Spinning Bricks and the Bending of Light by Gravity
Geoffrey A. Landis
writes:
>>>General relativity predicts a wee bit of dispersion
>>[wavelength dependence of the bending of light by gravity.]
>
>>but this must be a very tiny, very high order effect, because the
>>experimentalists all take gravitational lensing to be completely
>>wavelength independent.
>
>Oh, of course. The effect must be ridiculously small in the case you
>mention. I guess it's proportional to the wavelength of light divided by the
>radius of the galaxy that's doing the lensing, or some power of that, or
>something like that. To make the effect big we want long wavelengths of
>light and a very small massive gravitating body... but I don't expect
>we'll ever actually see it.
>
>I'm sorry if I caused any confusion here. Remember, I have a certain
>fascination for issues of principle, so I like to point out effects that
>must exist even if they are much too small for any sane person to worry
>about.
Say, no problem. I think it's important, too. What causes the effect?
>Once on sci.physics I noted that a spinning brick in the depths of
>intergalactic space would eventually slow to a halt due to the emission
>of gravitational radiation. To me this is fascinating simply because
>it's not true in classical mechanics.
Actually, that *is* rather fascinating.
>But I got some flack for it,
>because it would take a ridiculously long time to happen.
>
>In fact, someone had fun trying to work out what would actually make it
>spin down first: the emission of gravitational radiation, tidal coupling
>to galaxies, or friction with the occaisional atom wandering about in
>intergalactic space. I forget which effect won.
I'm sure that it's friction with the occasional atom. The other effects
are so small that, well, they redefine the meaning of small.
>Here's a silly question I just thought of. Say you have a brick in a
>perfect vacuum, at some very low temperature, but not absolute zero.
>Wouldn't the random motion of its molecules occaisionally concentrate
>enough kinetic energy in one near the surface to make that molecule
>break free and fly off? So, wouldn't the brick very very gradually
>dissipate?
Sure, as long as the brick is connected to a heat source to replace the
energy lost (the effect you are calculating is evaporation, and connected
to that is the related effect of evaporative cooling.) You don't need
any particularly sophisticated calculation to figure how much; it comes
directly out of the statistical mechanics.
Say that the binding energy of an atom to the brick's surface is Eb.
Call it 2 electron volts. Then the fraction of atoms with energy of Eb
or greater is Exp[-Eb/kT]. If the brick is in thermal equilibrium with
the cosmic background radiation of 3 K, then kT is 0.25 meV, so this
fraction is Exp[-8000]. For reference, Exp[-8000] is about 10^-3478.
Energy is redistributed in roughly the amount of time it takes a phonon
to cross the atom, which is something like the phonon wavelength divided
by the speed of sound v. Speed of sound will be on the order of a
km/sec. lambda = hc/energy, so t = lambda/v = hc/vkT comes to something
like ten to the minus 14 seconds, if I did the calculation correctly.
(Actually, it's irrelevant whether I did the calculation correctly or
not, since a factor of 10^-3500 trumps any possible algebra error except
dividing by zero]. So the time constant for the brick to evaporate by a
factor of e will look like Exp[-8000]*10^-14 seconds, or about 10^-3464
seconds. That's 10^3457 years.
Unfortunately, on a time scale of 10^3457 years, the cosmic background
temperature is not constant. It is left as an exercise to the student to
calculate whether the brick evaporation converges if the background
temperature is modelled as (2.7 K) times (t * R(0)/R(t)).
>The question is not whether this happens anytime soon (it obviously
>doesn't), but whether it *ever* happens. If it did happen, maybe the
>spinning brick in intergalactic space would dissipate before it spun
>down. (The answer here might depend on the pressure of intergalactic
>gas.)
I will venture to say that the Poynting effect (momentum lost to doppler
shift of cosmic background radiation as it reflects from the brick) spins
it down in a much much faster time scale than either gravitational
damping or evaporation.
____________________________________________
Geoffrey A. Landis,
Ohio Aerospace Institute at NASA Lewis Research Center
physicist and part-time science fiction writer
john baez
>In article <4l15hs$6...@guitar.ucr.edu> john baez, ba...@guitar.ucr.edu
>writes:
>>>>General relativity predicts a wee bit of dispersion
>>>[wavelength dependence of the bending of light by gravity.]
>>>but this must be a very tiny, very high order effect, because the
>>>experimentalists all take gravitational lensing to be completely
>>>wavelength independent.
>What causes the effect?
Since MTW turns out to have nothing to say about this, I am afraid I'd
have to do some scary calculations to say more about what the effect is
like in detail! I may even be seriously confused about it. But my
idea was this: if you consider situations where we are scattering light
off the gravitational field of a point mass (for example a black hole,
or star), if the wavelength of the light is a lot smaller than the
Schwarzschild radius, we can use a geometrical optics approximation and
treat the light as particles following a geodesic indepedent of their
wavelength. MTW does show how to do this. But the geometric optics
approximation breaks down when spacetime is curving a lot at distance
scales comparable to the wavelength of the light. I would expect that
as it begins to break down, we would get some slight wavelength-
dependent aspects of how the light was scattered. This would show up
when its wavelength was comparable to the Schwarzschild radius.