Home
Lecture blog
Exams
Exercises
Course PM
Test exam
SI2371 Special Relativity
Lecture blog

This page will contain brief summaries of what has been done in the lectures.

The approach taken during the lectures will mainly follow these lecture notes: PDF.
Lecture note erratum: (these are the known errors, please reports others that you find)
  • Page 29: Snell's law is sin(θo) = (1/n) sin(θi), not the cosine relation stated.
  • Page 35: At the end of the page, Mμν is correctly defined but then minus signs are missing. It is correct again on the next page.
  • Page 39: There is an ω missing in the expression for the Poynting vector at the bottom of the page.
The conceptual problems posed during the lectures can be found here.

Lecture 1: Introduction
We discussed the foundations of special relativity, mainly the special relativity principle and the fact that the speed of light is constant and equal in all directions. Starting from this and the assumption that space is homogenous, we derived the general form of a coordinate transformation between two inertial frames. We saw that the assumption of a universal time then leads to the Galilei transformation, while the assumption of the existence of a speed which is invariant under the transformations leads to the Lorentz transformation.

Lecture 2: Tensors and 4-velocity
We had a crash course reminder/primer in tensor analysis and discussed the properties of tensors. We introduced the metric tensor and Minkowski space, which is what we use to describe space-time in special relativity. We discussed the properties of world-lines through Minkowski space and the concept their length, which is the eigentime of an observer following that world-line. This leads to the concept of a 4-velocity, which is the tangent to the world-line.

Lecture 3: Space-time diagrams and paradoxes
We discussed some of the mathematical properties of the Lorentz transformations, leading to hyperbolic geometry and the construction of space-time diagrams as a tool for visualising how a physical process appears in different inertial frames. We discussed time-dilation as well as length contraction and two of the paradoxes associated with these: the twin paradox and the pole-in-a-garage paradox.

Lecture 4: Relativistic kinematics
We covered the relativistic generalization of kinematical quantities such as acceleration, momentum, energy, and force. We saw that all of these concepts essentially generalize to different parts of 4-vectors or were related to the square of a 4-vector. We saw how the conservation of 4-momentum component by component results in the conservation of energy and momentum in the classical setting.

Lecture 5: Electromagnetism
Starting from the assumption that a force field should be pure and the corresponding force at most dependent on the 4-velocity (and not higher derivatives of the world line), we deduced that the simplest possible force field corresponds to a anti-symmetric rank 2 tensor. Writing down the simplest possible field equations for such a field, we saw that this corresponds to Maxwell's equations. We discussed invariants of the force field as well as the 4-potenial and the source term appearing in the field equations (the 4-current).

Lecture 6: Waves
After discussing the electromagnetic force on a charged particle, we moved on to discussing the geometry of surfaces in Minkowski space. We then applied this to the concept of waves in Minkowski space as being described by the phase function φ(xμ). We discussed the interpretation of the phase velocity and the de Broglie interpretation of particles as waves.

Lecture 7: Photons and relativistic optics
We had a brief discussion on the photon 4-momentum relating the photon energy and momentum in the same was as that of the de Broglie interpretation of particles as waves. We discussed relativistic optics, in particular the Doppler effect and how to compute it as well as how light behaves in moving media and when reflected by relativistic objects. Finally, we discussed the phenomenon of relativistic light abberation.

Lecture 8: Particle kinematics
Topics in relativistic particle kinematics were raised and we discussed how experimental particle physics relate to collider experiments, such as the Large Hadron Collider. We saw how kinematical relations can be deduced from simply using the conservation of 4-momentum as well as basic tensor manipulations. An important point is that different scalars may be computed in different inertial systems without problems. The phenomenon of threshold effects, i.e., the energy necessary for a reaction to occur, was discussed in detail. We also saw the application of the Lorentz transformation to the decay of particles in flight and the resulting distribution of decay products.

Lecture 9: The electromagnetic stress energy tensor
Starting by noting that forces and counter forces must be located in the same space-time point, we related the electromagnetic field to a symmetric rank-2 tensor - the stress-energy tensor. We looked at its properties and saw that the force equation Fμ = -∂ν Mμν take the form of continuity equations for the energy and momentum densities of the electromagnetic field. Correspondingly, we identified:
  • σ -- electromagnetic energy density
  • g -- electromagnetic energy current density and momentum density (with c=1, in general, they are proportional with c2 as proportionality constant)
  • pij -- momentum current density (flow in direction j of momentum density in direction i)
Finally, we saw the general plane wave solution to the source free electromagnetic field equations in Lorentz gauge. We identified the wave vector and the polarization vectors in this gauge.

Lecture 10: Relativistics of a continuum
After discussing the form of the fields of an electromagnetic wave and the corresponding energy tensor, we generalized the concept of a relativistic continuum to other situations than EM fields. In particular, we discussed perfect fluids, which is locally isotropic in the rest-frame. We found that the energy tensor of such a fluid is diagonal Tμν = diag(ρ0,p,p,p) = (ρ0 + p) Uμ Uν - p ημν, where p is the pressure, ρ0 the rest frame energy density, and Uμ the 4-velocity of the rest frame of the fluid. We briefly discussed how ρ0 and p relate for different types of fluids, in particular:
  • p = ρ0/3 --- Photon gas (gas of massless particles)
  • p = 0 --- Dust / matter gas (T << m, where m is the mass of the constituent particles)
  • p = -ρ0 --- Cosmological constant (relevant for general relativity)

Lecture 11: Energy and momentum of a continuum
We integrated the energy tensor over an instantaneous surface and saw that the resulting tensor could be interpreted as the total 4-momentum of the continuum and that it was conserved under the assumption that there are no external forces acting on the continuum. We also showed that it does not matter in which system the total 4-momentum is defined as long as there is no external force. If there is an external force, then it will act on the fluid at different times in different inertial systems, which will lead to a difference. In addition, we also discussed the angular momentum tensor of the continuum, which is an asymmetric rank-2 tensor. We saw that the spatial part of the angular momentum tensor corresponds to the usual 3-angular momentum and that its change is related to the total torque on the continuum. When looking at the time-space components, we were able to derive the equation of motion for the center-of-energy for the continuum in terms of its energy, momentum, and the force acting on it.


This concludes the lectures of 2013. Good luck on the exam!


Last updated: Dec 18, 2013