Video Lectures - RelativityDoctor

The first ten chapters of the book are now covered in 20 videos which have been posted here and on

Classical Differential Geometry, the basis of Riemannian Geometry of General Relativity, is covered in 19 additional videos which have been posted here and on

You Tube!

General Relativity is covered in 27 additional videos which have also been posted here and on

You Tube!

The first set of videos begin with an overview of the textbook.

The series of videos ends with a demonstration that the full set of Maxwell’s equations follows from Gauss’ Law and the Lorentz Transformations of special relativity. The wave equation of electrodynamics is also derived there and discussed.

You can start with the overview and then continue to all the lectures. You can also search on “john kogut” in the YouTube search box and some of the lectures will come up (with a lot of other irrelevant stuff, too!). Anyway, below are all the video lectures.

Please view them on a LARGE screen so you can read the equations! You can also follow along in the textbook.

The Table of Contents of the first 20 Video Lectures, with links to each Video Lecture on YouTube:

Introduction and Overview lecture

Lecture 1 Newton’s World

Lecture 2 Space time Measurements in Einstein’s World

Lecture 3 Part 1 Minkowski Diagrams

Lecture 3 Part 2 Doppler Effect and the Twin Paradox

Lecture 4 Part 1 Lorentz Transformations

Lecture 4 Part 2 The Metric of Minkowski Diagrams

Lecture 5 Part 1 Relativistic Energy and Momentum

Lecture 5 Part 2 The Relativistic Force and Energy Conservation

Lecture 5 Part 3 Four Vectors

Lecture 5 Part 4 Four Vector Operators

Lecture 5 Part 5 Collisions and Conservation Laws

Lecture 6 Acceleration and Forces in Relativity. The Birth of Fields

Lecture 7 Part 1 The Electric Field of a Moving Charge

Lecture 7 Part 2 Transforming Electric and Magnetic Fields between Inertial Frames

Lecture 8 Part 1 Gauss’ Law and Current Conservation

Lecture 8 Part 2 Discovering Maxwell’s Equations

Lecture 8 Part 3 The Wave Equation

Lecture 9 Part 1 Magnetism, Lorentz Contraction and the Discovery of Relativity

Lecture 9 Part 2 Covariant Electromagnetism and the Way Forward

Second Video Lecture Series on YOU TUBE

Classical Differential Geometry for Physics Students

This Video Lecture series covers and expands upon the Classical Differential Geometry 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 Classical Differential Geometry can be found here,

Contents of the Video Lectures

The Video Lecture series begins with concepts from Euclidean geometry in two dimensions, then turns to concepts in Spherical geometry and then finally to a full and systematic presentation of the classical differential geometry of surfaces embedded in three dimensional Euclidean space. The first and second fundamental forms are introduced. The Gauss and Weingarten maps are introduced, developed and applied to specific surfaces. The Gaussian curvature is discussed intuitively and formally. Gauss’ Theorem Egregium is proved and applied. Geodesics, covariant differentiation and parallel transport are introduced and illustrated in general and for specific surfaces. The Gauss Bonnet Theorem is proved and applied to special cases. Geodesic deviation and Jacobi fields are introduced and developed. Holonomy is introduced. Extra special topic lectures include: 1. Triangulated surfaces, the Gauss Bonnet Theorem and global topology, 2. Spherical Geometry of the Projective plane and Hyperbolic Geometry of the Poincare Disk, and 3. Curves and Torsion in Three Dimensions.

Supplementary Lecture #9 can be read synchronously with the videos.

Supplementary Lectures #2, #4 and #8 are also covered in the later videos.

The Table of Contents of the Video Lectures

Lecture 1.a Overview and the Beginnings of Euclidean Geometry . Lines and curves on the plane.

Lecture 1.b Euclidean Geometry continued. The curvature of planar curves is introduced and is seen to control the rate at which a curve deviates from its tangent. The curvature is also related to the turning angle along a curve, a result that will generalize to the Gauss Bonnet Theorem on curved surfaces.

Lecture 1.c Euclidean Geometry completed. The Parallel Postulate of Euclidean geometry is reviewed. The rate of deviation of intersecting lines is presented as a precursor to the idea of geodesic deviation that will appear on curved surfaces when geodesics intersect.

Lecture 2.a Spherical Geometry introduced. Great Circles are shown to be the geodesics, the “straightest” possible curves, on a sphere. The areas of “double lunes” are computed and used to derive the spherical version of the Gauss Bonnet Theorem on a sphere. The sphere’s natural length scale, provided by its Gaussian curvature, implies that similar geodesic triangles are congruent, in contrast to Euclidean geometry.

Lecture 2.b Spherical Geometry continued. The geodesic deviation is introduced and illustrated on a sphere and a simple differential equation for the evolution of the geodesic deviation illustrates the importance of the Gaussian curvature. The metric on the sphere is studied and the first short distance deviations from Euclidean geometry are seen to be controlled by the surface’s Gaussian curvature. The covariant derivative is introduced and the idea of parallel transport is illustrated. The path dependence of parallel transport is discovered and is seen to arise from the curvature of the surface.

