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Roko comments on Open Thread June 2010, Part 3 - Less Wrong
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I'd like to pose a sort of brain-teaser about Relativity and Mach's Principle, to see if I understand them correctly. I'll post my answer in rot13.
Here goes: Assume the universe has the same rules it currently does, but instead consists of just you and two planets, which emit visible light. You are standing on one of them and looking at the other, and can see the surface features. It stays at the same position in the sky.
As time goes by, you gradually get a rotationally-shifted view of the features. That is, the longitudinal centerline of the side you see gradually shifts. This change in view could result from the other planet rotating, or from your planet revolving around it while facing it. (Remember, both planets emit light, so you don't see a different portion being in a shadow like the moon's phases.)
Question: What experiment could you do to determine whether the other planet is spinning, or your planet is revolving around it while facing it?
My answer (rot13): Gurer vf ab jnl gb qb fb, orpnhfr gurer vf ab snpg bs gur znggre nf gb juvpu bar vf ernyyl unccravat, naq vg vf yvgreny abafrafr gb rira guvax gung gurer vf n qvssrerapr. Gur bayl ernfba bar zvtug guvax gurer'f n qvssrerapr vf sebz orvat npphfgbzrq gb n havirefr jvgu zber guna whfg gurfr gjb cynargf, juvpu sbez n onpxtebhaq senzr ntnvafg juvpu bar bs gurz pbhyq or pbafvqrerq fcvaavat be eribyivat.
The universe adheres to General Relativity, not Newton's laws. What does GR say about the effect of spinning and revolving bodies?
Relativity says that as motion becomes very much slower than the speed of light, behavior becomes very similar to Newton's laws. Everyday materials (and planetary systems) and energies give rise to motions very very much slower than the speed of light, so it tends to be very very difficult to tell the difference. For a mechanical experimental design that can accurately described in a nontechnical blog post and that you could reasonably imagine building for yourself (e.g., a Foucault-style pendulum), the relativistic predictions are very likely to be indistinguishable from Newton's predictions.
(This is very much like the "Bohr correspondence principle" in QM, but AFAIK this relativistic correspondence principle doesn't have a special name. It's just obvious from Einstein's equations, and those equations have been known for as long as ordinary scientists have been thinking about (speed-of-light, as opposed to Galilean) relativity.)
Examples of "see, relativity isn't purely academic" tend to involve motion near the speed of light (e.g., in particle accelerators, cosmic rays, or inner-sphere electrons in heavy atoms), superextreme conditions plus sensitive instruments (e.g., timing neutron stars or black holes in close orbit around each other), or extreme conditions plus supersensitive instruments (e.g., timing GPS satellites, or measuring subtle splittings in atomic spectroscopy).
And the example I posited is a superextreme condition: the two bodies in question make up the entire universe, which amplifies the effects that are normally only observable with sensitive instruments. See frame-dragging.
Amplifies? The Schwarzschild spacetime (which behaves like Newtonian gravitational field in large distance limit) needs only one point-like massive object. What do you expect as a non-negligible difference made by (non-)existence of distant objects?
The fact that there's no longer a frame against which to measure local rotation in any sense other than its rotation relative to the frame of the other body. So it makes a big difference what counts as "the rest of the universe".
People believed for a quite long period of time that the distant stars don't provide a stable reference frame. That it is the Earth which rotates was shown by Foucault pendulum or similar experiments, without refering to outer stellar frame.
(two points, one about your invocation of frame-dragging upstream, one elaborating on prase's question...)
point 1: I've never studied the kinds of tensor math that I'd need to use the usual relativistic equations; I only know the special relativistic equations and the symmetry considerations which constrain the general relativistic equations. But it seems to me that special relativity plus symmetry suffice to justify my claim that any reasonable mechanical apparatus you can build for reasonable-sized planets in your example will be practically indistinguishable from Newtonian predictions.
It also seems to me that your cited reference to wikipedia "frame-dragging" supports my claim. E.g., I quote: "Lense and Thirring predicted that the rotation of an object would alter space and time, dragging a nearby object out of position compared with the predictions of Newtonian physics. The predicted effect is small --- about one part in a few trillion. To detect it, it is necessary to examine a very massive object, or build an instrument that is very sensitive."
You seem to be invoking the authority of standard GR to justify an informal paraphrase of version of Mach's principle (which has its own wikipedia article). I don't know GR well enough to be absolutely sure, but I'm about 90% sure that by doing so you misrepresent GR as badly as one misrepresents thermodynamics by invoking its authority to justify the informal entropy/order/whatever paraphrases in Rifkin's Entropy or in various creationists' arguments of the form "evolution is impossible because the second law of thermo prevents order from increase spontaneously."
point 2: I'll elaborate on prase's "What do you expect as a non-negligible difference made by (non-)existence of distant objects?" IIRC there was an old (monastic?) thought experiment critique of Aristotelian "heavy bodies fall faster:" what happens when you attach an exceedingly thin thread between two cannonballs before dropping them? Similarly, what happens to rotational physics of two bodies alone in the universe when you add a single neutrino very far away? Does the tiny perturbation cause the two cannonballs discontinously to have doubly-heavy-object falling dynamics, or the rotation of the system to discontinously become detectable?
How would you measure the centrifugal force?
ETA: I'm not asking because I don't know the standard ways to measure cetrifugal force, I'm asking because the standard measurement methods don't work when the universe is just two planets.
Calculate the gravitational force on the surface of a planet of the same size and mass as yours and compare with what you actually measure.
What do you calibrate your equipment against?
The equipment is already calibrated. You have said that everything works in the same way as today, except the universe consists of two planets. Which I have interpreted like that the observer already knows the value of the gravitational constant in units he can use. If the gravitational constant has to be independently measured first, then it is more complicated, of course.
Right: you know the laws of physics. You don't know your mass though, and you don't know any object that has a known mass. I posit this because, in the history of science, they made certain measurements that aren't possible in a two-planet universe, and to assume you can calibrate to those measurements would assume away the problem.
But still, in the rotating scenario the attractive force wouldn't be perpendicular to the planet's surface, and this can be established without knowing the gravitational constant. If the planet is spherical and you already know what is perpendicular, of course.