Re: [Phys-l] definitions ... purely operational, or not

[John Denker <jsd@av8n.com>]

On 11/08/2010 09:39 PM, William Robertson wrote:
> The person who is freely falling "feels" weightless, but by  
> our definition of weight as the force of gravity, the person is not  
> weightless.

Astronauts in the space station *are* weightless in the
frame comoving with the space station.  They are not 
weightless in the terrestrial lab frame.

The notion of weight is frame-dependent.  The notion of
gravity is frame-dependent.  Get used to it.

>  When we are clear and unambiguous with students, they  
> might be conflicted for a while, but they are not dissatisfied. And  
> unless I've completely forgotten my general relativity, there is no  
> weight in that theory. There is mass and there is curvature of space- 
> time. Objects follow their path in space-time. Unless they are  
> obstructed by an object such as the Earth, they are not accelerating  
> and they experience no force of gravity. I am prepared to be wrong on  
> that, because it's been a while.

1) Yes, it is time to "Stop Faking" general relativity.  GR 
leaves no doubt that the concept of gravity is frame-dependent.
Einstein's principle of equivalence says (roughly) that to 
first order, a gravitational field is indistinguishable from 
an accelerating reference frame.  This is perhaps the #1 most 
central concept in general relativity.

2) As a separate matter, objects in space *do* accelerate under
the influence of the gravitational field (even if there are
no collisions, no electromagnetic interactions, et cetera).

Consider two travelers on the surface of a sphere.  They
start out side by side with equal velocities, and follow
geodesic (great circle) routes.  They *will* accelerate
relative to one another.  Similarly consider two satellites 
in earth orbit.  They start out at different altitudes with
equal velocities.  They *will* accelerate relative to one
another.

You can reconcile item (1) with item (2) by writing the
gravitational field as a Taylor series.  The first-order
term is the baseline gravity and can be made to go away 
by application of Einstein's principle of equivalence.
Meanwhile, the second-order term is the tidal stress, 
which does not go away.  Yet-higher-order terms don't 
go away either.

To repeat:  geodesic world-lines that start out parallel 
do not accelerate relative to each other to first order
(which essentially defines what we mean by "parallel")
but they do accelerate to second order (if there is any
curvature).  All of the interesting physics is in the
second-order (and higher) terms.