Galileo's ship

I've seen debates about whether Galileo actually conducted certain experiments or not.  On page 297 of The Tuscan Artist, Greco says that the ship experiment, in which an object is tossed straight upward on a steadily moving ship, only to fall at the feet of the person who tossed it, was performed in 1624. I hope it's true.  He cites some works be L. Conti (1990), which were written in Italian.

Einstein's Clocks, Poincare's Maps

I'm reading Einstein's Clocks, Poincare's Maps: Empires of Time by Peter Galison.  It's about the preludes to the theory of relativity.  I won't blog the book in detail, but I want to record a few observations so I remember what I read.

Time and space measurements were a big deal in the 19th century.  Telegraphs enabled people to communicate rapidly, which both meant that people could exchange clock readings and that they'd want to (so that they could time-stamp communications).  Trains needed precise timing, so that if a schedule said you'd arrive at noon you knew if that meant when the sun was overhead at the arrival station or at some central hub where the schedules were being set.  And longitude measurements (which required determining the time when you observed a celestial object at a particular position in the sky) were crucial for navigation and also for treaties between colonial empires.  As a result, mathematician Henri Poincare (who was also an engineer involved in a lot of issues of timing and longitude) put a lot of thought into the notion of times and distances being the products of defined human procedures rather than immutable and absolute features of the universe.

On page 200 I learned that Poincare delivered a presentation at a 1900 philosophy conference and questioned whether the science of mechanics needs reformulation.  He questioned absolute time, absolute position, simultaneity, and even whether Euclidean geometry was just a linguistic convention.  As strange as the last one sounds, he was heavily involved with geometry on spherical surfaces (navigation and surveying) so he was used to the idea of very real problems of a very real world requiring geometry on curved surfaces.

Furthermore, he was in dialogues with Lorentz and others, who were questioning whether electromagnetism could be fixed by redefining space and time.  The Lorentz transformations were known before Einstein, but people used these formulas in a conceptual framework that distinguished between absolute time and the timing of things relative to the ether.  Einstein's leap was to do away with ether, not to invent these formulas de novo.  I was aware of this part previously, but the book fleshes out some of the timeline and correspondence.

There's also a lot of geopolitics that I can't bring myself to care about.

A fun little argument in relativity

Confession:  I know next to nothing about general relativity.  My graduate work was in materials and optics.  My current research is mostly on optics and biophysics.  I enjoy the elegance of special relativity, but I never studied general relativity.

In a couple weeks I'm teaching students about Newtonian mechanics in non-inertial reference frames, and I felt like I should try to learn at least a few tidbits of general relativity.  I wanted to understand gravitational time dilation, so I came up with a nice little argument that I'm quite proud of.

Suppose that a pair of particles collide and produce two photons.  One photon goes left, the other goes right.  We use mirrors to send them upward (i.e. to a place to higher gravitational potential) and then recombine them.  The photons collide and produce a pair of particles of the same type as the original particles.  (Such things can happen, though the cross-sections are small.)  If the photons did not change their frequencies, i.e. did not lose some energy, then we have a new pair of particles at higher gravitational potential energy but with the same kinetic energy. We have gained energy. We can let those particles fall and extract energy from the system to power machines...for free.  We have thus produced energy from nothing, and that's not allowed.

The photons must thus lose some energy, i.e. must change their frequencies. Say that the kinetic energy of the new particles is zero, i.e. mgy(final)=KE(initial)

The frequency shift can be found via:

KE(initial) + 2m = 2*omega (initial) = 2m + 2mgy

And 2m must also equal 2*omega(final), since the two photons have just enough energy to produce the particles, so we get that 2*omega(initial) = 2*omega(final) + 2*omega(final)*gy

omega(final)*(1+gy) = omega(initial)

omega(final) = omega(initial)/(1+gy) or approximately omega(initial)*(1-gy) (to first order)

(We are working in units where hbar and c are equal to 1.)

So the fractional frequency shift has to be of order gy/c^2.  Once we have the frequency shift, we can argue that clocks based on oscillations of EM fields must run slower lower in the gravitational field, since the people above them are receiving consecutive ticks at longer intervals.

I will present this at the end of my lecture on Newtonian mechanics in non-inertial frames, along with the argument that a guy in a falling elevator sees light curve.