## MP352: Special Relativity

### Spring 2020 semester (2nd semester of 2019-2020)

Lecturer:   Masud Haque   (haque@thphys.nuim.ie)     Tutor is Aonghus Hunter-McCabe (Aonghus.HunterMccabe@mu.ie).

Final Exam, Thursday May 28th, 2020

Here is the Final Exam for 28th May, available after 2:15PM.

Problem sets

Problem set 11.     Due Thursday May 7th.

Problem set 10.     Due Tuesday April 28th.

Problem set 09.     Due Tuesday April 21st, after the Easter break.

Problem set 08.     Due Tuesday April 7th.

Problem set 07.     Due Tuesday March 31st.

Problem set 06.     Was due Tuesday March 24th; extended to Thursday March 26th. To be scanned and uploaded as pdf file.
For the Study Break: twice as long as the usual weekly problems.

Problem set 05.     Due Tuesday March 10th.

Problem set 04.     Due Monday March 2nd.

Partial solutions/hints for problem set 04.
The tutorial (where this problem set was supposed to be discussed) was replaced by a lecture; so I am posting some solutions.

Problem set 03.     Due Monday February 24th.

Problem set 02.     Due Monday February 17th.

Problem set 01.     Due Monday February 10th.

Practice Problems

Problem bank (problem set 12'')
This is marked as Problem set 12', but it is really a large collection of problems covering all the module material.
Should be useful for practicing material/concepts from the entire module, and for exam preparation.

Notes on Index/Tensor notation

Electromagnetism in relativistic notation

We missed lectures on April 28th and May 1st due to my illness.

During this week, the plan was to cover electromagnetism in tensor notation, and relativistic effects in electromagnetism. Please learn this material from Sections 5.2, 5.3, 5.4 of these lectures by Tong.

This is quite a lot of material. The main topics you should pick up from here are:
(1) the electromagnetic field tensor, $$F^{\mu\nu}$$. How is $$F^{\mu\nu}$$ defined in terms of the potential 4-vector $$A^{\mu}$$? What are the 16 elements of $$F^{\mu\nu}$$; how many independent elements are there?
(2) How Maxwell's equations are written in terms of $$F^{\mu\nu}$$, in tensor notation.
(3) How gauge transformations are written in tensor notation.
(4) How the Lorentz force law is written in tensor notation, as an equation relating 4-force and 4-velocity with $$F^{\mu\nu}$$.
(5) How the continuity equation is written in tensor notation.

Virtual lecture'' for Friday April 24th

In this lecture, we will complete our study of the Lorentz group, adding material on the rotation group and the Poincare group.

Here are updated notes on the groups.

The previous material on Lorentz groups is in Section 2 of this writeup and has also been extended slightly. When you work through this writeup, you should also be getting a review of Tuesday's material.

Virtual lecture'' for Tuesday April 21st

In this lecture, we will start on the Lorentz group and its structure.

As usual, I am most thankful to those who are pointing out typos to me. If you spot any errors, please let me know!

As supplementary reading, you could try
* pages 10-17 of these notes introducing the metric tensor and the Lorentz group. The first few pages (5-9) introduce index notation, which would be also good to review.
* up to page 7 of these notes discussing the rotation group and the Lorentz group.

Virtual lecture'' for Tuesday April 7th

At this point we should learn about index notation. I offhandedly used this notation sometimes in earlier lectures. This will become standard language later on, so let's get used to this!

Index notation is introduced concisely in Section 5.1 of these lectures by Tong.

In particular, please read Subsections 5.1.3 and 5.1.4. (Of course you would benefit from reading the first two Subsections as well.) The notation is very similar to what we have been using, except that he uses $$\eta_{\mu\nu}$$ instead of $$g_{\mu\nu}$$. Both notations are common.

-- Sections 5.5 and 5.6 of these notes.

Virtual lecture'' for Friday April 3rd

This lecture will be light on new material. I suggest first reviewing the material in the previous virtual lectures.

I have updated the notes describing a few 4-vectors.

The new part is in the last two subsections (last 3+1/2 pages). Two new 4-vectors are added: the 4-potential and the 4-current. These are no longer related to the kinematics/dynamics of a point object. Instead, they combine objects that you've learned about in electromagnetism.

