Special Relativity and Classical Field Theory

Author: Leonard Susskind & Art Friedman 
Publisher: Basic Books
Pages: 448
ISBN: 978-0465093342
Print: 0465093345
Audience: Physics enthusiasts
Rating: 5
Reviewer: Mike James

This is the third in the Theoretical Minimum series and it's about relativity and electromagnetism.

If you are a physics enthusiast, or a student then just go and buy a copy - it is good. But you do need to check that you are right for the book to get the most out of it.

It helps if you have already encountered the topics before and have been mistaught the ideas, as is so so common. Also don't bother with this book if your aim in life is to prove Einstein wrong. What the book does best of all is to explain why Einstein was, and is, spectacularly right. It doesn't try to avoid maths so you need to be happy with integral signs and so on.

You might be wondering why the topic is relativity and classical field theory? Surely just relativity is enough of a subject? The reason is that relativity had to be invented to account for the theory of electromagnetic fields and waves in particular. The two go together. Maxwell's equations are the cause of the replacement of the old physics of absolute time by the new view of time as an additional dimension. The problem is that Maxwell's equations predict that light is an electromagnetic wave travelling at a speed c, i.e. the speed of light. This would be fine if the equations also included the speed that the source was travelling at, but they don't. They imply that the speed of light is the same for everyone and this is where relativity has to start.

 

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The first few lectures go over the basics of relativity and make heavy use of space-time diagrams. This is not really any better than many of the other alternative introductions to relativity. I don't know how tough it would be if you hadn't done a previous course in relativity, but if you have it is a useful refresher, complete with historical asides and some insights into what is going on and why things have to be this way. On the way you are slowly introduced to four vectors and quantities such as proper time and so on. I would have liked a little more discussion of what these ideas mean, in particular more discussion of why it is natural for mass and energy to be the same thing - when clearly, to us, they are not. 

For me, things got much more interesting when we reach the principle of least action. The problem with thinking in terms of a Lagrangian and an integral equation is that this is not the physics. The motion of a particle can only be determined by its local conditions, and this makes a differential equation the natural way to describe it. An integral equation that involves every possible path the particle can take is not something that is easy to accept as an explanation of the motion. But we need not worry because the Euler-Lagrange equations provide an equivalent formulation as differential equations which seem much more likely to be the laws of motion. The argument seems to be that the least action principle may not be the intuitive formulation but it seems that behind every good law of motion there is a Lagrangian waiting to be discovered. Later on the book this is used as the principle way to find laws of motion that are consistent with relativity. 

 

 

This is probably the main claim to fame for the book. It is an account that is accessible from a fairly elementary level that explains the Lagrangian way of thinking about particle and field dynamics. In Lecture 3 we have relativistic particle dynamics derived from a relativistic action. This is extended in the next chapter to classical fields, and most of the work is in explaining how a Lagrangian becomes an Action density and the way time enters the problem.

This is all summed up in Lecture 7 where the grand principles of physical theory are explained - the action principle allows you to take account of symetries which lead to conservation laws, the local principle ensures that you don't get unlimited action at a distance, Lorentz invariance ensures that your theory fits in with relativity and finally the gauge principle... and I'm still not sure I understand the gauge principle. I understand the gauge principle as applied to electromagnetism, but I don't follow why it is important in the wider sense. I am urged to regard it as important, but to be honest the book doesn't really explain it. I know about gauge invariance from quantum field theory and still do not see the connection with gauge transformations in electromagnetism. I hope this will be resolved in a future book - the next in the series is on general relativity and my guess is that we will have to wait for a volume on quantum field theory to meet the gauge principle again.

From here we are told that scalars are invariant, so if you want an action that is invariant you have to restrict your attention to scalars that you can form from the quantities you have. Once you have a scalar function, you can use the modified Euler Lagrange to derive equations of motion and be sure that these are relativistically invariant. Applying this approach to electromagnetism leads us to deriving the Lorentz force law and Maxwell's equations from the action. This is, of course, the opposite way that things happened - the laws of electromagnetism were found experimentally and any action you create has to lead to them, but this approach makes it all much simpler.

By the end of Lecture 6 we have the field tensor, and it is clear that there is no magnetic field and no electric field just a unified quantity that is an anti-symmetric tensor. What I see as a magnetic field, someone else will see as an electric field.

There are a lots of other things you find out about on the way. The discussion of magnetic monopoles at the end of the book is well worth reading.

Overall this is an opportunity to see how a modern physicist thinks about classical field theory and relativity. At this point all that is left is for me to mention Groucho - and, to learn why I have to, you need to see the last page of the final lecture.

Oh, I forgot to mention that there is some humour all the way through the book.

Notes Added Later:

Things that I'm still thinking about as the result of reading the book.

  1. It is stated that there is no way to construct a Lagrangian in E and B and therefore the vector potential is required - is this proven?
  2. Do I believe any more in the vector potential if it was? Recall that it is E and B that are measurable. The vector potential is not a physical entity.
  3. It is stated that all physical laws have a least action principle - is this proven or is it a consequence of Noether's theorem, symmetries and the need for conservation laws? 

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The Theoretical Minimum

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Last Updated ( Saturday, 07 April 2018 )