3.1.19) with respect to z, after using (3.1.22), we have:

z2 z2 X2 = y F 1(t, x) + z F 2(t, x) + F 3(t, x) + z F 4(t, x) + f 4(t, x, y), 2 2 (3.1.23)

86 f 4(t, x, y) being a function of integration. Also, using (3.1.22), the value of X3 given in (3.1.20) can be rewritten as:

y2 X3 = − z F 1(t, x) + F 2(t, x) − y z F 3(t, x) + F 4(t, x) 2 z3 z2 + F 1(t, x) + F 2(t, x) + z F 5(t, x) + F 6(t, x). (3.1.24) 6 2

The following value of f 4(t, x, y) can be found if we use the values of X2 and X3 from Eqs. (3.1.23) and (3.1.24) in Eq. (3.1.21) and integrate it with respect to y:

y3 y2 f 4(t, x, y) = − F 1(t, x) − F 3(t, x) + y F 5(t, x) + F 7(t, x), (3.1.25) 6 2

F 7(t, x) being a function of integration. Putting back this value in Eq. (3.1.23), we have:

z2 z2 X2 = y F 1(t, x) + z F 2(t, x) + F 3(t, x) + z F 4(t, x) 2 2 y3 y2 − F 1(t, x) − F 3(t, x) + y F 5(t, x) + F 7(t, x). (3.1.26) 6 2

2 2 Using the identity (3.1.18) in Eq. (3.1.15), we get X,13 + mX,3 = 0. Using the values of X2 and X3 from (3.1.24) and (3.1.26) in this equation and then

i i diﬀerentiating it with respect to y and z, we obtain Fx(t, x) + mF (t, x) = 0, for i = 1, ..., 4. This equation can be solved easily to get F i(t, x) = e−mxGi(t), Gi(t) being function of integration for i = 1, ..., 4. Thus Eqs. (3.1.24) and (3.1.26) become:

z2 z2 X2 = y e−mx G1(t) + z G2(t) + e−mx G3(t) + z G4(t) 2 2 y3 y2 − e−mx G1(t) + G3(t) + y F 5(t, x) + F 7(t, x), 6 2

87 y2 X3 = − e−mx z G1(t) + G2(t) − y e−mx z G3(t) + G4(t) 2 z3 z2 + e−mx G1(t) + G2(t) + z F 5(t, x) + F 6(t, x). (3.1.27) 6 2

Integrating Eq. (3.1.11) with respect to z, after using (3.1.27), we obtain:

B2 z2 m X1 = − e2mx F 5(t, x) + z F 6(t, x) + z2F 5(t, x) + mzF 6(t, x) A2 2 x x 2 + f 5(t, x, y), (3.1.28) where the function f 5(t, x, y) arises in the integration process and it gets the following value if we integrate Eq. (3.1.10) with respect to y after using the above values of X1 and X2:

B2 y2 m f 5(t, x, y) = − e2mx F 5(t, x) + y F 7(t, x) + y2F 5(t, x) A2 2 x x 2 + my F 7(t, x) + F 8(t, x), (3.1.29) where F 8(t, x) denotes a functions of integration. Putting back the above value of f 5(t, x, y) in (3.1.28), we have:

B2 z2 m X1 = − e2mx F 5(t, x) + zF 6(t, x) + z2F 5(t, x) + mzF 6(t, x) A2 2 x x 2 B2 y2 m − e2mx F 5(t, x) + yF 7(t, x) + y2F 5(t, x) + myF 7(t, x) A2 2 x x 2 + F 8(t, x). (3.1.30)

Putting the value of X3 from (3.1.27) in Eq. (3.1.8) and integrating the resulting equation with respect to z, we have:

88 y2 z2 z2 X0 = −B2 emx G1(t) + zG2(t) + y G3(t) + zG4(t) 2 2 t t 2 t t z4 z3 z2 + B2 emx G1(t) + G2(t) + B2 e2mx F 5(t, x) + zF 6(t, x) 24 t 6 t 2 t t y2 z2 z2 − BB˙ emx G1(t) + zG2(t) + y G3(t) + zG4(t) 2 2 2 z4 z3 z2 + BB˙ emx G1(t) + G2(t) + BBe˙ 2mx F 5(t, x) + zF 6(t, x) 24 6 2 + f 6(t, x, y), (3.1.31) f 6(t, x, y) being a function of integration. Keeping the identity given in Eq. (3.1.18) in mind, if we diﬀerentiate Eq. (3.1.6) with respect to y and z, we i ˙ i get BGt(t) + BG (t) = 0, for i = 1, ..., 4. This equation can be easily solved

i ci i to get G (t) = B , where ci ∈ R for i = 1, ..., 4. Using these values of G (t) in Eqs. (3.1.27), (3.1.30) and (3.1.31), we obtain:

z2 X0 = B2e2mx F 5(t, x) + zF 6(t, x) 2 t t z2 + BBe˙ 2mx F 5(t, x) + zF 6(t, x) + f 6(t, x, y), 2 B2 z2 m X1 = − e2mx F 5(t, x) + z F 6(t, x) + z2 F 5(t, x) + mz F 6(t, x) A2 2 x x 2 B2 y2 m − e2mx F 5(t, x) + y F 7(t, x) + y2 F 5(t, x) + my F 7(t, x) A2 2 x x 2 + F 8(t, x), y c 1 c 1 c c X2 = e−mx 1 z2 + c z + e−mx 3 z2 + c z − e−mx 1 y3 + 3 y2 B 2 2 B 2 4 B 6 2 + y F 5(t, x) + F 7(t, x), 1 y2 1 c c X3 = − e−mx (c z + c ) + y (c z + c ) + e−mx 1 z3 + 2 z2 B 2 1 2 3 4 B 6 2 + z F 5(t, x) + F 6(t, x). (3.1.32)

89 The following value of f 6(t, x, y) can be found if we use the above system in Eq. (3.1.7) and integrate it with respect to y: y2 f 6(t, x, y) = B2 e2mx F 5(t, x) + y F 7(t, x) 2 t t y2 + BB˙ e2mx F 5(t, x) + y F 6(t, x) + F 9(t, x), (3.1.33) 2 where F 9(t, x) is a function of integration. Moreover, the conformal factor α gets the following value if we use the above system in Eq. (3.1.12): 1 c 1 c α = e−mx 1 z2 + c z − e−mx 1 y2 + c y + F 5(t, x). (3.1.34) B 2 2 B 2 3 Hence we have the following form of teleparallel conformal Killing vectors in terms of unknown functions of t and x. y2 + z2 X0 = B2 e2mx F 5(t, x) + zF 6(t, x) + yF 7(t, x) 2 t t t y2 + z2 + BB˙ e2mx F 5(t, x) + zF 6(t, x) + yF 7(t, x) + F 9(t, x), 2 B2 y2 + z2 m X1 = − e2mx F 5(t, x) + (y2 + z2)F 5(t, x) + zF 6(t, x) A2 2 x 2 x 7 6 7 8 + yFx (t, x) + mzF (t, x) + myF (t, x) + F (t, x), 1 c c c c X2 = e−mx 1 yz2 + c yz + 3 z2 + c z − 1 y3 − 3 y2 + yF 5(t, x) B 2 2 2 4 6 2 + F 7(t, x), 1 c c c c X3 = − e−mx 1 y2z + c yz + 2 y2 + c y − 1 z3 − 2 z2 + zF 5(t, x) B 2 3 2 4 6 2 + F 6(t, x), 1 c c α = e−mx 1 z2 + c z − 1 y2 − c y + F 5(t, x). (3.1.35) B 2 2 2 3 The above Xa (a = 0, 1, 2, 3) satisfy Eqs. (3.1.7), (3.1.8) and (3.1.10)-(3.1.14) identically. Substituting these values of Xa in Eqs. (3.1.5), (3.1.6) and (3.1.9)

90 and equating like terms of the resulting equations, we get c1 = 0 and the following integrability conditions are generated:

2 5 ˙ 5 ¨ ˙ 2 5 B Ftt + 3BBFt + BB + B F = 0, (3.1.36) 3 6 2 ˙ 6 ¨ ˙ 2 6 −3mx B Ftt + 3B BFt + B BB + B F − c2e = 0, (3.1.37) 3 7 2 ˙ 7 ¨ ˙ 2 7 −3mx B Ftt + 3B BFt + B BB + B F + c3e = 0, (3.1.38)

5 5 2 5 Fxx + 3mFx + 2m F = 0, (3.1.39)

3 6 3 6 2 3 6 2 −3mx B Fxx + 3mB Fx + 2m B F + c2A e = 0, (3.1.40)

3 7 3 7 2 3 7 2 −3mx B Fxx + 3mB Fx + 2m B F − c3A e = 0, (3.1.41) 5 5 ˙ ˙ 5 ˙ ˙ 5 2ABFtx + 3mABFt − AB − 3AB Fx − m AB − 4AB F = 0, (3.1.42) 6 6 ˙ ˙ 6 ˙ ˙ 6 2ABFtx + 3mABFt − AB − 3AB Fx − m AB − 4AB F = 0, (3.1.43) 7 7 ˙ ˙ 7 ˙ ˙ 7 2ABFtx + 3mABFt − AB − 3AB Fx − m AB − 4AB F = 0, (3.1.44)

9 5 Ft − F = 0, (3.1.45)

8 5 Fx − F = 0, (3.1.46) 2 8 9 ˙ 8 A Ft − Fx + AAF = 0, (3.1.47) where each F i is dependent on t and x, for i = 5, ..., 9. The teleparallel conformal Killing vectors, given in (3.1.35), can be written in the following com- y2+z2 pact form by introducing the new variables βi = (β1, β2, β3) = 2 , z, y , P i = (P 1,P 2,P 3) = (F 5,F 6,F 7), P 0 = F 8 and P 4 = F 9 :

0 2 2mx i ˙ 2mx i 4 X = B e βiPt + BBe βiP + P , B2 X1 = − e2mx β P i + mβ P i + P 0, A2 i x i

91 1 c c X2 = e−mx c yz + 3 z2 + c z − 3 y2 + yP 1 + P 3, B 2 2 4 2 1 c c X3 = − e−mx c yz + 2 y2 + c y − 2 z2 + zP 1 + P 2, B 3 2 4 2 1 α = e−mx (c z − c y) + P 1. (3.1.48) B 2 3

Similarly the integrability conditions (3.1.36)-(3.1.47) reduce to:

3 i 2 ˙ i ¨ ˙ 2 i −3mx B Ptt + 3B BPt + B BB + B P + kie = 0, (3.1.49)

3 i 3 i 2 3 i 2 −3mx B Pxx + 3mB Px + 2m B P − kiA e = 0, (3.1.50) i i ˙ ˙ i ˙ ˙ i 2ABPtx + 3mABPt − AB − 3AB Px − m AB − 4AB P = 0, (3.1.51)

4 1 Pt − P = 0, (3.1.52)

0 1 Px − P = 0, (3.1.53) 2 0 4 ˙ 0 A Pt − Px + AAP = 0, (3.1.54)

where ki = 0, −c2, c3, for i = 1, 2, 3 respectively. To obtain the explicit form of teleparallel conformal Killing vectors, we need to solve the above integrability conditions. Due to the high non linearity of the above equations, it is not easy to solve them directly. In the following cases, we present the solution of these integrability conditions for some particular choices of the metric functions.

Case 1: In first case we solve the integrability conditions (3.1.49)-(3.1.54) by taking B = const. and A = A(t) such that it satisfies the differential

¨ ˙ 2 c6t constraint AA − A = 0, which implies that A(t) = c5 e with c5 6= 0 and c6 6= 0. For simplicity, we choose c5 = c6 = 1. With these values, the metric

92 given in (3.1.1) takes the form:

ds2 = −dt2 + e2tdx2 + e2mx dy2 + dz2 . (3.1.55)

i −3mx Putting B = const. in Eq. (3.1.49), we get Ptt +kie = 0. Integrating this

i ki 2 −3mx i i equation twice with respect to t, we obtain P = − 2 t e +t G (x)+H (x), where Gi(x) and Hi(x) are functions of integration. Using this value of P i in

Eq. (3.1.51) and differentiating it twice with respect to t, we find that ki = 0 i −mx i −mx i and G (x) = ri e , where ri ∈ R. Hence P = ri t e + H (x). Putting i i −mx back this value of P in Eq. (3.1.51), we get H (x) = e (rimx + hi), i −mx where hi ∈ R. Finally we have P = e (rit + rimx + hi). Simplifying i Eq. (3.1.50) with the help of this value of P , we find that ri = 0, so that i −mx P = hi e . For consistency, let us denote h1 = c7, h2 = c8 and h3 = c9, 1 −mx 2 −mx 3 −mx 1 −mx then P = c7 e ,P = c8 e and P = c9 e . Using P = c7 e in Eqs. (3.1.52) and (3.1.53) and integrating them with respect to t and x

4 −mx 4 0 c7 −mx 5 respectively, we have P = c7t e + G (x) and P = − m e + G (t), where G4(x) and G5(t) are unknown functions which arise in integration process. Substituting these values of P 0 and P 4 in Eq. (3.1.54) and then

4 diﬀerentiating it with respect to t and x, we get c7 = 0,G (x) = c10x+c11 and 5 −2t −t 1 2 −mx G (t) = −c10 e + c12e . Summarizing, we have P = 0,P = c8 e , 3 −mx 4 0 −2t −t P = c9 e ,P = c10x + c11, and P = −c10 e + c12 e . All the integrability conditions are now identically satisﬁed and by putting these values in (3.1.48), we obtain:

0 X = c10x + c11,

1 −2t −t X = −c10 e + c12 e ,

2 −mx X = e (c4z + c9) ,

93 3 −mx −mx X = −e c4y e − c8 , α = 0. (3.1.56)

From above we see that the conformal factor α vanishes, which shows that the teleparallel conformal Killing vectors for the metric given in (3.1.55) are just teleparallel Killing vectors. The teleparallel Killing vectors turned out to be six in number, which can be expressed as:

−2t −t X1 = x ∂t − e ∂x,X2 = ∂t,X3 = e ∂x,

−mx −mx −mx X4 = e (z ∂y − y ∂z) ,X5 = e ∂y,X6 = e ∂z. (3.1.57)

Case 2: In this case we solve the integrability conditions (3.1.49)-(3.1.54) by considering A = const. and B = B(t) such that BB¨ + B˙ 2 = 0, which gives p B(t) = 2 (c5t + c6) with c5 6= 0. In this case the metric given in (3.1.1) takes the form:

2 2 2 2mx 2 2 ds = −dt + dx + e (c5t + c6) dy + dz . (3.1.58)

Using the same procedure as that of the previous case, we can solve the system of equation (3.1.49)-(3.1.54) for these values of the metric functions. In this case we see that the conformal factor α again vanishes, showing that there is no proper teleparallel conformal Killing vector and the teleparallel conformal Killing vectors in this case again reduce to teleparallel Killing vectors, which are given below:

0 X = c7 x + c8,

1 X = c7 t + c9,

94 e−mx c z 2c X2 = √ √4 − 10 , c5t + c6 2 c5 e−mx c y 2c X3 = −√ √4 + 11 . (3.1.59) c5t + c6 2 c5

Choosing c5 = 1, the above six teleparallel Killing vectors can be expressed as follows:

ze−mx ye−mx X1 = x∂t + t∂x,X2 = ∂t,X3 = ∂x,X4 = p ∂y − p ∂z, 2 (t + c6) 2 (t + c6) 2e−mx 2e−mx X5 = −√ ∂y,X6 = −√ ∂z. (3.1.60) t + c6 t + c6

Case 3: In this case we take A = B such that AA¨ + A˙ 2 = 0, so that p A(t) = 2 (c5t + c6), where c5 6= 0. Under these restrictions, the LRS Bianchi type V spacetimes metric given in (3.1.1) becomes:

2 2 2 2mx 2 2 ds = −dt + (c5t + c6) dx + e dy + dz (3.1.61) and the simultaneous solution of the system of equations (3.1.49)-(3.1.54) gives α = 0. This shows that the teleparallel conformal Killing vectors in this case are teleparallel Killing vectors, which are given below:

0 X = c7x + c8, 1 2c mz 2c my c c X1 = −√ 10 + 11 − √9 + 7 , c5t + c6 c5 c5 2 c5 1 c ze−mx 2 (c e−mx + c e−2mx) X2 = √ 4 √ − 12 11 , c5t + c6 2 c5 1 c ye−mx 2 (c e−mx + c e−2mx) X3 = −√ 4 √ + 13 10 . (3.1.62) c5t + c6 2 c5

95 In this case the LRS Bianchi type V spacetimes admit eight teleparallel

Killing vectors which can be written as (choosing c5 = 1 ):

1 2e−mx X1 = x∂t + ∂x,X2 = ∂t,X3 = p ∂x,X4 = −√ ∂z, 2 (t + c6) t + c6 2e−mx ze−mx ye−mx X5 = −√ ∂y,X6 = p ∂y − p ∂z, t + c6 2 (t + c6) 2 (t + c6) 2mz 2e−2mx 2my 2e−2mx X7 = −√ ∂x − √ ∂z,X8 = −√ ∂x − √ ∂y. t + c6 t + c6 t + c6 t + c6

3.2 Teleparallel Conformal Killing Vectors in Static Plane Symmetric Spacetimes

The most general form of static plane symmetric spacetimes metric is [1]:

ds2 = −e2A(x)dt2 + dx2 + e2B(x) dy2 + dz2 , (3.2.1) where the metric functions A and B are dependent on x only. Using the µ a relation (1.4.1.2), the tetrad ha and its inverse hµ for the metric (3.2.1) are given by:

µ A B B a −A −B −B ha = diag e , 1, e , e , hµ = diag e , 1, e , e . (3.2.2)

Using (3.2.2) in Eqs. (1.4.2.2) and (1.4.2.3), we get the following non zero components of torsion tensor:

0 0 2 3 0 T10 = A ,T12 = T13 = B , (3.2.3) where a prime on a metric function denotes its derivative with respect to x. The following set of partial diﬀerential equations is obtained by using the

96 above torsion tensor components and the metric functions in Eq. (3.1.2):

0 X,0 = α, (3.2.4)

1 2A 0 0 2A 0 X,0 − e X,1 − A e X = 0, (3.2.5)

2B 2 2A 0 e X,0 − e X,2 = 0, (3.2.6)

