In 1905, while only twenty-six years old, Albert Einstein published "On the Electrodynamics of Moving Bodies" and effectively extended classical laws of relativity to all laws of physics, even electrodynamics. In this course, we will take a close look at the special theory of relativity and also at classical field theory. Concepts addressed here will include four-dimensional space-time, electromagnetic fields, and Maxwell's equations.  

In the first lecture of the course Professor Susskind introduces the original principle of relativity - also known as Galilean Invariance - and discusses inertial reference frames and simultaneity. He then derives the Lorentz transformation of special relativity following the method in Einstein's original paper [check this], and introduces length contraction and time dilation, invariants, and space- and time-like intervals.

Professor Susskind starts with a brief review of the Lorentz transformation, and moves on to derive the relativistic velocity addition formula. He then discusses invariant intervals, proper-time and distance, and light cones.

Note: this is only a 1 hour lecture

Topics: 

• Relativistic velocity addition

• Double Lorentz transformations

• Proper time

• Light cones

• Four-vectors

• Four-velocity

References:

• Velocity addition formula

• Proper time

• Light cone

• Four-vector

• Four-velocity

Professor Susskind begins with a review of space- and time-like intervals, and explains how these intervals relate to causality and action at a distance. He then introduces space-time four-vectors and four-velocity in particular.

After presenting these concepts, Professor Susskind introduces relativistic particle mechanics. He presents the action principle for a particle in free space, and derives the Lagrangian for such a particle.

Building on these concepts, Professor Susskind derives the relativistic formulas for momentum and energy, and discusses relativistic mass, and how the conservation of momentum and energy are modified by relativity. He then shows the origin of Einstein's famous equation E = mc2.

The lecture concludes with a discussion of massless particles under relativity.

Topics: 

• Relativistic particle mechanics

• More space-like and time-like intervals

• Four-vectors and four-velocity

• Relativistic action and Lagrangian for the motion of a particle

• Relativistic momentum and energy

• Derivation of mass-energy equivalence: E = mc^2^

• Massless particles

References:

• Four-vector

• Four-velocity

• Relativistic mass

• Mass–energy equivalence

• Does the Inertia of a Body Depend Upon Its Energy-Content? by Albert Einstein

Professor Susskind moves on from relativity to introduce classical field theory. The most commonly studied classical field is the electromagnetic field; however, we will start with a less complex field - one in which the field values only depends on time - not on any spatial dimensions.

Professor Susskind reviews the action principle and the Lagrangian formulation of classical mechanics, and describes how they apply to fields. He then shows how the generalized classical Lagrangian results in a wave equation much like a multi-dimensional harmonic oscillator.

Next, professor Susskind brings in relativity and demonstrates how to create a Lorentz invariant action, which implies that the Lagrangian must be a scalar.

The lecture concludes with a discussion of how a particle interacts with a scalar field, and how the scalar field can give rise to a mass for an otherwise massless particle. This is the Higgs mass mechanism, and the simple time dependent field we started the lecture with is the Higgs field.

Topics: 

• Introduction to classical field theory

• Action and Lagrangian for a field in four-space

• Continuum mechanics

• Introducing relativity into the Lagrangian formulation for a field

• Particle interacting with a simple scalar field

• Higgs mechanism

References:

• Classical field theory

• Continuum mechanics

• Higgs mechanism

Professor Susskind begins with a discussion of how, in the case of charged particle in an electromagnetic field, the particle affects the field and vice-versa. This effect arises from cross terms in the Lagrangian. He then derives the action, Lagrangian, and equations of motion for this case, and shows that the equations of motion are wave equations with a singularity at the location of the particle.

Professor Susskind then introduces the contravariant and covariant four-vector notation and Einstein's summation conventions used in the study of relativity.

He then proves that scalar Lagrangians are Lorentz invariant.

