brianbec's WebLog : Physics

More Minkowski for Free

brianbec — Sat, 09 Sep 2006 16:15:00 GMT

Due to technical problems uploading to this site, I posted a new article over here: http://lorentzframe.blogspot.com/

Time Dilation and Lorentz Contraction

brianbec — Fri, 08 Sep 2006 04:34:00 GMT

Consider the following snapshot from a Geometry-Expressions (GX) (http://www.geometryexpressions.com) file:

The blue coordinate-frame lines represent our inertial frame, at rest with respect to us. The red coordinate-frame lines represent another inertial frame, one moving at speed v with respect to us. The red x-axis has slope v (actually v / c^2, but c = 1 here) and the red y-axis has slope 1/v, reflecting the space-time symmetry of the Special Theory of Relativity. B is an event in space-time, that is, a single, fixed point in space and time whose coordinates depend on the frame of reference in which the coordinates are measured. The cyan lines, parallel to the blue axes, measure the coordinates in our frame. The magenta lines, parallel to the red axes, measure the coordinates in the moving frame. Geometry Expressions calculates the Lorentz transformation for us. Minkowski geometry for free. Is that cool, or what? Here's the GX source file: http://home.comcast.net/~brianbec/Minkowski1.gx . (Speed of light = 1, and every place you see v * x, mentally substitute v * x / c^2.)

Space-Time Curvature for Mort

brianbec — Sat, 02 Sep 2006 14:10:00 GMT

Should have said last time that space-time "curvature" is relativity code-speak for "acceleration" or "gravitation." It's often something one can feel. Here's an imaginary example. Put a very precise atomic clock, call it A, on board a rocket. Put another, identical clock, call it B, in a lab on the ground. Start the clocks ticking. Now, let the rocket take off (ACCELERATION! Curvature difference! We can feel it!). Once the rocket stops accelerating, but while it's moving more-or-less in a straight line. let's do some experiments, say, measuring the decay rates of radioactive samples. A and B get the same answers, since at this time there is no curvature difference. Now, let A and B compare notes: let the ground clock A send some of its data to B. B sees A's clock running slow! Vice versa: if the rocket crew sends some data to A, A will reckon B's clock running slow. Can they both be right? YES. As long as the rocket keeps flying, they will both reckon each other's clocks running slow, but they're getting further and further apart and they can't compare notes forever unless the rocket turns around. ACCELERATION! Curvature difference! We can feel it! The rocket crew is actually, objectively, measurably different from the ground crew, and when they get back together, they will both agree that the clock on the rocket actually ran slower, over all, than then one on the ground.

Such experiments have been done many times with atomic clocks on planes, satellites, the space shuttle, and rockets. The details are complicated by the fact that clocks on the ground run slower because they're deeper in the Earth's gravitational field (CURVATURE! We can feel the difference!) and the fact that the projectiles travel non-straight-line paths (codespeak: "non-intertial" paths). In fact, the Earth-grav effect is easier to measure: put two clocks on different floors of a building and compare notes: they get different results because their curvature is different (grav field is stronger on lower floors, clocks actually run slower, things weigh more, we can feel the difference!). GPS satellites run at slightly different altitudes because of lumps (mountains, valleys) in the grav field, and it's necessary to correct for their mutual clock drifts to maintain accuracy.

This is really weird, but it's actually, objectively, measurably true. The theory, which is conceptually simple, though mathematically intricate, is very very accurate. No one has ever seen anything to refute the theory.

Weird as this is, it's NOTHING compared to the weirdness of quantum physics.

Theory of Relativity Misnamed?

brianbec — Fri, 01 Sep 2006 17:31:00 GMT

One of the groups that I have lunch with likes to discuss physics and astronomy. During one of these lunches, there was a bit of confusion about the Theory of Relativity (ToR), with people not being quite sure whether it means that measurements depend upon one's frame of reference. I was fortunately able to clarify this absolutely (pun intended). The axioms of the ToR are that

* if you do an experiment and I do the same experiment, and

* if the regions of space and time enclosing the experiment are small enough as to appear un-curved (or if they have the same curvature within our measurement precision)

you and I will get the same answers. These axioms are *really* a statement of the absolute and permanent nature of the laws of physics! For "small" experiments, we get the same answers, period, now and forever, with "small" meaning the above.

So why is it called Theory of Relativity and not the Theory of Absolutes? Because if I measure YOUR experiment from a distance, I might get a different answer than you get, depending on our mutual (relative) velocity and curvature differences. I will ALWAYS get the same answer on my copy of the experiment as you get on your copy of the experiment, but I might get a different answer looking at your experiment than you get on your experiment. The two ToR's tell us exactly how to account for the different answers. The Special ToR accounts for our mutual velocities, and the General ToR accounts for differences in spacetime curvature.

Feynman Says: “Newton implies Kepler, without Calculus!”

brianbec — Fri, 04 Aug 2006 04:22:00 GMT

Reference: “Feynman’s Lost Lecture, The Motion of Planets Around the Sun,” by David L. Goodstein and Judith R. Goodstein, W. W. Norton & Company, New York, 1996.

Richard Feynman was one of the most important physicists of all time. He had a peculiar way of taking ideas out of thin air and bringing them to bear on problems that seemed unrelated, thereby busting them wide open. He is perhaps best remembered for the path-integral formulation of relativistic quantum mechanics, which not only showed that the otherwise stupefyingly mysterious quantum mechanics is just another application of the old familiar principle of least action, but also revealed space-time symmetries that resolved conundrums about antiparticles and particles going backwards in time. Every word he ever spoke or wrote is worthy of study. I had the privilege of meeting with him several times over the years as a child, an older student, and as a professional physicist, and witnessing the magic at work.

The book referenced at the top records a wonderful demonstration by Feynman that Kepler’s laws follow from Newton’s laws plus ordinary, high-school plane geometry: no calculus required. No differential equations, no angular momentum, no constants of integration. This is Feynman at his best: reducing something seemingly big, complicated, and difficult to something small, simple, and easy. In this PDF file (http://home.comcast.net/~brianbec/Feynman.zip), I present an original reconstruction of this argument, accompanied by several documents for the application Geometry Expressions. This is a very powerful constraint-propagation and symbolic-manipulation program specialized to plane geometry. It was, in fact, that an old friend of mine, Van Warren of Warren Design Visions, brought Geometry Expressions to my attention that stimulated me to recall Feynman’s Lost Lecture and attempt this reconstruction.

My presentation is very short, on purpose, but omits no detail. To get a more lengthy presentation that delectates on every precious morsel, PLUS an audio CD recording of the man himself delivering the lost lecture, see the book referenced at the top. You won’t regret it.

Here are Kepler’s three laws, in the order that we prove them:

K1: A planet orbiting a star sweeps out equal areas in equal times
K2: The square of the period of a closed orbit is proportional to the cube of its longest axis
K3: A closed orbit is an ellipse (open orbits are parabolic or hyperbolic)

Now, here are Newton’s laws (specialized to the fixed-star assumption and numbered as we need them)

N1: A body (planet) in motion continues in straight-line motion unless acted on by a force
N2: The change in velocity of a planet over a time interval is proportional to the force applied (N1 is really a special case of N2 with force equal to zero)
N3: The force between a planet and a star acts along the line connecting them (central-force law)
N4: The force between a planet and a star is proportional to the inverse square of the distance between them (inverse-square law)

Now, get the downloads to get the rest of the story :)