Applications of NP Formalism and GHP Formalism in General Relativity

Preface

Einstein proposed general relativity theory in 1915, this theory most successfully describes gravity. A typical approach to analyze gravitational field of an object is to solve Einstein’s field equations, in this approach the assumptions are guided by observations and they help in understanding physical aspects. From mathematical stand point of view we may look for geometric tools. In recent times, tetrad formalisms like Newman-Penrose formalism and Geroch-Held-Penrose formalism are popular to analyze gravitational field. Tetrad formalisms are based on choosing a suitable set of basis vectors for the spacetime under consideration. For using a tetrad formalism the relevant quantities are projected on to the chosen basis and the equations satisfied by them are considered. In this thesis, we have developed some applications of the NP formalism and GHP formalism to electromagnetic gravitational analogies, (i) electric and magnetic parts of the Weyl tensor and (ii) Lanczos potential. The complication involved in the computations compelled us to pay some attention to work in computer aided algebraic computations.

The thesis contains mainly five chapters, a write-up of scopes of future work, an appendix and a bibliography. A brief summary of the chapters appearing in the thesis is given below.

Chapter 1 is comprised of a brief introduction of the general theory of relativity along with expressions and role of basic tensors appear in the theory. Two analogies between electromagnetism and gravity namely, (i) electric and magnetic parts of the Weyl tensor and (ii) Lanczos potential are introduced. Also, tetrad formalisms are introduced in brief.

Chapter 2 is dedicated to algebraic computations using computer algebra system (CAS) Mathematica in the general relativity. It is known that the computations in general relativity are lengthy, time consuming, complicated and there is no guarantee of getting error free results when doing it manually. This motivates us to use CAS for such computations. We have developed mathematica programs for the algebraic computations of various quantities appear in general relativity.

In chapter 3, we have expressed electric and magnetic parts of the Weyl tensor into spin coefficients and it helped in developing a time efficient algorithm for obtaining electric and magnetic parts of the Weyl tensor. These expressions give an easy proof of known theorems on some geometric quantities.

Chapter 4 presents an application of the NP formalism for obtaining the Lanczos potential for non-vacuum spacetimes namely, the Vaidya metric and the Van Stockum spacetime. These results may be useful in obtaining the Lanczos potential for other non-vacuum and/or non-Petrov type spacetimes.

Chapter 5 contains solutions of the Weyl-Lanczos relations for some non-vacuum spacetimes: the Gödel spacetime, the Generalized pp-waves spacetime and the Van Stockum spacetime using GHP formalism approach of method proposed by Parga et al. This formalism reduce diÿculties when compared to NP method of solving Weyl-Lanczos relations. These results presented here will help in obtaining Lanczos potential for other non-vacuum spacetimes using GHP method.

Some part of the work included in the thesis have been published and one paper has been communicated.

1. Hasmani, A.H., and Panchal, R. Algebraic Computations of General Ob-server Quantities using Mathematica. Astrophysics and Space Science 359, 1 (2015), 1-5. doi: 10.1007/s10509-015-2465-6. IF: 2.263.

2. Hasmani, A.H., and Panchal, R. Algebraic Computations of Complex Tetrad Components of Ricci Tensor. In Proceedings of Recent Trends in Mathe-matics and Statistics (2015), pp. 97-104.

3. Hasmani, A.H., Patel, B.N. and Panchal, R. On Algebraic Computations of Electric and Magnetic Parts of the Weyl Tensor. International Journal of Mathematics and Scientific Computing 6, 1 (2016), 7-12.

4. Hasmani, A.H., and Panchal, R. Lanczos Potential for Some Non-Vacuum Spacetimes. The European Physical Journal Plus, 131, 9 (2016), 1-6. doi: 10.1140/epjp/i2016-16336-7. IF: 1.521.

5. Hasmani, A.H., and Panchal, R. Electric and Magnetic parts of the Weyl Tensor and Spin Coeÿcients. Communicated, (2016).

