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.

 

The Reimann curvature tensor

I have harpooned the great white whale of mathematics (for me at least) the Reimann curvature tensor.  Even better, I understand what curvature is, and how the Reimann curvature tensor expresses it.  Below you’ll see the nightmare of notation by which it is expressed.

Start with curvature.  We all know that a sphere (e.g. the earth) is curved.  But that’s when you look at it from space.  Gauss showed that you could prove a surface was curved just be performing measurements entirely within the surface itself, not looking at it from the outside (theorem egregium).

Start with the earth, assuming that it is a perfect sphere (it isn’t because its rotation fattens its middle).  We’ve got longitude running from pole to pole and the equator around the middle.  Perfect sphere means that all points are the same distance from the center — e.g. the radius.  Call the radius 1.

Now think of a line from the north pole to the plane formed by the equator (radius 1).  Take the midpoint of that line and inscribe a circle on the sphere, parallel to the plane of the equator.  Its radius is the half the square root of 3 (or 1.73). This comes from the right angle triangle just built with hypotenuse is 1 and  one side 1/2.   The circumference of the equator is 2*pi (remember the sphere’s radius is 1).  The circumference of the newly inscribed circle is 1.73 * pi.

Now pick a point on the smaller circle and follow a longitude down to the equator.  Call this point down1.  Move in one direction by 1/4 of the circumference of the sphere (pi/2).  Call that point on the equator down then across

Now go back to the smaller circle at the first point you picked and move in the same direction as you did on the equator by absolute distance pi/2 (not by pi/2 radians).  Then follow the longitude down to the equator.  Call that point across then down.  The two will not be the same.  Across then down is farther from down 1 than down then across.

The difference occurs because the surface of the sphere is curved, and the difference in endpoints of the two paths is exactly what the Reimann curvature tensor measures.

Here is the way the Riemann curvature tensor is notated.  Hideous isn’t it?

If you’re going to have any hope of understanding general relativity (not special relativity) you need to understand curvature.

I used paths in the example, Riemann uses the slope of the paths (e.g derivatives) which makes things much more complicated.  Which is where triangles (dels), and the capital gammas (Γ) come in.

To really understand the actual notation, you need to understand what a covariant derivative actually is, which is much more complicated, but knowing what you know now, you’ll see where you are going when enmeshed in thickets of notation.

What the Riemann curvature tensor is actually saying is that the order of taking covariant derivatives (which is the same thing as the order of taking paths)  is NOT commutative.

The simplest functions we grow up with are commutative.  2 + 3 is the same as 3 + 2, and 5*3 = 3*5.  The order of the terms doesn’t matter.

Although we weren’t taught to think of it that way, subtraction is not.  5 – 3 is not the same as 3 – 5.  There is all sorts of nonCommutativity in math.  The Lie bracket is one such, the Poisson bracket  another, and most groups are nonCommutative.  But that’s enough.  I wish I’d known this when I started studying general relativity.

Goodbye to the blind watchmaker — take I

The Michelson and Morley experiment destroyed the ether paradigm in 1887, but its replacement didn’t occur until Einstein’s special relativity in 1905.  One can disagree with a paradigm without being required to come up with something to replace it. Unfortunately, we tend to think in dichotomies, so disagreeing with the blind watchmaker hypothesis for life itself tends to place you in the life was created by some sort of conscious entity.  “Hypotheses non fingo”  (Latin for “I feign no hypotheses”) which is what  Newton famously said  when discussing action at a distance which his theory of gravity entailed (which he thought was pretty crazy).

Here are  summaries of four previous posts (with links) showing why I have problems accepting the blind watchmaker hypothesis.  These are not arguments from faith which nowhere appears, but deduction from experimental facts about the structures and processes which make life possible. Be warned.  This is hard core chemistry, biochemistry and molecular biology.

First the 20,000 or so proteins which make us up, a nearly vanishing fraction of the possible proteins.  For how vanishing see — https://luysii.wordpress.com/2009/12/20/how-many-proteins-can-be-made-using-the-entire-earth-mass-to-do-so/.  Just start with 20 amino acids, 400 dipeptides, 8000 tripeptides.  Make one molecule of each and see how long a protein you wind up with making all possibilities along the way.  The answer will surprise you.

