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]
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.Minkowski... [more]
Einstein summation convention.
Definition of an infinitesimal distance element.
Definition of a tensor.
Metric tensor defines the distance element.
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.
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.A geodesic curve is defined as... [more]
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]
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]
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]
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.
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]
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]
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.