Re: Gravitational redshift (potential well) query?
- From: Greg Egan <gregegan@xxxxxxxxxxxxxxx>
- Date: Sun, 13 Apr 2008 18:49:15 +0000 (UTC)
On Apr 12, 9:30=A0pm, WG <wgilm...@xxxxxxxxxx> wrote:
If you take a sphere of a given Radius (R) and Density (D) and have a
photon travel out from the center, it will experience a gravitational
redshift according to Zg=4.19GDr^2/c^2 =A0(where =A0r is the distance
traveled from the center and is less than R at this point, =A0 r < R ).
This is a standard photon climbing out of a potential well problem
taken from physics textbooks.
{To better visualize it, lets say we are at the center of the earth
and shoot a photon back towards the surface through a open shaft}
Lets assume the sphere is embedded in a background density of D1.
{Which in the earth case would be the density of the universe, (i.e.
critical Density), ignoring local lumpiness} =A0Now distance R or
density D1 do not come into play here due to Gausses theorem. This is
reflected in the equation since neither is present. In other words we
should be able to vary R and D1 to any value and the equation should
still hold true.
Now my question is this:
Would we still get a redshift as D1 approaches D and eventually
equals D?
The answer appears to be "yes it has to" since the photon can never
know about (i.e. feel the effects of) values D1 or R due to Gausses
theorem!
But this gives rise to an interesting problem if D=D1.
You may see it already.
Bill Gilmour =A0wgilm...@xxxxxxxxxx
You're mixing Newtonian and General-Relativistic treatments, for a
problem that can only be treated precisely with GR.
If you want to use Newtonian physics, the background of density D1
can't be infinite. If you choose some finite size for it (e.g. a
large sphere centred on the smaller sphere), you'll get a result
perfectly consistent with Gauss's theorem.
If you want to use GR, then the universe can be infinite if you like,
but it must be expanding or contracting. The redshift in a perfectly
homogeneous and isotropic universe follows a simple formula: the
ratio of photon energies is the inverse of the ratio of length scales
for the universe at the times of measurement. Inserting a localised
body of higher density is a complication that would be messy to solve
for exactly, but at the transition that you're describing -- where the
extra density is only infinitesimally above the background -- it's
obvious that the change in redshift will be a continuous function of
the size of the density perturbation.
.
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