Re: Cosmological redshift

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

I'm sorry I have not had a chance to respond to this thread earlier due to a
too busy current schedule.  Let me point out that there was a very
interesting (IMO) discussion on phys-l about cosmological matters (Hubble
expansion, red shift, etc.) which raged from 5 JUL to 16 JUL 1996.  It went
under the subject titles of 'Old Stars' and 'Old Stars/Olbers Paradox'.

(I have compiled a collected and saved a chronological file of this
discussion.  If anyone is interested in it I'm willing to email you a copy.
It is much easier to read than trying to sort through the phys-l archives of
over a meg of 7/96 posts looking for each of the relevant posts.  This 'Old
Stars' file is about 112k long.)

Regarding Margaret Mazzolini's question:
> If the universe is expanding as a cosmic microwave background photon is 
> travelling through space-time, then I am happy to accept that the 
> photon is "stretched" (if not, would it be blue 
> shifted?)  However as space is being stretched, why isn't the metric 
> being stretched too, in which case a photon whose wavelength was 
> once one centimetre, for example, would still measure a centimeter now?
> (Because today's 1 cm would be bigger than the original 1 cm by the same 
> factor as the amount by which the photon has been stretched.)
> Analogies such as the expanding balloon don't help, because a grid on the 
> balloon stretches the way I imagine the metric to stretch - wrongly, no 
> doubt.  

Recall the definition of the meter: (1/299792458) of the distance
that light travels in 1 second.  The standard of length is tied to the
defined factor c. As the universe expands the value of c, being defined as
a particular constant, does not change.  Since c stays the same throughout
the expansion process, the length of the meter (and thus the centimeter
above) also remains constant.  Because the space through which the photon
is propagating *is* stretching the photon's wavelength is being stretched
en route.  Suppose the photon above once had a wave length of 1 cm, and
while it was propagating the scale factor of the universe increased to 1000
times its former size.  When the photon was young there would have been
29979245800 cycles of the photon over the distance that the photon covers
in one second.  After the universe aged so much that the scale factor is
1000 times the value it had when the photon was emitted, then the photon
would have only fit 29979245.8 of its stretched cycles over the distance
that it covers in 1 second.  Thus the wavelength has objectively increased
to 10 meters from its inital value of 1 centimeter.  The key to the solution
to Margaret's question is that the standard of length does *not* change as
the universe expands.

Leigh wrote:
> I'm afraid that I am not a cosmologist either, but I'll have a go at this.
> David Bowman can probably shoot it down where it is excessively naive, but
> I don't mind erecting exceedingly flammable straw men for him.

Sorry, but I elect to decline to torch Leigh's straw man.  I agree with
everything *except one thing* that Leigh said in his expanding box of
normal modes/receding piston analogy for the photons.  The only part where
I differ with Leigh is concerning the relevance of his analogy/model to the  
problem of the Hubble expansion of the universe.  In his analogy the
walls of the container are receding and regions of space that formerly were 
outside the container later become interior to the container.  The space in
the container grows via the transfer of space across the boundary walls
of the container (or equivalently, via the motion of the boundary walls
across the preexisting space).  In the case of the Hubble expansion of the
universe, however, there is no preexistant space external to the system
which is invading the system from the outside via the outward motion of
boundaries.  Rather, the unbounded (in the sense of having a boundary, not
in the sense of amount of total volume) system continually has evermore new
space being created in situ locally throughout the system.

> ....
> I must hasten to add (David will probably not like this) that photons can
> be written out of the script entirely. This effect is purely classical. It
> seems to be fashionable now to write in a part for photons these days even
> when it is gratuitous.

What do you mean I will not like that?  I agree with this 100%. There is
absolutely no need for having the electromagnetic field possess only
quantized excitations when describing the effects of Hubble expansion. 
Leigh correct here; "the effect is purely classical".

> Any photon which is emitted by a radiative process after decoupling
> also shares in the space expansion. Thus a photon emitted early
> (one with a large red shift) and received by us some considerable
> time afterward will have its apparent wavelength increased by both
> the doppler effect due to its recessional velocity and by the
> expansion of the space itself. This latter part is what is referred
> to as a cosmological red shift.

I assume Leigh must allow a negative Doppler shift here corresponding to a
blue-shifted radial velocity of approach as well as the recessional velocity
indicated above (relative to the source and receiver locations expressed in 
comoving coordinates, i.e. either an increasing or decreasing difference in
comoving coordinates between the source's and the receiver's locations).
The Doppler effect (due to relative motions of the source and receiver wrt
fixed comoving coordinates) can add either negatively or positively to the
redshift due to the overall Hubble stretching of space whose essential
property is that the distance between two spatially separated points grows
when those points have fixed locations as expressed in comoving coordinates.

BTW, an object is thought to have *fixed* comoving (spatial) coordinates if
the cosmic background radiation appears essentially isotropic in the local
frame for which the object is at rest.

It should also be noted that the Hubble expansion of the space of the
universe is not spatially homogeneous.  The idea of uniform stretching is
only an artifact of the homogeneous dust-filled Friedman-Robertson-Walker
models used to describe the universe on the largest length scales.  The
Hubble expansion of space only takes place in deep intergalactic space far
from the local concentrations of matter in the galaxies which produce their
own local gravitational fields whose corresponding spacetime curvatures that
dominate over the weak Hubble spacetime curvature producing the expansion of
space through time and act to locally counteract the overall expansion.  The
model is *not* one of spots painted (with latex elastomeric paint) on a
stretching rubber sheet or surface of an inflating balloon whose spot size
grows uniformly as the rubber stretches and the distance between spots grows
with time.  The model is more one of an inhomogeneous rubber sheet or balloon
whose thickness is mostly uniform but for which there are many small
separated local concentrations of thick rigid rubber that are not elastic and
do not stretch.  It is these thick spots that are to be painted as the
galaxies.  Actually, a (still imperfect) 2-d (in space) model which sort of
has the local clustering of matter built in which produces the original
galaxies is one where the rubber sheet/balloon is initially painted with a
paint that drys with a hard inelastic texture.  As the underlying rubber
stretches this paint cracks and ruptures into small local painted regions
that separate as unpainted regions of stretching rubber grow between them.
As the stretching process continues these pieces continue to fragment into
smaller pieces.  (A down side to this model is that we can't allow for any
paint chips to flake off of the stretching rubber surface.)

David Bowman
dbowman@gtc.georgetown.ky.us