Which way does light travel?

How fast does light travel? This question is not nearly as old the more fundamental question: Which way does light travel?

Believe it of not for most of history people argued it traveled not from the object to your eye– but from your eye to the object. This seems absurd to us, living in post-Cartesian scientific world. Compared to the heliocentric vs. geocentric battles, this one is lesser known to us.

But I’ve realized that relativity has an interesting take on the question. In relativity, the space-time interval for light is minimal. In the screwy Minkowski metric of 4-dimensional spacetime (which very loosely puts the time dimension as imaginary coordinate axis) the pythagorean distance between two events is:

sqrt[dx^2 - c^2*dt^2]

Which looks like the Pythagorean theorem except that the time signature is negative, -cdt^2. Since dx=c*dt for light, clearly this interval is always zero. In some sense the “path” between object and eye is connected by an interval (in 4d spacetime) of zero length. So the question of which direction the photons travel is moot. The eye and object have zero interval between them as far as the light ray’s journey is concerned. Thus it is just as meaningful (or meaningless) to say light travels from object to eye, as eye to object.

I realize part my point is just taking advantage of some loose semantics in relativity, but I found it interesting.

ps, for a good visual explanation of this “zero” interval along lightray paths, look at Roger Penrose’s wonderful book, A Complete Guide to the Laws of the Universe.

Circular motion (for Greeks)

I’m really thinking of writing (someday) a retro physics textbook for greeks. No trig. Just similar triangles, etc. Doodling w/ circular motion, and some old text I put together a few things. E.g., the centripetal acceleration:

a = v^2 / r

which always seemed a bit odd, it just an obvious consequence of a geometric construction w/ triangles of any circular motion. Here’s webcam shot of my charkboard:

my chalkboard derivation

Bottom line: the triangles swept out by position vector (r) and velocity vector (v) are always perpendicular and similar. So it immediately follows that:

ratio: delta_v / v

is same as

ratio: delta_r / r

Dividing both sides of ration by delta_t gives:

a/v = v/r

or

a = v^2 / r
Q.E.D.

As fun corrollary the a/v = v/r step means that v is the geometric mean between a and r. In more modern notation it means v = sqrt(a*r), or v is the “mean” which harmonizes the lengths a and r. Geometric mean is old greek way of doing means or averages instead of our modern arithmetic mean of adding numbers and dividing by 2. Moderns would say the mean of a and r is (a+r)/2, using arithmetic operation (+/-) instead of geometric operations (viz., multiplication of a and r, then rooting by 2, instead of dividing by 2). My son, Ray, was learning this mean in his freshman highschool geometry class, so I got to rediscover this while helping him w/ homework. How soon we sweep this math under rug to make room for calc and trig.

Perhaps an alternative *definition* of circular motion is: that ideal motion where the velocity is the geometric mean between i) acceleration towards center and ii) distance from center.

High Mass and Energy

How many places in Houston– let alone the planet– can you experience in a 4hr span:

- A Latin Mass.

- One of the 10 great experiments in physical science.

I did this week. The usual noon mass at St. Thomas was in Latin today. Ten member choir Gregorian chanting every intercession, response, song and prayer. The priest said the his part in Latin also. It is an amazing slow, downshifted and lovely way to experience a mass.

Afterwards I wolfed some lunch and went to my General Physics lab. The experiment was a classic: The Mechanical Equivalent of Heat. We moderns (including me) not only don’t appreciate Latin, but don’t really appreciate the punch of the Conservation of Energy Law. We really didn’t have it until the 50years from Count Rumford’s famous military research paper on boring cannons (1798) to James Prescott Joule‘s measurement of “J” the equivalence (or conversion factor) between energy units and caloric (calorie) units. In our lab, we actually agitate water with mechanical energy and then measure the resulting heat. Dividing the two gives:

J = 4.186 Joules/cal

And one of my lab groups actually got the accepted value to within 0.05% error !

I’ve been alive about 400,000 hours on this planet. But in those mere 4hours (0.001%?!) I experienced such richness. Bless this new job.

Kepler’s laws and Atwood machines

Had great weekend. Saturday afternoon I “proved”
Kepler’s harmonic law from Newton’s laws and Euclid’s geometry (similar triangles crucial). Quite a pump to my ego. My brain felt like a stallion!

Last night I went on to other chapters, which had these problems with pulleys and weights on pulleys. I spent all night and couldn’t solve a simple pulley! My brain struggled– how could this simple machine be more complex than the solar system!!! (Atwood machine: 2 masses on pulley, where the pulley itself has a mass and moment of inertial so problem isn’t totally trivial). I finally solved it at 1am, and realized how beautiful this machine really is. It’s supposedly how/why elevators in buildings work, among I imagine 100′s other machines going back to Alexander and his armies, etc.