Re: [Phys-l] how to explain relativity

[John Denker <jsd@av8n.com>]

On 06/19/2010 08:33 AM, Jeffrey Schnick wrote:

>   I think this is actually a doable experiment.
> Tunnel into a mountain to a location, point B, at which the acceleration
> reading on an accelerometer at rest relative to the mountain is the same
> as it is at the top.  Given two identical clocks sitting side by side on
> a lab bench,  simultaneously take one to the top of the mountain and one
> to point B.  Leave them there for a year.  Bring them both back to the
> lab bench and compare readings.  

I reckon this is TOTALLY doable.  Very clever.

If it hasn't already been done, it would be fun to do.

> I think they'll be the same.  

I think not.

Here's the easy way to think about it:  First, the mountain
is not even necessary.  The analysis is easier without the
mountain, although the actual experiment is easier with.

So imagine a spherical earth, with a tower going up and a
borehole going down.  Ignore the mass of the tower and
the (missing) mass of the borehole.  For any given height
of tower, there will be a depth of hole such that the
gravitational acceleration will be the same at both places.
Outside the earth the acceleration goes like 1/r^2 whereas
inside the earth it goes like r.  The |g| field is greatest
right at the surface, and it decreases if you go up the 
tower or down the hole.

Meanwhile, the potential follows a slightly different law.
Outside the earth the potential goes like 1/r, whereas
inside it goes like r^2.

  Remember that inside the earth, the field and potential
  depend on the mass /inside/ radius r;  the shell of mass
  outside this radius contributes nothing.

Theory says the gravitational redshift depends directly
on the potential -- not on the acceleration per se.  In
the scenario under discussion, matching the accelerations
will not match the potentials;  not even close.

>  Leave them there for a year.

I reckon you would see a convincing signal in the first
week.
  http://leapsecond.com/great2005/tour/

> Also, which clock will be ahead of the other?

The clock that is higher in the potential will gain time
relative to the lower one.

It may help to visualize photons falling the potential,
down from the upper clock to the lower clock.  Each 
such photon will be blue-shifted.  If the photon stream 
is modulated to indicate the upper clock's notion of 
time, à la WWV, then the lower observer will conclude 
that /everything/ happening upstairs is happening faster.

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

Using a mountain saves you the cost of building a tower.
Also the mass of the mountain helps a little bit.

If you can't find a pre-existing tunnel at exactly
the right height, you should be able to bracket the
desired height and interpolate.