2B 3 2A 0 e X,0 − e X,3 = 0, (3.2.7)

1 X,1 = α, (3.2.8)

2B 2 1 0 2B 2 e X,1 + X,2 + B e X = 0, (3.2.9)

2B 3 1 0 2B 3 e X,1 + X,3 + B e X = 0, (3.2.10)

2 X,2 = α, (3.2.11)

2 3 X,3 + X,2 = 0, (3.2.12)

3 X,3 = α. (3.2.13)

We diﬀerentiate Eqs. (3.2.6), (3.2.7) and (3.2.12) with respect to z, y and t respectively to get the following system:

2B 2 2A 0 e X,03 − e X,23 = 0, (3.2.14)

2B 3 2A 0 e X,02 − e X,23 = 0, (3.2.15)

2 3 X,03 + X,02 = 0. (3.2.16)

Adding Eqs. (3.2.14) and (3.2.15) and using Eq. (3.2.16) in their sum, we have:

0 2 3 X,23 = X,03 = X,02 = 0. (3.2.17)

Similarly diﬀerentiating Eqs. (3.2.9), (3.2.10) and (3.2.12) with respect to z, y and x respectively, we obtain:

1 X,23 = 0. (3.2.18)

97 1 2 Subtracting Eq. (3.2.8) from Eq. (3.2.11), we get X,1 = X,2. If we diﬀeren- tiate this equation with respect to y and z and then use the identity (3.2.18)

2 in the obtained equation, we ﬁnd that X,223 = 0, which yields:

2 1 2 X,3 = y f (t, x, z) + f (t, x, z), (3.2.19) where f 1(t, x, z) and f 2(t, x, z) are unknown functions which arise in the process of integration. Integrating Eq. (3.2.12) with respect to y, after using the above value, we get:

y2 X3 = − f 1(t, x, z) − y f 2(t, x, z) + f 3(t, x, z), (3.2.20) 2

3 2 3 f (t, x, z) being a function of integration. The above values of X,3 and X can be rewritten in the following form by using the identity (3.2.17):

2 1 2 X,3 = y F (x, z) + F (x, z), (3.2.21) y2 X3 = − F 1(x, z) − y F 2(x, z) + f 3(t, x, z). (3.2.22) 2

The functions F 1(x, z) and F 2(x, z) appearing above are functions of integration. Comparing Eqs. (3.2.11) and (3.2.13), we have:

2 3 X,2 = X,3 . (3.2.23)

Diﬀerentiating Eq. (3.2.23) once with respect to z and twice with respect

1 2 3 1 to y, we ﬁnd that Fzz(x, z) = Fzz(x, z) = 0 and fzz(t, x, z) = F (x, z), which are second order partial diﬀerential equations and can be solved easily to obtain F 1(x, z) = z G1(x) + G2(x), F 2(x, z) = z G3(x) + G4(x) and

3 z3 1 z2 2 3 4 i f (t, x, z) = 6 G (x) + 2 G (x) + z F (t, x) + F (t, x), where G (x) and F j(t, x) are functions of integration for i = 1, ..., 4 and j = 3, 4. Substituting

98 these values in Eq. (3.2.21) and integrating it with respect to z, we have:

z2 z2 X2 = y G1(x) + z G2(x) + G3(x) + z G4(x) + f 4(t, x, y), (3.2.24) 2 2 f 4(t, x, y) being a function of integration. Also Eq. (3.2.22) takes the form:

y2 z3 X3 = − z G1(x) + G2(x) − y z G3(x) + G4(x) + G1(x) 2 6 z2 + G2(x) + z F 3(t, x) + F 4(t, x). (3.2.25) 2

Integrating Eq. (3.2.23) with respect to y, after using Eqs. (3.2.24) and

4 y3 1 y2 3 3 5 (3.2.25), we obtain f (t, x, y) = − 6 G (x) − 2 G (x) + y F (t, x) + F (t, x), where F 5(t, x) is a function of integration. Putting back this value in Eq. (3.2.24), we have:

z2 z2 y3 X2 = y G1(x) + z G2(x) + G3(x) + z G4(x) − G1(x) 2 2 6 y2 − G3(x) + y F 3(t, x) + F 5(t, x). (3.2.26) 2

The following value of X0 can be found if we put the value of X3 from Eq. (3.2.25) in Eq. (3.2.7) and then integrate it with respect to z:

z2 X0 = e2(B−A) F 3(t, x) + zF 4(t, x) + f 5(t, x, y), (3.2.27) 2 t t where the function f 5(t, x, y) arises in the integration process. Keeping the identity (3.2.18) in mind, if we diﬀerentiate Eq. (3.2.10) with respect to y, it gives:

3 0 3 X,12 + B X,2 = 0. (3.2.28)

Diﬀerentiating Eq. (3.2.28) with respect to y and z, after using the value of

3 i 0 i X from Eq. (3.2.25), we have the diﬀerential equation Gx(x)+B G (x) = 0,

99 i −B for i = 1, ..., 4. We can solve this diﬀerential equation to get G (x) = ci e , for i = 1, ..., 4. Putting back these values in Eqs. (3.2.25) and (3.2.26), we have: c c c X2 = ye−B 1 z2 + c z + e−B 3 z2 − y2 − 1 y3 + c z 2 2 2 6 4 + y F 3(t, x) + F 5(t, x), y2 c c X3 = − e−B (c z + c ) − ye−B (c z + c ) + e−B 1 z3 + 2 z2 2 1 2 3 4 6 2 + z F 3(t, x) + F 4(t, x). (3.2.29)

Using the above system in Eq. (3.2.10) and then integrating it with respect to z, we obtain: z2 z2 X1 = −e2B F 3(t, x) + zF 4(t, x) − B0e2B F 3(t, x) + zF 4(t, x) 2 x x 2 + f 6(t, x, y), (3.2.30) f 6(t, x, y) being a function of integration. Substituting the above obtained values of X0,X1 and X2 in Eqs. (3.2.6) and (3.2.9) and then integrating them with respect to y, we have: y2 f 5(t, x, y) = e2(B−A) F 3(t, x) + y F 5(t, x) + F 6(t, x), 2 t t y2 f 6(t, x, y) = −e2B F 3(t, x) + y F 5(t, x) 2 x x y2 − B0e2B F 3(t, x) + y F 5(t, x) + F 7(t, x), (3.2.31) 2 where the functions F 6(t, x) and F 7(t, x) arise in the process of integration. As far as the conformal factor α is concerned, it can be written in the following form by using Eq. (3.2.13): nc o α = e−B 1 z2 − y2 − c y + c z + F 3(t, x). (3.2.32) 2 3 2 100 Summarizing, we have the following teleparallel conformal Killing vectors and conformal factor in terms of unknown functions of t and x:

y2 + z2 X0 = e2(B−A) F 3(t, x) + zF 4(t, x) + yF 5(t, x) + F 6(t, x), 2 t t t y2 + z2 X1 = −e2B F 3(t, x) + zF 4(t, x) + yF 5(t, x) 2 x x x y2 + z2 − B0e2B F 3(t, x) + zF 4(t, x) + yF 5(t, x) + F 7(t, x), 2 c c c X2 = ye−B 1 z2 + c z + e−B 3 z2 − y2 − 1 y3 + c z 2 2 2 6 4 + y F 3(t, x) + F 5(t, x), y2 c c X3 = − e−B (c z + c ) − ye−B (c z + c ) + e−B 1 z3 + 2 z2 2 1 2 3 4 6 2 + z F 3(t, x) + F 4(t, x), c α = e−B 1 z2 − y2 − c y + c z + F 3(t, x). (3.2.33) 2 3 2

Substituting the above system in Eqs. (3.2.4), (3.2.5), (3.2.8) and then comparing the like terms on both sides of the resulting equations, we get c1 = 0 and the following integrability conditions are obtained:

3 0 3 00 02 3 Fxx(t, x) + 3B Fx (t, x) + B + 2B F (t, x) = 0, (3.2.34) 4 0 4 00 02 4 −3B Fxx(t, x) + 3B Fx (t, x) + B + 2B F (t, x) + c2e = 0, (3.2.35) 5 0 5 00 02 5 −3B Fxx(t, x) + 3B Fx (t, x) + B + 2B F (t, x) − c3e = 0, (3.2.36)

3 0 0 3 2Ftx(t, x) + (3B − A ) Ft (t, x) = 0, (3.2.37)

4 0 0 4 2Ftx(t, x) + (3B − A ) Ft (t, x) = 0, (3.2.38)

5 0 0 5 2Ftx(t, x) + (3B − A ) Ft (t, x) = 0, (3.2.39)

3 Ftt(t, x) = 0, (3.2.40)

4 2A−3B Ftt(t, x) − c2e = 0, (3.2.41)

101 5 2A−3B Ftt(t, x) + c3e = 0, (3.2.42)

6 3 Ft (t, x) − F (t, x) = 0, (3.2.43)

7 3 Fx (t, x) − F (t, x) = 0, (3.2.44)

7 2A 6 0 2A 6 Ft (t, x) − e Fx (t, x) − A e F (t, x) = 0. (3.2.45)

Introducing the new variables P i = (P 1,P 2,P 3) = (F 3,F 4,F 5), P 0 = F 6,

4 7 y2+z2 P = F and βi = (β1, β2, β3) = 2 , z, y , the teleparallel conformal Killing vectors given in (3.2.33) can be written in the following form:

0 2(B−A) i 0 X = e βiPt + P ,

1 2B i 0 2B i 4 X = −e βiPx − B e βiP + P , nc o X2 = β P i + e−B 3 z2 − y2 + c yz + c z , i,2 2 2 4 nc o X2 = β P i + e−B 2 z2 − y2 − c yz − c y , i,3 2 3 4 −B 1 α = e (c2z − c3y) + P . (3.2.46)

Similarly the integrability conditions (3.2.34)-(3.2.45) in these new variables are now reduced to:

i 0 i 00 02 i −3B Pxx + 3B Px + B + 2B P − kie = 0, (3.2.47)

i 0 0 i 2Ptx + (3B − A ) Pt = 0, (3.2.48)

i 2A−3B Ptt + kie = 0, (3.2.49)

0 1 Pt − P = 0, (3.2.50)

4 1 Px − P = 0, (3.2.51)

1 2A 0 0 2A 0 Pt − e Px − A e P = 0. (3.2.52) where ki = 0, −c2, c3 respectively for i = 1, 2, 3. To ﬁnd the explicit form of teleparallel conformal Killing vectors given in (3.2.46), we need to solve

102 equations (3.2.47)-(3.2.52). In the following three cases, we solve these equations by imposing some conditions on the metric functions A and B.

Case 1: Here we assume that A0 6= 0 and B0 = 0. For simplicity we take B = 0. If we put these values in Eqs. (3.2.47) and (3.2.49) for i = 1, we get

1 1 1 Ptt = Pxx = 0. Solving these equations, we have P = c5tx + c6x + c7t + c8. Putting this value of P 1 in Eqs. (3.2.50) and (3.2.51) and integrating them with respect to t and x respectively, we have:

c c P 0 = 5 t2x + c tx + 7 t2 + c t + R1(x), 2 6 2 8 c c P 4 = 5 tx2 + 6 x2 + c tx + c x + R2(t), (3.2.53) 2 2 7 8 where R1(x) and R2(t) are functions of integration. Also substituting the

1 0 above value of P in Eq. (3.2.48), we obtain 2c5 − A (c5x + c7) = 0. Now if we suppose that c5x + c7 6= 0, then diving the last equation by c5x + c7 and 2 integrating it with respect to x, we get A(x) = ln c9 (c5x + c7) . Substituting the values of P 0 and P 4 from system (3.2.53) in Eq. (3.2.52) and then diﬀerentiating the resulting equation with respect to t and x, we get c5 = 0 ⇒ A0 = 0, which is a contradiction because in this case we have assumed that

0 A 6= 0. This contradiction arises because of our supposition that c5x+c7 6= 0, so that we must have c5x + c7 = 0 ⇒ c5 = c7 = 0. Consequently the values of P 0,P 1 and P 4 become:

0 1 P = c6tx + c8t + R (x),

1 P = c6x + c8, c P 4 = 6 x2 + c x + R2(t). (3.2.54) 2 8

103 Using these values in Eq. (3.2.52) and diﬀerentiating it with respect to t and 2 c11 1 −A x, we get R (t) = c10, A = ln and R (x) = c12 e . Hence (3.2.54) c6x+c8 becomes:

0 −A P = c6 tx + c8 t + c12e ,

1 P = c6 x + c8, c P 4 = 6 x2 + c x + c . (3.2.55) 2 8 10

The values of the functions P 2 and P 3 are still left to be determined. Putting i = 2 in Eq. (3.2.47) and integrating it twice with respect to x, we obtain

2 c2 2 3 4 3 4 P = − 2 x + x R (t) + R (t), where R (t) and R (t) are functions of integration. If we put this value of P 2 in Eq. (3.2.49) and diﬀerentiate it with

3 4 respect to t, we see that Rttt(t) = Rttt(t) = 0, which on integration gives

3 c13 2 4 c16 2 R (t) = 2 t + c14t + c15 and R (t) = 2 t + c17t + c18. Substituting back these values in eq. (3.2.49) and diﬀerentiating it twice with respect to x, we

2 find that c2 = c13 = c16 = 0. Thus we have P = c14tx + c15x + c17t + c18. 2 The constants c14 and c17 vanish if we put this value of P in Eq. (3.2.48) 2 and differentiate it with respect to x. Finally we have P = c15x + c18. To find P 3, we repeat the same steps for i = 3 in Eqs. (3.2.47)-(3.2.49).

3 Consequently we get c3 = 0 and P = c19x + c20. Summarizing, we have:

0 −A P = c6 tx + c8 t + c12e ,

1 P = c6 x + c8,

2 P = c15 x + c18,

3 P = c19 x + c20, c P 4 = 6 x2 + c x + c . (3.2.56) 2 8 10

104 Using all these values from (3.2.56) in (3.2.46), we have the following explicit form of teleparallel conformal Killing vectors:

0 −A X = c6tx + c8t + c12e , c X1 = 6 x2 − y2 − z2 − c z − c y + c x + c , 2 15 19 8 10 2 X = c6xy + c8y + c4z + c19x + c20,

3 X = c6xz + c8z − c4y + c15x + c18,

α = c6x + c8. (3.2.57)

For the considered values of the metric functions A and B, the metric of static plane symmetric spacetimes given in (3.2.1) becomes:

c 2 ds2 = − 11 dt2 + dx2 + dy2 + dz2. (3.2.58) c6x + c8

From (3.2.57), we see that the above static plane symmetric metric admits nine teleparallel conformal Killing vectors, which can be expressed as:

1 X = tx ∂ + x2 − y2 − z2 ∂ + xy ∂ + xz ∂ , 1 t 2 x y z −A X2 = t ∂t + x ∂x + y ∂y + z ∂z,X3 = e ∂t,X4 = −z∂x + x∂z,

X5 = −y∂x + x∂y,X6 = z∂y − y∂z,X7 = ∂x,X8 = ∂y,X9 = ∂z. (3.2.59)

In the above set of nine teleparallel conformal Killing vectors, X1 and X2 are proper teleparallel conformal Killing and proper teleparallel homothetic vectors respectively. The remaining seven are teleparallel Killing vectors.

Case 2: In this case we assume that A = const. and B = B(x) such that

00 02 p B + 2B = 0, so that B = ln 2 (c5x + c6), where c5 6= 0. For these values

105 of A and B, the metric given in (3.2.1) after suitable rescaling takes the form:

2 2 2 2 2 ds = −dt + dx + (c5x + c6) dy + dz . (3.2.60)

The procedure of solving Eqs. (3.2.47)-(3.2.52) in this case is basic and similar to the previous case, we therefore omit to write the details here. Finally the following teleparallel conformal Killing vectors are obtained in this case:

0 X = c7t + c8x + c9, 1 X1 = − c c y2 + z2 − c (c y + c z) + c t + c x + c , 2 5 7 5 10 11 12 7 13 2 −B c14 −B X = c4z e + c7y − e + c10, c5 3 −B c15 −B X = −c4y e + c7z − e + c11, c5

α = c7. (3.2.61)

From above, it can be seen that the conformal factor α becomes constant. Thus the teleparallel conformal Killing vectors in this case are just teleparallel homothetic vectors which are ten in number and can be written as follows

(by choosing c5 = 1): 1 X = t ∂ + x − y2 + z2 ∂ + y ∂ + z ∂ ,X = x ∂ ,X = ∂ , 1 t 2 x y z 2 t 3 t −B X4 = −y ∂x + ∂y,X5 = −z ∂x + ∂z,X6 = t ∂x,X7 = ∂x,X8 = −e ∂y,

−B −B −B X9 = ze ∂y − ye ∂z,X10 = −e ∂z. (3.2.62)

In the above set of ten teleparallel homothetic vectors, X1 is a proper teleparallel homothetic vector and the remaining nine are teleparallel Killing vectors admitted by the static plane symmetric metric given in (3.2.60).

106 Case 3: In this case we solve the integrability conditions (3.2.47)-(3.2.52)

00 02 p by taking A = B = A(x) such that A + 2A = 0 ⇒ A = ln 2 (c5x + c6), where c5 6= 0. For these values of A and B, the metric given in (3.2.1) becomes: 2 2 2 2 2 ds = dx + (c5x + c6) −dt + dy + dz , (3.2.63) which is a conformally ﬂat metric in the context of general relativity. Taking c5 = 1 and solving Eqs. (3.2.47)-(3.2.52) for the above metric, we get the following teleparallel conformal Killing vectors. √ 0 −A X = c2tz − c3ty + c7z + c8y − 2 2 c9t e , √ 1 A X = −c10z − c11y + c2z − c3y − 2 2 c9 e + c12, c √ X2 = c yz + 3 z2 − y2 − t2 + c z − 2 2 c y + c t − c x − c e−A 2 2 4 9 13 3 14 A + c3 e + c11, c √ X3 = −c yz + 2 z2 − y2 + t2 − c y − 2 2c z + c t − c x − c e−A 3 2 4 9 7 2 15 A − c2 e + c10, √ −A α = c2z − c3y − 2 2 c9 e , (3.2.64)

where each ci is a constant. From above we can see that the static plane symmetric metric given in (3.2.63) admits twelve teleparallel conformal Killing vectors. The constants c2, c3 and c9 correspond to three proper teleparallel conformal Killing vectors and the remaining nine constants correspond to teleparallel Killing vectors admitted by the metric (3.2.63). There exist no proper teleparallel homothetic vector in this case. This is a surprising situation because we know that a conformally ﬂat metric admits ﬁfteen conformal Killing vectors in general relativity [17] but here in the context of teleparallel

107 gravity, we got twelve teleparallel conformal Killing vectors for the conformally ﬂat metric (3.2.63). This decrease in the number of conformal Killing vectors occurred because of presence of non vanishing torsion and vanishing curvature in the spacetime.

