Re: Sink or swim?

["John S. Denker" <jsd@AV8N.COM>]

On 08/09/2003 02:16 PM, Tucker Hiatt wrote:
> Perhaps time dilation also plays a role in this relativistic buoyancy
> (!) problem.

Yes, it does.

> The buoyant force of the water on the sub depends partially on the
> collision rate of the water molecules with the sub's hull, doesn't
> it?

That's true as stated, although I'm not sure
that's the easiest way to analyze the issue.

Instead I would recommend thinking in terms
of momentum whenever possible.  If it becomes
necessary to think in terms of force, be
careful to decide whether you mean d(p)/d(t)
or d(p)/d(tau).

> Therefore, wouldn't time dilation (alone) argue for a *decrease* in
> buoyant force?

Right.  So to first order that cancels the
effect of the density change, when the fluid
is analyzed in the submarine's frame.

> to say nothing of Bernoulli effects

There shouldn't be any Bernoulli effects.
Buoancy has to do with hydrostatic pressure,
which is not affected by the motion of the
fluid.  The altimeter on an airplane infers
the altitude from the static pressure, which
is independent of airspeed.  For details see
  http://www.av8n.com/how/htm/airfoils.html#sec-static-stagnation

To help clarify the issues, it may be helpful
to consider a stream of cars driving through
an underwater tube.  If the weight of the cars
is just right, the cars+tube system is neutrally
buoyant.

> At 9:44 PM -0400 8/7/03, Ian Ellis wrote:
>
> > For your consideration, from Nature magazine
> > http://www.nature.com/nsu/030728/030728-3.html

The "relativistic bouyancy" problem is very
tricky.  There are eleventeen things going
on.  The squib in Nature hardly even hints
at the real issues.  Anybody who is tempted
to look into this should get the Phys Rev D
article by Masdas.  It's downloadable form
arxiv.org.

As an interesting warm-up exercise, try the
following:

Suppose we have a fleet of N submarines,
all identical, length L, initially at rest,
single file, abutting nose-to-tail, so the
nose-to-nose distance is L. Each is painted
with 5 stripes, so from a distance we see
5 N equally-spaced stripes.  Then at time
t=0 in the lab frame, they all receive the
command to accelerate to velocity V.  They
accelerate smoothly and gradually, all the
same.  You might think that each stripe
would accelerate the same, so afterward the
5 N stripes would still be equally spaced,
but this is not the case.  By conservation
of submarines, in the lab frame their
nose-to-nose spacing is still L.  Their
proper length is still L.  So their length
as observed in the lab frame is less than
L.  That means there must be a gap between
the tail of one and the nose of the next.

Can you explain why a uniform acceleration
applied to an initially uniform stripe
pattern results in a non-uniform final
pattern?

Huge hint:  Draw the spacetime diagram for
N=2 submarines, stripes and all.

Hint:  Each sub is built out of mechanically
rigid materials.  There is no mechanical
couping between subs.

This warm-up exercise is necessary if we
are to make meaningful comparisons between a
neutrally-buoyant submarine at rest and the
"same" submarine in motion.

============

Overall, this is a seriously tricky problem.
I haven't checked it carefully enough to be
100% confident in Supplee's answer (which
Masdas reobtains).  But neither do I have
any particular reason to doubt it.