Topic outline

  • Announcements

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    • Module Announcements Forum
    • Student Forum

      Forum Description: This forum is available for everyone to post messages to. Students can raise questions or discuss issues related to the module. Students are encouraged to post to this forum and it will be checked daily by the module leaders. Students should feel free to reply to other students if they are able to.

  • Module structure

    Each week there will be 4 in-person lectures  in total, and each session is one hour. Sessions 1, 2 and 3 in each week are ordinary lectures where new material will be covered; Session 4 is a 1-hour tutorial.   In the tutorial session, students are encouraged to ask questions related to the material covered in the previous lectures or in the  coursework. The module organisers will discuss selected problems from the current week's coursework sheet, past exam papers and the lectures notes presenting a step by step solution. If possible, please submit questions ahead of the tutorial session via the module's forum.

    There will be 10 courseworks in total. These contain important practice questions. Each coursework is roughly associated to the material of the week. It is crucial that you engage with this coursework and that you attempt all problems. If you have any questions regarding a particular point in the coursework or you would like to have a particular problem discussed in Session 4, please let us know ahead of time, preferably via the module's forum. This coursework will not be submitted. However, we will be happy to provide feedback on the coursework. For this, please get in touch via email with the module organisers.

    In addition, there will be 2 online quizzes (on week 4 and week 11). These covers various aspects of the material discussed up to that point in the course. The quizzes are individual work and sufficiently time will be provided to reflect on the questions and to provide answers. The quizzes will contribute to 4% each to the final mark of the module. Feedback on the answers to the quiz will be provided once the window for solving it has closed. The purpose of the quizzes is to help you to engage with the module and to keep up with its content. Finally there will be an in-class test (on week 8) which will contribute to 12% to the final mark of the module.



    • Lecture notes File

      Full lecture notes for the module are given here. 

      Information about what is taught in each week can be seen in the sections for the individual weeks below.

      These notes are intended to be a definitive record of what is examinable.


      2.3 MB

  • Module description

    Brief description about this module:

    This module provides an introduction to Einstein's theories of special and general relativity. These theories are two cornerstones of modern theoretical and mathematical physics. The theory of special relativity revolutionised our understanding of space and time, intertwining them into a single entity known as space-time. In this module we will introduce this theory and some of its most salient physical consequences such as time dilation, length contraction and the famous formula E=mc2. Einstein's theory of general relativity revolutionised our understanding of gravity by describing it as the curvature of space-time, and therefore as pure geometry. To mathematically describe curved spaces, in this module we will introduce the differential geometry. As this course is intended to be an introductory one, we will only cover the basic aspects of  differential geometry that are needed for general relativity. Einstein's theory of gravity is covered in the last part of this module.  We will introduce the basics of the theory and we will focus on some of its most outrageous predictions,  such as black holes and the gravitational waves emitted from binary systems that have been directly detected for the first time in 2016 by LIGO.

    Prerequisites: the official prerequisites are MTH4101/MTH4201: Calculus II, MTH4102/MTH5123: Differential Equations and MTH5112: Linear Algebra I or MTH5212: Applied Linear Algebra. MTH5113: Introduction to Differential Geometry would be helpful but not essential. To do well in this module you will have to be fluent in differentiating and integrating simple functions. We will use basic concepts of calculus such as partial derivatives and gradient of a function


  • Syllabus

    Content:

    • Pre-relativistic physics: Galilean Relativity, Newton's Laws, Galilean transformations.
    • Special Relativity: Postulates of Special Relativity, Spacetime diagrams, Lorentz transformations, physical effects, hyperbolic form of Lorentz transformations, Minkowski spacetime, 4-vectors, photons, relativistic dynamics.
    • Differential Geometry: manifolds and coordinates, covariant and contravariant vectors and tensors, tensor algebra, manifolds with metric, derivatives and connections, parallel transport, Levi-Civita connection, metric geodesics, Riemann tensor, geodesic deviation, Bianchi identities, Ricci and Einstein tensors.
    • General Relativity: Einstein field equations, the Schwarzschild solution, Gravitational waves from binary systems.


    References:


  • Learning outcomes

    By the end of this module you will be able to:


    • Explain the principles of special relativity and the key steps leading to the Lorentz transformations.
    • Employ a geometrical approach to special relativistic effects by using Minkowski geometry and spacetime diagrams.
    • Use four-vectors in a variety of different settings relevant to relativistic dynamics and collisions.
    • Use the techniques of tensor algebra and tensor calculus in curved (Riemannian) spaces.
    • Define the notions of covariant derivatives, connections, parallel transport, geodesics and curvature in curved spaces.
    • Explain the importance of the metric tensor and the significance and applications of the geodesic equation.
    • Explain the significance of the terms in the Einstein field equations and understand the Newtonian (weak field) limit of the theory.
    • Explain the applications of the General Theory of Relativity, black holes, gravitational waves and cosmological spacetimes.