3.3 Teleparallel Conformal Killing Vectors in Non Static Plane Symmetric Spacetimes

In this section we extend our study of teleparallel conformal Killing vectors for non static plane symmetric spacetimes whose metric is of the form [1]:

ds2 = −e2A(t,x)dt2 + e2B(t,x)dx2 + e2C(t,x) dy2 + dz2 , (3.3.1) where A, B and C are functions of t and x only. Using the relation (1.4.1.2), µ a the tetrad ha and its inverse hµ for the above metric are given by:

µ A B C C a −A −B −C −C ha = diag e , e , e , e , hµ = diag e , e , e , e . (3.3.2)

The following non zero components of torsion tensor are obtained by using (3.3.2) in Eqs. (1.4.2.2) and (1.4.2.3):

0 0 1 ˙ 2 3 ˙ 2 3 0 T10 = A ,T01 = B,T02 = T03 = C,T12 = T13 = C , (3.3.3) where a dot and a prime on a metric function denotes its derivative with respect to t and x respectively. Using the above torsion tensor components and the metric given in (3.3.1) in Eq. (3.1.2), we obtain the following set of coupled partial diﬀerential equations:

˙ 0 0 AX + X,0 = α, (3.3.4) 2B 1 ˙ 2B 1 2A 0 0 2A 0 e X,0 + B e X − e X,1 − A e X = 0, (3.3.5)

108 2C 2 ˙ 2C 2 2A 0 e X,0 + C e X − e X,2 = 0, (3.3.6) 2C 3 ˙ 2C 3 2A 0 e X,0 + C e X − e X,3 = 0, (3.3.7)

0 1 1 B X + X,1 = α, (3.3.8)

2C 2 2B 1 0 2C 2 e X,1 + e X,2 + C e X = 0, (3.3.9)

2C 3 2B 1 0 2C 3 e X,1 + e X,3 + C e X = 0, (3.3.10)

2 X,2 = α, (3.3.11)

2 3 X,3 + X,2 = 0, (3.3.12)

3 X,3 = α. (3.3.13)

Like the previous two spacetimes, ﬁrst we solve the above ten equations to obtain teleparallel conformal Killing vectors in terms of some unknown functions of t and x. The process is explained below. If we diﬀerentiate Eqs. (3.3.6) and (3.3.9) with respect to z, Eqs. (3.3.7) and (3.3.10) with respect to y and Eq. (3.3.12) with respect to t and x, then after some simple algebraic calculations in the obtained equations, we have:

0 1 X,23 = X,23 = 0. (3.3.14)

Subtracting Eq. (3.3.8) from Eq. (3.3.11), we obtain the following relation:

2 0 1 1 X,2 − B X − X,1 = 0. (3.3.15)

Diﬀerentiating Eq. (3.3.15) with respect to y and z and then using the

2 identity (3.3.14) in the resulting equation, we get X,223 = 0, which implies:

2 1 2 X,3 = y f (t, x, z) + f (t, x, z), (3.3.16) where f 1(t, x, z) and f 2(t, x, z) are functions of integration. Using this value

2 of X,3 in Eq. (3.3.12) and integrating it with respect to y, we have:

109 y2 X3 = − f 1(t, x, z) − y f 2(t, x, z) + f 3(t, x, z), (3.3.17) 2 f 3(t, x, z) being a function of integration. Now subtracting Eq. (3.3.4) from Eq. (3.3.13), we obtain:

3 ˙ 0 0 X,3 − AX − X,0 = 0. (3.3.18)

Keeping the identity (3.3.14) in mind, if we diﬀerentiate Eq. (3.3.18) with

3 respect to y and z, it gives X,233 = 0, which by using Eq. (3.3.17) yields f 1(t, x, z) = z F 1(t, x) + F 2(t, x) and f 2(t, x, z) = z F 3(t, x) + F 4(t, x). The functions F i(t, x), for i = 1, ..., 4, arise in the integration process. Putting back these values in Eq. (3.3.17), we get:

y2 X3 = − z F 1(t, x) + F 2(t, x) − y z F 3(t, x) + F 4(t, x) + f 3(t, x, z). 2 (3.3.19) Also using the above obtained values of f 1(t, x, z) and f 2(t, x, z) in Eq. (3.3.16) and integrating it with respect to z, we have:

z2 z2 X2 = y F 1(t, x) + z F 2(t, x) + F 3(t, x) + z F 4(t, x) + f 4(t, x, y), 2 2 (3.3.20) where f 4(t, x, y) is an integration function. Subtracting Eq. (3.3.11) from Eq. (3.3.13), we get the following relation:

3 2 X,3 − X,2 = 0. (3.3.21)

If we diﬀerentiate Eqs. (3.3.12) and (3.3.21) with respect to y and z respec-

3 3 tively and then add them, we obtain X,22 + X,33 = 0, which by using Eq. 3 1 2 (3.3.19) gives fzz(t, x, z) = z F (t, x) + F (t, x). Integrating this equation

110 twice with respect to z, we get: z3 z2 f 3(t, x, z) = F 1(t, x) + F 2(t, x) + z F 5(t, x) + F 6(t, x), (3.3.22) 6 2 F 5(t, x) and F 6(t, x) being functions of integration. Similarly the following value of f 4(t, x, y) can be found by diﬀerentiating Eqs. (3.3.12) and (3.3.21) with respect to z and y respectively and then using Eq. (3.3.20) in their sum: y3 y2 f 4(t, x, y) = − F 1(t, x) − F 2(t, x) + y F 7(t, x) + F 8(t, x), (3.3.23) 6 2 where F 7(t, x) and F 8(t, x) are functions of integration. Putting back the values of f 3(t, x, z) and f 4(t, x, y) from Eqs. (3.3.22) and (3.3.23) in Eqs. (3.3.19) and (3.3.20), we have: z2 z2 X2 = y F 1(t, x) + zF 2(t, x) + F 3(t, x) + zF 4(t, x) 2 2 y3 y2 − F 1(t, x) − F 2(t, x) + y F 7(t, x) + F 8(t, x), 6 2 y2 X3 = − z F 1(t, x) + F 2(t, x) − y z F 3(t, x) + F 4(t, x) 2 z3 z2 + F 1(t, x) + F 2(t, x) + z F 5(t, x) + F 6(t, x). (3.3.24) 6 2 Using the system (3.3.24) in Eq. (3.3.10) and integrating it with respect to z, we obtain: y2z2 y2z yz2 X1 = e2(C−B) F 1(t, x) + F 2(t, x) + F 3(t, x) + yzF 4(t, x) 4 x 2 x 2 x x z4 z3 z2 − F 1(t, x) − F 2(t, x) − F 5(t, x) − zF 6(t, x) 24 x 6 x 2 x x y2z2 y2z yz2 + C0e2(C−B) F 1(t, x) + F 2(t, x) + F 3(t, x) + yzF 4(t, x) 4 2 2 z4 z3 z2 − F 1(t, x) − F 2(t, x) − F 5(t, x) − zF 6(t, x) + f 5(t, x, y), 24 6 2 (3.3.25)

111 f 5(t, x, y) being a function of integration. Substituting this value of X1 in

i 0 i Eq. (3.3.14), we get Fx(t, x) + C F (t, x) = 0, for i = 1, ..., 4. Solving this equation, we have F i(t, x) = e−C Gi(t), for i = 1, ..., 4. Putting back these values in (3.3.24) and (3.3.25), we have: z2 z2 X1 = −e2(C−B) F 5(t, x) + zF 6(t, x) + C0F 5(t, x) + z C0F 6(t, x) 2 x x 2 + f 5(t, x, y), z2 z2 X2 = y e−C G1(t) + zG2(t) + e−C G3(t) + zG4(t) 2 2 y3 y2 − e−C G1(t) + G2(t) + y F 7(t, x) + F 8(t, x), 6 2 y2 X3 = − e−C z G1(t) + G2(t) − y e−C z G3(t) + G4(t) 2 z3 z2 + e−C G1(t) + G2(t) + z F 5(t, x) + F 6(t, x). (3.3.26) 6 2 The following value of f 5(t, x, y) is found by putting the above system in Eq. (3.3.9) and then integrating it with respect to y: y2 f 5(t, x, y) = −e2(C−B) F 7(t, x) + yF 8(t, x) 2 x x y2 − C0 e2(C−B) F 7(t, x) + yF 8(t, x) + F 9(t, x), (3.3.27) 2 where F 9(t, x) is a function of integration. Moreover, using the system (3.3.26) in Eq. (3.3.7), diﬀerentiating it with respect to y and using the rela-

i tion (3.3.14) in the obtained equation, we ﬁnd that G (t) = ci, for i = 1, ..., 4. Substituting back these values in Eq. (3.3.7) and integrating it with respect to z, we have: z2 z2 X0 = e2(C−A) F 5(t, x) + zF 6(t, x) + CF˙ 5(t, x) + z CF˙ 6(t, x) 2 t t 2 + f 6(t, x, y), (3.3.28)

112 f 6(t, x, y) being a function of integration. Substituting the values of X2 and X0 from (3.3.26) and (3.3.28) in Eq. (3.3.6) and then integrating it with respect to y, we get: y2 y2 f 6(t, x, y) = e2(C−A) F 7(t, x) + yF 8(t, x) + CF˙ 7(t, x) + yCF˙ 8(t, x) 2 t t 2 + F 10(t, x), (3.3.29) where the function F 10(t, x) arises in the process of integration. Also using the system (3.3.26) in Eq. (3.3.21), we get F 5(t, x) = F 7(t, x). Summarizing all the obtained data, we have: y2 + z2 y2 + z2 X0 = e2(C−A) F 5(t, x) + CF˙ 5(t, x) + z F 6(t, x) 2 t 2 t ˙ 6 8 ˙ 8 10 + z CF (t, x) + y Ft (t, x) + y CF (t, x) + F (t, x), y2 + z2 y2 + z2 X1 = −e2(C−B) F 5(t, x) + C0 F 5(t, x) + z F 6(t, x) 2 x 2 x 0 6 8 0 8 9 + z C F (t, x) + y Fx (t, x) + y C F (t, x) + F (t, x), c c c X2 = e−C 1 yz2 + c yz + 3 z2 − y2 + c z − 1 y3 + y F 5(t, x) 2 2 2 4 6 + F 8(t, x), c c c X3 = e−C − 1 y2z − c yz + 2 z2 − y2 − c y + 1 z3 + z F 5(t, x) 2 3 2 4 6 + F 8(t, x). (3.3.30)

From above we see that the teleparallel conformal Killing vectors are obtained in terms of some unknown functions of t and x. As far as the conformal factor α is concerned, it gets the following form by using the system (3.3.30) in Eq. (3.3.13): nc o α = e−C 1 z2 − y2 + c z − c y + F 5(t, x). (3.3.31) 2 2 3 113 Substituting the values of Xa and α from (3.3.30) and (3.3.31) in Eqs. (3.3.4), (3.3.5) and (3.3.8) and then equating like terms on both sides of the resulting equations, we get c1 = 0 and the following integrability conditions are obtained:

5 0 0 5 0 ˙ ˙ 0 ˙ 0 ˙ 0 5 ˙ ˙ 5 2Ftx − (A − 3C ) Ft − A C + BC − 2C − 4CC F − B − 3C Fx = 0, (3.3.32) 6 0 0 6 0 ˙ ˙ 0 ˙ 0 ˙ 0 6 ˙ ˙ 6 2Ftx − (A − 3C ) Ft − A C + BC − 2C − 4CC F − B − 3C Fx = 0, (3.3.33) 8 0 0 8 0 ˙ ˙ 0 ˙ 0 ˙ 0 8 ˙ ˙ 8 2Ftx − (A − 3C ) Ft − A C + BC − 2C − 4CC F − B − 3C Fx = 0, (3.3.34)

5 0 0 5 0 0 02 00 5 Fxx − (B − 3C ) Fx − B C − 2C − C F = 0, (3.3.35) 6 0 0 6 0 0 02 00 6 2B−3C Fxx − (B − 3C ) Fx − B C − 2C − C F + c2e = 0, (3.3.36) 8 0 0 8 0 0 02 00 8 2B−3C Fxx − (B − 3C ) Fx − B C − 2C − C F − c3e = 0, (3.3.37) 5 ˙ ˙ 5 ¨ ˙ 2 ˙ ˙ 5 Ftt + 3C − A Ft + C + 2C − AC F = 0, (3.3.38) 6 ˙ ˙ 6 ¨ ˙ 2 ˙ ˙ 6 2A−3C Ftt + 3C − A Ft + C + 2C − AC F − c2e = 0, (3.3.39) 8 ˙ ˙ 8 ¨ ˙ 2 ˙ ˙ 8 2A−3C Ftt + 3C − A Ft + C + 2C − AC F + c3e = 0, (3.3.40)

0 9 9 5 B F + Fx − F = 0, (3.3.41) ˙ 10 10 5 AF + Ft − F = 0, (3.3.42) 2B 9 ˙ 2B 9 2A 10 0 2A 10 e Ft + Be F − e Fx − A e F = 0. (3.3.43)

It can be noticed that the components of the teleparallel conformal Killing vectors given in (3.3.30), the conformal factor and the integrability conditions (3.3.32)-(3.3.43) obtained for non static plane symmetric spacetimes reduce to those of the static plane symmetric spacetimes given in (3.2.33)-(3.2.45) if

114 we change the metric (3.3.1) into static plane symmetric spacetimes metric by taking all the metric functions in (3.3.1) to be independent of t. Introducing the variables P i = (P 1,P 2,P 3) = (F 5,F 6,F 8), P 0 = F 10, P 4 = F 9 and y2+z2 βi = (β1, β2, β3) = 2 , z, y , the teleparallel conformal Killing vectors given in (3.3.30) along with their conformal factor can be rewritten in the following form:

0 2(C−A) i 2(C−A) ˙ i 0 X = e βi Pt + e C βi P + P ,

1 2(C−B) i 2(C−B) 0 i 4 X = −e βi Px − e C βi P + P , c X2 = β P i + e−C c yz + 3 z2 − y2 + c z , i,2 2 2 4 c X3 = β P i + e−C −c yz + 2 z2 − y2 − c y , i,3 3 2 4

−C 1 α = e (c2z − c3y) + P . (3.3.44)

Similarly the integrability conditions (3.3.32)-(3.3.43) become:

i 0 0 i 0 0 02 00 i 2B−3C Pxx − (B − 3C ) Px − B C − 2C − C P + kie = 0, (3.3.45) i ˙ ˙ i ¨ ˙ 2 ˙ ˙ i 2A−3C Ptt + 3C − A Pt + C + 2C − AC P − kie = 0, (3.3.46)

0 4 4 1 B P + Px − P = 0, (3.3.47) ˙ 0 0 1 AP + Pt − P = 0, (3.3.48) 2B 4 ˙ 2B 4 2A 0 0 2A 0 e Pt + Be P − e Px − A e P = 0, (3.3.49) i 0 0 i 0 ˙ ˙ 0 ˙ 0 ˙ 0 i ˙ ˙ i 2Ptx − (A − 3C ) Pt − A C + BC − 2C − 4CC P − B − 3C Px = 0, (3.3.50) where ki = 0, c2, −c3 for i = 1, 2, 3 respectively. The teleparallel conformal Killing vectors given in (3.3.44) will get their ﬁnal form if we solve the system of equations (3.3.45)-(3.3.50). In the following cases, we solve these equations

115 by imposing certain restrictions on the metric functions A, B and C. The calculation involved in the solution of these equations is simple, therefore we only present the ﬁnally obtained results in each case.

Case 1: In this we assume that A = const., B = B(t) and C = const. Solving the system of equations (3.3.45)-(3.3.50) under these assumptions, c7 the metric function B gets the value B = ln , where c5 6= 0 and c5t+c6 c7 6= 0. Thus after a suitable rescaling, the line element (3.3.1) becomes: c 2 ds2 = −dt2 + 7 dx2 + dy2 + dz2 (3.3.51) c5t + c6 and the following teleparallel conformal Killing vectors are obtained: c c X0 = 5 8 t2 + y2 + z2 + c c t + c z + c y + c , 2 6 8 9 10 11 1 X = c8x (c5t + c6) ,

2 X = c8y (c5t + c6) + c4z + c10t + c12,

3 X = c8z (c5t + c6) − c4y + c9t + c13,

α = c8 (c5t + c6) . (3.3.52)

From above we see that the metric (3.3.51) admits seven teleparallel conformal Killing vectors, which by choosing c5 = c6 = 1, can be expressed as: 1 X = t2 + y2 + z2 + 2t ∂ + x (t + 1) ∂ + y (t + 1) ∂ + z (t + 1) ∂ , 1 2 t x y z

X2 = z∂t + t∂z,X3 = y∂t + t∂y,X4 = z∂y − y∂z,

X5 = ∂t,X6 = ∂y,X7 = ∂z, (3.3.53) where X1 represents a proper teleparallel conformal Killing vector admitted by the metric (3.3.51) and the remaining six are teleparallel Killing vectors.

116 No proper teleparallel homothetic vector exists in this case.

Case 2: Here we take A = const., B = const. and C = C(t) such that ¨ ˙ 2 p C + 2C = 0 ⇒ C = ln 2 (c5t + c6), where c5 6= 0. In this case the metric (3.3.1) gets the form:

2 2 2 2 2 ds = −dt + dx + (c5t + c6) dy + dz . (3.3.54)

Choosing c5 = 1 and solving equations (3.3.45)-(3.3.50) for the above metric, we obtain: c X0 = 7 y2 + z2 + c t + c x + c y + c z + c , 2 7 8 9 10 11 1 X = c7x + c8t + c12,

2 −C X = c4z e + c7y + c9,

3 −C X = −c4y e + c7z + c10,

α = c7. (3.3.55)

Thus in this case the teleparallel conformal Killing vectors are reduced to teleparallel homothetic vectors, which can be expressed as: 1 X = y2 + z2 + 2t ∂ + x ∂ + y ∂ + z ∂ , 1 2 t x y z

X2 = x ∂t + t ∂x,X3 = y ∂t + ∂y,X4 = z ∂t + ∂z,

−C −C X5 = ∂t,X6 = ∂x,X7 = z e ∂y − y e ∂z, (3.3.56)

where X1 represents a proper teleparallel homothetic vector and the remaining six are teleparallel Killing vectors admitted by the metric (3.3.54).

