Welcome to the Space-Time-Gravity (PHY-308) Home Page

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There are three lectures a week, sometimes one will take the form of an example class. Lectures will start at five minutes past the hour and end at five minutes to the next. Punctuality is important because latecomers miss material and disturb others. Attendance at lectures is compulsory; an attendance sheet is circulated and an attendance record kept. Make sure that you sign this sheet: you will be deemed not to have attended if you do not. It is a disciplinary offence to sign on behalf of anyone else or to get others to sign for you.

Contact: if you need extra help.

Attendance at lectures is monitored as poor attendance is strongly correlated with poor performance in a course. There is a computerised system to register absences and excuses (IDCheck).


The aim of the course is to give an introduction to the theory of general relativity, and to instil an understanding of some of its consequences.


By the end of the course, the successful student should be able to:

  • Explain the concepts of special and general relativity;
  • Draw space-time diagrams representing various phenomena in flat and curved space-times;
  • Transform between various co-ordinate systems;
  • Calculate the effects of various space-time geometries on the measurement of time and distance;
  • Calculate Christofell symbols and Curvatures and use Einstein's equations.



  1. Fundamentals of Special Relativity. Events, space-time diagrams, inertial observers, Relativity Principle, measuring space-time, simultaneity, Lorentz transformations, time-dilation, length contraction, transformation of velocity, Doppler effect.
  2. The Geometry of Spacetime. Invarial interval, proper time, timelike, null, and spacelike intervals, the light-cone and causality.
  3. Relativistic Mechanics. Four-vectors, invariant scalar product, timelike, null and space-like 4-vectors, 4-velocity, 4-momentum, 4-momentum conservation, centre of momentum frame, threshold energies.
  4. Specifying Geometry. Two dimensional dimensional surfaces, R2, S2, coordinates, line element, geodesics, Euler-Lagrange equations.
  5. General Relativity. Inertial and gravitational mass, the Equivalence Principle, tidal forces, local inertial frames, consequences of the EP: bending of light, gravitational red-shift. Curved spacetime: coordinates, line element, metric, Einstein's equation, timelike, null and spacetime curves, light cones, geodesics.
  6. The Schwartschild Solution. The Schwarzschild line element, asymptotic flatness, gravitational redshift, particle orbits: effective potential, qualitative behaviour. Solar system tests of General Relativity: perihelion precession of Mercury, light bending by the Sun. Gravitational collapse, Schwarzschild black hole:  light cones, event horizon, curvature singularity.
  7. Cosmology. Matter and radiation, cosmic microwave background radiation, dark matter. Expansion of the Universe: Hubble's Law, the Big Bang. Spatial homogeneity, and isotropy, the Cosmological Principle. Flat, closed, and open Friedman-Robertson-Walker spacetimes. Cosmological redshift, derivation of Hubble's Law. First Law of Thermodynamics. Equations of state: pressureless dust, radiation and vacuum energy. The Friedman equation: matter, radiation and vacuum energy dominated Universes. Critical density, Ω, present composition of the Universe, past and future of the Universe.

The Exam

The exam will consist of two sections. In section A, the basic section, you will be asked to answer ALL questions for a total mark of 50. In section B, the challenging section, you will be asked to answer 2 out of 4 questions for a total mark of 50.

Many of the questions in the exam will be very similar to problems on the exercise sheets, so you are encouraged to make sure that you have studied, and understood, the solutions to the exercises.

Past exam papers are available from the QM Library website.


An exercise sheet for each week will be available beginning on week 3.

Answers, clearly marked with your name, must be posted in the box outside the Departmental Secretary's room on the first floor of the Physics building, by the deadline given for each sheet. Late returns will not be accepted without good reason.

You are encouraged to work together on these exercises: much scientific research is advanced by discussion amongst collaborating scientists. The work you hand in, however, must ultimately be your own, whether it is the result of such discussions or your own isolated effort. You must not copy another students work or allow someone else to copy yours; it is a disciplinary offence to do either.


The Department keeps an attendance record for all courses. You may miss no more than one lecture, without prior permission or a subsequent acceptable written explanation. You must also hand in a genuine attempt at all the exercise assignments. If you do not meet these requirements, you are liable to be de-registered from the course: any de-registration is displayed on the transcript of courses eventually supplied by the College. If you are in danger of de-registration, you will receive one e-mail warning. [I remind you that e-mail is an official means of communication for this and all other aspects of the course; you must access your e-mail regularly.]

If you have a genuine and acceptable reason for absence, or for not handing in an assigment, you may register this via an absence notification form or you may contact the Senior Tutor directly.


The summer written examination will count 85% towards the course assessment.

The exercise marks will count 15%.


See the QMplus website for up to date homeworks.




Summary of course structure

Introduction. The Equivalence Principle.

Recap of Special Relativity. 

Basic notions in the description of curved geometries.

Geometric description of curved spacetime.

Einstein's theory of gravity: intro. 

The Einstein equations.

The Schwarzchild metric.

Tests of General Relativity. 

Black Holes. 


Course revision.


Various Past students have Sonets based in style on Shakespere's Sonnets

but inspired by the content of this course:

Sonnet 116

Let me not the laws of nature bind

Some frame dependence. Laws and not laws

That alter when they alteration find

Or bend under a Lorentzian cause.

Oh no! They are in tensorial forms

That can change frames and are never shaken

They will stand alone under all transforms

And when solved they are never mistaken

Laws are not space-time's fools, though g mu nu

May warp and change under space-time's curve;

Faced with Riemann and Christoffel symbols too

They bear it out and forever preserve.

If this be error and upon me proved 

Dr. Berman's course will soon be removed.

Ajina Hussein

Sonnet 18

Shall i compare thee to the curvature of spacetime

Thou art more tensorial and covariant

Rough winds do shake the Ricci tensor to its core

And our world lines hath all too short a date

Sometimes too long the Riemann tensor seems

And often do the indices confuse

And every field from field sometimes creates

By Einstiens changing path revealed

But thy eternal wrapping shall not fade

Rather, to field equations will lead

a lack of superposition

And in geometry lies conservation

So long as men can breath, gravity too

So long lives this, and this gives life to thee

Syeda Razzak

Sonnet 130

My space-time's curve are nothing like ample

There are simpler tensors readily read

If space be flat, my tensor is simple

If time be absolute, no frame need said

I have seen geodesic light at night

But no bending see I close to me

And in some space-times is a scary sight

Where light bends around blackholes endlessly

I love to solve the metric of flat space

Yet I know space is all too entagled

I grant I never solved for a curved case

Although no sheer force leaves me so mangled

GR is no sight I will ever see

Though c equals one, too slow is my spree

Kim L Dang

Sonnet 154

The Ricci Tensor lying once asleep

Laid by his side his G mu nu metric

Whilst many spaces vow'd flatten curves to keep

Came tripping by; in a tensor trick

The fairest physicist took up that flame

Which many Christoffel symbols had solved

And so begins in one's reference frame

Was the covariant transform resolved

This tensor quenched in dimensions high

Which from space-time's curve took coordinates

Growing a curve on where matter may lie

For space-time bent; but I, my frame creates,

Came there a curve, and this by that I prove

Space, time and gravity, one's soul, does move

Kim L Dang