Supplementary Lecture Series 11
This Lecture series covers and expands upon the General Relativity topics in Chapters 11 and 12 of the textbook “Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein”.
This is a junior or senior undergraduate course for physics, astronomy and math majors.
The video lectures on General Relativity for Physics Students follow these lectures closely and could be studied together.
Contents of the Lectures
The Lecture series begins with a discussion of the Equivalence Principle, gravity and apparent (“virtual”) forces. Concepts are illustrated using the metric of a relativistic rotating coordinate system.
Tidal forces of Newtonian mechanics are reviewed and are seen to be an indicator of curved space time and non-Euclidean geometry.
The idea of the tangent space, local inertial reference frames, in a curved four dimensional space time manifold is introduced. The natural physical limitations of the Equivalence Principle in environments of non-uniform gravitational fields are discussed.
The gravitational redshift is introduced as an application of the Equivalence principle, and the Twin Paradox is resolved as a problem in accelerating reference frames using the Equivalence Principle.
The similarities and differences in the theories of electromagnetism and gravity are discussed. Polarization properties of electromagnetic waves (photons) are contrasted with those of gravity waves (gravitons).
The equation for a geodesic in Riemannian space time is developed and applied to massive point particles. The covariant derivative is introduced to realize a coordinate independent measure of the rate of change of vector and tensor fields. Christoffel symbols are introduced to describe the space time dependence of sets of basis vectors. Metric Compatibility and the Equivalence Principle are discussed and are used to find an expression for Christoffel coefficients in terms of the metric. The curvature of space time is discussed and the Einstein Field Equations are introduced. The geodesic equation of motion of point particles are rederived from the field equations.
Linearized gravity is introduced to systematically study environments of weak gravitational fields.
The Schwarzschild metric is derived and its black hole is studied. Orbital motion around a static spherical mass is explored and the non-linearities that distinguish General Relativity from Newtonian gravity are explored.
Relativistic tidal effects are discussed in the context of the equation for the Geodesic Deviation.
Gravitational waves are studied and the LIGO experiment is discussed and illustrated as a detector of traveling gravitational tidal effects.
The Cosmological constant and Dark Energy are introduced in a brief look at modern puzzles of gravitational physics.
In a set of “Special Topic” lectures, rotating stars, frame-dragging, and the Lense-Thirring effect are studied as manifestations of velocity dependent effects, gravito-magnetism, in General Relativity. The Kerr metric is introduced and its event horizons and surfaces of infinite redshift are presented. Frame-dragging is illustrated by considering light rays propagating inside a rotating star’s ergosphere.
Symmetries and conservation laws are discussed and applied to an exact treatment of the gravitational redshift.
Special Relativity problems involving accelerating clocks and reference frames are discussed. The twin paradox is reconsidered as a problem in observing accelerating clocks and their hyperbolic motion in Minkowski space time. The Rindler Wedge is introduced and its metric is obtained and discussed from the perspective of the Equivalence Principle.
Geodesic precession of gyroscopes in orbits around static stars is studied and illustrated.