Case 3: In this case we consider A = A(x), B = B(t) and C = const.

117 The solution of equations (3.3.45)-(3.3.50) under these conditions on the metric functions yields A = ln x and B = ln t. Thus the metric (3.3.1) can be written as: ds2 = −x2 dt2 + t2 dx2 + dy2 + dz2. (3.3.57)

For the above metric, we have the following teleparallel conformal Killing vectors:

c c c c X0 = e−2A 5 x y2 + z2 + c xz + c xy + 5 xt2 + c t + 9 + 10 ln x, 2 7 8 2 6 x x c c c c X1 = −e−2B 5 t y2 + z2 + c tz + c ty + 5 tx2 + c x + 11 + 10 ln t, 2 7 8 2 6 t t 2 X = c4z + c5txy + c6y + c8tx + c12,

3 X = −c4y + c5txz + c6z + c7tx + c13,

α = c5xt + c6. (3.3.58)

It shows that the metric given in (3.3.57) admits ten teleparallel conformal Killing vectors, out of which one is each a proper teleparallel conformal Killing vector and a proper teleparallel homothetic vector. The remaining eight are teleparallel Killing vectors. In generators form, the above ten teleparallel conformal Killing vectors can be written as:

1 1 X = e−2A x y2 + z2 + e2At2 ∂ − e−2B t y2 + z2 + e2Bx2 ∂ 1 2 t 2 x

+ txy ∂y + txz ∂z,

−2A −2B X2 = t ∂t + x ∂x + y ∂y + z ∂z,X3 = e xz ∂t − tz e ∂x + tx ∂z, 1 1 X = e−2A xy ∂ − ty e−2B∂ + tx ∂ ,X = ∂ ,X = ∂ , 4 t x y 5 x t 6 t x 1 1 X = ln x ∂ + ln t ∂ ,X = z∂ − y ∂ ,X = ∂ ,X = ∂ . 7 x t t x 8 y z 9 y 10 z (3.3.59)

118 In the above set, X1 and X2 represent proper teleparallel conformal Killing and proper teleparallel homothetic vectors respectively, while X3,X4, ..., X10 represent teleparallel Killing vectors admitted by the metric (3.3.57).

Case 4: If we assume that A = A(x), B = const. and C = C(t) such ¨ ˙ 2 p 00 that C + 2C = 0 ⇒ C = ln 2 (c5t + c6) and A = 0 ⇒ A(x) = c7x + c8, where c5 6= 0 and c7 6= 0, then after a suitable rescaling the metric (3.3.1) becomes: 2 2x 2 2 2 2 ds = −e dt + dx + (c5t + c6) dy + dz . (3.3.60)

It is straightforward to solve Eqs. (3.3.45)-(3.3.50) for the above metric and obtain:

0 −A X = c9 e ,

1 X = c10, √ 2 −C X = e c4z − 2 2 c11 , √ 3 −C X = −e c4y + 2 2 c12 , α = 0, (3.3.61) which shows that the teleparallel conformal Killing vectors in this case are just teleparallel Killing vectors, which are ﬁve in number and can be written as follows:

−A −C −C X1 = e ∂t,X2 = ∂x,X3 = ze ∂y − ye ∂z, √ √ −C −C X4 = −2 2 e ∂y,X5 = −2 2 e ∂z. (3.3.62)

Case 5: Here we assume that A = const., B = B(t) and C = C(x) such

119 00 02 p that C + 2C = 0 ⇒ C = ln 2 (c5x + c6), where c5 6= 0. During the procedure of solving Eqs. (3.3.45)-(3.3.50), the function B gets the value 3 B = ln c7 (c8t + c9) with c7 6= 0, c8 6= 0 and the conformal factor α vanishes. Thus the metric (3.3.1) can be written as:

2 2 6 2 2 2 ds = −dt + (c8t + c9) dx + (c5x + c6) dy + dz . (3.3.63)

Like the previous case, the teleparallel conformal Killing vectors in this case are just teleparallel Killing vectors which are given below:

0 2C X = c8 (y + z) e + c10x + c11, Z 1 0 2(C−B) −B −B X = −C e (y + z)(c8t + c9) + e c10 e dt + c12 , √ 2 −C c7 X = e c4z − 2 2 + c8t + c9, c5 √ 3 −C c13 X = −e c4y + 2 2 + c8t + c9. (3.3.64) c5 From above we can see that there are ﬁve teleparallel Killing vectors which, by choosing c5 = 1, can be expressed as: Z −B −B −B X1 = x ∂t + e e dt ∂x,X2 = ∂t,X3 = e ∂x, √ −C −C −C X4 = z e ∂y − y e ∂z,X5 = −2 2 e ∂z. (3.3.65)

Case 6: In this case we take B = B(t) and A = C = A(x) such that

00 02 p A + 2A = 0 ⇒ A(x) = ln 2 (c5x + c6), where c5 6= 0. Solving the integrability conditions (3.3.45)-(3.3.50) under these restrictions, the value c7 of the metric function B becomes B = ln , where c7 6= 0 and c8 6= 0. c8t+c9 So the metric (3.3.1) becomes: 2 2 2 c7 2 2 2 ds = − (c5x + c6) dt + dx + (c5x + c6) dy + dz . (3.3.66) c8t + c9

120 Choosing c5 = c7 = 1 and solving Eqs. (3.3.45)-(3.3.50) for the above metric, the teleparallel conformal Killing vectors given in (3.3.44) get the form: √ 0 −A 2 2 2 X = 2 e −c8c14 t + y + z − 2 (c9c14t + c10y + c11z) + c12 , √ 1 −B A X = e −2 2 c14 e + c13 , √ 2 −A c4 −B X = 2 e √ z − 2c14 y e − 2 (c10t + c15) , 2 √ 3 −A c4 −B X = 2 e −√ y − 2c14 z e − 2 (c11t + c16) , 2 √ −(A+B) α = −2 2 c14 e . (3.3.67)

It shows that the metric (3.3.66) admits eight teleparallel conformal Killing vectors which, by taking c8 = c9 = 1, can be written as: √ √ −A 2 2 2 (A−B) X1 = − 2 e t + y + z + 2t ∂t − 2 2 e ∂x √ √ −(A+B) −(A+B) − 2 2 y e ∂y − 2 2 z e ∂z, √ √ √ −A −A −A −B X2 = −2 2 y e ∂t − 2 2 t e ∂y,X3 = 2 e ∂t,X4 = e ∂x, √ √ √ −A −A −A X5 = −2 2 z e ∂t − 2 2 t e ∂z,X6 = −2 2 e ∂y, √ −A −A −A X7 = −2 2 e ∂z,X8 = z e ∂y − y e ∂z. (3.3.68)

In the above set, X1 denotes a proper teleparallel conformal Killing vector and X2, ..., X8 represent teleparallel Killing vectors admitted by the metric (3.3.66). No proper teleparallel homothetic vector exist in this case.

Case 7: In this we consider A = A(x) and B = C = B(t) such that ¨ ˙ 2 p B + 2B = 0 ⇒ B = ln 2 (c5t + c6), where c5 6= 0. After some simple algebraic manipulations in Eqs. (3.3.45)-(3.3.50), the function A becomes c7 A = ln , where c7 6= 0 and c8 6= 0. Thus the metric (3.3.1) takes the c8x+c9

121 form: 2 2 c7 2 2 2 2 ds = − dt + (c5t + c6) dx + dy + dz . (3.3.69) c8x + c9

For this metric we have the following eight teleparallel conformal Killing vectors: √ 0 −A B X = e −2 2 c11 e + c10 , √ 1 −B 2 2 2 X = 2 e c8c11 y + z − x + 2 (−c9c11x + c12y + c13z) + c14 , √ 2 −B c4 −A X = 2 e √ z − 2c11 y e − 2 (c12x + c15) , 2 √ 3 −B c4 −A X = 2 e −√ y − 2c11 z e − 2 (c13x + c16) , 2 √ −(A+B) α = −2 2 c11 e . (3.3.70)

In generators form, we can write these eight teleparallel conformal Killing vectors as follows: √ √ −B 2 2 2 (B−A) X1 = 2 e y + z − x − 2x ∂x − 2 2 e ∂t √ √ −(A+B) −(A+B) − 2 2 y e ∂y − 2 2 z e ∂z, √ √ √ −B −B −B −A X2 = 2 2 y e ∂x − 2 2 x e ∂y,X3 = 2 e ∂x,X4 = e ∂t, √ √ √ −B −B −B X5 = 2 2 z e ∂x − 2 2 x e ∂x,X6 = −2 2 e ∂y, √ −B −B −B X7 = −2 2 e ∂z,X8 = z e ∂y − y e ∂z, (3.3.71)

where X1 is a proper teleparallel conformal Killing vector and X2, ..., X8 are teleparallel Killing vectors admitted by the metric (3.3.69). Like the previous case, no proper teleparallel homothetic vector exist in this case.

122 Case 8: Finally we solve the integrability conditions (3.3.45)-(3.3.50) by ¨ ˙ 2 p taking A = B = C = A(t) such that A + A = 0 ⇒ A = ln 2 (c5t + c6), where c5 6= 0. Under these restrictions on the functions A, B and C, the metric (3.3.1) after a suitable rescaling becomes:

2 2 2 2 2 ds = −dt + (c5t + c6) dx + dy + dz , (3.3.72) which is the well known conformally ﬂat FRW metric and it admits ﬁfteen conformal Killing vectors in general relativity [17]. Solving Eqs. (3.3.45)- (3.3.50) for the above metric, we obtain: c X0 = 9 x2 + y2 + z2 + c t + c x + c y + c z − c y eA + c z eA 2 9 18 17 13 3 2 ZZ −3A 2 + c8 e dt + c19, Z Z 1 −A −3A −A −A X = e (c2xz − c3xy − c10z − c14y) + c8x e dt + c18e e dt

−A + c20e + c9x, c X2 = 3 x2 − y2 + z2 e−A + (c x + c z + c yz + c ) e−A 2 14 4 2 15 Z Z Z −3A −3A −3A + c8y e dt + c16 e dt − c3 te dt + c9y + c17, c X3 = − 2 x2 + y2 − z2 e−A + (c x − c y − c yz + c ) e−A 2 10 4 3 11 Z Z Z −3A −3A −3A + c8z e dt + c12 e dt + c2 te dt + c9z + c13, Z −A −3A α = (c2z − c3y) e + c8 e dt + c9. (3.3.73)

From above we can see that the FRW metric (3.3.72) admits sixteen teleparallel conformal Killing vectors in the context of teleparallel gravity. Like the previous cases, we can say that this increase of one extra teleparallel conformal Killing vector in FRW spacetime occurred because of the presence of torsion in the spacetime.

123 3.4 Summary

In this chapter we have investigated teleparallel conformal Killing vectors in LRS Bianchi type V, static plane symmetric and non static plane symmetric spacetimes in the context of teleparallel gravity. For Bianchi type V spacetimes, it is observed that the conformal factor vanishes and hence these spacetimes do not admit any proper teleparallel conformal Killing vectors. For static plane symmetric spacetimes, we have solved the teleparallel conformal Killing’s equations in the cases when (i) A0 6= 0,B0 = 0, (ii) A0 = 0,B0 6= 0 and (iii) A = B. When A0 6= 0 and B0 = 0, we have obtained nine teleparallel conformal Killing vectors, out of which one is a proper teleparallel conformal Killing vector and eight are teleparallel Killing vectors. For A0 = 0 and B0 6= 0, the conformal factor becomes constant and hence the teleparallel conformal Killing vectors are reduced to teleparallel homothetic vectors, which are ten in number with one proper teleparallel homothetic vector and nine teleparallel Killing vectors. Finally when A = B, the static plane symmetric spacetime becomes conformally flat and in such a case it admits fifteen conformal Killing vectors in general relativity. However, in the context of teleparallel gravity we have found twelve teleparallel conformal Killing vectors admitted by this metric, out of which three are proper teleparallel conformal Killing vectors and the remaining nine are teleparallel Killing vectors. Finally, we have solved the teleparallel conformal Killing’s equations of non static plane symmetric spacetimes for eight different values of the metric functions. In three cases, either the conformal factor vanishes or it be-

124 comes constant, giving no proper teleparallel conformal Killing vector. In four cases, a proper teleparallel conformal Killing vector is found to be admitted by these spacetimes. In the last case, the plane symmetric spacetimes metric becomes conformally flat FriedmannRobertsonWalker (FRW) metric which admits fifteen conformal Killing vectors in general relativity. It was expected that this metric would also admit fifteen conformal Killing vectors in the context of teleparallel gravity but we have seen that it admits sixteen teleparallel conformal Killing vectors. Perhaps, the increase of an additional conformal Killing vector in the context of teleparallel gravity is because of the non zero torsion and vanishing curvature.

125 Chapter 4

Teleparallel Killing and Homothetic Vectors of Kantowski-Sachs and LTB Metrics

In this chapter we discuss teleparallel Killing and homothetic vectors of Kantowski-Sachs and Lemaitre-Tolman-Bondi (LTB) metrics in teleparallel gravity. A non diagonal tetrad is chosen in case of Kantowski-Sachs spacetimes, while for LTB metric we have considered a diagonal tetrad. The whole procedure to obtain these symmetries in the above mentioned spacetimes is to ﬁrst obtain ten partial diﬀerential equations with the help of Eq. (1.5.3), then to impose certain restrictions on the metric functions in two cases, that is for α 6= 0 and α = 0, giving explicit form of teleparallel homothetic and teleparallel Killing vectors respectively.

126 4.1 Proper Teleparallel Homothetic Vectors in Kantowski-Sachs Spacetimes

Kantowski-Sachs spacetimes are spherically symmetric, homogenous and anisotropic cosmological models possessing a 3-dimensional isometry group G3 acting on two-dimensional spacelike orbits of positive curvature and admitting an additional spacelike Killing vector which does not lie in these orbits. The general form of Kantowski-Sachs spacetimes metric in spherical coordinates (t, r, θ, φ) is given as [80]:

ds2 = −dt2 + A2(t) dr2 + B2(t) dθ2 + sin2 θ dφ2 , (4.1.1) where A and B are nowhere zero functions of t only. In the context of general relativity, the above metric admits the following four independent Killing vectors [80]:

∂r, ∂φ, cos φ ∂θ − cot θ sin φ ∂φ, sin φ ∂θ + cot θ cos φ ∂φ . (4.1.2)

The homothetic vectors for Kantowski-Sachs spacetimes have been discussed by Iqbal [81] in the context of general relativity. The authors concluded that these spacetimes admit a proper homothetic vector, in addition to the above mentioned four Killing vectors, for some particular values of the metric functions. In order to see the effect of torsion tensor on homothetic symmetry, we intend to investigate teleparallel homothetic vectors for Kantowski-Sachs spacetimes. It is worth noticing here that different choices of tetrad fields may give different results. In particular, the choice of a non diagonal tetrad may produce more teleparallel Killing vectors as those given by a diagonal tetrad [82]. For

127 this reason we choose a non diagonal tetrad for our study of teleparallel homothetic vectors in Kantowski-Sachs metric. Using the procedure presented in [83], we get the following non diagonal tetrad and its inverse for the metric (4.1.1).

  1 0 0 0     µ 0 A sin θ cos φ B cos θ cos φ −B sin θ sin φ ha =   (4.1.3)   0 A sin θ sin φ B cos θ cos φ B sin θ cos φ    0 A cos θ −B sin θ 0   1 0 0 0    1 1 1  a 0 A sin θ cos φ A sin θ sin φ A cos θ  hµ =   (4.1.4)  1 1 1  0 B cos θ cos φ B cos θ sin φ − B sin θ   1 1 0 − B cos θ sin φ B cos θ cos φ 0

The corresponding non zero components of torsion tensor turned out to be:

A˙ B˙ A T 1 = ,T 2 = T 3 = ,T 2 = T 3 = − , (4.1.5) 01 A 02 03 B 12 13 B where a dot on a metric function signiﬁes its derivative with respect to t. Using the above torsion tensor components and the metric functions from Eq. (4.1.1) in Eq. (1.5.3), we have the following system of equations:

0 X,0 = α, (4.1.6) 2 1 0 ˙ 1 A X,0 − X,1 + AAX = 0, (4.1.7) 2 2 0 ˙ 2 B X,0 − X,2 + BBX = 0, (4.1.8) 2 2 3 0 ˙ 2 3 B sin θ X,0 − X,3 + BB sin θ X = 0, (4.1.9)

1 X,1 = α, (4.1.10)

128 2 2 2 1 2 B X,1 + A X,2 − AB X = 0, (4.1.11)

2 2 3 2 1 2 3 B sin θ X,1 + A X,3 − AB sin θ X = 0, (4.1.12)

2 1 BX,2 + AX = α B, (4.1.13)

2 2 3 X,3 + sin θ X,2 = 0, (4.1.14)

2 3 1 B cot θ X + BX,3 + AX = α B, (4.1.15) where α ∈ R. We try to solve the above system for α 6= 0. Integrating Eqs. (4.1.6) and (4.1.10) with respect to t and r respectively, we have:

X0 = α t + f 1(r, θ, φ),

1 2 X = α r + fθ (t, θ, φ), (4.1.16)

1 2 where the functions f (r, θ, φ) and fθ (t, θ, φ) arise in the process of integration. Using the above value of X1 in Eq. (4.1.13) and integrating it with respect to θ, we get: A A X2 = α θ − α r θ − f 2(t, θ, φ) + f 3(t, r, φ), (4.1.17) B B f 3(t, r, φ) being a function of integration. Similarly using the value of X0 from (4.1.16) in Eq. (4.1.9) and solving it, we obtain: 1 Z 1 1 X3 = csc2 θ f 1(r, θ, φ) dt + f 4(r, θ, φ), (4.1.18) φ B B B where f 4(r, θ, φ) is a function of integration. Now putting the values of X0 and X1 from (4.1.16) in Eq. (4.1.7) and diﬀerentiating it with resect to r, we get: 1 ˙ frr(r, θ, φ) = α AA = c1, (4.1.19)

1 2c1 2 where c1 is a separation constant. Solving Eq. (4.1.19), we get A = α t + c2

1 c1 2 1 2 1 2 and f (r, θ, φ) = 2 r + r F (θ, φ) + F (θ, φ), F (θ, φ) and F (θ, φ) being

129 functions of integration. Substituting back these values in Eq. (4.1.7), we

2 1 1 R 1 1 3 4 obtain f (t, θ, φ) = F (θ, φ) A A dt + A F (θ, φ) + F (t, φ). The functions F 3(θ, φ) and F 4(t, φ) appearing here arise during the process of integration. Using all these obtained values in Eqs. (4.1.16)-(4.1.18), we get the following system:

c X0 = α t + 1 r2 + r F 1(θ, φ) + F 2(θ, φ), 2 1 Z 1 1 X1 = α r + F 1(θ, φ) dt + F 3(θ, φ), θ A A A θ A 1 Z 1 1 A X2 = α θ − α r θ − F 1(θ, φ) dt − F 3(θ, φ) − F 4(t, φ) B B A B B + f 3(t, r, φ), csc2 θ Z 1 1 X3 = r F 1(θ, φ) + F 2(θ, φ) dt + f 4(r, θ, φ). (4.1.20) B φ φ B B

If we use the above system in Eq. (4.1.11) and diﬀerentiate it with respect to r and θ, it gives α = 0. It shows that the Kantowski-Sachs metric, given in (4.1.1), do not admit any proper teleparallel homothetic vector for the chosen non diagonal tetrad.

