Special Relativity and Classical Field Theory: The Theoretical Minumum

To the reader: The authors of this book intersperse the text with conversations between Art and Lenny. The conversations are cute, fun, and give the text a playful feel. With that in mind, please meet my dear friend Andy I. Andy is a (real) theoretical physicist and holds a Ph. D. in physics. He’s been my friend for decades and I’m thrilled to have him join us.

David: Hi Andrew, glad we could meet for coffee today. How you doing?

Andy: Fine. Got more work than I can imagine. What have you been up to?

D: I’m reading the book by Susskind and Friedman on special relativity. You know I’m an engineer, but what the hey? I can do physics, too.

A: Let me see it. Do you have it with you?

D: Here you go.

Andy takes the book and flips through it as if he’s read it a thousand times and it’s a good friend.

A: Do you see these differential equations? The book is filled with them. Know what they are?

D: The authors explain they are Lagrangians. The derivatives, some partial, some complete derivatives, describe how fields can behave.

A: Yes, that’s right. In physics, once you have these differential equations set, say for relativity, the problem is solved. The rest is math.

D: Don’t people solve the equations to see what the fields actually do?

A: Look through the book… do you see solutions?

David flips through the text. There are lots of differential equations, few solutions.

D: You’re right.

A: Of course I am. Now, tell me about the book.

D: Well, this book is part of a series called The Theoretical Minimum. There is a book on physics in general, one on quantum mechanics, and a book on classical mechanics. They go along with a video course Dr. Susskind teaches. I’ve not seen the videos though. But the book I have should tell us the basics of special relativity and field theory.

A: Does it do that?

David gets up and heads to Starbucks. “Let’s get some coffee.” 

They each return with Grande sized coffees.

D: To your question, Yes, the book does explain relativity nicely. The authors give a good discussion of Newton’s approach to motion and what simultaneous events would be to Newton. Then they tell you how Einstein defined simultaneous events with clocks based the arrival of light pulses and what led to his thinking. The explanation is gentle so I followed it effortlessly. That’s something, as you would say, for an engineer.

A: Did they mention Lorenz transformations?

D: You mean the way we transform coordinates from different frames of reference, such as at rest frames and moving frames? Yes, they do and they give lots of neat examples. For an engineer, I followed that discussion nicely.

A: Physics has some fascinating history. Did these guys mention any of that?

D: Why yes. It seems Einstein was not the first to use Lorenz transformations to show Lorenz contraction. Lorenz and George FitzGerald thought super-fast moving objects (that’s how I describe motion close to the speed of light) would contract, just like Einstein. But, these guys thought the contraction would be due to motion in the ether with pressure from the ether causing the contraction. They didn’t see it as a result of the change of coordinates.

A: It’s always good to know the context of a discovery. The history lets you understand how great was the scientist.

D: The authors say no one saw the beauty of working with inertial frames like Einstein. Although, Henri Poincaré derived the Lorentz transformations from constraining Maxwell’s equations to have the same form in every inertial frame. Poincaré is my hero from chaos theory, but that will take more coffee.

A: I am sure the book explains how to add velocities.

D: Yes, and there is another part to relativity that I learned, on top of many other topics.

A: Tell me one.

D: There’s an invariant in relativity. A form that is constant for any reference frame.

A: Wait, let me guess.

Andy takes out his pen, grabs a napkin and writes: \[ r^2 = t^2 - \left( x^2 + y^2 + z^2 \right)\]

And under that equation: \[ s^2 = -t^2 + \left( x^2 + y^2 + z^2 \right)\]

Andy points to the top equation: This one is proper time, the bottom one I call its twin, the space-time interval. These quantities do not change under the Lorenz transformation.

D: Right you are, Andy! The authors show you how we can use these in our Lagrangians to glean new insights.

A: Care to tell me more?

D: You can read the book, too.

Andy drinks more coffee.

A: This coffee is excellent, shall we take some home?

D: I’m drinking a different brand at home these days. I only drink Starbucks with you, it’s a bit special that way.

