Fall, 2008
General relativity, or the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1916 and the current description of gravitation in modern physics. General relativity generalises special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. (Source: Wikipedia)
(Image credit: Ute Kraus, Max-Planck-Institut für Gravitationsphysik, Golm, and Theoretische Astrophysik, Universität Tübingen)
Lectures in this Course
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1
Newtonian Gravity and the equivalence principle
This lecture starts with the tidal effects of Newtonian gravity. Tidal effects are due to a nonuniform gravitational field. A person in the freely falling Einstein's elevator experiences weightlessness, and if the elevator is small enough, no tidal... [more] -
2
Tidal forces and curvature
Review preliminary mathematics. Einstein: the laws of nature in a gravitational field are equivalent to the laws in an accelerated frame. Study bending of light due to curvature of space. Tidal forces and curvature cannot be transformed away. -
3
Essential tools: tensors and the metric
Einstein summation convention. Definition of an infinitesimal distance element. Definition of a tensor. Contravariant/covariant transformations. Metric tensor defines the distance element. -
4
Tensor mechanics
Tensor indices. Index contraction. Inverse of the metric tensor, the Kronecker delta. The metric tensor is symmetric. Raising and lowering indices. Tensors must have the same transformation properties if they are to be added. Proper time. -
5
Covariant differentiation and geodesics
Transformation properties of tensors are most important. Covariant differentiation transforms covariantly. The covariant derivative of the metric tensor vanishes. Construct the Christoffel symbol from the metric tensor. -
6
The flat space of special relativity
The Minkowski metric. For flat space, there always exists a coordinate system for which the metric tensor is constant. For flat space, the Christoffel symbols vanish. For flat space, parallel transport moves a vector along a space curve without... [more] -
7
The Riemannian curvature tensor
Parallel transport in curved space. A gyroscope parallel transports it's axis of spin. Define the Riemann curvature tensor and the Ricci tensor. In general relativity, mass alters geometry, and curved geometry deflects mass from moving in a straight... [more] -
8
Equations of motion in curved space
Define the covariant derivative. Define the Riemann curvature tensor through the commutation of the covariant derivative. Two types of curvature are intrinsic and extrinsic Determine the equation of motion given by the covariant derivative of the... [more] -
9
Gravitation in the Newtonian approximation
The metric tensor is smooth, indefinite, symmetric and invertible. Derive the field equations of relativity in the Newtonian approximation. The Einstein tensor. Einstein's equation relating curvature and the energy momentum tensor. -
10
Energy-momentum tensor and Einstein's equations
The covariant divergence of the energy momentum tensor vanishes. The covariant derivative of the metric tensor vanishes. Einstein's equation in the Newtonian approximation. Wave equation for a scalar field in curved space Energy momentum tensor for... [more] -
11
Accelerated coordinates
The integrated curvature depends only upon the topology of spacetime. (Euler number) An accelerated observer coordinate drawn in a spacetime diagram traces hyperbolas. Rindler coordinates describe a uniformly accelerated coordinate frame. Rindler... [more] -
12
World lines and Schwarzschild solution
World lines of accelerated motion in space-time diagrams. Light cone and accelerated motion. The Schwarzschild solution for a point mass. The central singularity of the Schwarzschild solution cannot be transformed away. The event horizon.