Next the improbability of a protein having a single shape (or a few shapes) for some chemical arguments about this — see https://luysii.wordpress.com/2010/08/04/why-should-a-protein-have-just-one-shape-or-any-shape-for-that-matter/

After that — have a look at https://luysii.wordpress.com/2010/10/24/the-essential-strangeness-of-the-proteins-that-make-us-up/.

The following quote is from an old book on LISP programming (Let’s Talk LISP) by Laurent Siklossy.“Remember, if you don’t understand it right away, don’t worry. You never learn anything, you only get used to it.”   Basically I think biochemists got used to thinking of proteins have ‘a’ shape or a few shapes because that’s what they found when they studied them.

If you think of amino acids as letters, then proteins are paragraphs of them, but to have biochemical utility they must have ‘meaning’ e.g. a constant shape.

Obviously the ones making us do have shapes, but how common is this in the large universe of possible proteins.  Here is an experiment which might show us (or not)– https://luysii.wordpress.com/2010/08/08/a-chemical-gedanken-experiment/.

From a philosophical point of view, the experiment is quite specific.  From a practical point of view quite possible to start, but impossible to carry to completion.

Well this is a lot of reading to do (assuming anyone does it) and I’ll stop now (although there is more to come).

Why do this at all?  Because I’ve been around long enough to see authoritative statements (by very authoritative figures) crash and burn.  Most of them I didn’t believe at the time — here are a few

l. The club of Rome’s predictions

2. The population bomb of Ehrlich

3. Junk DNA

4. We are 98% Chimpanzee because our proteins are that similar.

5. Gunther Stent, very distinguished molecular biologist, writing that we were close to the end of our understanding of genetic biology.  This in 1969.

The links elaborate several reasons why I find the Blind Watchmaker hypothesis difficult to accept.  There is more to come.

“Hypotheses non fingo”

A mathematical kludge and its repair

If you are in a train going x miles an hour and throw a paper airplane forward at x feet per second (or x * 3600/5280 miles per hour, relative to someone outside the train sees the plane move a bit faster than x miles an hour.  Well that’s the whole idea of the Galilean transformation.  Except that they don’t really see velocities adding that way for really fast velocities (close to the speed of light).

Relativity says that there are no privileged sites of observation and that no matter how fast two observer frames are moving relative to each other light will zing past both at the same speed (3 x 10^8 meters/second, 186,000 miles/second).

All of Newton’s mechanics and force laws obeys the Galilean transformation (e.g. velocities add).  Maxwell conceived a series of 4 laws linking electricity and magnetism together, which predicted new phenomena (such as radio waves, and the fact that light was basically a form of wave traveling through space).

Even though incredibly successful, Maxwell’s laws led to an equation (the wave equation) which didn’t obey the Galilean transformation.  This led Lorentz to modify it so the wave equation did obey Galileo.  If you’ve got some mathematical background an excellent exposition of all this is to be found in “The Geometry of Spacetime” by James J. Callahan pp. 22 – 27.

The Lorentz transformation is basically a kludge which makes things work out.  But he had no understanding of why it worked (or what it meant).  The equations produced by the Lorentz transformation are ugly.

Here are the variables involved.

t’ is time in the new frame, t in the old, x’ is position in the new frame x in the old. v is the constant velocity at which the two observation frames are moving relative to each other. c is the common symbol for the velocity of light.

Here are the two equations

t’ =  ( t – vz/c^2 )/ sqrt (1 – v^2/c^2)

x’ = ( z – vt ) /  sqrt (1 – v^2/c^2)

Enter Einstein — he derived them purely by thought.  I recommend Appendix 1 in Einstein’s book “Relativity”.  Amazingly you do not need tensors or even calculus to understand his derivation — just high school algebra (and not much of that — no trigonometry etc. etc.)  You will have the pleasure of watching the power of a great mind at work.

One caveat.  The first few equations won’t make much sense if you hit the appendix without having read the rest of the book (as I did).

Light travels at c miles/hour, so multiplying c by time gives you where it is after t seconds.  In equations x = ct.  This is also true for another reference frame x’ = ct’.

