Tag Archives: Galilean relativity

Fiber bundles at last

As an  undergraduate, I loved looking at math books in the U-store.  They had a wall of them back then, now it’s mostly swag.  The title of one book by a local prof threw me — The Topology of Fiber Bundles.

Decades later I found that to understand serious physics you had to understand fiber bundles.

It was easy enough to memorize the definition, but I had no concept what they really were until I got to page 387 of Roger Penrose’s marvelous book “The Road to Reality”.  It’s certainly not a book to learn physics from for the first time.  But if you have some background (say just from reading physics popularizations), it will make things much clearer, and will (usually) give you a different an deeper perspective on it.

Consider a long picket fence.  Each fencepost is just like every other, but different, because each has its own place.  The pickets are the fibers and the line in the ground on which they sit is something called the base space.

What does that have to do with our 3 dimensional world and its time?

Everything.

So you’re sitting at your computer looking at this post.  Nothing changes position as you do so.  The space between you and the screen  is the same.

But the 3 dimensional space you’re sitting in is different at every moment, just as the pickets are different at every position on the fence line.

Why?  Because you’re siting on earth.  The earth is rotating, the solar system is rotating about the galactic center, which is itself moving toward the center of the local galactic cluster.

Penrose shows that this is exactly the type of space implied by Galilean relativity. (Yes Galileo conceived of relativity long before Einstein).   Best to let him speak for himself.   It’s a long quote but worth reading.

“Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping toward the stern, although while the drops are in the air the ship runs many spans. The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other. The cause of all these correspondences of effects is the fact that the ship’s motion is common to all the things contained in it, and to the air also. That is why I said you should be below decks; for if this took place above in the open air, which would not follow the course of the ship, more or less noticeable differences would be seen in some of the effects noted.”

I’d read this many times, but Penrose’s discussion draws out what Galileo is implying. “Clearly we should take Galileo seriously.  There is no meaning to be attached to notion that any particular point in space a minute from now is to be judged as the same point in space that I have chosen. In Galilean dynamics we do not have just one Euclidean 3-space as an arena for the actions of the physical world evolving with time, we have a different E^3 for each moment in time, with no natural identification between these various E^3 ‘s.”

Although it was obvious to us that the points of our space retain their identity from one moment to the next, they don’t.

Penrose’s book is full of wonderful stuff like this.  However, all is not perfect.  Physics Nobelist Frank Wilczek in his review of the book [ Science vol. 307 pp. 852 – 853 notes that “The worst parts of the book are the chapters on high energy physics and quantum field theory, which in spite of their brevity contain several serious blunders.”

However, all the math is fine, and Wilczek says “the discussions of the conformal geometry of special relativity and of spinors are real gems.”

Since he doesn’t even get to quantum mechanics until p. 493 (of 1049) there is a lot to chew on (without worrying about anything other than the capability of your intellect).

 

The Pleasures of Reading Feynman on Physics – IV

Chemists don’t really need to know much about electromagnetism.  Understand Coulombic forces between charges and you’re pretty much done.   You can use NMR easily without knowing much about magnetism aside from the shielding of the nucleus from a magnetic field by  charge distributions and ring currents. That’s  about it.  Of course, to really understand NMR you need the whole 9 yards.

I wonder how many chemists actually have gone this far.  I certainly haven’t.  Which brings me to volume II of the Feynman Lectures on Physics which contains over 500 pages and is all about electromagnetism.

Trying to learn about relativity told me that the way Einstein got into it was figuring out how to transform Maxwell’s equations correctly (James J. Callahan “The Geometry of Spacetime” pp. 22 – 27).  Using the Galilean transformation (which just adds velocities) an observer moving at constant velocity gets a different set of Maxwell equations, which according to the Galilean principle of relativity (yes Galileo got there first) shouldn’t happen.

Lorentz figured out a mathematical kludge so Maxwell’s equations transformed correctly, but it was just that,  a kludge.  Einstein derived the Lorentz transformation from first principles.

Feynman back in the 60s realized that the entering 18 yearolds had heard of relativity and quantum mechanics.  He didn’t like watching them being turned off to physics by studying how blocks travel down inclined planes for 2 or more years before getting to the good stuff (e. g. relativity, quantum mechanics).  So there is special relativity (no gravity) starting in volume I lecture 15 (p. 138) including all the paradoxes, time dilation length contraction, a very clear explanation fo the Michelson Morley experiment etc. etc.

