General Relativity and Cosmology using Mathematica
Robert L. Zimmerman
Institute of Theoretical Science
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.
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:
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:
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
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 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
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 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
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
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
Chapter Seventeen
Radiation Dominated Cosmology
17.1 Radiation Dominated Models
17.1.1 3K Radiation
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
17.3.1 Radiation-Matter Cosmologies
17.4 Other Solutions
17.4.1 Bianchi Solutions
17.4.2 Vacuum Solutions
Appendix
A1.
Problem solutions