Re: curvature of large buckets of water

["John S. Denker" <jsd@MONMOUTH.COM>]

At 03:13 PM 7/6/99 -0400, Robert A Cohen wrote:
> Why is it possible to tell if the bucket is off-center but it isn't
> possible to tell (if one is stuck inside a box) where one is relative to
> the axis of rotation (without more information given)?

*** It depends on the size of the box. ***

As the cute saying goes:

 * To zeroth order, all surfaces are horizontal.
 * To first order, all surfaces are flat.
 * To second order, all surfaces are paraboloids.

In this case:
 * At any particular point in space and time, all you can tell is that you are not in a free-falling reference frame.  This could be due to gravity, or due to brute-force acceleration of the box, or some combination;  you can't tell.
 * Given a slightly larger box and/or more time, you can detect nonuniformities in the field.  Again, these could be due to a peculiar distribution of gravitating sources, or to a non-rectilinear acceleration, or some combination.



Specifically:  It is highly edificational to consider the job of astronomer confined to a planet with a totally opaque atmosphere.  I claim such a person could determine the (mass and distance) of the (sun and moon), by measuring such things as the precession of gyroscopes and the height of the tides.



For extra credit:  Explain why the cute saying given above is not strictly true.  Give an example where that cute notion would be inconsistent with a modern, prize-winning achievement in physics.