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Part III (MMath/MASt)

Please take this page in conjunction with the Part III Guide to Courses Relativity and Cosmology section.

Please read in particular Introduction and prerequisites given in the Guide to Courses.

For the General Relativity lecture course, successful Part III students have told us that they found it extremely useful to work through the Differential Geometry part of the course (for example the chapter on manifolds and tensors in these notes by Prof. Reall) during the summer. If you draw pictures to get an intuition for these more abstract concepts and definitions, you will find it much easier to understand once you get to the lectures. The book Spacetime and Geometry: An Introduction to General Relativity by Sean Carroll (illustrated, revised edition, Pearson Education, Limited, 2013) has similar information in its Manifolds chapter, though slightly less advanced.

## Special Relativity

You will need this for the following Part III courses:

• all Relativity and Gravitation courses but primarily General Relativity and Black Holes
• High Energy Physics courses.

Relevant undergraduate courses are (for relevant schedules, example sheets and exam questions, refer to the General Resources):

• Part IA Dynamics and Relativity

### Reality check

You should be confident in the area of Special Relativity. Of course you can remind yourself and get back into it with the material provided here. See e.g. Part IA Dynamics and Relativity. You might also want to look at the corresponding entry on the Particle Physics, Quantum Fields and Strings page. You can check if you are at the required level by doing the following exercises: Special Relativity exercise sheet (provided by Markus Kunesch, thank you!).

## Euler-Lagrange Equations

You will need this for the following Part III courses:

• all Relativity and Gravitation courses but primarily General Relativity and Black Holes

Relevant undergraduate courses are (for relevant schedules, example sheets and exam questions, refer to the General Resources):

• (Part IA Differential Equations)
• Part IB Variational Principles
• Part IB Methods
• Part II Classical Dynamics

### Reality check

• Index notation and summation convention. See e.g. Part IA Vectors and Matrices and Part IA Vector Calculus (VC). You can check if you are at the required level by doing the following exercises: Index Gymnastics exercise sheet (provided by Markus Kunesch, thank you!).
• Fluency in calculating line integrals and surface/volume integrals (VC); you can check if you are at the required level by doing the following exercises: VC Example Sheet 1 Questions 3,5,8,9,12. Past Exam question: Paper 3 from Part IA 2005, Question 12. Note that this is a good exercise in getting the Jacobian and all limits done in the right order: if you get it slightly differently it might get messy. So if your solution is messy, rethink!

### Prerequisites

Fluency with methods pertaining to variational problems and classical dynamics, in particular:

• Euler-Lagrange equations and geodesics

You can use these exercises to check your present level and to guide your study.

### Useful books and resources

• Part IB Variational Principles lecture notes by Prof Townsend from 2016.
• Part IB Variational Principles Example Sheets 1, 2 and 3.
• Part IB Methods lecture notes in 4 parts: Part 1, Part 2, Part 3 , Part 4, all by Prof Josza from 2013.
• Part IB Methods Example Sheets 1, 2, 3 and 4.
• Part II Classical Dynamics Example Sheets 1, 2, 3 and 4.

## General Relativity

This is not strictly needed, but very helpful for the following Part III courses:

• all Relativity and Cosmology courses but primarily General Relativity and Black Holes

As mentioned above: for the General Relativity lecture course, successful Part III students have told us that they found it extremely useful to work through the Differential Geometry part of the course (for example the chapter on manifolds and tensors in these notes by Prof. Reall) during the summer. If you draw pictures to get an intuition for these more abstract concepts and definitions, you will find it much easier to understand once you get to the lectures. The book Spacetime and Geometry: An Introduction to General Relativity by Sean Carroll (illustrated, revised edition, Pearson Education, Limited, 2013) has similar information in its Manifolds chapter, though slightly less advanced.

Relevant undergraduate courses are (for relevant schedules, example sheets and exam questions, refer to the General Resources):

• Part II General Relativity

Helpful might be in particular topics such as:

• Equivalence principles
• Structure of GR (qualitative)
• Euler-Lagrange equations in a GR context, and geodesics
• Manifolds

The exercises on Euler-Lagrange above include some with a GR context for you to try.

### Useful books and resources

• Part II General Relativity lecture notes by Prof Gibbons from 2004.
• Part II General Relativity Example Sheets 1, 2, 3 and 4.
• Part III General Relativity lecture notes by Prof. Reall from 2012.
• Spacetime and Geometry: An Introduction to General Relativity by Sean Carroll (illustrated, revised edition, Pearson Education, Limited, 2013)