A more advanced course in the four-vector and tensor description of the space-time structure of Special Relativity with applications to mechanics, waves, and electromagnetic phenomena.

Level HE1 and HE2 of Physics degree, or equivalent, including a first course in Relativity.

To develop understanding and manipulation skills in applying four-vector and tensor techniques in Special Relativity. To introduce the four-vector and four-tensor approach to Special Relativity and show how this can be used to describe mechanics, wave and electromagnetic phenomena and their transformation properties, with applications to problems in these areas.

Following the course students should be able to:

(a)   Define and use the machinery of four-vectors and their invariants and tensors to transform problems in spacetime, including velocity, momentum, force, waves, spacetime derivatives and the current density,

(b)   Recognise and be able to express and manipulate the Maxwell Equations in four-space form and be able to transform the E and B vector fields in simple cases.

Mathematical foundations: Review of transformations and vectors. A mathematical introduction to covariant and contravariant quantities, four-vectors, invariants, tensors and the metric tensor.

Relativistic mechanics: Review of the Einstein postulates of special relativity, inertial frames, the standard clock, distance measurement and synchronization. Derivation of the Lorentz Transformation for reference frames in relative velocity along the x-axis (L1) and its generalization to an arbitrary velocity (L3). Consequences of the transformation equations, Lorentz contraction, time dilation. The concept of proper time and the transformation of velocities, the four-velocity, four momentum and their invariants. Proper (rest) mass, rest energy, kinetic energy, four-momentum conservation and their use in centre of mass transformations and applications. Force, acceleration and their transformation properties. Concept of longitudinal and transverse masses.

Wave phenomena: The wave four-vector, transformation of waves, and its invariant. Aberration and the relativistic Doppler effect, the searchlight effect, and reflection at a moving interface. The photon and its four-momentum.

Electrodynamics: The Maxwell equations and the space-time derivatives as a covariant vector. Transformation rules for the vector fields E, B and their invariants. The current four-vector, the Lorentz force, fields due to a moving charge and the electromagnetic potentials A and V. The four-divergence, equation of continuity, the four-potential. The use of tensor notation and the E.M. field tensors, the four cross product as a tensor. Use of the four-curl to express the Maxwell equations in tensor form, the angular momentum in tensor form, leading to the Minkowski tensor and its dual.

Lecture course with associated problems to introduce a range of applications.

Recommended reading:

1.      1. W. Rindler, Introduction to Special Relativity, Oxford University Press (Dec 1982),

2.         ISBN-13: 978-0198531821

2. R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures on Physics,

    Vol. I, Chapters 15,16,17,24, Vol. II, Chapters 18, 25,26,27,42, Addison-Wesley,

3. E.F. Taylor and J.A. Wheeler, Spacetime Physics, W.H.Freeman & Co Ltd (Aug 1971),

    ISBN-13: 978-0716703365

3.      4. A.P. French, Special Relativity,  CRC Press Inc; 1 edition (30 Sep 1968)

    ISBN-13: 978-0748764228

Further and background reading:

5. D.F. Lawden, Elements of Relativity Theory, Wiley  

6. D.F. Lawden, Tensor Calculus and Relativity,   Methuen   

7. W.G.V. Rosser, Classical Electromagnetism via Relativity ,  Butterworths  

8. A. Einstein, Relativity, Methuen  

9. A. Einstein, The Meaning of Relativity,  Science Paperbacks   

10. W.H. McCrea, Relativity Physics,   Methuen   

11. M. Born, Einstein's theory of Relativity,   Dover   

12. C. Moller, The Theory of Relativity,   Oxford   

13. B. Spain, Tensor Calculus,  Oliver and Boyd   

14. A.I. Miller, Special theory of Relativity,  Addison-Wesley