4.2 Teleparallel Killing Vectors in Kantowski- Sachs Spacetime

The teleparallel Killing vectors in Kantowski-Sachs spacetime for a diagonal tetrad are already discussed by Shabbir et. al. [60]. In order to ﬁnd teleparallel Killing vectors in the same spacetime for a non diagonal tetrad, we solve Eqs. (4.1.6)-(4.1.15) by taking α = 0. To have a complete classiﬁcation, we consider the following cases depending on the nature of the metric functions.

130 (I) A = A(t),B = const. (II) A = const., B = B(t) (III) A = B = A(t) (IV) A = const., B = const. and A 6= B (V) A = (t), B = B(t) and A 6= B.

Case (I): Here we consider A = A(t) and B = const. = β, where β 6= 0. Un- der these restrictions, the Kantowski-Sachs metric given in (4.1.1) becomes:

ds2 = −dt2 + A2(t) dr2 + β2 dθ2 + sin2 θ dφ2 . (4.2.1)

Now solving Eqs. (4.1.6) and (4.1.10), we have:

X0 = f 1(r, θ, φ),

X1 = f 2(t, θ, φ), (4.2.2) where f 1(r, θ, φ) and f 2(t, θ, φ) are functions of integration. Using the above system in Eqs. (4.1.8) and (4.1.9) and then integrating them with respect to t, we obtain:

t X2 = f 1(r, θ, φ) + f 3(r, θ, φ), β2 θ t X3 = csc2 θ f 1(r, θ, φ) + f 4(r, θ, φ), (4.2.3) β2 φ where the functions f 3(r, θ, φ) and f 4(r, θ, φ) arise during the process of integration. Also, if we use the values of X0 and X1 from the system (4.2.2) in Eq. (4.1.7) and diﬀerentiate it with respect to r, it gives:

f 1(r, θ, φ) = r F 1(θ, φ) + F 2(θ, φ), 1 Z 1 1 f 2(t, θ, φ) = F 1(θ, φ) dt + F 3(θ, φ), (4.2.4) A A A θ

131 Here F 1(θ, φ),F 2(θ, φ) and F 3(θ, φ) are functions of integration. Substituting back these values from (4.2.4) in (4.2.2) and (4.2.3), we have:

X0 = r F 1(θ, φ) + F 2(θ, φ), 1 Z 1 1 X1 = F 1(θ, φ) dt + F 3(θ, φ), A A A θ t X2 = r F 1(θ, φ) + F 2(θ, φ) + f 3(r, θ, φ), β2 θ θ t X3 = csc2 θ r F 1(θ, φ) + F 2(θ, φ) + f 4(r, θ, φ). (4.2.5) β2 φ φ

Using the above system in Eq. (4.1.13) and then diﬀerentiating it twice with respect to t, we get F 1(θ, φ) = 0,F 2(θ, φ) = θ G1(φ) + G2(φ) and

3 1 3 4 1 2 f (r, θ, φ) = − β F (θ, φ) + F (r, φ). The functions G (φ) and G (φ) become constant if we use these values and the system (4.2.5) in Eq. (4.1.14) and

1 2 diﬀerentiate it with respect to t and θ. Let G (φ) = c1 and G (φ) = c2. Hence the system (4.2.5) reduces to:

0 X = c1θ + c2, 1 X1 = F 3(θ, φ), A θ c t 1 X2 = 1 − F 3(θ, φ) + F 4(r, φ), β2 β X3 = f 4(r, θ, φ). (4.2.6)

Using the above system in Eq. (4.1.11) and doing some simple algebraic manipulations, we obtain F 3(θ, φ) = sin θ G4(φ) − cos θ G5(φ) + β G3(φ),

4 3 F (r, φ) = G (φ) and c1 = 0. Moreover, if we use the system (4.2.6) in Eq. (4.1.12) and then diﬀerentiate the resulting equation with respect to t,

4 1 2 4 5 we have f (r, θ, φ) = β csc θ cos θ Gφ(φ) + sin θ Gφ(φ) . With all these

132 obtained values, the system (4.2.6) becomes:

0 X = c2, 1 X1 = cos θ G4(φ) + sin θ G5(φ) , A 1 X2 = − sin θ G4(φ) − cos θ G5(φ) , β 1 X3 = csc2 θ cos θ G4 (φ) + sin θ G5 (φ) . (4.2.7) β φ φ

Using the above system in Eq. (4.1.15) and then diﬀerentiating it with

4 5 respect to θ, we get G (φ) = c3φ + c4 and G (φ) = c5 cos φ + c6 sin φ. The constant c3 vanishes if we simplify Eq. (4.1.14) by using the system (4.2.7). Finally we get the following system, satisfying Eqs. (4.1.6)-(4.1.15) for α = 0.

0 X = c2, 1 X1 = c cos θ + c sin θ sin φ + c sin θ cos φ , A 4 6 5 1 X2 = − c sin θ − c cos θ sin φ − c cos θ cos φ , β 4 6 5 1 X3 = csc θ (c cos φ − c sin φ) , (4.2.8) β 6 5 where c2, c4, c5, c6 ∈ R. Thus the metric (4.2.1) admits four teleparallel Killing vectors, whose generators can be expressed as follows:

cos θ sin θ X = ∂ ,X = ∂ − ∂ , 1 t 2 A r β θ sin θ sin φ cos θ sin φ csc θ cos φ X = ∂ + ∂ + ∂ , 3 A r β θ β φ sin θ cos φ cos θ cos φ csc θ sin φ X = ∂ + ∂ − ∂ . (4.2.9) 4 A r β θ β φ

Comparing these results with those obtained for a diagonal tetrad in [60], we see that the number of teleparallel Killing vectors for both types of tetrad

133 are same with a slight change in their generators.

Case (II): In this case we take A = const. and B = B(t). After a suitable rescaling of r coordinate, the metric of Kantowski-Sachs spacetimes given in (4.1.1) becomes:

ds2 = −dt2 + dr2 + B2(t) dθ2 + sin2 θ dφ2 . (4.2.10)

By similar procedure as that of the previous case, the complete solution of Eqs. (4.1.6)-(4.1.15) for the above metric yields the following system:

0 X = c1,

1 X = c2 cos θ − c3 sin θ sin φ + c4 sin θ cos φ, 1 X2 = − c sin θ + c cos θ sin φ − c cos θ cos φ , B 2 3 4 1 X3 = − csc θ (c cos φ + c sin φ) , (4.2.11) B 3 4 where c1, c2, c3, c4 ∈ R. Thus similar to the above case, the metric (4.2.10) admits four teleparallel Killing vectors which can be written as:

sin θ X = ∂ ,X = cos θ ∂ − ∂ , 1 t 2 r B θ cos θ sin φ csc θ cos φ X = − sin θ sin φ ∂ − ∂ − ∂ , 3 r B θ B φ cos θ cos φ csc θ sin φ X = sin θ cos φ ∂ + ∂ − ∂ . (4.2.12) 4 r B θ B φ

Comparing the above results with those obtained in [60] for the metric (4.2.10), we see that the number of teleparallel Killing vectors is same for diagonal and non diagonal tetrad with a slight change in their generators.

134 Case (III): If we take A = B = A(t), then the Kantowski-Sachs metric (4.1.1) reduces to:

ds2 = −dt2 + A2(t) dr2 + dθ2 + sin2 θ dφ2 (4.2.13) and the solution of Eqs. (4.1.6)-(4.1.15), for α = 0, yields:

0 X = c1, 1 X1 = c cos θ − c sin θ sin φ + c sin θ cos φ , A 2 3 4 1 X2 = − c sin θ + c cos θ sin φ − c cos θ cos φ + c er sin φ − c er cos φ , A 2 3 4 5 6 1 X3 = − c csc θ cos φ + c csc θ sin φ + c er cot θ cos φ A 3 4 5 r r + c6e cot θ sin φ − c7e , (4.2.14) where ci ∈ R, for i = 1, ..., 7. It shows that the metric (4.2.13) admits seven teleparallel Killing vectors. The generators of these seven teleparallel Killing vectors are listed below. cos θ sin θ er X = ∂ ,X = ∂ − ∂ ,X = ∂ , 1 t 2 A r A θ 3 A φ sin θ sin φ cos θ sin φ csc θ cos φ X = − ∂ − ∂ − ∂ , 4 A r A θ A φ sin θ cos φ cos θ cos φ csc θ sin φ X = ∂ + ∂ − ∂ . 5 A r A θ A φ er sin φ er cot θ cos φ X = − ∂ − ∂ , 6 A θ A φ er cos φ er cot θ sin φ X = ∂ − ∂ . (4.2.15) 7 A θ A φ It is interesting to see that the metric given in (4.2.13) admits four teleparallel Killing vectors in case of a diagonal tetrad [60], while with a non diagonal tetrad we got three extra teleparallel Killing vectors admitted by the same

135 metric.

Case (IV): Here we assume that both the metric functions are diﬀerent non zero constants, say A = γ 6= 0, B = β 6= 0 and γ 6= β. In this case the metric (4.1.1), after a suitable rescaling of r coordinate, takes the form:

ds2 = −dt2 + dr2 + β2 dθ2 + sin2 θ dφ2 , (4.2.16)

Taking α = 0 and solving the system of Eqs. (4.1.6)-(4.1.15) for the above metric, we obtain:

0 X = c1,

1 X = c2 cos θ − c3 sin θ sin φ + c4 sin θ cos φ,

2 1 r X = (−c sin θ − c cos θ sin φ + c cos θ cos φ) + c e β cos φ β 2 3 4 5 r + c6 e β sin φ,

3 1 r X = (−c csc θ cos φ − c csc θ sin φ) − c e β cot θ sin φ β 3 4 5 r r + c6 e β cot θ cos φ + β c7 e β , (4.2.17) where ci ∈ R, for i = 1, ..., 7. Thus the metric (4.2.16) admits seven teleparallel Killing vectors which can be written as:

sin θ r X = ∂ ,X = cos θ ∂ − ∂ ,X = β e β ∂ , 1 t 2 r β θ 3 φ cos θ sin φ csc θ cos φ X = − sin θ sin φ ∂ − ∂ − ∂ , 4 r β θ β φ cos θ cos φ csc θ sin φ X = sin θ cos φ ∂ + ∂ − ∂ . 5 r β θ β φ r r X6 = e β cos φ ∂θ − e β cot θ sin φ ∂φ,

r r X7 = e β sin φ ∂θ + e β cot θ cos φ ∂φ. (4.2.18)

136 Note that the same metric (4.2.16) admits six teleparallel Killing vectors for a diagonal tetrad [60], while for a non diagonal tetrad we got an extra teleparallel Killing vector.

Case (V): Finally we consider A = A(t), B = B(t) and A 6= B. In this case the metric of Kantowski-Sachs spacetime is same as given in (4.1.1). It is straightforward to solve the system of equations (4.1.6)-(4.1.15) for α = 0 and obtain the following teleparallel Killing vectors:

0 X = c1, 1 X1 = c cos θ − c sin θ sin φ + c sin θ cos φ , A 2 3 4 1 X2 = − c sin θ + c cos θ sin φ − c cos θ cos φ , B 2 3 4 1 X3 = − csc θ (c cos φ + c sin φ) , (4.2.19) B 3 4 where c1, c2, c3, c4 ∈ R. The above result shows that the Kantowski-Sachs metric (4.1.1) admits four teleparallel Killing vectors, when A(t) 6= B(t). In generators form, the above four teleparallel Killing vectors can be expressed as:

cos θ sin θ X = ∂ ,X = ∂ − ∂ , 1 t 2 A r B θ sin θ sin φ cos θ sin φ csc θ cos φ X = − ∂ − ∂ − ∂ , 3 A r B θ B φ sin θ cos φ cos θ cos φ csc θ sin φ X = ∂ + ∂ − ∂ . (4.2.20) 4 A r B θ B φ

Comparing these results with those obtained for a diagonal tetrad in [60], we see that the number of teleparallel Killing vectors for both types of tetrad are same with a slight change in their generators.

137 We conclude this section with the remarks that Kantowski-Sachs spacetime metric admits more teleparallel Killing vectors for the choice of a non diagonal tetrad as compared to the teleparallel Killing vectors admitted by the same metric for a diagonal tetrad.

4.3 Proper Teleparallel Homothetic Vectors in LTB Metric

The Lemaitre-Tolman-Bondi (LTB) metric is an anisotropic cosmological model representing spherically symmetric and inhomogeneous dust solution of Einstein’ ﬁeld equations, which provides a toy model for an inhomogeneous universe. From mathematical point of view, the LTB cosmological model is based on a spacetime whose metric in spherical coordinates (t, r, θ, φ) is given as [84]:

ds2 = −dt2 + A2(t, r) dr2 + B2(t, r) dθ2 + sin2 θ dφ2 , (4.3.1) where the metric functions A and B are nowhere zero functions which are dependent on t and r only. To investigate teleparallel homothetic vectors in this metric, we use a diagonal tetrad which is found along with its inverse as given below by using (4.3.1) in Eq. (1.4.1.2): µ ha = diag 1,A(t, r),B(t, r), sin θ B(t, r) , a −1 −1 −1 −1 hµ = diag 1,A (t, r),B (t, r), sin θ B (t, r) . (4.3.2)

138 Using (4.3.2) in Eqs. (1.4.2.2) and (1.4.2.3), we have the following non zero components of torsion tensor:

A˙ B˙ B0 T 1 = ,T 2 = T 3 = ,T 2 = T 3 = ,T 3 = cot θ, (4.3.3) 01 A 02 03 B 12 13 B 23 where a dot and a prime on a metric function denotes its partial derivative with respect to t and r respectively. Using the torsion tensor components from (4.3.3) and the metric functions from Eq. (4.3.1) in Eq. (1.5.3), we have the following system of ten coupled partial diﬀerential equations:

0 X,0 = α, (4.3.4) 2 1 0 ˙ 1 A X,0 − X,1 + AAX = 0, (4.3.5) 2 2 0 ˙ 2 B X,0 − X,2 + BBX = 0, (4.3.6) 2 2 3 0 ˙ 2 3 B sin θ X,0 − X,3 + BB sin θ X = 0, (4.3.7)

0 1 1 A X + AX,1 = α A, (4.3.8)

2 2 2 1 0 2 B X,1 + A X,2 + BB X = 0, (4.3.9)

2 2 3 2 1 0 2 3 B sin θ X,1 + A X,3 + BB sin θ X = 0, (4.3.10)

2 X,2 = α, (4.3.11)

2 2 3 3 X,3 + sin θ X,2 + sin θ cos θ X = 0, (4.3.12)

3 X,3 = α, (4.3.13) where α ∈ R. In order to ﬁnd proper teleparallel homothetic vectors in LTB metric, we try to solve the above system for non zero value of α. The following system can be easily found by solving Eqs. (4.3.4), (4.3.8), (4.3.11) and (4.3.13):

X0 = αt + f 1(r, θ, φ),

139 α Z 1 X1 = A dr + f 2(t, θ, φ), A A X2 = α θ + f 3(t, r, φ),

X3 = α φ + f 4(t, r, θ), (4.3.14) where f 1(r, θ, φ), f 2(t, θ, φ), f 3(t, r, φ) and f 4(t, r, θ) are functions of integration. Using the above system in Eq. (4.3.12) and diﬀerentiating it with respect to θ and φ respectively, we get α = 0. Hence the LTB metric given in (4.3.1) admit no proper teleparallel homothetic vector.

4.4 Teleparallel Killing Vectors in LTB Met- ric

In this section, we classify the LTB metric according to teleparallel Killing vectors by solving the system of Eqs. (4.3.4)-(4.3.13) for α = 0. In order to achieve a complete classification, we impose certain restrictions on the metric functions A and B and solve Eqs. (4.3.4)-(4.3.13) in each case. Three main cases arise when LTB metric admits 4, 5 or 6 teleparallel Killing vectors. We give the detailed calculations in the first case and present the final form of teleparallel Killing vectors in the remaining cases directly.

Case 1: Four Teleparallel Killing Vectors For the following choices of the metric functions, the LTB metric admits four teleparallel Killing vectors: (i) A = A(t, r) 6= B = B(t, r) (ii) A = A(t, r), B = B(r) (iii) A = A(t, r), B = B(t) (iv) A = A(t, r), B = const.

