General Relativity and Cosmology using  Mathematica

Robert L. Zimmerman

Institute of Theoretical Science

# Eugene, OR 97403

Pick up the current issue of any general relativity or cosmology journal and you will find statements like "The calculations were performed using the computer algebra system...".  Calculations in Riemannian geometry are extremely demanding so the need for computer algebra is a necessity. This book uses Mathematica  to visualize and display concepts, to perform tensor calculus, and to generate numerical and graphical solutions. This book provides a basic introduction to General Relativity and Cosmology.

What makes this book unique is that the calculations are done using Mathematica.  It is the only book that develops the tools for Computer Algebra as well as laying the foundations for General Relativity and Cosmology.  It covers the material contained in a senior year or graduate course in Physics. The book is designed to be studied sequentially as a whole, in a one-year course, but it can be shortened to accommodate a half-year course.  This book provides the reader with a sound understanding of the basic theory and the Mathematica tools needed to do the calculations.

Each chapter starts with the basic concepts and then applies  the  concepts to explicit calculations using user-defined procedures constructed with. Exercises that accompany the chapters reinforce the concepts and fill in gaps that were omitted in the chapter. The book is divided into five parts and contains a total of seventeen chapters:

Part one:  The beginning of the book lays the foundation for the basic physical principles, tensor calculations, and develops the appropriate user-defined Mathematica  procedures.

Part two:  This section explores the spacetime around the Schwarzschild solution. The tensor procedures are used to explore the properties of the solution and the graphic commands and numerical procedures are used to illustrate geodesics and horizons.

Part three: This section studies other exact solutions like the Kerr and other  metrics.

Part four:  Gravitational radiation is covered in this section of the book. Numerical procedures, graphics and specialized commands are developed to calculate gravitational radiation for sources like colliding black holes,  collapsing binary stars, rotating pulsars and etc.

Part five:  Cosmology is covered in this section of the book. Mathematical procedures are use to calculate magnitude-redshift diagrams, look back times, and etc for cosmological models with and without the cosmological constant.

### Part I  Foundations of General Relativity

Chapter One  Special  Relativity

1.1 Absolute Space and Time and Newtonian Motion

1.2 The Principle of Equivalence

1.2.1 Equivalence of Inertial and Passive mass

1.2.2 Equivalence of Active and Passive mass

1.2.3 Consequences of the Principle of Equivalence

1.3 Galilean Transformations and Inertial Frames

1.3.1 Mechanics and Inertial Frames

1.3.2  Electrodynamics and  Ether

1.4  Foundations of Special Relativity

1.4.1 Postulates of Special Relativity

1.4.2 Space and Time and the Speed of Light

1.5  Lorentz Transformations

1.5.1 Conditions for a General Lorentz Transformation

1.5.2  Lorentz Boosts

Example  1.5.1:  Product  of  Lorentz  Boosts

Example  1.5.2:  Space Rotations

1.6 Relativistic Tensors

1.6.1  Tensors

1.6.2  Four-velocity and Momentum Vectors

1.6.3  Relativistic Doppler Shift

1.7 Relativistic Kinematics

1.7.1 Decay of a particle

1.7.2 Two-particle collision

1.7.3 Compton  scattering

1.8 Electromagnetism

1.8.1 Noncovariant Formulation of Maxwell's Equations

1.8.2 Covariant Formulation of Maxwell's Equations

1.9  Problems

Chapter Two  Tensors

user-Defined  Procedures

Example 2.1.1:  {r, f}  Polar Coordinates

2.2 Tensor Transformations

2.3 Tensor Properties

Example 2.3.1: Product of  symmetric and antisymmetric tensors

2.4 Parallel Transport and Christoffel Symbol

Example 2.4.1: Christoffel Symbol  for Two-dimensional Polar Coordinates

Example 2.4.2: Christoffel Symbol for Spherical Pseudo-Euclidean Spacetime

2.5 Covariant Derivatives

Example 2.5.1

2.6  Geodesics

Example 2.6.1: Geodesic in Spherical Coordinates

2.7  Isometries

2.8    Problems

Chapter Three  Curvature and Gravity

User-Defined  Procedures

Example 3.1.1: Curvature of a Two- sphere

Example 3.1.2 : Interchange of  Covariant Derivatives

Example 3.1.3: Flat Pseudo-Euclidean Metric in Spherical Coordinates

3.2  Symmetries of the Curvature Tensor

3.3  Ricci, Einstein, and Weyl Tensors

Example 3.3.2: Conformally Flat Metric in Spherical Coordinates

3.4  Weak Gravity and the Metric

3.5  Gravitational Redshift

3.6  Principle of General Covariance

3.7  Einstein Field equations

3.8 Problems:

### Part II  Schwarzschild Solution

Chapter Four  Line Element and Coordinates

User-Defined  Procedures:

4.1  Spherically Symmetric Vacuum Solution

4.1.1 Metric for a Static Spherically Symmetric Spacetime

4.1.2  Christoffel Symbol and Curvature Tensor

4.1.3 Einstein Tensor  and Vacuum Solution

4.2  Killing Vectors

4.2.1 Rotational Killing Vectors

4.2.2 Timelike  Killing Vector

4.3 Gravitational Redshift

4.3.1  Static Observer and Source

4.3.2  Moving Observer and Source

4.4  Isotropic Coordinates

4.4.1 Isotropic Spherical Coordinates

4.4.2 Isotropic Cartesian Coordinates

4.5 PPN Metric

4.5  Problems:

Chapter Five  Time-Like Geodesics

User-Defined  Procedures: Christoffel Symbol G, Geodesic,  Schwarzschild Metric, Killing Rules

5.1 Timelike Geodesics

5.1.1 Geodesics

5.1.2 Killing Constants

5.2  Potential Analysis

5.2.1 Types of Trajectories

5.2.2 Behavior of Extremum Points as a Function of jz

5.3  Numerical Solution

5.3.1 Numerical Procedure

5.3.2  Precessing Ellipses

5.3.3  Loitering Orbit

5.4  Falling into a Black Hole

5.6  Circular Orbits

5.7  Precession of the Perihelia

5.7.1  Equation for u''[f]

5.7.2  Perturbative Solution for Precessing Ellipses

5.7.3  Precession of the Planets

5.8  Problems:

Chapter Six     Null Geodesics

User-Defined  Procedures:  Schwarzschild Metric

6.1 Null Geodesics

6.2  Potential Analysis

6.4  Bending of Light by the Sun

Chapter Seven  Coordinates and Singularities

User-Defined  Procedures:

7.1 Singularities and Curvature Invariants

7.2  Eddington-Finkelstein  Coordinates:

7.3 Kruskal Coordinates

7.3.1 Exterior Black Hole, Region  I

7.3.2 Exterior White Hole, Region  III

7.3.3 Interior Black Hole, Region  II

7.3.4 Interior White Hole, Region  IV

7.3.5 Complete Kruskal Diagram

Chapter Eight   Interior Solution

User-Defined  Procedures:

8.1 Non-vacuum Field Equations

8.2  Solution for the Metric Component

8.3  Pressure Equation

8.4  Solution for the Metric Component

8.5  Behavior of the Pressure and Singularities

8.6  Physical Density and Binding Energy

8.7 Problems:

### Part III Other Solutions

Chapter Nine   Charged Black Hole

User-Defined  Procedures

9.1 Einstein-Maxwell Equations for a Point Charge

9.2 Reissner-Nordstrom Solution

9.3 Event Horizons

9.4 Potential Analysis for Timelike Geodesics

9.6 Problems:

Chapter Ten      Rotating Black Holes

User-Defined Procedures:

10.1 Kerr Geometry

10.1.1 Kerr Metric

10.1.2  Killing Vectors

10.1.3 Other Coordinates and the Ring Singularity

10.2 Inertial Frames

10.2.1  Frame Dragging

10.2.2 Static Limit

10.3 Horizons

10.3.1 Killing Horizons

10.3.2 Event Horizons

10.3.3  Behavior of the Horizons as a Function of the Rotation

10.4 Black Hole Dynamics

10.4.1 Extraction of  Energy from the Ergosphere

10.4.2 Black Hole Thermodynamics

10.5 Geodesics

10.5.1  Equations for t'[s] and f'[s]

10.5.2  Equations for  and

10.5.3 Equatorial Orbits

10.5.4 Numerical Solution

10.5.4.1  User-Defined Plotting Functions

10.5.4.2  Example: Precessing Ellipses

10.5.4.3   Example :  Loitering Orbits

10.5.5 Circular Geodesics

10.6  Problems:

Chapter Eleven  Wyle Solutions

User-Defined  Procedures:

11.1  Wyle Canonical Coordinates

11.2 Curzon Solution

11.3 Superposition of Two Curzon solutions Schwarzschild Solution Superposition of two Schwarzschild

Particles 11.6 Problem

Chapter Twelve  Linearized Gravity

User-Defined  Procedures:  hwave, makeTT,Eloss, Lloss

12.1 Linearized Equations

12.1.1 Linearized Curvature

12.1.2 Linearized Field Equations

12.2 Coordinate Systems

12.2.1 Gauge Transformation

12.2.2 Harmonic Gauge

12.2.3  Plane waves

12.2.4  Transverse-Traceless Gauge

12.3.1 Gravitational Amplitude and the Quadrupole Moment

12.3.2  Energy and Angular Momentum  Loss

12.4.1   Oscillating Masses

12.4.2    Masses oscillating from the End of a Bar

12.4.3  Particle falling into a black hole

12.5  Order of magnitude Estimates

12.4.5.1 Supernova  Estimates

12.4.5.2  Coalescing Binary Estimates

12.4.5.3  Pulsar Estimates

12.4.5.4 Cosmological Sources

12.6 Problems

Chapter Thirteen  Sources for Gravitational Radiation

User-Defined Procedures:

13.1 The Search for Gravity Waves

13.1.1  Sources for Gravity Waves

13.1.2 Bars

13.1.3 LIGO (Laser Interferometer Gravitational Wave Observatory)

13.1.4 Doppler tracking

13.1.5  LISA

13.2  Binary Stars in Circular orbits

13.2.1 Wave Amplitude for Circular Orbits

13.2.2  Orbital Decay

13.2.3 Radiation from Two Compact Solar Mass Stars

13.2.4 Power of gravitational radiation from the planets

13.3  Binary Stars in Elliptical orbits

13.3.1 Energy and Angular Momentum Loss

13.3.2 Orbital Decay

13.4.1  PSR 1913 +16

13.4.2 Gravitational Radiation from specific Binaries

13.4.1 Rotating ellipsoids

13.4.2  Crab and Vela Pulsars

13.4.3 Wobbling Pulsars

13.4.4 Pulsars as gravity wave detectors

13.6   Binary Pulsars

13.6.1 Relativistic Orbit Parameters

13.6.1.1  Binary Pulsar PSR 1913 +16

13.6.1.2  Binary Pulsar  PSR 2127+11C

13.6.1.3  Binary Pulsar   PSR 1534+12

13.7  Problems

### Part V    Cosmology

Chapter Fourteen_Friedmann-Robertson-Walker Geometry

User-Defined Procedures:

14.1 Foundations

14.1.1 Introduction

14.1.2  Fundamental Assumptions

14.1.3 Comoving Coordinates

14.2 Geometry

14.2.1 Spaces with Constant Curvature

14.2.2  Coordinate Systems

14.3  Observables

14.3.1 Redshifts

14.3.2 Time and Distance

14.3.3 Proper Distance and Horizons

14.3.4 Luminosity and Angular Distances

14.3.5 Number Count

14.5  Slowly varying Scale Factor, R[t]

14.5.1 Hubble and Deceleration Parameters

14.5.2  Redshift

14.5.2.1 Example: Look back time to 3C273  and 3C48

14.5.3 Luminosity and Angular Distances

14.5.3.1 Example: Quasar distance

14.5.4 Magnitude-Redshift Diagram

14.5.4.1 Example: Virgo Cluster and the value of Ho

14.5.5  Number Count

14.6  Problems

Chapter Fifteen    Matter Dominated Models with Zero l

User-Defined Procedures

15.1 General Field Equations

15.1.1 Field Equations

15.1.2 Equation of State and Conservation Laws

15.1.3 Parameterized Form of Field Equations

15.2 Matter Dominated Equations

15.2.1  Matter Dominated  Field Equations

15.2.2 The Fate of the Universe: The Density of Matter

15.3  Properties of Matter-Dominated Models

15.3.1  Age

15.3.2  Cosmic Time and Coordinate Distance

15.3.3  Luminosity Distance

15.3.4  Angular-Size Distance

15.3.4.1 Example:  Angular-size of Compact Radio Sources

15.3.5 Distance Modulus

15.4  Euclidean Model: (Wo=1, k=0, Einstein-de Sitter model)

15.4.1  Scale Factor and Age

15.4.2  Redshift and Time

15.4.3  Distance

15.4.3.1 Example: Large Redshift Quasars

15.4.4 Summary

15.5 Problems

Chapter Sixteen    Models with Non-Zero l

User-Defined Procedures:

16.1 Basic Equations

16.1.1 Introduction

16.1.2 Field Equation

16.2  Qualitative Analysis of Models

16.2.1 Types or Models for k =1

16.2.1.1 Einstein static model (E)

16.2.1.2 Asymptotic models (A1&A2)

16.2.1.3  Singular oscillating models (o)

16.2.1.4 Singular Monotonic Models  (M1)

16.2.1.5  Non-singular monotonic models ( M2 )

16.2.2  Types of Models for k=0 and -1

16.3 Models in terms of  Wl and Wm

16.3.1 Parameterized Equations

16.3.2 Types of Models

16.4  General Properties

16.4.1 Look Back Time

16.4.2 Model Age

16.4.3 Distance

16.4.4 Luminosity Distance

16.5 Problems

17.1.2  Dynamical Equations

1.7.1.3 Temperature in the Early Universe

1.7.1.4  Period of Recombination

17.2 Very Early Universe

17.2.1 Temperature Time Line and Early Periods

17.2.2 Nuclear Synthesis During  the First Few Minutes

17.2.3 Neutrino Decoupling

17.2.4 Matter anti matter

17.2.4 Phase Transitions and the electroweak force

17.2.5 Cosmic Strings & other Topological

17.2.6 Phase Transitions and the Grand Unification

17.2.7 Types of Phase Transitions and Consequences

17.2.8 Inflation

17.3   Mixture of Radiation and Matter Solutions

17.4 Other Solutions

17.4.1 Bianchi Solutions

17.4.2  Vacuum Solutions

Appendix

A1. Problem solutions  