Module Specification
The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
1. Module Title Relativity
2. Module Code MATH326
3. Year Session 2023-24
4. Originating Department Mathematical Sciences
5. Faculty Fac of Science & Engineering
6. Semester First Semester
7. CATS Level Level 6 FHEQ
8. CATS Value 15
9. Member of staff with responsibility for the module
Dr SL Parameswaran Mathematical Sciences Susha.Parameswaran@liverpool.ac.uk
10. Module Moderator
11. Other Contributing Departments  
12. Other Staff Teaching on this Module
Dr TM Mohaupt Mathematical Sciences Thomas.Mohaupt@liverpool.ac.uk
13. Board of Studies
14. Mode of Delivery
15. Location Main Liverpool City Campus
    Lectures Seminars Tutorials Lab Practicals Fieldwork Placement Other TOTAL
16. Study Hours 36

   12

       48
17.

Private Study

 102
18.

TOTAL HOURS

 150
 
    Lectures Seminars Tutorials Lab Practicals Fieldwork Placement Other
19. Timetable (if known)            
 
20. Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH122 NEWTONIAN MECHANICS; MATH228 CLASSICAL MECHANICS; MATH102 CALCULUS II; MATH101 Calculus I; MATH103 Introduction to Linear Algebra
21. Modules for which this module is a pre-requisite:

 
22. Co-requisite modules:

 
23. Linked Modules:

 
24. Programme(s) (including Year of Study) to which this module is available on a mandatory basis:
25. Programme(s) (including Year of Study) to which this module is available on a required basis:
26. Programme(s) (including Year of Study) to which this module is available on an optional basis:
27. Aims
 

(i) Develop a firm grasp of the physical principles behind Special and General Relativity and their main consequences;

(ii) Develop technical competence in the mathematical framework of the subjects - Lorentz transformation, coordinate transformations and geodesics in Riemann space;

(iii) Develop understanding of some of the classical tests of General Relativity - perihelion shift, gravitational deflection of light;

(iv) Develop understanding of basic concepts of black holes.
 
28. Learning Outcomes
 

(LO1) Understand why space-time forms a non-Euclidean four-dimensional manifold.
 

(LO2) Develop technical competency in calculations involving Lorentz transformations, energy-momentum conservation, and the Christoffel symbols.
 

(LO3) Understand the arguments leading to the Einstein's field equations and how Newton's law of gravity arises as a limiting case.
 

(LO4) Develop technical competency in calculations of the trajectories of bodies in a Schwarzschild space-time.
 

(S1) Problem solving skills
 

(S2) Numeracy
 
29. Teaching and Learning Strategies
 

Material is presented during lectures (3 hours per week). Tutorials (1 hour per week) are used for consolidation and practice, and for help with individual questions.
 
30. Syllabus
   

Newtonian mechanics and its limitations.

Principles of special relativity. Lorentz transformation: derivation, properties.

Relativistic kinematics: length contraction, time dilation, velocity addition. Minkowski space formulation.

Relativistic particle mechanics: energy-mass relation, four-momentum conservation, scattering.

Riemann space, Properties of tensors. Parallel displacement, geodesics, covariant derivatives. Curvature tensor and scalar, Ricci tensor.

Equivalence principle, gravitational time dilation, non-Euclidean space-time. Freely falling bodies, weak field limit. Field equations, cosmological constant.

Schwarzschild solution and its geodesics. Classical tests of General Relativity. Black holes.
 
31. Recommended Texts
  Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
 

Assessment
32. EXAM Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
  final assessment 120 70
33. CONTINUOUS Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
  Homework 1 0 15
  Homework 2 0 15