140 The procedure of solving Eqs. (4.3.4)-(4.3.13) for α = 0 in the sub case (i) is explained below: Solving Eqs. (4.3.4), (4.3.8), (4.3.11) and (4.3.13), we have:

X0 = f 1(r, θ, φ), 1 X1 = f 2(t, θ, φ), A X2 = f 3(t, r, φ),

X3 = f 4(t, r, θ), (4.4.1) where f 1(r, θ, φ), f 2(t, θ, φ), f 3(t, r, φ) and f 4(t, r, θ) are functions of integration. Using this system in Eq. (4.3.12) and diﬀerentiating it with respect

3 3 1 2 to φ, we get fφφ(t, r, φ) = 0 ⇒ f (t, r, φ) = φ F (t, r) + F (t, r), where F 1(t, r) and F 2(t, r) are functions of integration. Hence Eq. (4.3.12) implies f 4(t, r, θ) = − csc θ ln | csc θ − cot θ| F 1(t, r) + csc θF 2(t, r). Substituting the values of f 3(t, r, φ) and f 4(t, r, θ) in system (4.4.1), we get:

X0 = f 1(r, θ, φ), 1 X1 = f 2(t, θ, φ), A X2 = φ F 1(t, r) + F 2(t, r),

X3 = − csc θ ln | csc θ − cot θ| F 1(t, r) + csc θF 2(t, r). (4.4.2)

Using the above system in Eq. (4.3.6) and diﬀerentiating it with respect to

1 1 3 4 θ, we ﬁnd that fθθ(r, θ, φ) = 0 ⇒ f (r, θ, φ) = θ F (r, φ) + F (r, φ), where F 3(r, φ) and F 4(r, φ) are functions of integration. Putting back this value in

3 Eq. (4.3.6) and then diﬀerentiating it with respect to φ, we get Fφφ(r, φ) = 0 3 1 2 1 G1(r) R 1 G3(r) ⇒ F (r, φ) = φ G (r)+G (r) and F (t, r) = B B dt+ B . Simplifying 2 G2(r) R 1 G4(r) Eq. (4.3.6) with these obtained value, we have F (t, r) = B B dt+ B .

141 With all these values, the system (4.4.2) takes the form:

X0 = θ φ G1(r) + G2(r) + F 4(r, φ), 1 X1 = f 2(t, θ, φ), A G1(r) Z 1 G3(r) G2(r) Z 1 G4(r) X2 = φ dt + + dt + , B B B B B B G1(r) Z 1 G3(r) X3 = − csc θ ln | csc θ − cot θ| dt + B B B G2(r) Z 1 G4(r) + csc θ dt + . (4.4.3) B B B

The functions G1(r) and G2(r) vanish if we use the above system in Eq. (4.3.7) and then diﬀerentiate it twice with respect to θ. Substituting back G1(r) = G2(r) = 0 in Eq. (4.3.7), we obtain F 4(r, φ) = G5(r). Thus the system (4.4.3) becomes:

X0 = G5(r), 1 X1 = f 2(t, θ, φ), A G3(r) G4(r) X2 = φ + , A A G3(r) G4(r) X3 = − csc θ ln | csc θ − cot θ| + csc θ . (4.4.4) A A

2 If we diﬀerentiate Eq. (4.3.9) with respect to θ, it gives fθθ(t, θ, φ) = 0 ⇒ f 2(t, θ, φ) = θ F 5(t, φ)+F 6(t, φ), where F 5(t, φ) and F 6(t, φ) are functions of integration. Moreover, if we diﬀerentiate Eq. (4.3.5) with respect to θ,

5 5 6 6 we ﬁnd that Ft (t, φ) = 0 ⇒ F (t, φ) = G (φ), G (φ) being a function of integration. Putting back this value in Eq. (4.3.5) and then diﬀerentiating it with respect to φ, we get F 6(t, φ) = G7(t) + G8(φ) and hence we have f 2(t, θ, φ) = θ G6(φ) + G7(t) + G8(φ). Substituting this value of f 2(t, θ, φ) in

142 (4.4.4) and then using the resulting system in Eq. (4.3.9), we have:

3 4 6 φ B Gr(r) + BGr(r) + AG (φ) = 0. (4.4.5)

Diving each term of the above equation by B and then diﬀerentiating it with respect to t, we get G6(φ) = 0. Putting back G6(φ) = 0 in Eq. (4.4.5) and

3 4 solving this equation, we ﬁnd that G (r) = c1 and G (r) = c2. Hence the system (4.4.4) becomes:

X0 = G5(r), 1 X1 = G7(t) + G8(φ) , A c φ c X2 = 1 + 2 , B B c c X3 = − 1 csc θ ln | csc θ − cot θ| + 2 csc θ. (4.4.6) B B

8 Simplifying Eq. (4.3.10) with the help of above system, we get Gφ(φ) = 0 8 7 5 ⇒ G (φ) = c3. Also, if we simplify Eq. (4.3.5), we have AGt (t)−Gr(r) = 0 7 5 7 5 ⇒ Gt (t) = Gr(r) = 0 ⇒ G (t) = c4 and G (r) = c5. With all these values, the system (4.4.6) gets the following form, satisfying Eqs. (4.3.4)-(4.3.13) for α = 0.

0 X = c5, c X1 = 6 , A c φ c X2 = 1 + 2 , B B c c X3 = − 1 csc θ ln | csc θ − cot θ| + 2 csc θ, (4.4.7) B B where c6 = c3 + c4. This result reveals that the LTB metric, given in (4.3.1), admits four teleparallel Killing vectors for A = A(t, r) 6= B = B(t, r). The

143 generators of these four teleparallel Killing vectors can be written as:

1 1 X = ∂ ,X = ∂ ,X = ∂ + csc θ ∂ , 1 t 2 A r 3 B θ φ 1 X = φ ∂ − csc θ ln | csc θ − cot θ| ∂ . (4.4.8) 4 B θ φ

Moreover, one can easily check that if A = B = A(t, r), then the teleparallel Killing vectors and their corresponding generators in this case are respectively same as given in (4.4.7) and (4.4.8) with a slight diﬀerent that B is replaced by A. Similarly the teleparallel Killing vectors and their associated generators in the sub cases (ii) and (iii) are similar to those given in (4.4.7) and (4.4.8) with a diﬀerence of the dependence of the metric functions as mentioned in the respective cases (ii) and (iii). For the last sub case, that is when A = A(t, r) and B = const., the solution of Eqs. (4.3.4)-(4.3.13) (for α = 0) yields the following four teleparallel Killing vectors:

0 X = c1, c X1 = 2 , A 2 X = c4 φ + c3, 3 X = − csc θ c4 ln | csc θ − cot θ | − c3 , (4.4.9) whose generators can be expressed as:

1 X = ∂ ,X = ∂ ,X = ∂ + csc θ ∂ , 1 t 2 A r 3 θ φ

X4 = φ ∂θ − csc θ ln | csc θ − cot θ| ∂φ. (4.4.10)

Case 2: Five Teleparallel Killing Vectors Here we consider those values of the metric functions in Eq. (4.3.1) for

144 which the LTB metric admits ﬁve teleparallel Killing vectors. The following possibilities arise as sub cases of this case. (i) A = const., B = B(r) (ii) A = const., B = B(t, r) (iii) A = const., B = B(t) (iv) A = A(r), B = const. (v) A = A(t), B = B(t, r) (vi) A = A(r), B = B(t, r) (vii) A = (t), B = const. For the ﬁrst three sub cases, the solution of Eqs. (4.3.4)-(4.3.13) with α = 0 give the same result which is given below:

0 X = c1 r + c2,

1 X = c1 t + c3, 1 X2 = c φ + c , B 5 4 csc θ X3 = − c ln | csc θ − cot θ | − c , (4.4.11) B 5 4 which shows that the LTB metric admits ﬁve teleparallel Killing vectors for the choices of the metric functions as mentioned in part (i), (ii) and (iii) of the above list. The generators of these ﬁve teleparallel Killing vectors are listed below:

1 X = ∂ ,X = ∂ ,X = r ∂ + t ∂ ,X = ∂ + csc θ ∂ , 1 t 2 r 3 t r 4 B θ φ 1 X = φ ∂ − csc θ ln | csc θ − cot θ| ∂ . (4.4.12) 5 B θ φ

For each of the sub cases (iv)-(vii), the simultaneous solution of the system of Eqs. (4.3.4)-(4.3.13) with α = 0 produce ﬁve teleparallel Killing vectors. However, the generators of teleparallel Killing vectors in all these four sub cases are diﬀerent from each other. In the following four systems, we

145 summarize the obtained teleparallel Killing vectors and their corresponding generators for the sub cases (iv)-(vii) respectively. Z 1 X0 = c A dr + c ,X1 = (c t + c ) , 1 2 A 1 3 2 3 X = c5 φ + c4,X = − csc θ c5 ln | csc θ − cot θ | − c4 , Z t 1 X = A dr ∂ + ∂ ,X = ∂ ,X = ∂ ,X = ∂ + csc θ ∂ , 1 t A r 2 t 3 A r 4 θ φ

X5 = φ ∂θ − csc θ ln | csc θ − cot θ| ∂φ. (4.4.13)

c Z 1 c X0 = c r + c ,X1 = 1 dt + 3 , 1 2 A A A 1 csc θ X2 = c φ + c ,X3 = − c ln | csc θ − cot θ | − c , B 5 4 B 5 4 1 Z 1 1 1 X = r ∂ + dt ∂ ,X = ∂ ,X = ∂ ,X = ∂ + csc θ ∂ , 1 t A A r 2 t 3 A r 4 B θ φ 1 X = φ ∂ − csc θ ln | csc θ − cot θ| ∂ . (4.4.14) 5 B θ φ

Z 1 X0 = c A dr + c ,X1 = (c t + c ) , 1 2 A 1 3 1 csc θ X2 = c φ + c ,X3 = − c ln | csc θ − cot θ | − c , B 5 4 B 5 4 Z t 1 1 X = A dr ∂ + ∂ ,X = ∂ ,X = ∂ ,X = ∂ + csc θ ∂ , 1 t A r 2 t 3 A r 4 B θ φ 1 X = φ ∂ − csc θ ln | csc θ − cot θ| ∂ . (4.4.15) 5 B θ φ

c Z 1 c X0 = c r + c ,X1 = 1 dt + 3 , 1 2 A A A 2 3 X = c5 φ + c4,X = − csc θ c5 ln | csc θ − cot θ | − c4 , 1 Z 1 1 X = r ∂ + dt ∂ ,X = ∂ ,X = ∂ ,X = ∂ + csc θ ∂ , 1 t A A r 2 t 3 A r 4 θ φ

X5 = φ ∂θ − csc θ ln | csc θ − cot θ| ∂φ. (4.4.16)

146 Case 3: Six Teleparallel Killing Vectors For only one choice of the metric functions in (4.3.1), the LTB metric admits six teleparallel Killing vectors which is the case when A = A(t) and B = B(r). Taking α = 0 and solving Eqs. (4.3.4)-(4.3.13) for these choices of the metric functions, we obtain:

0 X = c1 r + c2, c Z 1 c X1 = 1 dt + 3 , A A A 1 X2 = c φ + c , B 5 4 csc θ X3 = − c ln | csc θ − cot θ | − c . (4.4.17) B 5 6

In generators form, the above six teleparallel Killing vectors can be expressed as:

1 Z 1 1 1 X = r ∂ + dt ∂ ,X = ∂ ,X = ∂ ,X = ∂ , 1 t A A r 2 t 3 A r 4 B θ 1 csc θ X = φ ∂ − csc θ ln | csc θ − cot θ| ∂ ,X = ∂ . (4.4.18) 5 B θ φ 6 B φ

4.5 Summary

In order to study Killing and homothetic symmetries in the presence of torsion and with vanishing curvature, we have investigated proper teleparallel homothetic and teleparallel Killing vectors in Kantowski-Sachs and LTB spacetimes metrics. The ﬁrst outcome of our study is that the Kantowski- Sachs spacetimes do not admit any proper teleparallel homothetic vector.

147 Comparing this result with that obtained in general relativity [81], we conclude that the proper homothetic vector in Kantowski-Sachs spacetimes dis- appear because of the presence of non zero torsion and vanishing curvature in these spacetimes. Thus non zero torsion and vanishing curvature have a strong effect on proper homothetic symmetry in this case. Moreover our study reveals that Kantowski-Sachs spacetimes admit four or seven teleparallel Killing vectors for the choice of a non diagonal tetrad, in contrast to four or six teleparallel Killing vectors admitted by the same spacetimes for a diagonal tetrad [60]. The number of teleparallel Killing vectors for both choices of tetrad turned out to be same if the two metric functions in Kantowski-Sachs spacetimes are different or one of them becomes constant (Section 4.2, Cases 1,2,3). In the cases when both metric functions of Kantowski-Sachs spacetimes become constant or when both metric functions are equal, the choice of a non diagonal tetrad produces one or three extra teleparallel Killing vectors respectively (Section 4.2, Cases 4,5). Further, we have shown that the LTB metric do not admit any proper teleparallel homothetic vector and the number of teleparallel Killing vectors for this metric turned out to be 4, 5 or 6 for different choices of the metric functions. In the context of general relativity, the LTB metric admits three or four Killing vectors for some particular values of the metric functions [35]. Com- paring this result with our obtained result of teleparallel gravity, we conclude that the vanishing curvature and presence of non zero torsion in LTB metric enhance the number of Killing symmetries it possesses.

148 Chapter 5

Teleparallel Killing and Homothetic Vectors of 3-dimensional Static Circularly Symmetric Spacetimes

This chapter is devoted to study teleparallel Killing and homothetic vectors of 3-dimensional static circularly symmetric spacetimes in teleparallel gravity. Six teleparallel Killing and homothetic equations are obtained by using the metric functions and non zero torsion tensor components of the mentioned spacetimes metric in Eqs. (1.5.2) and (1.5.3) respectively. In both cases, these equations are solved to get the explicit form of teleparallel Killing and homothetic vectors respectively. The obtained results are compared to those of general relativity and the eﬀect of non zero torsion and vanishing curvature on teleparallel Killing and homothetic vectors is observed.

149 5.1 Teleparallel Killing Vectors of 3-dimensional Static Circularly Symmetric Spacetimes

A 3-dimensional spacetime M is a 3-dimensional smooth manifold with a smooth Lorentz metric of signature (−, +, +). Berredo-Peixoto obtained static circularly symmetric solutions in 3-dimensional gravity coupled to a scalar ﬁeld, according to which the metric of a 3-dimensional static circularly symmetric spacetimes can be written as [33]:

ds2 = − eA(r) dt2 + eB(r) dr2 + r2 dθ2, (5.1.1) where the metric functions A and B are dependent on r only. Bokhari et. al. [33] explored Killing vectors for the above metric in the context of general relativity and concluded that it admits 2,3,4 or 6 Killing vectors. In order to see the eﬀect of non zero torsion and vanishing curvature in 3-dimensional static circularly symmetric spacetimes, we investigate teleparallel Killing vectors for the same metric by solving the teleparallel Killing equation (1.5.2). µ a The following tetrad ha and its inverse hµ is found by using the metric (5.1.1) in Eq. (1.4.1.2): µ A B a − A − B 1 h = diag e 2 , e 2 , r , h = diag e 2 , e 2 , . (5.1.2) a µ r Using (5.1.2) in Eqs. (1.4.2.2) and (1.4.2.3), we get the following non zero components of torsion tensor: A0 2 T 0 = ,T 2 = , (5.1.3) 10 2 12 r where a prime on a metric function denotes its derivative with respect to r. Using these torsion tensor components and the metric functions from Eq.

150 (5.1.1) in Eq. (1.5.2), we have the following system of six coupled partial diﬀerential equations:

0 X,0 = 0, (5.1.4)

B 1 A 0 0 A 0 2e X,0 − 2e X,1 − A e X = 0, (5.1.5)

2 2 A 0 r X,0 − e X,2 = 0, (5.1.6)

0 1 1 B X + 2 X,1 = 0, (5.1.7)

2 2 B 1 2 r X,1 + e X,2 + 2r X = 0, (5.1.8)

1 2 X − r X,2 = 0. (5.1.9)

In the above set of equations, each Xa is dependent on t, r and θ only. If we integrate Eqs. (5.1.4) and (5.1.7) with respect to t and r respectively, we get:

X0 = F 1(r, θ),

1 − B 2 X = e 2 F (t, θ), (5.1.10) where F 1(r, θ) and F 2(t, θ) are functions of integration. Using the above value of X1 in Eq. (5.1.9) and integrating the resulting equation with respect to θ, we obtain: Z 2 1 − B 2 3 X = e 2 F (t, θ) dθ + F (t, r), (5.1.11) r F 3(t, r) being a function of integration. If we assume that F 2(t, θ) 6= 0, then substituting the above values of X1 and X2 in Eq. (5.1.8) and diﬀerentiating it with respect to θ, we have the following relation: F 2 (t, θ) r θθ = − e−B 1 − B0 = γ, (5.1.12) F 2(t, θ) 2 where γ is a separation constant. The case when F 2(t, θ) = 0 will be discussed later. From Eq. (5.1.12), the following three possibilities arise:

151 (I) γ > 0 (II) γ < 0 (III) γ = 0

Case (I): If γ = k2 > 0, where k 6= 0, then putting this value of γ in Eq.

2 2 2 (5.1.12), we have Fθθ(t, θ) − k F (t, θ) = 0. Solving this partial diﬀerential equation, we get F 2(t, θ) = cosh kθ G1(t) + sinh kθ G2(t), where G1(t) and G2(t) are arbitrary functions of t coordinate. Using the value of F 2(t, θ) in Eqs. (5.1.10) and (5.1.11), we obtain:

X0 = F 1(r, θ), 1 − B 1 2 X = e 2 cosh kθ G (t) + sinh kθ G (t) , 2 1 − B 1 2 3 X = e 2 sinh kθ G (t) + cosh kθ G (t) + F (t, r). (5.1.13) rk

Using the above system in Eq. (5.1.6) and then diﬀerentiating it with respect

1 2 3 to t, we get Gtt(t) = Gtt(t) = Ftt(t, r) = 0. Solving these equations, we have 1 2 3 3 4 3 G (t) = c1t + c2, G (t) = c3t + c4 and F (t, r) = t G (r) + G (r), G (r) and G4(r) being functions of integration. Substituting back these values in Eq. (5.1.6) and integrating the resulting equation with respect to θ, we have r B 1 −(A+ 2 ) 2 −A 3 5 F (r, θ) = k2 e c1 cosh kθ+c3 sinh kθ +r θ e G (r)+G (r), where G5(r) is a function of integration. Thus the system (5.1.13) can be rewritten as: 0 r −(A+ B ) 2 −A 3 5 X = e 2 c cosh kθ + c sinh kθ + r θ e G (r) + G (r), k2 1 3 1 − B X = e 2 (c1t + c2) cosh kθ + (c3t + c4) sinh kθ , 2 1 − B 3 4 X = e 2 (c t + c ) sinh kθ + (c t + c ) cosh kθ + tG (r) + G (r). rk 1 2 3 4 (5.1.14)

152 Now since γ = k2, so Eq. (5.1.12) implies: r 1 − B0 + k2eB = 0. (5.1.15) 2 Using Eq. (5.1.15) and the system (5.1.14) in Eq. (5.1.8) and then diﬀerenti-

3 3 4 4 ating it with respect to t, we get rGr(r)+2G (r) = 0 and rGr(r)+2G (r) = 0.

