PHYM006 Relativity and Cosmology 2023-24
Prof. T.J. Harries

Delivery Weeks: T2:01-11
Level: 7 (NQF)
Credits: 15 NICATS / 7.5 ECTS
Enrolment: 32 students (approx)

### Description

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.

### Module Aims

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)

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:
1. explain to non-specialists the basis of one of the corner-stones of 20th century physics;
• Personal and Key Transferable / Employment Skills and Knowledge:
1. locate, retrieve and evaluate relevant information from the WWW;
2. meet deadlines for completion of work to be discussed in class by developing appropriate time-management strategies.

### Syllabus Plan

1. Introduction
2. Recap of key aspects of special relativity
1. Galilean and Lorentz transformations
2. Length contraction and time dilation
3. Doppler effect
4. Relativistic mechanics
3. 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
4. Curved spaces
1. Intrinsic and extrinsic curvature
2. Parallel transport
3. Riemannian curvature
4. Ricci tensor and scalar
5. 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
6. 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
7. Gravitational waves
1. Linearised gravity
2. Wave equation
3. Weak gravitational waves
4. The motion of a test particle
5. Detecting gravitational waves
8. Cosmology
1. The cosmological principle
2. Robertson-Walker metric
3. Red-shift distance relation
4. The Friedmann equations
5. Inflation
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

Description Study time KIS type
20×1-hour lectures 20 hours SLT
2×1-hour problems/revision classes 2 hours SLT
5×6-hour self-study packages 30 hours GIS
4×4-hour problem sets 16 hours GIS
Reading, private study and revision 82 hours GIS

#### Assessment

Weight Form Size When ILOS assessed Feedback
0% Guided self-study 5×6-hour packages Fortnightly 1-12 Discussion in class
0% 4 × Problems sets 4 hours per set Fortnightly 1-12 Solutions discussed in problems classes.
100% Final Examination 2 hours 30 minutes May/June 1-12 Mark via MyExeter, collective feedback via ELE and solutions.

### Resources

The following list is offered as an indication of the type & level of information that students are expected to consult. Further guidance will be provided by the Module Instructor(s).

Core text:

Supplementary texts:

ELE:

### Further Information

#### Prior Knowledge Requirements

Pre-requisite Modules Vector Mechanics (PHY1021), Introduction to Astrophysics (PHY1022) and Mathematics with Physical Applications (PHY2025) none

#### Re-assessment

Re-assessment is not available except when 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

Notes: See Physics Assessment Conventions.

#### KIS Data Summary

Learning activities and teaching methods
SLT - scheduled learning & teaching activities 22 hrs
GIS - guided independent study 128 hrs
PLS - placement/study abroad 0 hrs
Total 150 hrs
Summative assessment
Coursework 0%
Written exams 100%
Practical exams 0%
Total 100%

#### Miscellaneous

 IoP Accreditation Checklist N/A this is an optional module Availability MPhys only Distance learning NO Keywords Physics; Theory; Spaces; Curvature; Time; Curves; General theory; Shifts; Cosmological; Equation; Inertial frame. Created 01-Oct-10 Revised 27-Jun-19