## Pre-requisite modules

## Co-requisite modules

- None

## Prohibited combinations

- None

Students taking this module should have taken the prerequisite modules listed above (Quantum Physics II - PHY00032I or Mathematics II - PHY00030I) or the appropriate equivalent modules.

Occurrence | Teaching cycle |
---|---|

A | Autumn Term 2022-23 |

This module builds on the lectures given in the second term of the first year. The object of special relativity is the description of the physical world and its laws in such a way that they take the same 'form' in all inertial frames. This is known as Lorentz or 'form' invariance. At an early stage in the module the Minkowski Rotation Matrix is introduced as a more useful form of the Lorentz-Einstein transformations, and is subsequently used in all further developments of the subject of Special Relativity. We shall also look at the concept of world-lines and Minkowski space time diagrams- tools which we will continue to employ as we go on to consider the topic of General Relativity.

In the second part of the course we shall go on to consider the General theory of Relativity, in which gravity is incorporated as the curvature of space-time. We will show that freely falling frames are the equivalent of inertial frames in a gravitational field. We will consider the Schwarzschild description of a spherical gravitational system, and look at the so called “Radar time delay” in which the time taken for radiation to propagate is increased in the presence of a massive body. We shall consider the bending of light in a gravitational field, and the Schwarzschild picture of a black-hole, alongside some experimental evidence for General Relativity, such as the precession of the perihelion of Mercury. Finally we shall briefly look at the concept of Hawking radiation—and the suggestion that black holes are not really black at all!

You will not be required to derive the Lorentz/ Einstein transformations, however it is hoped that you will commit their form to memory. As the Minkowski rotation matrix is just another form of the Lorentz transformations it is also advisable to commit this to memory. You will not be required to implement tensor calculus, and the second half of the course will focus on illustrating some key results in General Relativity, using simplified mathematical descriptions, rather than upon rigorous mathematical derivation.

At the end of this module successful students will be able to:

__Advanced Special Relativity__

- Apply the Lorentz transformations to the solution of problems which relate the appearance of events in one inertial frame to their appearance in another.
- Be familiar with the use of the various four vectors (which may be examined).
- Develop a quantitative understanding of the subject and to be able to solve numerical examples in all areas of the module. Problems involving the use of the relation of relativistic energy and momentum may be set under examination conditions. As the derivation of the various relationships between physical variables is normally straightforward in special relativity these can also form a part of an examination question.

__General Relativity__

- Show a clear qualitative understanding of the principle of equivalence, with some simple mathematical under-pinning
- Demonstrate the “radar time delay” effect by considering the effect of setting d =0 in the Schwarzschild metric
- Derive an expression for the curved path of a photon in the presence of a gravitational field, by making appropriate simplifications and apply this to real physical scenarios (e.g. photon passing sun)
- Give a simplified quantitative under-pinning for effects such as the precession of the perihelion of Mercury, and the pericentre shift of a binary pulsar, and otherwise discuss experimental evidence that supports Einstein’s general theory.
- Show a thorough qualitative understanding of the behaviour of light in the presence of a black hole, in terms of the tilting of light cones
- Understand the qualitative basis for Hawking radiation

**Syllabus**

__Advanced Special Relativity__

•Space/Time: Inertial reference frames, the synchronisation of clocks and Einstein's derivation of the Lorentz Transformations.

•Interval or Extension: The four dimensional nature of space-time. The absolute nature of the time ordering of events.

•The Minkowski rotation matrix and the imaginary nature of the fourth dimension.

•Proper Time and Four Vectors: The invariant time interval between events, the elemental four vector and its differentiation with respect to the proper time to give the four velocity. The four velocity transformation into a second inertial frame using the Minkowski rotation matrix to give the relativistic velocity transformation.

•Four Vectors: The product of the four-velocity with mass to give four-momentum, and the transformation of four momentum. The four force, the transformation of the three-force between inertial frames.

•Four Momentum: The application of the conservation of the four momentum to the scattering of photons by electrons (the Compton effect) and the scattering of charged particles by photons (the inverse Compton effect)

•Mass-Energy-Momentum: The application of the equation relating the total energy of a particle and its energy and momentum in particle collisions.

•Space-Time Diagrams: The application of Minkowski diagrams to the causal connection of events, the light cone, future, past and present; length contraction and time dilation.

__General Relativity__

•The principle of equivalence: The principle of equivalence in General relativity, including a quantitative illustration of the principle of equivalence. The weak and strong statements of the principle of equivalence.

•The Schwarzschild Metric: A description of the Schwarzschild metric, using spherical co-ordinates, centered upon a gravitating body. Spherical and static solutions of Einstein’s equations of General Relativity.

•The radar-time delay: In a Schwarzschild geometry, the behaviour of a radially propagating photon is considered. The radar time-delay in gravitational fields is arrived at.

•Geodesic equations.

•The curvature of light: Considering a photon propagating obliquely in a Schwarzschild-like universe, the deflection from a straight-line path is calculated. The curvature of light in the presence of a massive body is thereby demonstrated. Evidence of this effect is shown in astronomical observations

•Observational effects: The precession of the perihelion of Mercury, and the pericentre shift of a binary pulsar.

•Black holes: The behaviour of space-time in regions of high curvature (Black Holes), including an overview of the Schwarzschild black hole and the use of Kruskal Szekeres coordinates. Tilting of light cones in the presence of black-holes, and the effects of tidal gravity upon material bodies falling through the event horizon.

•Hawking radiation and black hole thermodynamics: consideration of quantum electrodynamical effects in the presence of black-holes gives rise to Hawking radiation, and the “White hole”. Following from this we shall also show how a temperature can be ascribed to a black-hole by treating it as a black-body radiator.

Task | Length | % of module mark |
---|---|---|

Essay/courseworkRelativity Assignment 1 |
N/A | 40 |

Essay/courseworkRelativity Assignment 2 |
N/A | 60 |

None

Task | Length | % of module mark |
---|---|---|

Essay/courseworkRelativity Assignment 1 |
N/A | 40 |

Essay/courseworkRelativity Assignment 2 |
N/A | 60 |

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Schutz B, A first course in General Relativity (Cambridge)