3 c5 4 c6 These equations can be easily solved to ﬁnd that G (r) = r2 and G (r) = r2 , 3 4 where c5, c6 ∈ R. With these values of G (r) and G (r), the system (5.1.14) becomes: 0 r −(A+ B ) −A 5 X = e 2 c cosh kθ + c sinh kθ + c θ e + G (r), k2 1 3 5 1 − B X = e 2 (c1t + c2) cosh kθ + (c3t + c4) sinh kθ , 2 1 − B c5t c6 X = e 2 (c t + c ) sinh kθ + (c t + c ) cosh kθ + + . rk 1 2 3 4 r2 r2 (5.1.16)

Using Eq. (5.1.15) and the above system in Eq. (5.1.5), we have:

0 − B B r 0 c5θ 0 A 5 A A 5 (c cosh kθ + c sinh kθ) e 2 2e + A + A −e G (r)− e G (r) = 0. 1 3 2k2 2 r 2 (5.1.17) If we diﬀerentiate the above equation with respect to θ, we obtain: − B B r 0 c5 0 k (c sinh kθ + c cosh kθ) e 2 2e + A + A = 0. (5.1.18) 1 3 2k2 2 Diﬀerentiating Eq. (5.1.18) with respect to θ, we get: r (c cosh kθ + c sinh kθ) 2eB + A0 = 0. (5.1.19) 1 3 2k2 Eq. (5.1.19) gives rise to the following two possibilities:

B r 0 B r 0 (a) 2e + 2k2 A 6= 0 (b) 2e + 2k2 A = 0

153 B r 0 Case (Ia): If 2e + 2k2 A 6= 0, then Eq. (5.1.19) yields c1 = c3 = 0. 0 Substituting c1 = c3 = 0 in Eq. (5.1.18), we have c5 A = 0 which gives two possibilities according as A0 = 0 or A0 6= 0. In the former case, Eq. (5.1.17)

5 5 gives Gr(r) = 0 ⇒ G (r) = c7, where c7 ∈ R. Hence the system (5.1.16) gets the form:

0 X = c5θ + c7, 1 − B X = e 2 c2 cosh kθ + c4 sinh kθ , 2 1 − B c5t c6 X = e 2 c sinh kθ + c cosh kθ + + . (5.1.20) rk 2 4 r2 r2

The system of Eqs. (5.1.4)-(5.1.9) is identically satisfied by the above system, which shows that the 3-dimensional static circularly symmetric spacetimes admits five teleparallel Killing vectors in this case. The generators of these five teleparallel Killing vectors can be expressed as: − B sinh kθ 1 X = e 2 cosh kθ ∂ + ∂ ,X = ∂ ,X = ∂ , 1 r rk θ 2 t 3 r2 θ − B cosh kθ t X = e 2 sinh kθ ∂ + ∂ ,X = θ ∂ + ∂ . (5.1.21) 4 r rk θ 5 t r2 θ

0 In the latter case, that is when A 6= 0, we must have c5 = 0. Substituting 5 0 5 c1 = c3 = c5 = 0 in Eq. (5.1.17), we get 2 Gr(r) + A G (r) = 0. Solving this 5 − A equation, we ﬁnd that G (r) = c7 e 2 , where c7 ∈ R. With these values, system (5.1.16) reduces to:

0 − A X = c7 e 2 , 1 − B X = e 2 c2 cosh kθ + c4 sinh kθ , 2 1 − B c6 X = e 2 c sinh kθ + c cosh kθ + . (5.1.22) rk 2 4 r2

154 The above system satisﬁes the system of Eqs. (5.1.4)-(5.1.9), which shows that the 3-dimensional static circularly symmetric spacetimes metric, given in (5.1.1), admits four teleparallel Killing vectors in this case whose generators can be written as: − B sinh kθ − A X = e 2 cosh kθ ∂ + ∂ ,X = e 2 ∂ , 1 r rk θ 2 t − B cosh kθ 1 X = e 2 sinh kθ ∂ + ∂ ,X = ∂ (5.1.23) 3 r rk θ 4 r2 θ

B r 0 0 Case (Ib): If 2e + 2k2 A = 0, then Eq. (5.1.18) implies c5 A = 0. Now 0 B r 0 B if A = 0, then since 2e + 2k2 A = 0 ⇒ e = 0, which is absurd. Hence 0 we must take A 6= 0, so that c5 = 0. Using this value in Eq. (5.1.17), 5 A0 5 we obtain Gr(r) + 2 G (r) = 0. We can easily solve this equation to ﬁnd 5 − A that G (r) = c7 e 2 , where c7 ∈ R. Substituting all these values in system (5.1.16), we have the following explicit form of the components of teleparallel Killing vectors: 0 r −(A+ B ) − A X = e 2 c cosh kθ + c sinh kθ + c e 2 , k2 1 3 7 1 − B X = e 2 (c1t + c2) cosh kθ + (c3t + c4) sinh kθ , 2 1 − B c6 X = e 2 (c t + c ) sinh kθ + (c t + c ) cosh kθ + . (5.1.24) rk 1 2 3 4 r2

There are six teleparallel Killing vectors admitted by the static circularly symmetric spacetimes metric, given in (5.1.1), provided that the metric func-

B r 0 tions A and B satisfy the relation 2e + 2k2 A = 0. In generators form, we can write the above six teleparallel Killing vectors as follows: − B r −A t X = e 2 e cosh kθ ∂ + t cosh kθ ∂ + sinh kθ ∂ , 1 k2 t r rk θ

155 − B r −A t X = e 2 e sinh kθ ∂ + t sinh kθ ∂ + cosh kθ ∂ , 2 k2 t r rk θ − B 1 X = e 2 cosh kθ ∂ + sinh kθ ∂ , 3 r rk θ − B 1 X = e 2 sinh kθ ∂ + cosh kθ ∂ , 4 r rk θ

− A 1 X = e 2 ∂ ,X = ∂ . (5.1.25) 5 t 6 r2 θ

Case (II): Here we take γ = −k2 < 0, where k 6= 0. With this value of γ,

2 2 2 Eq. (5.1.12) gives Fθθ(t, θ) + k F (t, θ) = 0. Solving this equation, we ﬁnd that F 2(t, θ) = cos kθ G1(t) + sin kθ G2(t), where G1(t) and G2(t) are functions of integration. Rest of the analysis of this case is same as that of the previous case with a slight diﬀerence that hyperbolic functions are replaced by respective trigonometric functions.

Case (III): Finally we discuss the case when γ = 0. Substituting γ = 0

2 r 0 in Eq.(5.1.12), we get Fθθ(t, θ) = 0 and 1 − 2 B = 0. Solving these equations, we have F 2(t, θ) = θ G1(t) + G2(t) and B(r) = ln r2, where G1(t) and G2(t) are functions of integration. Using these values in Eqs. (5.1.10) and (5.1.11), we have:

X0 = F 1(r, θ), 1 X1 = θ G1(t) + G2(t) , r 1 1 X2 = θ2 G1(t) + θ G2(t) + F 3(t, r). (5.1.26) r2 2 Using the above system in Eq. (5.1.6) and diﬀerentiating it with respect

1 2 3 to t and θ respectively, we obtain Gtt(t) = Gtt(t) = Ftt(t, r) = 0. These

156 1 2 equations can be easily solved to get G (t) = c1t + c2,G (t) = c3t + c4 and F 3(t, r) = t G3(r)+G4(r). The functions G3(r) and G4(r) arise in the process of integration. Putting back these values in Eq. (5.1.6) and integrating the resulting equation with respect to θ, we obtain:

c c F 1(r, θ) = e−A 1 θ3 + 3 θ2 + r2 θ e−A G3(r) + G5(r), 6 2

G5(r) being a functin of integration. Thus the system (5.1.26) can be rewritten as:

c c X0 = e−A 1 θ3 + 3 θ2 + r2 θ e−A G3(r) + G5(r), 6 2 1 X1 = (c t + c )θ + c t + c , r 1 2 3 4 1 1 X2 = θ2 (c t + c ) + θ (c t + c ) + t G3(r) + G4(r). (5.1.27) r2 2 1 2 3 4

Using the above system in Eq. (5.1.8) and then diﬀerentiating it with respect

3 3 4 4 to t, we get r Gr(r)+2 G (r)+c1 = 0 and r Gr(r)+2 G (r)+c2 = 0. Solving

3 c5 c1 4 c6 c2 these two equations, we ﬁnd that G (r) = r2 − 2 and G (r) = r2 − 2 , where c5, c6 ∈ R. Hence the system (5.1.27) becomes: c c c c X0 = e−A 1 θ3 + 3 θ2 + r2 θ e−A 5 − 1 + G5(r), 6 2 r2 2 1 X1 = (c t + c )θ + c t + c , r 1 2 3 4 1 1 c c c c X2 = θ2 (c t + c ) + θ (c t + c ) + t 5 − 1 + 6 − 2 . (5.1.28) r2 2 1 2 3 4 r2 2 r2 2

Using the system (5.1.28) in Eq. (5.1.5) and then diﬀerentiating it with respect to θ, we get:

A0 A0 c c θ2 + c θ + − 1 r2 + c + c r = 0. (5.1.29) 2 1 3 2 2 5 1 157 Diﬀerentiating Eq. (5.1.29) with respect to θ, we have:

0 A (2c1θ + c3) = 0. (5.1.30)

Equation (5.1.30) gives rise to two possibilities, namely A0 6= 0 and A0 = 0.

0 If A 6= 0, then Eq. (5.1.30) implies c1 = c3 = 0. Putting c1 = c3 = 0 in

Eq. (5.1.29), we get c5 = 0. Substituting all these values in Eq. (5.1.5), we 5 0 5 5 − A have 2 Gr(r) + A G (r) = 0. Solving this equation, we get G (r) = c7 e 2 . Hence the system (5.1.28) becomes:

0 − A X = c7 e 2 , 1 X1 = (c θ + c ) , r 2 4 1 c c X2 = 2 θ2 − 2 r2 + c θ + c . (5.1.31) r2 2 2 4 6

The above system satisﬁes the system of equations (5.1.4)-(5.1.9). This result reveals that in this case the 3-dimensional static circularly symmetric metric admits four teleparallel Killing vectors whose generators can be written as:

2 2 − A θ θ − r X = e 2 ∂ ,X = ∂ + ∂ , 1 t 2 r r 2r2 θ 1 θ 1 X = ∂ + ∂ ,X = ∂ . (5.1.32) 3 r r r2 θ 4 r2 θ

In the later case, that is when A0 = 0, Eq. (5.1.30) is identically satisﬁed and

0 Eq. (5.1.29) gives c1 = 0. Substituting A = c1 = 0 in Eq. (5.1.5), we obtain

5 5 c3 2 c3r − Gr(r) = 0 ⇒ G (r) = 2 r + c7, where c7 is a constant of integration. Hence the system (5.1.28) reduces to:

c X0 = 3 r2 + θ2 + c θ + c , 2 5 7

158 1 X1 = c θ + c t + c , r 2 3 4 1 c c X2 = 2 θ2 − 2 r2 + c tθ + c θ + c t + c . (5.1.33) r2 2 2 3 4 5 6

All the equations (5.1.4)-(5.1.9) are identically satisﬁed by the above system, which shows that in this case the 3-dimensional static circularly symmetric spacetimes metric admits six teleparallel Killing vectors which can be expressed as:

r2 + θ2 t t θ t X = ∂ + ∂ + ∂ ,X = θ ∂ + ∂ ,X = ∂ , 1 2 t r r r2 θ 2 t r2 θ 3 t θ θ2 − r2 1 θ 1 X = ∂ + ∂ ,X = ∂ + ∂ ,X = ∂ . (5.1.34) 4 r r 2r2 θ 5 r r r2 θ 6 r2 θ

Finally we discuss the case when F 2(t, θ) = 0 in Eqs. (5.1.10) and (5.1.11). In a such a case we have:

X0 = F 1(r, θ),

X1 = 0,

X2 = F 3(t, r). (5.1.35)

Using the above system in Eq. (5.1.8) and integrating it with respect to r,

3 1 1 1 we get F (t, r) = r2 G (t), where G (t) is a function of integration. Using this value of F 3(t, r) and the system (5.1.35) in Eq. (5.1.6), we obtain:

1 A 1 Gt (t) − e Fθ (r, θ) = 0. (5.1.36)

1 If we diﬀerentiate the above equation with respect to t, it gives Gtt(t) = 0 1 ⇒ G (t) = c1t + c2. Putting back this value in Eq. (5.1.36) and integrating

159 1 −A 2 2 it with respect to θ, we have F (r, θ) = c1θ e + G (r), where G (r) is a function of integration. Hence the system (5.1.35) can be rewritten in the following form:

0 −A 2 X = c1 θ e + G (r), X1 = 0, 1 X2 = (c t + c ) . (5.1.37) r2 1 2

Simplifying Eq. (5.1.5) with the use of above system, we obtain:

0 −A 2 0 2 c1 θ A e − 2 Gr(r) − A G (r) = 0. (5.1.38)

0 Diﬀerentiating this equation with respect to θ, we obtain c1A = 0. Now if 0 2 2 A = 0, then Eq. (5.1.38) gives Gr(r) = 0 ⇒ G (r) = c3. Hence the system (5.1.37) becomes:

0 X = c1 θ + c3, X1 = 0, 1 X2 = (c t + c ) . (5.1.39) r2 1 2

This system satisﬁes Eqs. (5.1.4)-(5.1.9), which shows that in this case the 3-dimensional static circularly symmetric spacetimes admit three teleparallel Killing vectors, with the following generators:

t 1 X = θ ∂ + ∂ ,X = ∂ ,X = ∂ . (5.1.40) 1 t r2 θ 2 t 3 r2 θ

0 Moreover if A 6= 0, then we must have c1 = 0. Putting c1 = 0 in Eq. 2 0 2 (5.1.38), we have 2 Gr(r) + A G (r) = 0. Solving this equation, we ﬁnd

160 2 − A that G (r) = c3 e 2 . With these values, the system (5.1.37) reduces to the following form satisfying Eqs. (5.1.4)-(5.1.9):

0 − A X = c3 e 2 , X1 = 0, c X2 = 2 . (5.1.41) r2 In this case the 3-dimensional static circularly symmetric spacetimes metric admits only two teleparallel Killing vectors, which can be written as:

− A 1 X = e 2 ∂ ,X = ∂ . (5.1.42) 1 t 2 r2 θ We conclude this section with the comments that 3-dimensional static circularly symmetric spacetimes admit 2,3,4,5 or 6 teleparallel Killing vectors in diﬀerent cases.

5.2 Proper Teleparallel Homothetic Vectors for 3-dimensional Static Circularly Sym- metric Spacetimes

In this section we explore proper teleparallel homothetic vectors in 3-dimensional static circularly symmetric spacetimes by solving Eq. (1.5.3). The metric and non vanishing components of torsion tensor for the above mentioned spacetimes are already given in the previous section, using which, Eq. (1.5.3) produces the following six partial diﬀerential equations.

0 X,0 = α, (5.2.1)

B 1 A 0 0 A 0 2 e X,0 − 2 e X,1 − A e X = 0, (5.2.2)

161 2 2 A 0 r X,0 − e X,2 = 0, (5.2.3)

0 1 1 B X + 2 X,1 = 2α, (5.2.4)

2 2 B 1 2 r X,1 + e X,2 + 2r X = 0, (5.2.5)

1 2 X − r X,2 = −α r. (5.2.6)

To ﬁnd the explicit form of proper teleparallel homothetic vectors, we need to solve the above system of equations for α 6= 0. The following system can be easily obtained by solving Eqs. (5.2.1), (5.2.3) and (5.2.6):

X0 = αt + F 1(r, θ), t X1 = −αr + eA F 1 (r, θ) + r F 2(r, θ), r θθ θ t X2 = eA F 1(r, θ) + F 2r, θ), (5.2.7) r2 θ ( where F 1(r, θ) and F 2(r, θ) are functions of integration, to be found using the remaining three equations. Using the above system in Eq. (5.2.2) and then diﬀerentiating it with respect to t, we get αA0 = 0 ⇒ A0 = 0, as α 6= 0. For simplicity, we take A = 0. Moreover, diﬀerentiating Eq. (5.2.4) with respect to t, using the system (5.2.7) in the resulting equation and then integrating it

1 − B 1 2 3 twice with respect to θ, we ﬁnd that F (r, θ) = r e 2 G (θ)+θ G (r)+G (r). The functions G1(θ), G2(r) and G3(r) arise in the process of integration. With these obtained values, the system (5.2.7) can be rewritten as:

0 − B 1 2 3 X = αt + r e 2 G (θ) + θ G (r) + G (r),

B 1 − 2 1 2 X = −αr + t e Gθθ(θ) + r Fθ (r, θ), 2 t − B 1 2 2 X = r e 2 G (θ) + G (r) + F (r, θ). (5.2.8) r2 θ

162 The following value of F 2(r, θ) is obtained by using the above system in Eq. (5.2.4) and then integrating it with respect to r and θ respectively: Z 2 αθ − B B 1 − B 4 5 F (r, θ) = αθ + e 2 e 2 dr + e 2 G (θ) + G (r), (5.2.9) r r G4(θ) and G5(r) being functions of integration. Substituting this value of F 2(r, θ) in system (5.2.8), we have:

0 − B 1 2 3 X = αt + r e 2 G (θ) + θ G (r) + G (r),

B B Z B B 1 − 2 1 − 2 2 − 2 4 X = t e Gθθ(θ) + α e e dr + e Gθ(θ), Z 2 t − B 1 2 αθ − B B X = r e 2 G (θ) + G (r) + αθ + e 2 e 2 dr r2 θ r

1 − B 4 5 + e 2 G (θ) + G (r). (5.2.10) r If we use the above system in Eq. (5.2.2) and then diﬀerentiate the resulting equation with respect to θ, we obtain:

1 −B r 0 1 − B 2 G (θ) − e 1 − B G (θ) − e 2 G (r) = 0 (5.2.11) θθθ 2 θ r Diﬀerentiating Eq. (5.2.11) with respect to θ and then rearranging, we have the following relation:

1 Gθθθθ(θ) −B r 0 1 = e 1 − B = γ, (5.2.12) Gθθ(θ) 2 where γ is a separation constant and it can get the following three values:

(I) γ > 0 (II) γ < 0 (III) γ = 0

2 1 2 1 Case (I): If γ = k > 0, where k 6= 0, then we have Gθθθθ(θ) − k Gθθ(θ) = 0.

