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General Relativity Online Course
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------ The Universe Around Us: An Integrative View of Science & Cosmology
--------Preface
--------Chapter 1: The Big Questions
--------Chapter 2: Understanding the Universe
--------Chapter 3: The Physical World
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--Overview
--Tensors and Relativity: Chapter 1
----Contents
----Pre- relativistic physics
----The equations of electromagnetism
----The principle of Special Relativity
----Spacetime diagrams and the Lorentz transformations
----The spacetime interval
----Minkowski spacetime
----The null cone
----Consequences of the Einstein postulates
----Time dilation
----Length contraction
----The twin paradox
----Velocity composition law
----Index
----About this document ...
--Tensors and Relativity: Chapter 2
----Contents
----Four- vectors
----Four- velocity, momentum and acceleration
----Relativistic Doppler shift
----Four- acceleration
----Relativistic dynamics
----Example 2.1
----Example 2.2
----Index
----About this document ...
-- Tensors and Relativity: Chapter 3
----Contents
----Metrics and forms
----One- forms
----Gradients
----The metric as a mapping of vectors onto one- forms
----More general tensors
----Tensors of type 0/2
----Tensors of type 0/N
----Tensors of type 2/0
----Tensors of type M/0
----Tensors of type M/N
----Index ``raising'' and ``lowering''
----General properties of tensors
----Tensor derivatives and gradients
----Index
----About this document ...
-- Tensors and Relativity: Chapter 4
----Contents
----The energy- momentum tensor
----General fluids
----Conservation of energy- momentum
----Conservation of particles
----Perfect fluids
----The conservation equations
----The Electromagnetic tensor
----Index
----About this document ...
-- Tensors and Relativity: Chapter 5
----Contents
----The story so far
----The gravitational redshift experiment
----Non- existence of an inertial frame at rest on earth
----Mass in Newtonian theory
----The principle of equivalence
----The principle of equivalence in action
----Effect of gravity on light
----Effect of gravity on time
----Towards spacetime curvature
----Index
----About this document ...
-- Tensors and Relativity: Chapter 6
----Contents
----Manifolds, tangent spaces and local inertial frames
----Covariant derivatives and Christoffel symbols
----Calculating from the metric
----Tensors in polar coordinates
----Parallel transport and geodesics
----The variational method for geodesics
----The principle of equivalence again
----The curvature tensor and geodesic deviation
----The curvature tensor
----Properties of the Riemann curvature tensor
----Geodesic deviation
----The Bianchi identities; Ricci and Einstein tensors
----The Ricci tensor
----The Einstein Tensor
----Index
----About this document ...
-- Tensors and Relativity: Chapter 7
----Contents
----Introduction
----The non- vacuum field equations
----The weak field approximation
----Index
----About this document ...
-- Tensors and Relativity: Chapter 8
----Contents
----The line element
----The Christoffel symbols
----The Ricci Tensor
----The vacuum field equations
----The Schwarzschild solution
----General discussion of the Schwartzschild solution
----Time dilation in a gravitational field
----Length contraction in a gravitational field
----Orbits in Schwartzschild spacetime
----Solution for timelike orbits and precession
----The bending of light
----Index
----About this document ...
-- Tensors and Relativity: Assignment 1
----Problem
----About this document ...
----Hints 1
-- Tensors and Relativity: Assignment 2
----Problem
----About this document ...
----Hints 2
-- Tensors and Relativity: Assignment 3
----Problem
----About this document ...
----Hints 3
-- Tensors and Relativity: Assignment 4
----Problem
----About this document ...
----Hints 4
-- Tensors and Relativity: Assignment 5
----Problem
----About this document ...
----Hints 5
-- Tensors and Relativity: Assignment 6
----Problem
----About this document ...
----Hints 6
-- Tensors and Relativity: Assignment 7
----Problem
----About this document ...
