Horace Heffner wrote:
On Jan 23, 2006, at 5:12 AM, Stephen A. Lawrence wrote:
No, I haven't, because, as stated elsewhere, clocks in GR are
apparently affected by gravitational _potential_ but not by the local
intensity of the gravitational _field_.
From a QM point of view this is utter nonsense. Field potential is
merely a calculation device.
Absolutely. I agree. It's not a "field" in relativity either, and the
"gravitational potential" doesn't behave like a sensible "potential".
It's just a convenient way to think of it, and it works pretty well in
the low-curvature (Newtonian) limit.
A "field" is something that can be represented as a mathematical tensor
field and in GR, gravity certainly can't be. Einstein tried hard to
make that work before he abandoned it, or so I've been led to believe.
[regarding that spherical hole cut in a larger uniform sphere:]
[SAL]
I don't know if I understand you. Light should be redshifted as it
crosses the chamber, in proportion to the intensity of the field in
the chamber, right?
[HH]
Since the amount of red shift is a function of g, the change in red
shift is a function of the change in g as movement occurs.
If that's what you're saying, I agree.
[HH]
I'm not sure we even agree on the g field in the bubble being uniform.
As you move across the bubble you become "outside" a larger and larger
sphere of material.
:-)
Yes but you're outside a larger and larger "bubble", too. It's a very
cute example; I ran across it here:
http://www.geocities.com/physics_world/gr/grav_cavity.htm
Unfortunately it looks like the article has suffered an editing error
(or six!) since the last time I saw it. It's completely illegible, at
least in my browser, and the main illustration's gotten roached. Darn.
(The page author's been having a rough time of it lately, I'm afraid,
and hasn't had a lot of energy to worry about the state of his website.)
Luckily I had stashed a copy of the (undamaged) page, to which I just
referred to refresh my weak memory of the proof. The way to work this
one out is to look at the field from an "intact" sphere, and _subtract_
the field due to the sphere we cut out to make the chamber. By the
principle of superposition this is legit in Newtonian gravitation
(doesn't quite work in GR, of course).
The field at any point inside a uniform sphere of density rho is
F = -(4/3)*pi*G*rho*R
where "R" is the _radius vector_ from the center of the sphere to the
point where we're finding the field.
For the big sphere, let the radius vector be R1. For the small
(cut-out) sphere let the radius vector be R2. (Note that they point
from different origins, but that's OK, all we care about are the
direction and length.) Then the net field anywhere inside the small
(cut-out) sphere will be
F(total) = -(4/3)*pi*G*rho*(R1 - R2)
But (R1 - R2) is a _constant_, and is just the vector from the center of
the big sphere to the center of the small sphere.
So the force is also a constant, proportional to the distance between
the spheres' centers, pointing along the line which connects the small
sphere's center to the big sphere's center.
According to GR the exact field won't be _quite_ uniform, I'm sure.
