## Details

Code 00971

Level of study Third/Final year

Credit value 10

Semester 2

## Module description

This module is a first step to the summit of classical physics, the Theory of Relativity. The core of the course is a mathematical description of the physical structure of 'flat' spacetime based on Einstein's relativity principle and the invariance of the speed of light. A natural language for such a description is that of tensors which, being by itself very elegant and powerful, would be necessary in extending (in General Theory of Relativity in the 4th year) the relativity principle to 'curved' spacetime and thus the laws of gravitation.
Combining the relativity principle with the invariance of electric charge and using a fully 'covariant' tensor description, the entire Maxwell's electrodynamics, including the emission of radiation, will be "derived" step by step from Gauss' electrostatics.

Syllabus

• Introduction: incompatibility of Maxwell's equations and Galileo's relativity principle; the Michelson-Morley and Trouton-Noble experiments; the absence of long-range interaction.
• Einstein's postulates: the principle of relativity and invariance of the speed of light. The invariance of the interval as a mathematic expression of Einstein's postulates. The Lorentz transformations between inertial frames as a consequence of this invariance. Relativistic invariance and covariance.
• Space-time in special relativity. 4-vectors and 4-tensors. The Minkowski metric and Minkowski space. Proper time. Spacelike and timelike intervals. The Minkowski diagram and world lines. Simple relativistic effects in the space-time description: the time dilation and the Lorentz-FitzGerald contraction. "Paradoxes'' of relativity.
• Vectors and tensors in a metric space. Contra- and co-variant vectors and tensors and linear transformations of coordinates. Scalar invariants. Tensors contractions; Einstein's summation convention. Metric tensor and a distance between points; a scalar product of two vectors.
• Relativistic mechanics. Energy-momentum 4-vector. The equivalence of mass and energy. 4-forces. 4-momentum conservation law.
• Electrostatics and Lorentz invariance. Coulomb's law and the electrical Lorentz force. 4-potential and 4-current density; the continuity equation. The Lorentz transformation of an electrostatic field to a moving frame: magnetic fields must exist.
• Relativistic covariance and invariance of electric charge as the building blocks for electrodynamics.
• Fields transformations. Relations between electromagnetic fields and potentials. The electromagnetic field tensor. The Lorentz transformations rules for the electromagnetic field as a consequence of its tensor nature. The electromagnetic-field invariants.
• Homogeneous and inhomogeneous Maxwell's equations in standard and 4-vector form. The dual electromagnetic field tensor. Integral form of Maxwell's equations.
• Charges in electromagnetic fields. The Lorentz force in 3- and 4-vector form. Motion of an electric charge in uniform electric and magnetic fields.
• Energy and momentum of the electromagnetic field. Poynting vector. The energy-momentum tensor of the field. Conservation laws for the field.
• Constant electromagnetic fields. Electrostatic and ``magnetostatic'' energies of charges and currents. The field of a uniformly moving charge. Dipole and multipole electric and magnetic moments.
• Electromagnetic waves. Gauge invariance and gauge transformations. The wave equation. Plain and monochromatic plain waves. Doppler's effect.
• Electromagnetic radiation. The retarded potentials. The field of moving particles. Relativistically covariant Larmor's formula for the emission of radiation. Energy radiation in the nonrelativistic limit. Dipole radiation.

Aims:

To learn how how to formulate and apply the laws of mechanics and electrodynamics in relativistically invariant four-tensor form.