----Hints 7
Home
About Us
Vision
About
Undergraduate
Overview
First Year Course List
Overview and MAM Hotseat
Mathematics
Applied Mathematics
Engineering
Commerce
Second Year Course List
Overview
Mathematics
Applied Mathematics
Engineering
Third Year Course List
Overview
Mathematics
Applied Mathematics
Engineering
Forms & Documents
Postgraduate
Message from the Dean
Message from the Science Postgraduate Students' Association
Honours
Masters
Overview
AST5003F
PhD
Recent Graduates & Theses
Thesis Submission
Applications
Application to Tutor in the Department of Mathematics
Finances
Enquiries
Links
Contact Us
Forms & Documents
Staff
Forms & Documents
Permanent Academic Staff
Temporary Academic Staff
Administration
Emeritus Professors & Honorary Research Associates
Research
Research Areas
Centre for Research in Computational & Applied Mechanics
Cosmology & Gravity Group
Laboratory for Discrete Mathematics and Theoretical Computer Science
Laboratory for Quantum Gravity & Strings
Marine Resource Assessment & Management Group
Topology & Algebra Research Group
Introduction
Bernhard Banaschewski
Members
Activities
Output
Links
Conferences
Contact Us
National Astrophysics & Space Science Programme (NASSP)
Rangeland Modelling Group
Objectives
Future Projects
Publications
Contact Us
Recent Graduates & Theses
Outreach
Mathematical Digest
Overview
Archive
Articles
Problems
Humour
Links
Order
Contact
UCT Mathematics Competition
Overview
About the Competition
Maths Competition 2018
Challenge & UCT Mathematics Olympiad
Awards
Training Material
Entry Form
Registration & Invigilation
Winners
Photos
In the News
Credits
Contact Us
Mathematics Olympiads
International Mathematical Olympiad (IMO)
Pan African Mathematics Olympiad PAMO)
SA Mathematical Talent Search
Seminar Talks & Lectures
Employment
Contact Us
General Relativity Online Course
Overview
Tensors and Relativity: Chapter 1
Contents
Pre- relativistic physics
The equations of electromagnetism
The principle of Special Relativity
Spacetime diagrams and the Lorentz transformations
The spacetime interval
Minkowski spacetime
The null cone
Consequences of the Einstein postulates
Time dilation
Length contraction
The twin paradox
Velocity composition law
Index
About this document ...
Tensors and Relativity: Chapter 2
Contents
Four- vectors
Four- velocity, momentum and acceleration
Relativistic Doppler shift
Four- acceleration
Relativistic dynamics
Example 2.1
Example 2.2
Index
About this document ...
Tensors and Relativity: Chapter 3
Contents
Metrics and forms
One- forms
Gradients
The metric as a mapping of vectors onto one- forms
More general tensors
Tensors of type 0/2
Tensors of type 0/N
Tensors of type 2/0
Tensors of type M/0
Tensors of type M/N
Index ``raising'' and ``lowering''
General properties of tensors
Tensor derivatives and gradients
Index
About this document ...
Tensors and Relativity: Chapter 4
Contents
The energy- momentum tensor
General fluids
Conservation of energy- momentum
Conservation of particles
Perfect fluids
The conservation equations
The Electromagnetic tensor
Index
About this document ...
Tensors and Relativity: Chapter 5
Contents
The story so far
The gravitational redshift experiment
Non- existence of an inertial frame at rest on earth
Mass in Newtonian theory
The principle of equivalence
The principle of equivalence in action
Effect of gravity on light
Effect of gravity on time
Towards spacetime curvature
Index
About this document ...
Tensors and Relativity: Chapter 6
Contents
Manifolds, tangent spaces and local inertial frames
Covariant derivatives and Christoffel symbols
Calculating from the metric
Tensors in polar coordinates
Parallel transport and geodesics
The variational method for geodesics
The principle of equivalence again
The curvature tensor and geodesic deviation
The curvature tensor
Properties of the Riemann curvature tensor
Geodesic deviation
The Bianchi identities; Ricci and Einstein tensors
The Ricci tensor
The Einstein Tensor
Index
About this document ...
Tensors and Relativity: Chapter 7
Contents
Introduction
The non- vacuum field equations
The weak field approximation
Index
About this document ...
Tensors and Relativity: Chapter 8
Contents
The line element
The Christoffel symbols
The Ricci Tensor
The vacuum field equations
The Schwarzschild solution
General discussion of the Schwartzschild solution
Time dilation in a gravitational field
Length contraction in a gravitational field
Orbits in Schwartzschild spacetime
Solution for timelike orbits and precession
The bending of light
Index
About this document ...
Tensors and Relativity: Assignment 1
Problem
About this document ...
Hints 1
Tensors and Relativity: Assignment 2
Problem
About this document ...
Hints 2
Tensors and Relativity: Assignment 3
Problem
About this document ...
Hints 3
Tensors and Relativity: Assignment 4
Problem
About this document ...
Hints 4
Tensors and Relativity: Assignment 5
Problem
About this document ...
Hints 5
Tensors and Relativity: Assignment 6
Problem
About this document ...
Hints 6
Tensors and Relativity: Assignment 7
Problem
About this document ...
Hints 7
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