A: Did you know physicists like to have their own notation. It’s funky but that’s how we are.

D: Know it, this book is filled with funky notations. There are tensors galore and the authors explain them but then they shorten derivatives to single variables with subscripts and superscripts. It’s quite weird and I found it difficult to follow. There is index notation, covariants, upstairs indexes, downstairs indexes, and some others. Frankly, I found these confusing. The equations are simplified, but they left me behind.

A: Welcome to physics. If you had used that notation before reading the book, it’d be second nature.

D: I guess. The book goes beyond relativity. Want to hear more?

A: If you can do more, sure, I’d like to hear more.

David swallows the rest of his coffee.

D: The last part of the book is about field theory. Here we find electric fields, magnetic fields, Poynting vectors, and least action principles. Oh, there’s gauge invariance.

A: Care to talk about these a bit?

D: I do, but not a lot for today. Unfortunately the text seems readable for these concepts, but there seems to be an underlying assumption the reader already knows the topics. It’s frustrating and I think readers deserve better.

David pauses to think.

D: Let me tell you something I found interesting.

A: I’m listening. I’ll be impressed if you got just a bit from the book.

D: Electric charges can exist but magnetic charges cannot exist.

A: Is that right? You know our friend, the particle physicist Mike? He wrote his dissertation on the search for monopoles. These are magnetic particles with one pole. Magnets come with two poles, north and south. Monopoles don’t. If Mike had found one, well, he’d be a Nobel laureate.

D: Monopoles can exist. Mike’s quest wasn’t in vain.

A: Really? how so?

D: The authors tell you how at the very end of the book. I’ll try to paraphrase what they write. A monopole is an isolated charge. A magnetic monopole would be surrounded by a magnetic field similarly to an electric monopole that is surrounded by an electric field.

David takes a napkin and writes: \[\nabla \cdot E = \rho \]

D: This equation says the electric charge is the source of an electric field. The divergence, or vector flow out of the field, is \(\rho\). We can solve this for a point source to derive the Coulomb field for an electric monopole. I remember this from school, vaguely.

A: What about magnetic fields?

D: Well, for a magnet field we have: \[ \nabla \cdot B = 0 \] which means the magnetic field has no sources. Magnetic monopoles would be non-existent.

A: An equation wouldn’t stop a physicist. We just ignore them. In quantum mechanics we ignore infinities and call it normalization.

D: Right, and you could modify this equation to be: \[ \nabla \cdot B = \sigma \] And we’d have an analog to the electric field. It’s not that simple though.

A: It’s getting late.

D: Okay, let me be quick. Let’s say you had a magnet, like solenoid. It has a north and south pole and magnetic field. Now, the magnetic field comes out the positive pole and returns through the negative pole. The field goes through the solenoid, too. Make the solenoid long and thin. Really long and thin as if you stretch it to infinity, or nearly so. The now isolated north pole looks like an isolated magnetic charge. The authors go into more detail but that’s my take. This isolated pole would act like a monopole.

A: Not bad.

D: So, Mike’s search was real. He never found any monopoles but he got his doctorate.

Andy gets out of his chair.

A: Thanks for the talk. Now sum up the book.

D: It’s like you said earlier today. If you know what the book is about, you can understand it. If you don’t know the material already you may struggle to learn it from this book. I did struggle but the book seems readable if not fully understandable. I learned a few new ideas from it. I can’t ask for more.

A: Don’t be offended, but can you prove you actually read the entire book?

D: Yes. Remember the Marx brothers? Harpo, Groucho, Zeppo, and Chico?

A: Sure, great comedians.

D: Yep, I liked them. So did the authors.

A: How do you know that?

D: Read the book.

On the last page of this book, the authors ask reviewers to mention Groucho Marx to let the authors know the reviewer read the entire book.

David S. Mazel is a practicing engineer in Washington, DC. He welcomes your thoughts and feedback. He can be reached at mazeld at gmail dot com.