Virtual lecture'' for Tuesday March 31st

In this lecture, we will continue learning about four-vectors:
(1) We will learn what it means for a 4-vector to be time-like, space-like, or light-like;
(2) We will introduce a few physical 4-vectors.

Here are notes describing a few 4-vectors.

As usual, I am most thankful to those who are pointing out typos to me. If you spot any errors, please let me know!

In this lecture we concentrate on 4-vectors associated with the motion of an object or particle. In the next lecture we will continue encountering a few other 4-vectors.

This material is very standard, but the notation varies widely. I hope I have given enough hints about varying notation that you should be able to read another source with reasonable comfort.
For extra reading, I can suggest
this page on 4-velocity & 4-acceleration that actually puts time as the 4th dimension instead of 0-th,
Section 1 (1.1 to 1.5) of this scholarpedia page.

Virtual lecture'' for Friday March 27th

In this lecture, we will introduce four-vectors.

First, please review the last page of the notes for the lecture of March 23rd. There, we motivated the need for 4-vectors --- objects that transform like spacetime coordinates or intervals. We also know two examples of 4-vectors already: (1) spacetime coordinates/intervals and (2) 4-momentum or energy-momentum.

For this lecture: Here are my typed-up notes for the lecture of March 27th.

First (Section 1), we re-examine our understanding of what vectors' and scalars' really mean in ordinary (Euclidean) mechanics.
Thus prepared, we are ready to introduce 4-vectors, and some of their properties (Section 2).

As usual, if you spot any typographical or other errors, please let me know. A big thanks to those who spotted errors in the the notes of March 23rd and pointed them out to me.

To supplement my notes, you could read sections 6.1, 6.2, 6.3, 6.4 of these notes.

In the next lectures, we will construct/list various physical 4-vectors (4-velocity, 4-force, density-current-density, 4-potential in electromagnetism, etc). We will also introduce the index notation for 4 vectors and their inner products.

Virtual lecture'' for Tuesday March 23rd

For this lecture, we want to
(1) review and extend the expressions for energy and momentum;
(2) Introduce mass-less particles or photons;
(3) Take a first look at force in relativity;
(4) Figure out the transformation of energy and momentum under a boost.

Points (2) and (3) above will point to the need for introducing 4-vectors, which we will do in the following lecture.

Here are notes for the lecture of March 23rd.
(If you spot any typographical or other errors, please let me know.)

Review: momentum & energy

Sections 1 through 4 reviews some of what we've learned about momentum and energy, with some extensions:

• I am not sure if I introduced Eq.(5) already? It's an important equation. Please derive it from Eqs.(3,4).
• Section 4 is a more extended discussion of the collision that we treated in class before the shutdown. You will have to work on more complicated collisions in the coming weeks, so I suggest working through this simple collision in detail. E.g., write down momentum conservation equations and energy conservation equations in both frames.

The amazing photon

Section 5 is about massless particles which move at the speed of light: photons. It argues how relativistic equations allow meaningfully assigning momenta and energies to particles having zero mass!

Subsection 5.1 is a practice of things you have learned.

Force

Section 6 shows and derives the expressions for force in special relativity. These are ugly expressions!!

Transformations of energy and momentum

Considering a boost, we find out that the energy and momentum together transform the same way as time and position.

This motivates 4-vectors, for the friday class....

Continuous Assessment and the purpose of problem sets

The plan is to assign one problem set every week. They will be posted on this webpage. The assignments will be due mondays, in the drawer marked 352 near the front door of the Theoretical Physics department.

We will not be able to mark all the assignments. We will not announce beforehand which assignments are to be marked, so it will be to your advantage to submit every assignment.

Assignment marks (continuous assessment') will be counted toward the final module mark only if they are to the students' advantage.
(The policy is module-dependent and varies within the Mathematical Physics department. In some modules, continuous assessment always counts toward the final module mark.)

HOWEVER: Do not think of the assignments as voluntary or optional. Without working on each assignment set, you are likely to get lost quickly, as the module will build on the assignments and assume that you have learned the material you are supposed to learn through doing the assignments.