  • Week 1 - Introduction

    • Checklist for Week 1:

      • Acquaint yourself with this QMPlus page
      • Read Week 1 typeset lecture notes
      • Attend the  lectures (Monday and Thursday)
      • By the end of the week, start solving the problems in Coursework 1

  • Week 2 : Special Relativity

    • Checklist for Week 2:

      • Read Week 2 typeset lecture notes
      • Attend the  lectures (Monday and Thursday)
      • Attend the tutorial session on Thursday. Post requests of problems beforehand
      • By the end of the week, start solving the problems in Coursework 2

  • Week 3: Special Relativity and tensors

    Checklist for Week 3:

    • Read Week 3 typeset lecture notes
    • Attend the  lectures (Monday and Thursday)
    • Attend the tutorial session on Thursday. Post requests of problems beforehand
    • By the end of the week, start solving the problems in Coursework 3

  • Module Description

    What is the maths behind the black holes, ravitational waves, and the big bang? To answer these questions we need Einstein's theories of special and general relativity; two cornerstones of modern theoretical and mathematical physics. In the first part of the module we will focus on Einstein’s theory of special relativity which is about the “strange” dynamics that take place when we consider speeds comparable to the speed of light. Next we develop mathematical tools to describe the famous curvature of space-time and the subtle effects of gravity which characterise Einstein’s general theory of relativity. To enjoy this module you will need to be comfortable with introductory Calculus modules. The second year Introduction to Differential Geometry module would also be a helpful prerequisite.

  • Week 4 - Special Relativity and Differential geometry

    Checklist for Week 4:

    • Read Week 4 typeset lecture notes
    • Attend the  lectures (Monday and Thursday)
    • Attend the tutorial session on Thursday. Post requests of problems beforehand
    • By the end of the week, start solving the problems in Coursework 4

  • Week 5: Differential Geometry

    • Read Week 5 typeset lecture notes 
    • Attend the lectures (Monday and Tuesday)
    • Post any questions in the module's forum
    • Attend the tutorial on Tuesday. Post requests of problems beforehand
    • By the end of the week, start solving the problems in  Coursework 5

  • Week 6: Differential Geometry

    Checklist for Week 6:

    • Attend the lectures on Monday and Thursday
    • Attend the tutorial session on Thursday. Post requests of problems beforehand.
    • Start solving the problems in  Coursework 6 .

  • Week 7: Study Week

  • Week 8: Differential Geometry and General Relativity

    Checklist for Week 8:

    • Attend the lectures on Monday and Thursday
    • Attend the tutorial session on Thursday. Post requests of problems beforehand.
    • In class assessed coursework in the second hour of the Thursday lectures (i.e., 11:00am)

  • Week 9: General Relativity

    Checklist for Week 9:

    • Attend the lectures on Monday and Thursday
    • Attend the tutorial session on Thursday. Post requests of problems beforehand.

  • Week 10: Black holes

    • Attend the lectures on Monday and Thursday
    • Attend the tutorial session on Thursday. Post requests of problems beforehand.

  • Week 11: Gravitational waves

    • Attend the lectures on Tuesday and Thursday
    • Attend the tutorial session on Thursday. Post requests of problems beforehand.

  • Syllabus

    • Content:

      • Pre-relativistic physics: Galilean Relativity, Newton's Laws, Galilean transformations.
      • Special Relativity: Postulates of Special Relativity, Spacetime diagrams, Lorentz transformations, physical effects, hyperbolic form of Lorentz transformations, Minkowski spacetime, 4-vectors, photons, relativistic dynamics.
      • Differential Geometry: manifolds and coordinates, covariant and contravariant vectors and tensors, tensor algebra, manifolds with metric, derivatives and connections, parallel transport, Levi-Civita connection, metric geodesics, Riemann tensor, geodesic deviation, Bianchi identities, Ricci and Einstein tensors.
      • General Relativity: Einstein field equations, the Schwarzschild solution, experimental tests of general relativity, black holes.
      • Linearised theory and gravitational waves from binary systems.

      References:

      • Ray D'Inverno, "Introducing Einstein's Relativity", Claredon Press (1992), Oxford
      • James B Hartle, "Gravity: An Introduction to Einstein's General Relativity", Cambridge University Press (2021).
      • Sean M Carroll, "Spacetime and Geometry", Cambridge University Press (2019).
      • Bernard Schutz, "A First Course in General Relativity", 3rd Edition, Cambridge University Press (2022).
      • Robert M Wald, "General Relativity", the University of Chicago Press (1984)

  • Assessment

    • Add information here.

  • Coursework

      • 2 assessed online quizzes on Weeks 4 and 11.
      • An assessed in-class test on Week 8.
      • 10 non assessed coursework sheets

  • Exam papers

  • Week 12: Gravitational waves and revision

  • Q-Review

  • Online Reading List

  • Assessment information

    • Assessment Pattern -  The assessment pattern of this module is 20% coursework + 80% final exam

      Format and dates for the in-term assessments - 2 online quizzes on Weeks 4 and 11 (worth 4% each). An in-class test on Week 8 (worth 12%).

      Format of final assessment - Your final examination will be on campus.  It will be 3 hours in duration with SpLD accommodations handled separately.  You will be allowed a non-programmable calculator and 3 sheets of handwritten A4 notes to bring to the exam.

      Link to past papers - https://qmplus.qmul.ac.uk/mod/data/view.php?id=2443216 

      Description of Feedback - Discussion of the main difficulties encountered in the quizzes.