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Study information

# Relativity and Cosmology - 2023 entry

MODULE TITLE CREDIT VALUE Relativity and Cosmology 15 PHYM006 Prof Tim Harries (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11
 Number of Students Taking Module (anticipated) 32
DESCRIPTION - summary of the module content

This module is an introduction a cornerstone of 20th century physics, the general theory of relativity, Einstein's geometric theory of gravity. The module begins with a recap of special relativity. Subsequently, the mathematical tools (tensor analysis and differential geometry) that underpin general relativity are presented, and students will require a good level level of mathematical fluency and intuition in order to engage with material. Topics include Einstein's field equation, Schwarzschild's solution and black holes, gravitational waves, and the Robertson-Walker metric and cosmology.

AIMS - intentions of the module
The module aims to develop an understanding of Einstein's theory of general relativity (GR). The module starts with a recap of special relativity and then introduces the principles of equivalence, covariance and consistency that lead Einstein to the general theory. The mathematics of tensors and differential geometry are presented in the context of Einstein's field equation. This is followed by a detailed derivation of Schwarzchild's solution and its implication for time and space around a black hole. The module concludes by examining the use of GR in cosmology.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)
A student who has passed this module should be able to:

Module Specific Skills and Knowledge:
1. give coherent explanations of the principles associated with: special relativity, general relativity, and cosmology;
2. interpret observational data in terms of the standard model of the evolution of the Universe;
3. describe experiments and observational evidence to test the general theory of relativity, explain how these support the general theory and can be used to criticise and rule-out alternative possibilities;
4. apply tensors to the description of curved spaces;
5. solve problems by applying the principles of relativity;
6. deduce the Friedmann equations describing the evolution of the Universe.
7. explain what is meant by: intrinsic and extrinsic curvatures, the curvature of space, local inertial reference frame, and Riemannian coordinates/geometry;
8. describe world lines of particles and photons in a curved space-time;
9. describe the cosmological principle and the Robertson-Walker metric;

Discipline Specific Skills and Knowledge:
10. explain to non-specialists the basis of one of the corner-stones of 20th century physics;

Personal and Key Transferable / Employment Skills and Knowledge:
11. locate, retrieve and evaluate relevant information from the WWW;
12. meet deadlines for completion of work to be discussed in class by developing appropriate time-management strategies.
SYLLABUS PLAN - summary of the structure and academic content of the module
I. Introduction
II. Recap of key aspects of special relativity
1. Galilean and Lorentz transformations
2. Length contraction and time dilation
3. Doppler effect
4. Relativistic mechanics
III. Tensor analysis
1. Covariant and contravariant tensors
2. Reciprocal basis vectors
3. Tensor algebra
4. The metric tensor
5. Christoffel symbols and covariant differentiation
6. The geodesic equation
IV. Curved spaces
1. Intrinsic and extrinsic curvature
2. Parallel transport
3. Riemannian curvature
4. Ricci tensor and scalar
V. Einstein's field equation
1. The stress-energy tensor
2. Einstein's field equation
3. The weak field limit
4. Schwarzschild's solution
5. Black holes and singularities
VI. Black holes
1. Geodesic equations, orbital shape equation
2. Falling into a black hole
3. Eddington-Finkelstein coordinates
4. Rotating black holes and the Kerr metric
5. Frame dragging and ergosphere
VII. Gravitational waves
1. Linearised gravity
2. Wave equation
3. Weak gravitational waves
4. The motion of a test particle
5. Detecting gravitational waves
VIII. Cosmology
1. The cosmological principle
2. Robertson-Walker metric
3. Red-shift distance relation
4. The Friedmann equations
5. Inflation
IX. Additional Topics
1. Eotvos experiments
2. Observational tests of GR
3. A recap of special relativity
4. An introduction to tensor mathematics
5. Derivation of the Friedmann equations from the Robertson-Walker metric
LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study 22 128
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning & teaching activities 20 hours 20×1-hour lectures Scheduled learning & teaching activities 2 hours 2×1-hour problems/revision classes Guided independent study 30 hours 5×6-hour self-study packages Guided independent study 16 hours 4×4-hour problem sets Guided independent study 82 hours Reading, private study and revision

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Guided self-study (0%) 5×6-hour packages (fortnightly) 1-12 Discussion in class
4 × Problems sets (0%) 4 hours per set (fortnightly) 1-12
Solutions discussed in problems classes.

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 0 100
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Final Examination 100% 2 hours 30 minutes (May/June) 1-10 Mark via MyExeter, collective feedback via ELE and solutions.

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
Whole module Written examination (100%) 1-12 August/September assessment period

Re-assessment is not available except when required by referral or deferral.

RE-ASSESSMENT NOTES
An original assessment that is based on both examination and coursework, tests, etc., is considered as a single element for the purpose of referral; i.e., the referred mark is based on the referred examination only, discounting all previous marks. In the event that the mark for a referred assessment is lower than that of the original assessment, the original higher mark will be retained.

Physics Modules with PHY Codes
Referred examinations will only be available in PHY3064, PHYM004 and those other modules for which the original assessment includes an examination component - this information is given in individual module descriptors.
RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener
ELE:

Reading list for this module:

Type Author Title Edition Publisher Year ISBN
Set Lambourne, R. Relativity, Gravitation and Cosmology Cambridge UP 2010 9-7805-2113-1384
Extended Coles, P. and F. Lucchin Cosmology - the Origin and Evolution of Cosmic Structure 2nd edition Wiley 2002 978-0-471-48909-2
Extended Hartle, J.B. Gravity: An Introduction to Einstein's General Relativity Addison-Wesley 2003 9-7808-0538-6622
Extended Kenyon, I. General Relativity Oxford University Press 1990
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES PHY1021, PHY1022, PHY2025
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 7 No Thursday 15th December 2011 Thursday 26th January 2023
KEY WORDS SEARCH Physics; Theory; Spaces; Curvature; Time; Curves; General theory; Shifts; Cosmological; Equation; Inertial frame.

Please note that all modules are subject to change, please get in touch if you have any questions about this module.