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Relativity & Cosmology - PHY00077H

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  • Department: Physics
  • Module co-ordinator: Dr. John Pasley
  • Credit value: 20 credits
  • Credit level: H
  • Academic year of delivery: 2024-25
    • See module specification for other years: 2023-24

Module summary

This module provides an introduction to the ideas and concepts of Einstein’s special and general theories of relativity and considers the dynamics and evolution of the universe as a whole, with an examination of experimental observables.

The module will begin by describing 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.

By examining the theory of General Relativity 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!

This will be followed by an introduction to properties of space and cosmological models. Finally we will look at the early phases, and the main epochs, in the development of the universe from the Big Bang to the present as well as links with particle physics.

Related modules

Pre-requisites:  Y1 Electromagnetism and Relativity     OR  other module containing the equivalent introductory special relativity content

Module will run

Occurrence Teaching period
A Semester 2 2024-25

Module aims

This module provides an introduction to the ideas and concepts of Einstein’s special and general theories of relativity and considers the dynamics and evolution of the universe as a whole, with an examination of experimental observables.

The module will begin by describing 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.

By examining the theory of General Relativity 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!

This will be followed by an introduction to properties of space and cosmological models. Finally we will look at the early phases, and the main epochs, in the development of the universe from the Big Bang to the present as well as links with particle physics.

Module learning outcomes

Relativity

  • Utilise space-time diagrams and four-vector based descriptions of special relativity

  • Understand the results demonstrated by the four classical experimental tests of general relativity

  • Demonstrate a semi-quantitative understanding of black-hole physics including black hole thermodynamics

Cosmology

  • Quantitatively describe the evolution of this and other possible universes using

cosmological models

  • Understand the inflationary Big Bang theory and the evidence for it

Shared

Qualitatively understand gravitational wave astronomy and be able to describe some ways in which it can be employed to enhance our understanding of relativity and cosmology

Module content

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 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 coordinates, centred 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.

Cosmology

  • Galaxy distribution and large-scale structure
  • Observations and relevance to cosmology
  • Cosmological models using Newtonian gravity and General Relativity
  • Curvature and the geometry of the Universe
  • The Cosmological Constant
  • The Big Bang Model and problems
  • Primordial Nucleosynthesis and Inflation
  • The Cosmic Microwave Background and Precision cosmology
  • Gravitational Wave Cosmology

Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Relativity & Cosmology		 3 hours 80
Essay/coursework
Essay : Assignment		 N/A 20

Special assessment rules

None

Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Relativity & Cosmology		 3 hours 80

Module feedback

'Feedback’ at a university level can be understood as any part of the learning process which is designed to guide your progress through your degree programme. We aim to help you reflect on your own learning and help you feel more clear about your progress through clarifying what is expected of you in both formative and summative assessments.

A comprehensive guide to feedback and to forms of feedback is available in the Guide to Assessment Standards, Marking and Feedback. This can be found at:

https://www.york.ac.uk/students/studying/assessment-and-examination/guide-to-assessment/

The School of Physics, Engineering & Technology aims to provide some form of feedback on all formative and summative assessments that are carried out during the degree programme. In general, feedback on any written work/assignments undertaken will be sufficient so as to indicate the nature of the changes needed in order to improve the work. Students are provided with their examination results within 25 working days of the end of any given examination period. The School will also endeavour to return all coursework feedback within 25 working days of the submission deadline. The School would normally expect to adhere to the times given, however, it is possible that exceptional circumstances may delay feedback. The School will endeavour to keep such delays to a minimum. Please note that any marks released are subject to ratification by the Board of Examiners and Senate. Meetings at the start/end of each semester provide you with an opportunity to discuss and reflect with your supervisor on your overall performance to date.

Our policy on how you receive feedback for formative and summative purposes is contained in our Physics at York Taught Student Handbook.

Indicative reading

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


The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.