Finally, Professor Susskind solves the wave equation for a particle in a field and demonstrates that the solutions are sums of plane waves. The Higgs boson is the case of a charged particle with zero mass, and the resulting field derived from the equations solved here is the Higgs field. The Higgs field is the origin of the electron mass.

After a review of Einstein & Minkowski notation and an introduction to tensors, Professor Susskind derives the relativistic Lorentz force law from the Lagrangian for a particle in a vector field. At the end of the lecture, he introduces the the four fundamental principles that apply to all of modern physics: stationary action, locality, Lorentz invariance, gauge invariance. 

Topics: 

• Review of Einstein & Minkowski notation

• Introduction to tensors and tensor notation

• Derivation of the Lorentz force law

• The fundamental principles of physical laws

References:

• Tensor

• Electromagnetism

• Lorentz force

Professor Susskind elaborates on the four fundamental principles that apply to all physical laws. He then reviews the derivation of the Lorentz force law as an example of the application of these principles. The lecture closes with an introduction to gauge invariance.

Topics: 

• Stationary action

• Locality

• Lorentz invariance

• Gauge invariance

• Review of the derivation of the Lorentz force law

References:

• Principle of stationary action

• Principle of locality

• Lorentz invariance

• Gauge theory

After a brief review of gauge invariance, Professor Susskind describes the introductory paragraph of Einstein's 1905 paper "On the Electrodynamics of Moving Bodies," and derives the results of the paragraph in terms of the relativistic transformation of the electromagnetic field tensor. This paragraph asks the fundamental question "what is the difference between a charge moving in a magnetic field, and a fixed charge in a changing magnetic field." The answer to this fundamental question must be "nothing" if the principle of relativity is true. This conclusion is what led Einstein to develop the special theory of relativity.

Professor Susskind then moves on to present Maxwell's equations. He discusses the definition of charge and current density that appear in them, and then derives the relationship between these quantities. This relationship is the continuity equation for charge and current, and represents the principle of charge conservation.

The lecture concludes with the presentation the first two Maxwell equations in relativistic notation. This single equation is the Bianchi identity, and this identity makes it clear that magnetic charge sources (monopoles) and magnetic current do not exist.

Topics: 

• Relativistic transformation of the electromagnetic field tensor

• Maxwell's equations

• Conservation of charge

• Maxwell's equations in relativistic notation

• Magnetic monopole

References:

• Electromagnetic tensor

• Maxwell's equations

• Continuity equation

• Charge conservation

• Four-current

• Magnetic potential

• Magnetic monopole

Professor Susskind begins the lecture by solving Maxwell's equations for electromagnetic plane waves. He then uses the principles of action, locality and Lorentz invariance to develop the Lagrangian for electrodynamics for the special case without charges or currents. Using the Euler-Lagrange equations with this Lagrangian, he derives Maxwell's equations for this special case. Finally, Professor Susskind adds the Lagrangian term for charges and currents by using the principle of gauge invariance, and again uses the Euler-Lagrange equations to derive Maxwell's equations in relativistic notation.

Topics: 

• Electromagnetic plane waves

• Choosing a Lagrangian for electrodynamics and deriving Maxwell's equations

• Adding charges and currents to the Lagrangian

References:

• Plane waves

Professor Susskind begins the final lecture with a review and comparison of the three different concepts of momentum: mechanical momentum from Newtonian mechanics, canonical momentum from the Lagrangian formulation of mechanics, and momentum that is conserved by symmetry under translation invariance from Noether's theorem. He then develops the connection between Lagrangian and Hamiltonian mechanics and field theory in more detail than in previous lectures.

Professor Susskind moves on to develop the concepts of energy and momentum density, and then applies these concepts to electromagnetic fields. He concludes the course with an introduction to energy and momentum flux, and the stress-energy tensor.

Topics: 

• Comparison of the three concepts of momentum

• Connection between classical mechanics and field theory

• Energy and momentum density

• Stress-energy tensor

References:

• Noethe's theorem

• Poynting vector

• Stress–energy tensor