1 c1 c2 Solving this equation, we get G (θ) = k2 cosh kθ + k2 sin kθ +c3θ +c4, where 1 c1, ..., c4 ∈ R. Putting this value of G (θ) in Eq. (5.2.11) and integrating it

163 2 2 R B with respect to r, we get G (r) = −c3 k e 2 dr + c5. Substituting these values of G1(θ) and G2(r) in Eq. (5.2.2) and integrating it with respect to r,

3 2 R B we obtain G (r) = −c4 k e 2 dr + c6. Hence the system (5.2.10) becomes: Z 0 − B c1 c2 2 B X = αt + r e 2 cosh kθ + sin kθ + c θ + c − c k θ e 2 dr k2 k2 3 4 3 Z 2 B + c5θ − c4 k e 2 dr + c6,

B B Z B B 1 − 2 − 2 2 − 2 4 X = t e (c1 cosh kθ + c2 sinh kθ) + α e e dr + e Gθ(θ), Z 2 t − B c1 c2 t 2 B X = e 2 sinh kθ + cosh kθ + c − c k e 2 dr − c r k k 3 r2 3 5 Z αθ − B B 1 − B 4 5 + αθ + e 2 e 2 dr + e 2 G (θ) + G (r). (5.2.13) r r Now since γ = k2, so Eq. (5.2.12) implies: r 1 − B0 = k2 eB. (5.2.14) 2 Using the system (5.2.13) and Eq. (5.2.14) in Eq. (5.2.5) and diﬀerentiating it with respect to t, θ and r respectively, we get α = 0. It shows that, in this case, the 3-dimensional static circularly symmetric spacetimes do not admit any proper teleparallel homothetic vector.

Case (II): Here we consider the case when γ = −k2 < 0, where k 6= 0. The analysis of this case is exactly same as that of the previous case with a slight diﬀerence that the hyperbolic functions involved in the above case are get replaced by the respective trigonometric functions. Consequently we get α = 0, showing that there exist no proper teleparallel homothetic vector in this case.

164 Case (III): Finally we consider the case when γ = 0. Solving Eq. (5.2.12)

2 1 c1 3 c2 2 for γ = 0, we obtain B(r) = ln r and G (θ) = 6 θ + 2 θ + c3θ + c4. using this value of G1(θ) in Eq. (5.2.11) and integrating it with respect to r, we

2 c1 2 get G (r) = 2 r + c5. Using the system (5.2.10) and these obtained values of G1(θ) and G2(r) in Eq. (5.2.2) and integrating it with respect to r, we

3 c2 2 have G (r) = 2 r + c6. Hence the system (5.2.10) becomes: c c c c X0 = αt + 1 θ3 + 2 θ2 + c θ + 1 r2θ + 2 r2 + c , 6 2 3 2 2 4 αr t 1 X1 = + c θ + c + G4(θ), 2 r 1 2 r θ t c c 3αθ 1 X2 = 1 θ2 + 1 r2 + c θ + c + + G4(θ) + G5(r), (5.2.15) r2 2 2 2 3 2 r2 where the constants c5 and c6 are merged in to c3 and c4 respectively. Sim- plifying Eq. (5.2.5) by using the above system, we obtain:

2 5 4 5 2c1tr + 3αrθ + r Gr(r) + r Gθθ(θ) + 2r G (r) = 0. (5.2.16)

If we diﬀerentiate the above equation with respect to t and r, it gives c1 = 0.

Putting back c1 = 0 in this equation and separating the variables, we have:

5 5 4 r Gr(r) + 2 G (r) = − Gθθ(θ) − 3αθ = c7, (5.2.17) where c7 is a separation constant. Eqs. (5.2.17) can be easily solved to get

4 α 3 c7 2 5 c10 c7 G (θ) = − 2 θ − 2 θ + c8θ + c9 and G (r) = r2 + 2 . Substituting these 4 5 values of G (θ), G (r) and c1 = 0 in system (5.2.15), we have the following system, satisfying Eqs. (5.2.1)-(5.2.6): c X0 = αt + 2 r2 + θ2 + c θ + c , 2 3 4 αr 3αθ2 1 X1 = − + c t − c θ + c , 2 2r r 2 7 8

165 3αθ αθ3 1 c c X2 = − + 7 r2 − 7 θ2 + c tθ + c θ + c t + c , (5.2.18) 2 2r2 r2 2 2 2 8 3 9 where we have merged the constant c10 in to c9. This result reveals that the 3-dimensional static circularly symmetric metric admits a proper teleparallel homothetic vector, in addition to six teleparallel Killing vectors. The proper teleparallel homothetic vector can be expressed as:

r 3θ2 3θ θ3 t ∂ + − ∂ + − ∂ . (5.2.19) t 2 2r r 2 2r2 θ

It is interesting to see that by putting α = 0 in the system (5.2.18), we get exactly same six teleparallel Killing vectors as presented in (5.1.33) for the same values of the metric functions A and B, with only a change of labeling of constants c2 → c3, c3 → c5, c4 → c7, c7 → −c2, c8 → c4 and c9 → c6. The generators of the above six teleparallel Killing vectors are same as given in (5.1.34).

5.3 Summary

In this chapter, we have explored teleparallel Killing and homothetic symmetries in 3-dimensional static circularly symmetric spacetimes in the framework of teleparallel gravity. Before us, Killing vectors for the same spacetimes were investigated in general relativity [33]. The authors concluded with the remarks that 3-dimensional static circularly symmetric spacetimes admit 2,3,4 or 6 Killing vectors in general relativity. Our investigations show that the same metric admits 2,3,4,5 or 6 teleparallel Killing vectors in teleparallel gravity. Comparing the results of both theories, the main diﬀerence observed is that 3-dimensional static circularly symmetric spacetimes do not admit ﬁve

166 Killing vectors in general relativity, while they admit ﬁve teleparallel Killing vectors in teleparallel gravity which are presented in (5.1.33). Moreover, it is observed that although the number of Killing vectors in the remaining cases is same for both the theories, their corresponding generators in two theories turned out to be diﬀerent. Finally we have shown that the spacetimes under consideration admit a proper teleparallel homothetic vector, in addition to six teleparallel Killing vectors. The proper teleparallel homothetic vector is presented in (5.2.19).

167 Chapter 6

Conclusion

This thesis has dealt with the problem of ﬁnding Killing, proper homothetic and conformal Killing vectors of some well known spacetimes in general relativity and teleparallel gravity. Conformal Killing vectors have been investigated for LRS Bianchi type V, static and non static plane symmetric spacetimes in the context of general relativity and teleparallel gravity, whereas Killing and proper homothetic vectors are explored for Kantowski- Sachs, LTB and 3-dimensional static circularly symmetric spacetimes in the framework of teleparallel gravity. To summarize the obtained results of each problem, a brief summary is presented at the end of each chapter. However an overall conclusion of the thesis is stated below. In chapter 1, we have presented some basic concepts of general relativity and teleparallel gravity. Conformal Killing vectors for LRS Bianchi type V, static and non static plane symmetric spacetimes in general relativity have been investigated in chapter 2. For all the three mentioned spacetimes, we have found the general solu-

168 tion of conformal Killing’s equations in terms of some unknown functions of t and x, given in (2.1.49), (2.3.27) and (2.5.40) respectively, which determine the conformal Killing vectors of LRS Bianchi type V, static and non static plane symmetric spacetimes in terms of unknown functions of t and x. For each of the three considered spacetimes, these conformal Killing vectors are obtained subject to a set of integrability conditions, which are presented in Eqs. (2.1.50)-(2.1.55), (2.3.28)-(2.3.33) and (2.5.41)-(2.5.46) respectively. To get the explicit form of conformal Killing vectors, the integrability conditions (2.1.50)-(2.1.55), (2.3.28)-(2.3.33) and (2.5.41)-(2.5.46) are solved for diﬀer- ent choices of the metric functions of the spacetime concerned, giving the ﬁnal form of conformal Killing vectors in LRS Bianchi type V, static and non static plane symmetric spacetimes respectively. In the following three tables, we summarize the obtained results of conformal Killing vectors for all three spacetimes by giving the metric functions and the number of proper conformal, proper homothetic and Killing vectors admitted by the corresponding spacetime metric.

Metric functions Proper CKVs Proper HVs KVs

A = mt + m2, B = m1 1 1 4 mt − m A = m1, B = m2e 1 0 1 5

Table 6.1: CKVs in LRS Bianchi type V Spacetimes

169 Metric functions Proper CKVs Proper HVs KVs

x A = x, B = ln(m1e + m2) 1 0 4

A = ln x, B = ln x(m1 − m2 ln x) 1 0 4 −x A = m1e + x + m2, B = x 1 0 4

A = m1, B = ln x 1 1 4 A = ln x2, B = ln x 1 1 4

Table 6.2: CKVs in Static Plane Symmetric Spacetimes

Metric functions Proper CKVs Proper HVs KVs √ A = ln x1+ 2, B = const., C = ln x 1 1 4 √ A = const., B = const., C = ln 2t 0 1 4 x x A = a , B = const., C = b 0 0 5 t A = const., B = a , C = const. 0 0 5 x A = const., B = const., C = a 8 0 7 t A = const., B = const., C = a 8 0 7 A = C = A(x), B = const. 8 0 7

Table 6.3: CKVs in Non Static Plane Symmetric Spacetimes

A special class of conformal Killing vectors, known as inheriting conformal Killing vectors, are also discussed for all the above mentioned spacetimes in the same chapter. Our analysis showed that if the metric functions A and B of LRS Bianchi type V spacetimes are not multiples of each other, then these spacetimes admit ﬁve inheriting conformal Killing vectors with one proper inheriting conformal Killing vector and four Killing vectors. These ﬁve in-

170 heriting conformal Killing vectors are given in (2.2.9). When A is a multiple of B, the LRS Bianchi tyep V spacetimes admit seven inheriting conformal Killing vectors, which are presented in (2.2.11). Out of these seven inheriting conformal Killing vectors, one is a proper inheriting conformal Killing vector and the remaining six are Killing vectors. In both the cases, no proper homothetic vector exist. Moreover, we have shown that each conformal Killing vector of static plane symmetric spacetimes is inheriting. The inheriting conformal Killing vectors for non static plane symmetric spacetimes are presented in (2.6.10), however they contain two unknown functions P 0 and P 4 which satisfy Eq. (2.6.9). In chapter 3, we have explored teleparallel conformal Killing vectors for LRS Bianchi type V, static and non static plane symmetric spacetimes in teleparallel gravity. Following the same method as that of chapter 2, here the teleparallel conformal Killing’s equations are solved to get a general solution of these equations in terms of some unknown functions of t and x, for each of the three mentioned spacetimes. These general solutions are presented in the systems (3.1.48), (3.2.46) and (3.3.44) for LRS Bianchi type V, static and non static plane symmetric spacetimes respectively and they determine the teleparallel conformal Killing vectors for these three spacetimes in terms of unknown functions of t and x. For each of the three considered spacetimes, these teleparallel conformal Killing vectors are found subject to a set of integrability conditions, which are given in Eqs. (3.1.49)-(3.1.54), (3.2.47)-(3.2.52) and (3.3.45)-(3.3.50) respectively. To ﬁnd the explicit form of teleparallel conformal Killing vectors of LRS Bianchi type V, static and non static plane symmetric spacetimes, we have solved each set of these integra-

171 bility conditions for diﬀerent values of the metric functions of the associated spacetimes. All the obtained results are summarized in the following three tables.

Metric functions Proper TP CKVs Proper TP HVs TP KVs A = et, B = const. 0 0 6 p A = const., B = 2(c5t + c6) 0 0 6 p A = B = 2(ct + c6) 0 0 8

Table 6.4: TP CKVs in LRS Bianchi type V Spacetimes

Metric functions Proper TP CKVs Proper TP HVs TP KVs A = ln c11 , B = const. 1 1 7 c6x+c8 p A = const., B = ln 2(c5x + c6) 0 1 9 p A = B = ln 2(c5x + c6) 3 0 9

Table 6.5: TP CKVs in Static Plane Symmetric Spacetimes

Metric functions Proper Proper TP KVs TP CKVs TP HVs A = const., B = ln c7 , C = const. 1 0 6 c5t+c6 p A = const., B = const., C = ln 2(c5t + c6) 0 1 6 A = ln x, B = ln t, C = const. 1 1 8

172 Metric functions Proper Proper TP KVs TP CKVs TP HVs p A = c7x + c8, B = const., C = ln 2(c5t + c6) 0 0 5 3 p A = const., B = ln(c8t + c9) , C = ln 2(c5x + c6) 0 0 5 p c7 A = C = ln 2(c5x + c6), B = ln 1 0 7 c5t+c6 c7 p A = ln , B = C = ln 2(c5t + c6) 1 0 7 c8x+c9 p A = B = C = ln 2(c5t + c6) 3 1 12

Table 6.6: TP CKVs in Non Static Plane Symmetric Spacetimes

In chapter 4, we have investigated teleparallel Killing and homothetic vectors for Kantowski-Sachs and LTB metrics in teleparallel gravity. In case of Kantowski-Sachs spacetimes, we have chosen a non diagonal tetrad, while a diagonal tetrad is considered for LTB metric. For Kantowski-Sachs spacetimes, we have shown that for the choice of a non diagonal tetrad, these spacetimes do not admit any proper teleparallel homothetic vector. If we compare this result with that of general relativity [81], we observe that the vanishing curvature and presence of non zero torsion tensor disappeared proper teleparallel homothetic vector. As far as teleparallel Killing vectors are concerned, they turned out to be four or seven, in contrast to the teleparallel Killing vectors of the same spacetimes in case of diagonal tetrad, where this number is either four or six [60]. Further, it is shown in the same chapter that the LTB metric do not admit any proper teleparallel homothetic vector, while it admits 4, 5 or 6 teleparallel Killing vectors for diﬀerent choices of the metric functions. In general relativity, the same metric admits three or four Killing vectors for some particular values of the metric functions [35]. Comparing the results of both theories, we have concluded that the vanishing curvature

173 and presence of non zero torsion in LTB metric enhanced the number of Killing symmetries it possess. The following tables show the summary of teleparallel Killing vectors admitted by Kantowski-Sachs and LTB metric for diﬀerent choices of the metric functions. In chapter 5, we have explored teleparallel Killing and homothetic

Metric functions No. of TP KVs A = A(t), B = const. 4 A = const., B = B(t) 4 A = B = A(t) 7 A = const., B = const., A 6= B 7 A = (t), B = B(t), A 6= B 4

Table 6.7: TP KVs in Kantowski-Sachs Spacetimes

Metric functions No. of TP KVs (i) A = A(t, r) 6= B = B(t, r) (ii) A = A(t, r),B = B(r) 4 (iii) A = A(t, r),B = B(t) (iv) A = A(t, r),B = const. (i) A = const., B = B(r) (ii) A = const., B = B(t, r) 5 (iii) A = const., B = B(t) (iv) A = A(r),B = const. (v) A = A(t),B = B(t, r) (vi) A = A(r),B = B(t, r) (vii) A = A(t),B = const. A = A(t),B = B(r) 6

Table 6.8: TP KVs in LTB Metric

174 vectors in 3-dimensional static circularly symmetric spacetimes in teleparallel gravity. As a consequence of our analysis, we have obtained a proper teleparallel homothetic vector for these spacetimes in only once case, which is presented in (5.2.19). The number of teleparallel Killing vectors in teleparallel gravity for the above mentioned spacetimes turned out to be 2,3,4,5 or 6, in contrast to general relativity where the same spacetimes admit 2,3,4 or 6 Killing vectors [33]. In the following table, we present the number of teleparallel Killing vectors along with the constraints satisﬁed by the metric functions of 3-dimensional static circularly symmetric spacetimes under which they admit the teleparallel Killing vectors.

Constraints on Metric Functions No. of TP KVs A0 6= 0, B = B(r) 2 A0 = 0, B = B(r) 3 A0 6= 0, B = ln r2 4

0 B r 0 A 6= 0, 2e + 2k2 A 6= 0 4 0 B r 0 A = 0, 2e + 2k2 A 6= 0 5 0 B r 0 A 6= 0, 2e + 2k2 A = 0 6 A0 = 0, B = ln r2 6

Table 6.9: TP KVs in 3-dimensional Static Circularly Symmetric Spacetimes

Comparing these results with those of general relativity [33], we have observed that 3-dimensional static circularly symmetric spacetimes do not admit ﬁve Killing vectors in general relativity for any choice of the metric

175 functions but they do admit the same number of teleparallel Killing vectors in teleparallel gravity. We conclude with the remarks that spacetime symmetries including Killing, homothetic and conformal Killing vectors are strongly eﬀected by vanishing curvature and introducing non zero torsion in spacetimes. As spacetimes symmetries have a close link with conservation laws in Physics, consequently the presence of non zero torsion and vanishing curvature will have aﬀect on the corresponding conservation laws. Although we have investigated Killing, homothetic and conformal Killing vectors for some well known spacetimes in general relativity and teleparallel gravity, our comparison could not establish any general relationship between these three types of symmetries in the two theories. The investigation of these three types of spacetime symmetries for some other spacetimes is under consideration in both theories, which may help us to make a conjecture about the general relationship of spacetime symmetries in general relativity and teleparallel gravity.

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