General Relativity and Cosmology using  Mathematica

Robert L. Zimmerman

Institute of Theoretical Science

University of Oregon

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

      5.1.3 Radial Equation and Potential

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

5.5 Numerical Radial Trajectories

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 

6.3  Radial  Light Paths 

6.4  Bending of Light by the Sun

6.5  Radar Time Delay

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  

9.5 Radial Geodesic Equation

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

            User-Defined Plotting Functions

            Example: Precessing Ellipses

             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

Part IV  Gravitational Radiation

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 Quadrupole Radiation 

       12.3.1 Gravitational Amplitude and the Quadrupole Moment

       12.3.2  Energy and Angular Momentum  Loss

12.4 Examples of Radiating Systems

       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 Supernova  Estimates  Coalescing Binary Estimates  Pulsar Estimates 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  Radiation from Binary Stars

          13.4.1  PSR 1913 +16

          13.4.2 Gravitational Radiation from specific Binaries

13.5  Radiation from Pulsars  

          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

            Binary Pulsar PSR 1913 +16

            Binary Pulsar  PSR 2127+11C

            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 

          Example: Look back time to 3C273  and 3C48

        14.5.3 Luminosity and Angular Distances

          Example: Quasar distance

        14.5.4 Magnitude-Redshift Diagram

          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  

        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

              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

       Einstein static model (E)

       Asymptotic models (A1&A2)

        Singular oscillating models (o)

       Singular Monotonic Models  (M1)

        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

Chapter Seventeen  Radiation Dominated Cosmology

17.1  Radiation Dominated Models

         17.1.1  3K Radiation

         17.1.2  Dynamical Equations
 Temperature in the Early Universe
  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

          17.3.1 Radiation-Matter Cosmologies

17.4 Other Solutions

           17.4.1 Bianchi Solutions

           17.4.2  Vacuum Solutions



A1. Problem solutions






many many