Which brings me to volume II, which is also crystal clear and contains all the vector calculus (in 3 dimensions anyway) you need to know.  As you probably know, moving charge produces a magnetic field, and a changing magnetic field produces a force on a moving charge.

Well and good but on 144 Feynman asks you to consider 2 situations

  1. A stationary wire carrying a current and a moving charge outside the wire — because the charge is moving, a magnetic force is exerted on it causing the charge to move toward the wire (circle it actually)

2. A stationary charge and a  moving wire carrying a current

Paradox — since the charge isn’t moving there should be no magnetic force on it, so it shouldn’t move.

Then Feynman uses relativity to produce an electric force on the stationary charge so it moves.  (The world does not come equipped with coordinates) and any reference frame you choose should give you the same physics.

He has to use the length (Fitzgerald) contraction of a moving object (relativistic effect #1) and the time dilation of a moving object (relativistic effect #2) to produce  an electric force on the stationary charge.

It’s a tour de force and explains how electricity and magnetism are parts of a larger whole (electromagnetism).  Keep the charge from moving and you see only electric forces, let it move and you see only magnetic forces.  Of course there are reference frames where you see both.

 

Read Einstein

Devoted readers of this blog (assuming there are any) know that I’ve been studying relativity for some time — for why see https://luysii.wordpress.com/2011/12/31/some-new-years-resolutions/.

Probably some of you have looked at writings about relativity, and have seen equations containing terms like ( 1 – v^2/c^2)^1/2. You need a lot of math for general relativity (which is about gravity), but to my surprise not so much for special relativity.

Back in the early 50’s we were told not to study Calculus before reaching 18, as it was simply to hard for the young brain, and would harm it, the way lifting something too heavy could bring on a hernia. That all changed after Sputnik in ’58 (but too late for me).

I had similar temerity in approaching anything written by Einstein himself. But somehow I began looking at his book “Relativity” to clear up a few questions I had. The Routledge paperback edition (which I got in England) cost me all of 13 pounds. Routledge is a branch of a much larger publisher Taylor and Francis.

The book is extremely accessible. You need almost no math to read it. No linear algebra, no calculus, no topology, no manifolds, no differential geometry, just high school algebra.

You will see a great mind at work in terms you can understand.

Some background. Galileo had a theory of relativity, which basically said that there was no absolute position, and that motion was only meaningful relative to another object. Not much algebra was available to him, and later Galilean relativity came be taken to mean that the equations of physics should look the same to people in unaccelerated motion relative to each other.

Newton’s laws worked out quite well this way, but in the late 1800’s Maxwell’s equations for electromagnetism did not. This was recognized as a problem by physicists, so much so that some of them even wondered if the Maxwell equations were correct. In 1895 Lorentz figured out a way (purely by trying different equations out) to transform the Maxwell equations so they looked the same to two observers in relative motion to each other. It was a classic kludge (before there even were kludges).

The equation to transform the x coordinate of observer 1 to the x’ of observer 2 looks like this

x’ = ( x – v*t) / ( 1 – v^2/c^2)^1/2)

t = time, v = the constant velocity of the two observers relative to each other, c = velocity of light

Gruesome no ?

All Lorentz knew was that it made Maxwell’s equations transform properly from x to x’.

What you will see on pp. 117 – 123 of the book, is Einstein derive the Lorentz equation from
l. the constancy of the velocity of light to both observers regardless of whether they are moving relative to each other
2. the fact that as judged from observer1 the length of a rod at rest relative to observer2, is the same as the length of the same rod at rest relative to observer1 as judged from observer2. Tricky to state, but this just means that the rod is out there and has a length independent of who is measuring it.

To follow his derivation you need only high school algebra. That’s right — no linear algebra, no calculus, no topology, no manifolds, no differential geometry. Honest to God.

It’s a good idea to have figure 2 from p. 34 in front of you

The derivation isn’t particularly easy to follow, but the steps are quite clear, and you will have the experience of Einstein explaining relativity to you in terms you can understand. Like reading the Origin of Species, it’s fascinating to see a great mind at work.

Enjoy