Programme And Module Handbook
 
Course Details in 2025/26 Session


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Module Title LM The General Theory of Relativity
SchoolPhysics and Astronomy
Department Physics & Astronomy
Module Code 03 00563
Module Lead Igor Lerner
Level Masters Level
Credits 10
Semester Semester 1
Pre-requisites LH Radiation and Relativity - (03 00971)
Co-requisites
Restrictions 03 00539 (LI Lagrangian and Hamiltonian Mechanics) is strongly advised as a prerequisite
Contact Hours Lecture-22 hours
Seminar-2 hours
Guided independent study-76 hours
Total: 100 hours
Exclusions
Description

The General Theory of Relativity is a major landmark in the development of classical Theoretical Physics. In 1905, Einstein presented his Special Theory of Relativity which we understand to be a theory of the relation between space and time in inertial frames of reference, which are in uniform relative motion with respect to each other with a constant velocity. The requirement that the laws of Physics be the same in all inertial frames leads to the idea that mass is a frame dependent quantity, and hence to a modification of Newton's Laws to take into account the new relations between the space and time coordinates of two inertial frames. In a strict sense it does not deal with accelerations.

 

The General Theory appeared some ten years later in 1915 and represents not only a theory which describes frames of reference in arbitrary motion with respect to each other, but also a theory of gravitation. When two frames of reference are indeed in uniform (unaccelerated) relative motion, it is required that the relations between two such frames of reference become those of the Special Theory. The Special Theory can be thought of as a theory of a flat four dimensional space-time. The General Theory requires that the four dimensional space time be curved, with a curvature determined by any matter which is present: we then experience this curvature as gravity. The analogue in the Newtonian view of physics, is the calculation of potential and fields from mass distributions by using Poisson's equation. In General Relativity objects experiencing gravitational forces are now required to move along optimal paths, called geodesics, in this curved space-time. In this new geometrical way of looking at things, the requirement that motion occurs along an optimal path, takes the place of Newton's second law.

 

In this course we will begin by examining the physical basis of the ideas lying behind General Relativity. These ideas are encapsulated in the Principle of Equivalence. We then need to develop the mathematics of tensor analysis in a curved space-time in order to be able to make accurate statements about the physical and geometrical ideas which lie behind the General Theory and from which predictions can be made. We then use the theory to predict phenomena such as the bending of light near massive objects, the precession of the perihelion of a planet and the gravitational redshift. These latter phenomena provided early experimental tests of the General Theory (tests which have been refined with time) and attracted much public attention at the time. The course then concludes by looking at the cosmological consequences of the General Theory and introduces the idea of Black Holes.

 

The General Theory of Relativity contains very profound physical ideas which are expressed through an elegant geometric structure. Its understanding requires high level skills in handling tensors and differential equations. This beautiful theory is part of the mainstream of physics. In 1932 Dirac married together the Special Theory of Relativity with Quantum Mechanics, but to this day there is no accepted theory of quantum gravity.

Learning Outcomes

By the end of the module the student should be able to:

 

  • use the Principle of Equivalence to make simple estimates of physical phenomena such as the bending of light and the gravitational red shift
  • verify the tensor properties of appropriate quantities, under arbitrary coordinate transformations
  • construct the metric tensor and metric of a Riemann space either from coordinate differentials or from the basis vectors
  • write down a Lagrangian and calculate the equation of geodesic for the metric ds²=g_{μν}dx^{μ}dx^{ν}
  • calculate the Christoffel symbols Γ^{μ}_{αβ} and Γ_{μαβ} either from their definitions in terms of derivatives of the metric tensor or from the canonical forms of the geodesic equations
  • construct a weak field metric by invoking Hamilton's Principle
  • calculate the gravitational red shift and Doppler shift when there is transverse motion, using the weak field metric
  • calculate proper time and proper distances from a Riemann metric in which the space like and time like coordinates are decoupled
  • calculate a covariant derivative and describe the parallel transport of a vector
  • calculate the Riemann tensor R^{μ}_{ναβ} (or R_{μναβ}) and use its symmetry properties
  • calculate the Ricci tensor R^{μ}_{ν} (or R_{μν}) and use its symmetry properties
  • calculate the scalar curvature R=R^{μ}_{μ} and identify coordinate or real singularities
  • calculate the tidal tensor and set up the Einstein field equations in a matter free region of spacetime
  • reproduce the Schwarzschild solution of the field equations in the region exterior to a spherically symmetric mass distribution
  • calculate the coordinate speed of light in the Schwarzschild metric
  • calculate the geodesics associated with the Schwarzschild metric
  • use the idea of an effective potential to obtain qualitative and some quantitative information about the orbits of material particles and photons in the Schwarzschild metric
  • calculate the orbits of planets in the Schwarzschild metric and calculate in particular the precession of the perihelion of Mercury
  • calculate the orbits of photons (light) near a massive gravitating body and calculate the bending of light from a distant star due to the Sun's gravitational field
  • explain the significance of the Schwarzschild radius as an event horizon and calculate some of the simple properties of a non rotating Schwarzschild black hole
  • relate observations of the Universe to the General Theory of Relativity
  • use simple arguments that suggest the Robertson Walker metric
  • calculate solutions to the Friedmann equations in simple geometries (if time permits)
Assessment 00563-01 : Exam : Exam (Centrally Timetabled) - Written Unseen (100%)
Assessment Methods & Exceptions Assessment:
2 hour Examination (100%)
Other None
Reading List