New intros to Maxwell's Equations, Special and General Relativity

Will Flannery

I've put quick intros to these subjects on my web site, and I'd like to invite readers to comment. The goal of the tutorials is to give as quick an introduction as possible with the miminum of unnecessary technicalities, and yet to get to the essence of the matter . No doubt improvment is possible.

Maxwell's Equations
www.berkeleyscience.com/maxwells.htm

What's unusual - I was always a little buffaloed by the curl and gradient operators, and I set out to give the reader a feel for these operators. I discovered I could present the essence of Maxwell's equations without using them at all, and I'm happy with the result.

Special relativity
www.berkeleyscience.com/relativity.htm

What's unusual - the derivation of the Lorentz transform is by experiment and not algebra. There is also a very quick proof of e=mcc.

General relativity
www.berkeleyscience.com/gr.htm

What's unusual - this you have to see to believe - no use of Einstein's notation so everything is written out - plus there is a special 2-d pedagogical solution to the field equations.

Bill Foster

Good links

captphoton

In your example in the section Curves in Euclidean 3 Space, I believe a(t) should be a(s) in the first equation. In the second equation, -(1/r) should be -(1/R).

It isn't clear to me why a'(s) = 1 in the second equation.
Is length(a(s)) = sqrt(x(s)^2+y(s)^2+z(s)^2) in general?

In the section Curved Surfaces in Euclidean 3 Space:

X(x0,x1) = (x(x0,x1), y(x0,x1), z(x0,x1))
should read
X(u,v) = (x(u,v), y(u,v), z(u,v)) to make it agree with the graphic.

That is all I have found so far. See if you agree. If so, I will post some more.

Will Flannery

>>> In your example in the section Curves in Euclidean 3 Space, I believe a(t) should be a(s) in the first equation. In the second equation, -(1/r) should be -(1/R).

Correct on both counts: the equations should be (well, they are now)
a(s) = (R*sin(s/R), R*cos(s/R), 0)
and
a'(s) = ((1/R)*R*cos(s/R), -(1/R)R*sin(s/R), 0)

>>> It isn't clear to me why a'(s) = 1 in the second equation.

The 's' parameter is arc length, so, by definition or convention, length(a'(s)) = 1

>>>X(x0,x1) = (x(x0,x1), y(x0,x1), z(x0,x1))
should read
X(u,v) = (x(u,v), y(u,v), z(u,v))

You're right. I'll have to relabel the drawing.

Thanks !

Greg Bernhardt

Well done, thanks!!

pellis

Question: While I've found this article very helpful in some respects (e.g. clarity re Christoffel Symbols), I'm confused about your intention to relabel the drawing.

a) Has it now been corrected, to what it now shows (i.e. did you relabel the (u,v) to (x0,x1)?,

OR

b) Is it still in need of correction, from (x0,x1) to (u,v)?

And if the latter, then how far down the article does the switch need to be made - just to the end of the paragraph, or all the remaining occurrences of little-x0 and little-x1 (little = non-bold) right through to halfway through the last section on Geodesics and the Curvature of Spacetime?

Advice appreciated - P

Will Flannery

Yes, I corrected it, changing the (u,v) to (x0, x1) which is the standard notation in the literature. It is used through the rest of the page.

pellis

Thanks for the reply to mine about your correction

Phrak

Will-
If you are still about, and monitoring this thread, perhaps you'd like to comment on the following.

In reviewing your chapters, I notice a common thread as to the mathematical notations. This is not criticism but observation.

In your section on electromagnetism you've managed to work around the div and curl, which are very suspicious objects to begin with. Now, to the new student of electromagnetism, there appears to be a large and arbitrary set of rules set forth. But all of Maxwell becomes so very simple in the notation of tensors using the Levi-Cevita symbol. In other words, the mathematics without tensors is not quite right. Even the tensors are somewhat suspicious without differential forms.

Likewise with special relativity. Gamma is all very nice, because it works, but the differential calculus leaves something untold until special relativity is recase in a form using the Lorentz transform matrix.

I get the same uneasy feeling with the Christoffell symbol. It just seems wrong in the sense of being the basis of a notational structure that somehow needs improvement.
Do you happen to know of any alternatives, or if the curvature tensor, and all, can be recase in terms of differential forms, or a reason they cannot?
- deCraig

Will Flannery

Do you happen to know of any alternatives, or if the curvature tensor, and all, can be recase in terms of differential forms,

I also managed to avoid tensors, which, I confess, I think is essential to an understandable introduction (i.e. it's essential to avoid tensors). I did call the curvature tensor the curvature tensor, but I didn't use any tensor operations, or even define tensor. It was at the point of reaching the curvature tensor that my intuition began to fail, and I was happy to be able to get through the field equations and to a solution just based on formula manipulations, really. So, I don't know of any alternatives, but I would agree that my intro doesn't give any kind of intuitive feel for why the curvature tensor does what it does, and that is a major shortcoming.