Lecture 2.c Spherical Geometry completed. The Gauss Bonnet Theorem on the sphere illustrates the path dependence of parallel transport. The commutator of components of the covariant derivative is computed on a sphere and is seen to be non-zero because of the local curvature of the surface. The generalization of these calculations to an arbitrary surface will lead to a discussion of Holonomy later in the course.

Lecture 3.a The metric on a general surface embedded in three dimensional Euclidean space is introduced .The Gauss map of Normals to a surface and its derivative are introduced and studied. The Weingarten map and the second fundamental form are introduced.

Lecture 3.b The Weingarten map is written as a 2×2 symmetric metrix and is solved in terms of the first and second fundamental forms of the surface. The eigenvalues (principle curvatures) and eigen-directions of the Weingarten map are introduced and are used to define the Gaussian and Mean curvatures.

Lecture 4.a Surfaces of Revolution. The Torus is introduced and is given a natural coordinate mesh. Its first and second fundamental forms and its Gaussian Curvature are calculated. General Surfaces of Revolution are considered and a local differential equation relating its shape to the surface’s Gaussian curvature is derived.

Lecture 4.b The second fundamental form is shown to indicate how fast a surface deviates from its tangent plane, generalizing the idea from curves in Euclidean space that the curvature indicates how quickly a curve deviates from its tangent. Gauss’ original geometric interpretation of the Gaussian curvature is derived: it is the ratio of the area on the unit sphere of the Normals to the area mapped from a patch on the original surface.

Lecture 4.c Gauss’ Theorem Egregium is derived showing that the Gaussian curvature is determined just in terms of a surface’s metric and its derivatives. So, the Gaussian curvature is invariant to bending, isometric mappings of the surface. The Christoffel symbols are introduced and the Gaussian curvature is written in terms of these symbols and their derivatives.

Lecture 5.a Parallel transport and covariant differentiation are introduced and illustrated on general surfaces and on the sphere. The relation of the generalization of the “turning angle” from flat, two dimensional spaces to curved surfaces is explained. This is the ground work for a derivation of the Gauss Bonnet Theorem, the subject of the next lecture.

Lecture 5.b The geodesic curvature of a curve embedded on a surface is related to the curve’s turning angle. This observation leads to a derivation of the Gauss Bonnet Theorem for simple, regular closed curves.

Lecture 6.a Gaussian geodesic polar coordinates are introduced for a general surface. The short distance deviations from the Euclidean metric are determined by the Gaussian curvature. Geodesic deviation is illustrated for this coordinate system.

Lecture 6.b The commutator of components of the covariant derivative is computed and shown to be proportional to the Gaussian curvature. The Jacobi differential equation is derived and applied to the geodesic deviation on the sphere.

Lecture 7.a The Gauss Bonnet Theorem is generalized to piecewise smooth, simple, closed curves. Triangulated surfaces are considered and the Gauss Bonnet Theorem is applied to surfaces of various Euler indices (and Genus). The topological invariance of the resulting classification is discussed.

Lecture 8.a Curves in three dimensions are described by a moving triad of unit vectors and the matrix differential equation describing their coupled propagation along the arc length of the curve is derived. The result is the Fernet-Serret equations. Two parameters, the curvature and torsion, describe the curve. The helix serves as an illustration.

Lecture 9.a We consider surfaces of revolution with Gaussian curvature -1. A differential equation for the surface’s shape is solved producing the pseudo-sphere. The surface has a sharp edge and is a flawed model of hyperbolic geometry, consistent with a no-go theorem of Hilbert. The geometry of the sphere is reexamined by stereographically projecting the upper hemisphere onto the equatorial plane. The mapping is shown to be conformal and the properties of the stereographic projection are pointed out.

Lecture 9.b The Poincare Disk model of hyperbolic geometry is introduced. it is mapped onto the Upper Half Plane model of hyperbolic geometry. Both models are conformal to the plane.The isometries of the models are found and are expressed as linear fractional transformations. The geodesics of the models are found and the Gauss Bonnet Theorem is used to study hyperbolic triangles. Parallel lines are discussed and illustrated. Special features of non-Euclidean geometries are presented and illustrated. The metric is written in terms of the hyperbolic distance and is compared to the metric of the sphere written in polar coordinates.

Video Lecture Series on YOU TUBE

General Relativity for Physics Students

This Video 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 are linked to the Table of Contents below.

This video lecture series follows Supplementary Lecture 11 closely. They could be studied synchronously.

Contents of the Video Lectures

The Video 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.

Supplementary Lecture #11 should be read synchronously with the videos.

The Table of Contents of the Video Lectures, each linked to YouTube

Lecture 1. Overview and the Contents of the Lecture Series.

Lecture 2. The Equivalence Principle, Virtual Forces and Gravity.

Lecture 3. Relativistic Rotating Reference Frames and Tidal Forces.

Lecture 4. Gravitational Redshift and The Twin Paradox.

Lecture 5. Contrasting Electromagnetism and Gravity.

Lecture 6. Covariant Differentiation and The Equation For Geodesics.

Lecture 7. Equivalence Principle, Metric Compatibility and Christoffel Symbols.

Lecture 8. Holonomy, Curvature and the Einstein Field Equation,

Lecture 9. Geodesic Equation as a Consequence of the Field Equation.