Notes/handouts

Here are some derivations of the Lorentz transformations for a standard boost.
If you spot any typos or errors, please let me know.

Material, sources

Special Relativity challenges our intuition and takes effort to digest; so I strongly suggest trying to read a fair bit every week.

For example, you could aim to work through two sections of Nash's notes (below) every week, or a similar amount of work from another text.

Nash notes:

lecture notes of Prof. Charles Nash, who taught MP352 some years ago
This has some overlap with the material covered now in MP352. Unfortunately, the ordering is different.

Textbooks:

There are many textbooks covering special relativity. Our library carries a number of these textbooks.
Special relativity is also covered in many textbooks on classical mechanics or electrodynamics/electromagnetism, and often summarized in the beginning of texts on general relativity or particle physics.
I list a selection below.

(I omit publisher and publication date; should be easy enough to find.)

• Zee, Einstein Gravity in a Nutshell
This book targets General Relativity, but on the way (in Part III) treats Special Relativity in detail. Great fun to read.

• Chapters 11 -- 13 of:     Morin, Introduction to Classical Mechanics
Very thorough treatment. Excellent problems and exercises. Covers the more basic half or so of what we will do in this module.

• The chapter on relativity in:     Griffiths, Introduction to Electrodynamics
Very clear and physical treatment.
(Relativity is in Chapter 10 in the old edition I own; chapter number varies with edition.)

• Chapters 12 -- 14 of:     Kleppner & Kolenkow, Introduction to Mechanics, 2nd ed.
Very pedagogical. Covers maybe the more basic one-third or so of what we do in this module. The level is intermediate between the Resnick-Halliday level and the level of this module.
There are several copies of the book in the university library.

• Chapters 15 -- 17 of:     The Feynman Lectures on Physics, Volume I
A classic. Free to read online, on this website.
Somewhat outdated terminology/convention, but should still be worthwhile to read. Covers the more basic one-third or so of what we do in this module.

• Cheng, Relativity, gravitation, and cosmology
Special relativity is reviewed as an prerequisite to General Relativity. Very clear treatment.
Special Relativity is treated in Chapter 2 in the 1st edition, but broken up into chapters 2 and 3 in the 2nd edition.

• Chapter 1 of:     Landau & Lifshitz, The Classical Theory of Fields
This is Volume 2 of the famous Course of Theoretical Physics'.

• Chapter 7 of:     Goldstein, Classical Mechanics
Concise treatment of some of the more advanced aspects.

• Chapters 1 -- 4 (and bits of chapters 5,6) of:     Woodhouse, Special Relativity
More mathematical.

• Chapters 1 -- 5 of:     Rindler, Relativity: Special, General, and Cosmological

• Chapters 29 -- 34 of:     Greiner, Classical Mechanics: Point Particles and Relativity

• Bais, Very Special Relativity: An Illustrated Guide
A very special and worthwhile read. Avoids formalism and teaches through pictures (through carefully analyzed spacetime diagrams).

• Steane, Relativity Made Relatively Easy
This textbook covers most of the material to be covered in this module.

• The chapter on relativity in:     Jackson, Classical Electrodynamics
Concise treatment of some of the more advanced aspects.

Material available online:

The wikipedia page on Lorentz Transformations contains material very relevant to this module. Highly recommended.

The following lecture notes' or other links are at various levels; mostly they cover the more elementary half of this module.

MP352 should make you comfortable with 4-vectors and Index notation. This is covered in many of the textbooks or notes linked above. In addition, the following links might help.

In MP352 we discuss the Lorentz group (together with the rotation group and the Poincare group); these are not covered in more elementary treatments or in Nash's notes. The following links might help.

Electromagnetism in special relativity is covered in some of the references given above. In addition:

Widely discussed basic material which are mathematically simple but cause conceptual confusions:

Solutions to previous exams   +   Sample Exams

Below are solutions to some past exams.
(The length of exams has changed since 2017.)

Below are sample exams for practice. They are in the style of previous (2017-2018) exams. Future exams may be structured slightly differently, but the material covered and the level of difficulty should be similar.