THE METRIC OF REALITY

On Mon, 15 Jul 2013 06:07:26 -0400, George Hammond
wrote:



In the first place I'm not positively sure that there "is no
Hubble shift in a conformal metric". This is a mathematical
question that I am still digging into.

George


NOTE: SWITCH TO FIXED FONT TO PRESERVE EQUATIONS
NOTE: SWITCH TO FIXED FONT TO PRESERVE EQUATIONS
NOTE: SWITCH TO FIXED FONT TO PRESERVE EQUATIONS
NOTE: SWITCH TO FIXED FONT TO PRESERVE EQUATIONS
NOTE: SWITCH TO FIXED FONT TO PRESERVE EQUATIONS

[George Hammond]
Hello Tom Roberts:
I have found a MATHEMATICAL PROOF that there IS NO HUBBLE
SHIFT in the conformal metric given by:

ds^2 = a(t)^2 [-dt^2 + dr^2]

While this might be intuitively obvious it is nice to have
a rigorous mathematical proof.
The proof follows James B. Hartle's derivation of the
Hubble Shift for the conventional FLRW metric which he gives
on pp 369-370 of his 2003 textbook entitled GRAVITY. No
doubt you have a copy of GRAVITY sitting on the bookshelf
behind you!
ON pp 369-370 Hartle says:
The standard FLRW metric is:

ds^2 = -dt^2 + a(t)^2 dr^2

Then he says for a light beam ds^2 = 0 so that dr = dt/a(t)
or integrating that we have:

R t0
/ /
/dr = R = /dt/a(t) (Integration)
/ /
0 te

where te is the time of light beam emission and t0 is the
time of light beam reception and R is simply the coordinate
distance to the star (comoving coordinate).
He then goes on to say on pp 370 that we can imagine a
series of light pulses emitted from the star spaced by short
time intervals dte (i.e. with frequency fe = 1/dte ). Then
he says the time interval between the pulses at reception,
dto, can be calculated using the above equation. He says
that since all the pulses travel the same coordinate
distance R, that we can write:

to+dto to
/ /
/dt/a(t) = R = /dt/a(t) (eqn 18.8 p 370)
/ /
te+dte te

expanding the first integral on the left, we have:

to to
/ /
/dt/a(t) + dto/a(to)-dte/a(te) = R = / dt/a(t)
/ /
te te

from which we immediately see that:

dto/a(to)-dte/a(te) = 0 so that

dto/dte = a(te)/a(to)

or in terms of frequencies:

fe/f0 = a(te)/a(to)

OKAY, THIS COMPLETES HIS SIMPLE DERIVATION OF THE STANDARD
HUBBLE COSMOLOGICAL REDSHIFT. AND THIS RESULT IS FAMILIAR
TO ALL COSMOLOGISTS.

NOW WHAT I WANT TO DO IS REPEAT THE SAME EXACT CALCULATION
FOR THE "CONFORMAL METRIC" GIVEN BY:

ds^2 = a(t)^2 [-dt^2 + dr^2]

and see if there is a redshift or not. We begin by setting
ds=0 for a light beam so that we get:

(a(t)dt = a(t)dr or: dt=dr

Integrating as above we get:

R t0
/ /
/dr = R = /dt = to-te
/ /
0 te

again we consider a train of light pulses separated by dte
to be emitted by tghe star and recieved at the origin, and
following Hartle we have as above:

to+dto to
/ /
/dt = R = /dt = to-te
/ /
te+dte te

and expanding the integral on the left we have:

to+dto-te-dte = to-te

or: dto-dte=0 or dto=dte

or again, in terms of frequencies, 1/dt, we have:

fo/fe = 1

IN OTHER WORDS, THERE IS NO HUBBLE FREQUENCY SHIFT IN THE
CONFOMAL METRIC GIVEN ABOVE !!

QED GEORGE HAMMOND, via JAMES B. HARTLE