This implies that both x – ct =  0 and x’ – ct’ = 0

Then Einstein claims that these two equations imply that

(x – ct) = lambda * (x’ – ct’) ; lambda is some nonzero number.

Say what?  Is he really saying  0 = lambda * 0.

This is mathematical fantasy.  Lambda could be anything and the statement lacks mathematical content.

Yes, but . . .

It does not lack physical content, which is where the rest of the book comes in.

This is because the two frames (x, t) and (x’ , t’) are said to be in ‘standard configuration which is a complicated state of affairs. We’ll see why y, y’, z, z’ are left out shortly

The assumptions of the standard configuration are as follows:

• An observer in frame of reference K defines events with coordinates t, x
• Another frame K’ moves with velocity v relative to K, with an observer in this moving frame K’ defining events using coordinates t’, x’
• The coordinate axes in each frame of reference are parallel
• The relative motion is along the coincident xx’ axes (y = y’ and z = z’ for all time, only x’ changes, explaining why they are left out)
• At time t = t’ =0, the origins of both coordinate systems are the same.

Another assumption is that at time t = t’ = 0 a light pulse is emitted by K at the origin (x = x’ = 0)

The only possible events in K and K’ are observations of the light pulse. Since the velocity of light (c) is independent of the coordinate system, K’ will see the pulse at time t’ and x’ axis location ct’, NOT x’-axis location ct’ – vt’ (which is what Galileo would say). So whenever K sees the pulse at time t and on worldline (ct, t), K’ will see the pulse SOMEWHERE on worldline (ct’, t’).

The way to express this mathematically is by (3) (x – ct) = lambda * (x’ – ct’)

This may seem trivial, but I spent a lot of time puzzling over equation (3)

Now get Einstein’s book and watch him derive the complicated looking Lorentz transformations using simple math and complex reasoning.

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

Help wanted

Just about done with special relativity. It is simply marvelous to see how everything follows from the constancy of the speed of light — time moving more slowly for a moving object (relative to an object standing still in its own frame of reference), a moving object shrinking (ditto), the increase in mass which occurs as an object begins to approach the speed of light, and how this leads to the equivalence of mass and energy. Special relativity is even sufficient to show how a gravitational field will bend light — although to really understand this, general relativity is required.

The one fly in the intellectual ointment is the Minkowski metric for the space time of special relativity. In all the sources I’ve been able to find, it appears ad hoc, or is defined analogously to the euclidean metric. I’d love to see an argument why this metric (time coordinates positive, space coordinates negative) must follow from the constancy of the speed of light. It is clear that the Minkowski metric is preserved under the hyperbolic transformation of space-time, but likely others are as well. Why this particular metric and not something else.

Consider the determinant function of an n by n matrix. It has a god awful mathematical form involving the sum of n ! terms. Yet all you need to get the (unique) formula are a few postulates — the determinant of the identity matrix is 1, the determinant is a linear function of its rows (or its columns), interchanging any two rows of the determinant reverses the sign of the determinant, etc. etc. This basically determines the (unique) formula of the determinant. I’d really like to see the Minkowski metric come out of something like that.

Can anyone out there shed light on this or give me a link?

The weirdness of gravity

We experience gravity every waking moment, so it’s hard to recognize just how strange the gravitational ‘force’ actually is. Push a toy sailboat, a rowboat, and a yacht with the same amount of force (effort). What happens?

The smaller the boat, the faster it moves. Physicists would say the acceleration (change in velocity over time e.g. from the boat not moving at all to moving somewhat) is inversely proportional to the mass of the boat. This is Newton’s famous second law force = mass * acceleration. This isn’t actually what he said which you’ll find at the end.

So in every force except gravity, the bigger the force the more the acceleration. In Galileo’s famous experiment (which Wikipedia says might actually not have occurred), he dropped 2 objects of different masses from the leaning tower of Pisa and found that they hit the ground at the same time, so the acceleration of both due to the ‘force’ of gravity is the for all objects regardless of their different masses.

This implies that gravity is a force that adjusts itself to the mass of the object it is pushing on to produce the same acceleration. Weird, but true.