Contents

Preface . . iii

Acknowledgments . . vi

1 Introduction . . 1

1.1 Geometry in Relativity . . 1

1.2 Analogies between Electromagnetism and Gravity . . 7

1.3 Tetrad Formalism . . 13

1.4 Newman-Penrose Formalism . . 15

1.5 GHP Formalism . . 17

2 Algebraic Computations in General Relativity using Mathematica . . 18

2.1 Introduction . . 18

2.2 Computer Algebra System: MATHEMATICA . . 20

2.3 Algebraic Computations of General Observer Quantities . . 21

2.4 Algebraic Computations of Complex Tetrad Components of Ricci Tensor . . 29

2.5 Algebraic Computations of Electric and Magnetic Parts of the Weyl Tensor . . 35

2.6 Conclusion . . 45

3 Electric and Magnetic parts of the Weyl Tensor and Spin Coefficients . . 46

3.1 Introduction . . 46

3.2 Spin Coefficients . . 48

3.3 Electric and Magnetic Parts of theWeyl Tensor and Spin Coefficients . . 51

3.4 Computing of Electric and Magnetic Parts of the Weyl Tensor . . 53

3.5 Pure Radiation Metric - an Example . . 55

3.6 Conclusion . . 58

4 Lanczos Potential for Some Non-Vacuum Spacetimes . . 59

4.1 Introduction . . 59

4.2 Lanczos Potential in NP formalism . . 61

4.3 Lanczos Potential for Vaidya Metric . . 63

4.4 Lanczos Potential for Van Stockum Spacetime . . 65

4.5 Conclusion . . 67

5 Lanczos Potential using GHP Formalism . . 69

5.1 Introduction . . 69

5.2 Lorentz Transformations for Various Geometric Quantities . . 70

5.3 GHP version of Weyl-Lanczos Relations . . 80

5.4 Solutions of GHP Weyl-Lanczos Relations . . 88

5.5 Conclusion . . 94

6 Future Work . . 95

A Directional Derivative of Null Tetrad . . 96

B Petrov Classification . . 98

Bibliography . . 100

Chapter 1

Introduction

The pioneer idea behind the Einstein’s general theory of relativity is geometrizing the gravitational force. In other words, mapping all properties of the gravitational force and its influence upon physical process onto the properties of four dimensional semi-Riemannian spacetime. Postulates of general theory of relativity include postulates of special relativity by setting gravity for accelerating frame of reference. The main principle behind this is principle of covariance. According to that [61] "All laws of Physics are to be written covariantly by the use of tensors, to ensure the equivalence, in principle, of all coordinate systems". In other words, it can be said that all laws of physics (or nature) should be in the same mathematical form in all frames of references. So, we do not need to worry about the coordinate system we are working with. For this, we require to know pseudo-Riemannian or semi-Riemannian geometry which is briefly explained in the next sections. The notations and terminologies used in this chapter are standard and can be found in any standard relevant material. However, we have used the books referred in serial numbers in the bibliography [3, 14, 52, 61].

1.1 Geometry in Relativity

Let M be a four dimensional spacetime. The infinitesimal distance ds between two neighboring points with co-ordinates in any co-ordinate system xi and xⁱ+dxⁱ by the quadratic differential formula,

ds² = g_ij dxⁱdx^j

i, j = 1, 2, 3, 4, (1.1)

is called line element or metric. Here and in the entire this thesis we have assumed Einstein’s summation convention which means summing over a repeating index; the range of summation being four. Now, gij is covariant rank-2 tensor called metric tensor or fundamental tensor, the components gij are arbitrary functions of the co-ordinates xi subject to the restriction

[Formulas are omitted from this preview]

This tensor describes topology of the spacetime under consideration. We have adopted signature of metric tensor to be −2, which means that, at any selected point, g_ij is reduced to the diagonal matrix (−1, −1, −1, 1) locally.

Also, the quantity ds² is invariant under a co-ordinate transformation and it may be positive, negative or zero. Accordingly, they are called timelike, null or spacelike.

1.1.1 Basics

The Christoffel symbols of first kind are denoted and defined by,

[Formulas are omitted from this preview] (1.2) (1.3)

are called Christoffel symbols of second kind. It should be noted that though components g_ij form a tensor, Christoffel symbols do not. In fact, partial derivative of a tensor (except scalar) is not necessarily a tensor. This forces us to modify concept of derivative in such a way that the modified differentiation must pro-duce tensor quantity and it must satisfy properties which partial differentiation satisfies. For example, addition, substraction and outer product of any two tensors (when possible) should obey the similar rules that of partial differentiation. It is possible to find a frame in which partial derivative of metric tensor and Levi-Civita tensor are zero (for example, flat space), the modified derivative should be defined in such a way that the fundamental tensors g _ij , g_ij and Levi-Civita tensor behave as constants as far as this derivative is concerned. Since this modified differentiation produces covariant quantities, it is known as covariant derivative and it is denoted by semicolon (whereas a partial derivative is de-noted by comma). The formal definition of covariant derivatives is given in the following way,

(A) Covariant derivatives of covariant and contravariant tensor of rank one:

[Formulas are omitted from this preview] (1.4) (1.5)

(B) Covariant derivatives of tensor of rank two:

The covariant derivatives of covariant, contravariant and mixed tensors of rank two are defined as follows,

[Formulas are omitted from this preview] (1.6) (1.7) (1.8)

(C) Covariant derivative of tensors (mixed) of higher rank:

In general, the covariant derivative of mixed tensor of rank (m, n) is,

[Formulas are omitted from this preview] (1.9)

1.1.2 Ricci Identity and Curvature Tensor

It is known that the second order covariant derivative of a covariant vector field A_j is non-commutative, their commutator is known as Ricci identity, [1]

