Special Relativity

How are new particles created in high energy collisions in powerful accelerators? Or if you wondered what it would be like to be on a spaceship journeying through the galaxy and see how the universe is seen by the travellers, then this research area is for you!

Special Relativity plays a unique role in modern physics. The theory has been with us for over a century but it still fascinates students, scientists and, generally, those with an interest in physics. New articles continue to be published, presenting the theory and discussing some of its results in different, more modern perspectives, or extending to the relativistic world the known solutions of old problems in classical dynamics.

The research topics cover a wide field! From looking back at the state of theoretical physics at the beginning of the 20th century and evaluating the successes and difficulties of the pre-relativistic physics to understanding how Einstein was guided to formulate such a novel theory that, to a large extent, reinvented the fundamental concepts of space, time, mass and energy that were around since the times of Newton.

We shall study in detail the concepts of relativistic kinematics, relativistic dynamics and relativistic optics, examine the well-known twin paradox in order to understand the relativity of time, the conversion of mass into energy and, discuss the Doppler effect and the aberration of light and some of their, perhaps unexpected, consequences.

Intended audience

This research area is suitable for anyone with a basic background of Special Relativity who would like to explore some remarkable, perhaps unexpected, phenomena that occur when bodies move at close to the speed of light. The knowledge acquired during the course and the individual research will help participants in further studies such as Relativistic Quantum Mechanics and General Relativity.

Previous knowledge

Students are assumed to have basic knowledge of special relativity, namely the Lorentz transformation, relativistic kinematics and relativistic dynamics, as well as a good understanding of Newtonian dynamics. Knowledge of Maxwell equations of electromagnetism and the non-relativistic motion of a particle in an electric or magnetic field is desirable though not essential.

Assessment

During the course of the programme, the supervisor will set exercises to check the student’s understanding and progress. These will make up 75% of the total assessment. The remaining 25% will be assessed via a written report, which should give an account of the material covered during the project. In particular, the report will contain a careful explanation of any of the pieces of theory that the student has used to solve the exercises.

Research Topics

The potential research topics you could pursue on the programme are listed below. The specific research focus of your project will be determined and confirmed with guidance from your supervisor. 

1. The interstellar traveller

The planning of a journey through the galaxy presents an opportunity to use the equations of Special Relativity to figure out how different the sky will be seen by an observer inside a spaceship that moves at relativistic speeds as compared to an observer at rest with respect to the stars. With the formulas for the aberration of light, the student will investigate, and try to numerically simulate, how the angular distribution of stars appears to the traveller as a function of the velocity of the spaceship. On the other hand, an astute use of the relation between the wavelengths of the light emitted by the stars as measured in the frame of the stars and in the frame of the spaceship (Doppler Effect) will determine which wavelengths are seen by the traveller and, as a consequence, what colour picture of the sky he will have.

The student will also consider diverse versions of the relativistic rocket and derive and solve the equations of motion for different types of propulsion. Using the conversion of mass into energy he or she will investigate the type of reactions that can be used to produce the necessary energy for the ship to accelerate

2. The visual appearance of rapidly moving objects

It has been known since the formulation of Special Relativity that the dimensions and the shape of an object are not absolute quantities. For example, if an object at rest has a characteristic length l0 and a certain shape, the same object will have, in general, a different characteristic length l and a different shape when observed in a reference frame where it is moving with a certain velocity. The first effect to explain these changes in size and shape is the Lorentz contraction. For some time, it was thought that the Lorentz contraction could be directly verified, if sufficiently advanced technology was available, for example by taking a photograph of a moving object, say a cube, and determining the new ratio between the length of an edge parallel to the direction of the motion and that of an edge perpendicular to it. Implicit in these considerations must be the rigorous definition of length of a moving object. The evaluation of the length of a moving rod, for instance, requires the simultaneous determination of the positions of its two extremities.

The Lorentz contraction, however, is insufficient to find how an object will appear in a photograph or in a film. The point is that, whereas to determine the length, width or depth of an object, one has to perform simultaneous measurements of its extremities, the procedure is very different when using a photographic or video camera. In a photograph, light emitted by all points of the surface of an object arrives at the same time at the camera. A ray coming from the rear extremity of an incoming object takes longer than one coming from the middle of the object. If the two rays are to arrive at the camera simultaneously, the one from the rear would have left the object before the one from the middle. Because what a photograph registers is the (angular) position of all points of the surface of the object at the time of the emission, one expects that due to this phenomenon the object will appear distorted, or more precisely, elongated.

In this project the student will investigate how objects with given shape and dimensions moving at relativistic speeds would appear in a photograph or sequence of photographs. Whereas in comparatively simple cases such as those of a cube or a sphere the calculations may be done exactly, for more complicated shapes a numerical simulation will be required.

The student may proceed to investigate how our perspective of the world would be different if the speed of light were much smaller than its actual value.

3. Modern versions of the twin paradox

The Twin Paradox, originally proposed by Langevin in 1911, is one of the most debated `paradoxes’ in modern science. Since the first formulation, several versions, with different degrees of sophistication, have appeared in the literature. These include, for instance, the Circular Twin Paradox and the Triple Twin Paradox. The emergence of these variants seems to suggest that the argument is not yet closed.

The student will compile, compare and analyse what has been published about the subject, including different formulations and explanations of the apparent paradox. They will then select which versions appear more `paradoxical’ and give complete explanations of what is really happening.

In the process the student will learn and discuss the subtleties of clock synchronisation, measurements of times and distances, the relativity of the simultaneity of two events and other peculiar features of the theory of Relativity.

The interested student will proceed to look at other at-first-sight paradoxes and try to give robust explanations to the apparent contradictions.

4. Relativistic orbits

In this project the student will investigate the motion of relativistic particles under several central potentials, including the harmonic oscillator potential and the Newtonian (1/r) potential and some of its higher order corrections, with different power laws. The study can be extended to include more complex phenomena such as tides and precession, understood in Newtonian physics but more convoluted when relativistic effects are taken into account.

In the process the student will be introduced to the covariant Lagrangian formulation of relativistic dynamics in terms of 4-vectors, how to write an action for a relativistic particle, how to deduce Lagrange’s equations, how to identify conserved quantities and how to deduce first integrals of the equations. The student will learn how to solve analytically the equations for the simplest potentials and will try to apply more advanced mathematical techniques and numerical integration algorithms in the most difficult cases.

5. Relativistic motion of a particle in an electromagnetic field

In this project the student will start by learning the covariant formulation of Maxwell’s equations, the electromagnetic action and how to derive from it the covariant equations of motion for a particle moving in an electromagnetic field. Once in possession of this essential knowledge, the student will attempt to obtain, analytically or numerically, the solutions of the equations of motion of a particle moving in some specific electric and magnetic fields.

We start with the simpler, but certainly not trivial, cases of uniform electric and magnetic fields in arbitrary directions, and proceed to the case of non-uniform fields, for instance the field inside a coaxial cable. The student will be invited to consider a system of his choice to derive and solve the equations of motion with the aim of producing accurate plots for the trajectory of the particle for different initial conditions and values of the relevant parameters.

Have your own research idea in this subject stream that you would like to pursue on the programme?

Please apply via the Open Stream.