Norman Albers

May I say that uncertainty applies more or less locally, and vice versa?

Indeed, "size" depends on context. Electrons in an S-state are exhibiting no signs of a bolero until we or Nature's interactions interrupt it. Swansont What energy/freq range are you describing?

Too much speed. I told him to go elsewhere and do something useful.

"...fifty-dollar Buick", that was a cool line. R. Feynman said that for sure he could only say the purpose of life is to do it again.

The last printing of my edition was 1975. I took the course in grad school at Stanford in 1972 from Schiffer. He was a short, punchy German, who would fill five blackboards, turn briefly to say, "Do you see?" and then continue full blast. There are many places I could quote, such as p.117: "When Minkowski first introduced the [math]F_{ab} [/math] tensor into electrodynamics, he had in mind that it should transform as a tensor only under Lorentz transformations. However, as we see here, no such restriction appears necessary, for the Maxwell equations go over very easily indeed into a completely covariant form. The reader should not be blinded by our mathematical transformations into assuming the that the statement "[math]F^{ab}[/math] is a tensor" is a purely mathematical one. It is a very important and far-reaching physical principle which can be motivated by mathematical elegance, but must also be tested by physical experiment. The fact that [math]F^{ab}[/math] is a tensor under Lorentz transformation embodies a large part of special relativity theory. Our assumption that it is a tensor in a general Riemannian space-time leads to important consequences for the electrodynamics of accelerated systems of reference. The methodological principle that laws of nature which appear in tensor form in a particular coordinate system should be interpreted as valid in every system is called the principle of covariance. Its philosophic motivation is the postulate that no coordinate system should be distinguished in the formation of physical laws. It is, however, mathematically somewhat ambiguous and comes in practice to the old principle that we should try to explain facts with the simplest and most aesthetically satisfactory theory." Indeed, I have suggested that perhaps inside an event horizon, physics is not at all the same.

Thank you, ajb. If you can, check out that book. I find it to be written with a clear mathematician's sense. This is most excellent, because one must understand all the assumptions made in creating mathematical physics. To my mind the best authors are those who show clearly their assumptions. They are leaving you the freedom, at a future date, to go differently.

I have never taken this sort of mathematics course, so I cannot yet follow the Wikipedia of 'diffeomorphism'. I learned all I know from the mathematically strong text, Introduction to General Relativity by Adler, Bazin, and Schiffer (McGraw-Hill). I am testing and seeking to expand my understandings. . . . . I can understand the first statement about a transform being differentiable...

Solid math, ajb. So, then, two transplanted vectors could stretch by differing amounts but their angle, defined I think by: [math]cos \theta=\frac{v\cdot u}{|v||u|} [/math], stays the same?

http://laps.noaa.gov/albers/physics/na Please read Photon Localization and Dark Energy. I've said pretty much what I have to in several threads: Quantum Mechanics (Open Question...), Relativity (Reissner-Nordstrom-?), and a few others.

Great fun, Bignose, I like it! Golf balls are made with a certain amount of stipling for this reason. I had my niece show me one of her curves, and how she gripped the ball. I showed her that there were not as many seams crossing the airstream, and she agreed to try it at right angles. Never got feedback, but this is fun physics.

I have a lovely niece who is currently third year at Harvard and a pitcher on their softball team. She is maybe 5'1" tall, of slight build, and in high school, pitched softballs clocked at 60 miles per hour!!! We had some great discussions. You can understand a curveball: say you give the ball counterclockwise spin looking down on it. The spin drags some air around with it, and on the right side this bucks the air trying to slide by, and pressure rises. The opposite happens on the left side, so the ball goes left. Same for right, up, or down. The knuckball has no spin (a boson), so there is no consistent pressure gradient. I'll guess and say, if it happens to wander rightward, there is pressure buildup back the other way, and no clear trajectory.

Elas, I appreciate your positive reply. I need to clean up my terminology about DeBroglie waves, as I was really implying Compton wavelength. DeBroglie waves are a subtle system. We assume the total energy of the particle or photon to be proportional to a frequency, and at rest for a mass there is still a frequency implied by its energy. Wavelength, however, is taken as proportional to momentum, so we are folding special relativity into the representation, at least once. So for a slowly moving particle there is a "effective momentum wavelenth", or Debroglie wavelength, that gets arbitrarily long; we can still speak of its characteristic Compton wavelength, defined as: [math]h/mc[/math]. This seems to me to suggest the usefulness of seeing mass as energy convinced to localize. Discussion of DeBroglie waves I guess is reserved for dynamics.

Good refs, Snail. I'll read up on my Alfven waves and explain them; it's been a few decades. For scale reference in relativity, a million mph is one or two parts in a thousand compared to lightspeed; [math]\beta[/math]~[math]10^{-3}[/math].

Another hero of mine is Hans Alfven and I just heard on the radio of a report on the solar wind implicating Alfven waves as an acceleration mechanism to get particles up to one or two million miles per hour speed.

My good friend and colleague solidspin is doing handsprings and telling me to get with this. Frustratingly I am not yet at this level of mathematics.

Yes! At any point "speed" means total velocity which is the vector sum of the two components. Horizontal velocity is assumed to not change, and vertical velocity is related through its square to energy and height.

The most disunified thing around here is us, unless we understand the sexual beauty of growing up in this universe. We try to guide these developments in our children, lest they choke on stones.

Clear and funny reference site! Yes, this guy ran 65 milliamps and warned against lengthy overcharging, measurable by voltage or acid rupture (after two days' time when a few hours are needed for the capacity).

These are the reasons we are so careful with our young children. I have a friend who is an experienced M.D., and a three-year old child of his died in the hospital after choking on a piece of broccoli in their home.

Is there a low amperage rate to maintain a moderate charging?

Doesn't inner product involve magnitudes and angle? Is the term 'conformal'?

Yes, depending on <Out of the inhomogeneous closet!> your eyesight and imagination, babe. To me it is a matter of Maxwell's equations allowed to express what might be properties of the vacuum.

The vacuum response is not nothing. Quite the opposite.

What produced the green cheese I had heard the moon was made of?