[Formulas are omitted from this preview] (1.10) (1.11)

is Riemann curvature tensor. The covariant derivative commute if and only if Riemann curvature tensor vanishes. In general relativity gravitation is attributed to the curvature of the spacetime, which means Riemann curvature tensor describes the gravitational field and hence analysis of the gravitational field is done using Riemann tensor. The necessary and sufficient condition for a spacetime to be flat is vanishing of Riemann tensor. In the case of flat spacetime, a coordinate system can be transformed into a Minkowski coordinate system (more details are given in [61]). In order to explore algebraic properties of the Riemann curvature tensor, instead of Rⁱ_jkl, it is more convenient to use its fully covariant form,

[Formulas are omitted from this preview] (1.12)

From (1.12), Riemann curvature tensor have following algebraic properties

(A) Antisymmetry:

[Formulas are omitted from this preview] (1.13)

(B) Symmetry:

[Formulas are omitted from this preview] (1.14)

(C) Cyclicity:

[Formulas are omitted from this preview] (1.15)

The Riemann curvature has total 4⁴ = 256 components. Due to (1.13)-(1.15), the maximum number of independent components reduce to 20. Besides these algebraic properties, Riemann tensor satisfies a differential property also, namely, Bianchi identity,

[Formulas are omitted from this preview] (1.16)

The Riemann tensor is a tensor of rank four, its contraction with metric tensor gives another tensor called the Ricci tensor (actually, there are three possible contractions, in which contraction of first two indices vanishes, whereas remaining two are with first and third; and with first and fourth indices. Both will reduce to the Ricci tensor) as follows,

[Formulas are omitted from this preview] (1.17)

and it has at most ten independent components. Further contraction of Ricci tensor gives a scalar called the Ricci scalar,)

[Formulas are omitted from this preview] (1.18)

The following is Einstein’s tensor,

[Formulas are omitted from this preview] (1.19)

This tensor plays a fundamental role in the theory of Relativity; this tensor is a symmetric tensor and has vanishing divergence G^ij _j = 0. Einstein used this fact to present gravitational field equations.

1.1.3 Einstein’s Field Equations

Einstein’s field equations relate geometry of spacetime with matter content, which are given by,

[Formulas are omitted from this preview] (1.20)

where k is relativistic gravitational constant and T_ij is energy-momentum tensor relevant to the material content of the spacetime (T _ij is source of gravitational field and g_ij is its potential. So, (1.20) is analogous to the limiting case of Poisson equation of Newton’s theory). Einstein’s field equations are symmetric in nature, which form a system of ten second order non-linear partial differential equations. As a consequence of Einstein’s field equations absence of matter is characterized by vanishing of Ricci tensor.

1.1.4 Weyl Tensor

Riemann tensor can be split into its trace part (that is, combination of Ricci tensor and Ricci scalar) and trace-free part (which defines a new tensor) as follows,

[Formulas are omitted from this preview] (1.21)

The newly introduced tensor C_hijk is known as the Weyl curvature tensor which is completely traceless, that is

[Formulas are omitted from this preview]

and has the same symmetry properties which Riemann curvature tensor satisfies. Thus, the Weyl tensor has ten independent components. From equation (1.21), it can be seen that in empty region, R_ij = 0, the Weyl tensor is same as Riemann tensor. In other words, the Weyl tensor describes curvature in a vacuum region. As mentioned earlier, gravitation is attributed to curvature (that is, Riemann tensor). Hence, C_hijk characterizes gravitation in vacuum region.

[Formulas are omitted from this preview] (1.22)

Conformally related metrics may have different Riemann curvature tensor, but they have the same Weyl tensor (see [3, 61]). Thus, the Weyl tensor is called the Weyl conformal tensor or the Conformal tensor. A spacetime is said to be conformally flat if C_ijkl = 0.

Due to symmetry properties satisfied by Weyl tensor, its left dual is same as the right dual,

[Formulas are omitted from this preview] (1.23) (1.24)

Classification of the Weyl tensor based on multiplicity of principal null directions is known as the Petrov classification. Since Weyl tensor describes gravitation, Petrov classification classifies the gravitation described by the spacetime under consideration. The information about the classification helps in simplifying the procedure of dealing with Weyl tensor (see appendix B).

1.2 Analogies between Electromagnetism and Gravity

Electromagnetic interactions and gravitational interactions are among four fundamental interactions; the other two are the strong nuclear force and the weak nuclear force. Many attempts are made to unify all forces. Electromagnetism and gravity have many analogies like, expressions in both Newton’s and Coulomb’s laws are in terms of inverse square form, Maxwell-like form of the gravitational field tensor, etc. In relativistic theory however electromagnetic field is nicely understood, we explore the possibility of analyzing gravitational field with reference to following analogies with electromagnetic field

(i) Electric and Magnetic parts of Weyl tensor [45]

(ii) Lanczos potential [41]

[...]

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[1] The tensors defined here are according to a monograph by S. Chandrasekhar [14].