General relativity says, that the motion must be considered not just in space and time, but in 4 dimensional space-time where space can become our conventional time and vice versa. Here all paths are as straight as possible — because the 4 dimensional space-time we inhabit has an intrinsic curvature, produced by the masses found within it.

What Newton said: “The change of motion is proportional to the motive force impressed and is made in the direction of the straight line in which that force is impressed” By motion Newton means what we call momentum — mass * velocity.

The change in momentum is of course a change in velocity — which is what acceleration actually is. Note that mass is assumed constant regardless of how fast the object is moving. This isn’t even true in special relativity (which doesn’t include gravity — that’s what general relativity is all about).

An old year’s resolution

One of the things I thought I was going to do in 2012 was learn about relativity.   For why see https://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/.  Also my cousin’s new husband wrote a paper on a new way of looking at it.  I’ve been putting him off as I thought I should know the old way first.

I knew that general relativity involved lots of math such as manifolds and the curvature of space-time.  So rather than read verbal explanations, I thought I’d learn the math first.  I started reading John M. Lee’s two books on manifolds.  The first involves topological manifolds, the second involves manifolds with extra structure (smoothness) permitting calculus to be done on them.  Distance is not a topological concept, but is absolutely required for calculus — that’s what the smoothness is about.

I started with “Introduction to Topological Manifolds” (2nd. Edition) by John M. Lee.  I’ve got about 34 pages of notes on the first 95 pages (25% of the text), and made a list of the definitions I thought worth writing down — there are 170 of them. Eventually I got through a third of its 380 pages of text.  I thought that might be enough to help me read his second book “Introduction to Smooth Manifolds” but I only got through 100 of its 600 pages before I could see that I really needed to go back and completely go through the first book.

This seemed endless, and would probably take 2 more years.  This shouldn’t be taken as a criticism of Lee — his writing is clear as a bell.  One of the few criticisms of his books is that they are so clear, you think you understand what you are reading when you don’t.

So what to do?  A prof at one of the local colleges, James J. Callahan, wrote a book called “The Geometry of Spacetime” which concerns special and general relativity.  I asked if I could audit the course on it he’d been teaching there for decades.  Unfortunately he said “been there, done that” and had no plans ever to teach the course again.

Well, for the last month or so, I’ve been going through his book.  It’s excellent, with lots of diagrams and pictures, and wide margins for taking notes.  A symbol table would have been helpful, as would answers to the excellent (and fairly difficult) problems.

This also explains why there have been no posts in the past month.

The good news is that the only math you need for special relativity is calculus and linear algebra.  Really nothing more.  No manifolds.  At the end of the first third of the book (about 145 pages) you will have a clear understanding of

l. time dilation — why time slows down for moving objects

2. length contraction — why moving objects shrink

3. why two observers looking at the same event can see it happening at different times.

4. the Michelson Morley experiment — but the explanation of it in the Feynman lectures on physics 15-3, 15-4 is much better

5. The Kludge Lorentz used to make Maxwell’s equations obey the Galilean principle of relativity (e.g. Newton’s first law)

6. How Einstein derived Lorentz’s kludge purely by assuming the velocity of light was constant for all observers, never mind how they were moving relative to each other.  Reading how he did it, is like watching a master sculptor at work.

Well, I’ll never get through the rest of Callahan by the end of 2012, but I can see doing it in a few more months.  You could conceivably learn linear algebra by reading his book, but it would be tough.  I’ve written some fairly simplistic background linear algebra for quantum mechanics posts — you might find them useful. https://luysii.wordpress.com/category/linear-algebra-survival-guide-for-quantum-mechanics/

One of the nicest things was seeing clearly what it means for different matrices to represent the same transformation, and why you should care.  I’d seen this many times in linear algebra, but seeing how simple reflection through an arbitrary line through the origin can be when you (1) rotate the line to the x axis by tan(y/x) radians (2) change the y coordinate to – y  — by an incredibly simple matrix  (3) rotate it back to the original angle .

That’s why any two n x n matrices X and Y represent the same linear transformation if they are related by the invertible matrix Z in the following way  X = Z^-1 * Y * Z

Merry Christmas and Happy New Year (none of that Happy Holidays crap for me)