RTfTrence: Re: Chomsky -- Put up or blah blah

Charles Brown CharlesB at CNCL.ci.detroit.mi.us
Fri Mar 31 09:09:05 PST 2000

Thanks for this one, doc.

>>> Les Schaffer <godzilla at netmeg.net> 03/31/00 10:43AM >>>
Chis Doss said:

> Once again, wait a second--doesn't the General Theory contradict the
> Special Theory on this issue? I thought that c was only a constant
> in non-rotating systems. This doesn't have anything to do with
> Newton being falsified or not, but I am curious.

Newton wasnt so much falsified really, his theory's limitations were understood and rectified.

re/ "c was only a constant" ==> one needs to be clear about how one measures the speed of something, ie, how do we measure space and time intervals (5 feet in 2 seconds is 2.5 ft/sec), assuming you know what you are doing when you say you've measured a 5 foot interval and a 2 second interval. einstein's special theory makes sure you sweat the details on this.

in particular, einstein established that the things we measure are space-time intervals, and that space and time seperately are only coordinates of that unified space-time.

re/ the special and general theory: the general theory removes the notion that spacetime is everywhere flat/cartesian-shaped, ie, the curious notion that matter somehow wouldnt affects spacetime's shape.


it blends perfectly with the special theory in the following manner: in every reference frame that is __freely falling__, that is, moving along the straightest possible lines in curved spacetime, locally in that frame, spacetime is almost Lorentzian (flat -- special relativity describes a flat space-time). Its like saying that if you look closely enough at a sphere, you can convince yourself that the small section you are looking as is not much different from a plane.

the general theory then dictates how one can make meaningful global comparisons of spacetime intervals in a way which honors the local properties we observe in laboratory tests: constancy of light speed, local flatness, etc. or to twist it a little differently, how do you add a bunch of planes together to build a sphere???

for fun, as a little experiment, grab a hold of a large spherical object and try to draw a small right triangle on its surface. see how close the pythagorean theorem holds:

hyp = sqrt( y^2 + x^2 )

or in a different format:

hyp^2 = y^2 + x^2

the equivalent result in special relativity is

tau^2 = c^2 t^2 - x^2

thats your "flat spacetime with constant speed of light 'c'", and just like in the preceding experiment of drawing a small triangle on a sphere and verifying pythagoreas, this "hyperbolic pythagoreas" holds true for small regions of curved space-time. that tau is called the space-time interval, and is what we "really" measure. but we are spoiled by flat cartesian geometries where coordinates and measurements share equal value numerically.

There is a great quote from Einstein about why it took "so long" to go from special relativity and his "happiest thought" -- imagining that someone falling freely in an elevator feels no gravitational force -- to his theory of general relativity:

"It is not so easy to free oneself from the idea that co-ordinates

must have an immediate metrical meaning,"

a buzzwordy summary of how special and general relativity mesh perfectly goes something like this: "flatness is a local property, curvature is a global property". in terms of gravity and einstein's happy thought, one person falling freely does not detect and gravity, but two people near each other falling freely can detect gravity by observing the interval between themselves. the 'one person' is the flat lorentz special relativity, and the 'two people' is the global check on spacetime curvature.

<aside> for more fun, draw triangles of increasing size on your sphere, and measure the sums of the interior angles. watch how the result slips away from 180 degrees as your triangles get bigger and bigger.

for pure joy, draw smaller and smaller circles on the sphere. comparing the ratio of their circumference to their radii, defined as the distance from the center of the circle to the circumference __along the curved sphere__. for "small enough" circles, you get 2*pi. </aside>

re/ non-rotating vs rotating systems. there is a general relativistic effect associated with rotating masses called 'frame dragging' and is currently being tested to higher and higher precision with satellite/laser-systems. briefly, a rotating mass not only warps spacetime in its vicinity, its spins it around a little bit. how don't know hot to explain this complicated effect in a simple manner.

incidentally, one can compute the anomolous precession of mercury's perihelion using epicycle theory and a first order approximation to einstein's general relativistic field equations. indeed this is the current way to patch up epicyclic theory: remove it from the realm of being a purely kinematical theory by considering dynamical perturbations from other bodies. this causes the parameters which describe the kinematics of the epicycles to slowly vary, slow, that is, relative to motion along the epicycle trajectory. so on one time scale, epicycles look okay, on a longer time scale, the epicycles gradually loose track of the motion, but in specific, often easily calculable ways.

then there are situations where epicycles just ain't no good.

btw: one result of general relativity was the more accurate "prediction" of the anomlous precession of Mercury's perihelion. the frame-dragging stuff has a parallel effect: the precession of a gyroscope's spin axis for orbiting gyroscopes close enough to __spinning__ masssive bodies.

gotta go... apologies to einstein and others for any quick slipups i mighta made above...

later maybe i'll do some more on that bad bad naughty false falsified really false bad epicycle theory and maybe hit the QM/Einstein quandry you raise. and michael pollak raises an interesting questions about climate change in history.

les 'observing business from left field' schaffer

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