Spoonfed wrote:
Ben, you successfully identified my model as an Omega = 0 model. To
first order, it matches the diagram in Ned Wright's Cosmology page
http://www.astro.ucla.edu/~wright/cosmo_02.htm
I'm glad to hear this, since it means that I might just understand it after all.
I disagree with his definition of the word "now" as using the event of
a distant galaxy reaching 13.7 billion years as a definition of our
"now" is completely at odds with Einstein's methods of defining
simultaneous events in Special Relativity.
This is the crux of the matter right here. The only point of talking about
simultaneous events in special relativity is to relate it to the Newtonian
worldview, where simultaneity is taken for granted. There are not multiple
notions of distant simultaneity in relativity -- there is *no notion of
distant simultaneity at all*.
I come back to this several times below.
As far as the FLRW metric goes, I meant a(t)=1 as in, it is constant.
If I understood right, Tom told me that the FLRW metric was the family
of solution to some differential equation when you assumed that the
cosmological constant was zero.
Yes, your function a(t) has to satisfy the equations given here, which work
even when Lambda =/= 0:
http://en.wikipedia.org/wiki/Friedmann_equation
But it gets even simpler than that. Not only do I assume that Omega =
0, but I also assume that of the many possible solutions available in
the family of FLRW metrics, I am choosing the very most simple one.
Then you are definitely wrong! FLRW cosmology is well understood. It has a
few adjustible parameters, which are constrained by astronomical
observations. If your theory is a particular parameterized version of FLRW,
then there cannot be anything new about it. Either it's excluded by the
evidence, or it's identical with the currently accepted big bang model.
If you take the FLRW general metric
ds^2 = dt^2 - a(t)(dr^2/(sqrt(1-k r^2)) + r^2
(d(theta)^2+sin^2(theta)d(phi)^2))
and set a(t)=1 and k=0, this becomes, (unless I've made a horrible
blunder)
ds^2 = dt^2 - dx^2 -dy^2 - dz^2
which is the definition of the differential space-time interval between
two differentially separated events.
But your model then violates one of the assumptions behind the FLRW
solution, namely that rho and p depend only on t, not on x, y, or z. In your
model rho is nonzero inside an expanding sphere and zero outside it.
There's a second flat FLRW solution, which you get by taking k = -1 and a(t)
= t. It is a different coordinate cover of the same (flat, SR) spacetime.
With respect to those coordinates, your rho and p *do* only depend on t (if,
as always, I understand your idea correctly).
Here's an SR conceptual question which may be pertinent. At one end of Main
Street is a clock tower. Alice is running along Main Street toward the clock
tower at a relativistic speed. Bob is standing stationary on Main Street,
looking at the clock tower. At the moment Alice passes Bob, they compare the
times they see on the clock face. Does Alice see an earlier time, a later
time, or the same time?
Alice and Bob see the same moment on the clock face. However, Alice
sees the clock-face further away, and measures that the event happened
longer ago than Bob measures it to have occurred.
I agree with the first sentence, but the second is iffy. Again, this is the
crux of the matter. What you see is physically real, but these inferences
about distance and time are to a large extent arbitrary artifacts of one's
choice of coordinates. I don't think you understand this yet. I didn't
really understand it until I took GR.
It is not by a conspiracy of length contraction and time dilation that Alice
and Bob see the same moment on the clock face. It is simply because they are
both detecting photons *locally*; they are in the same place, so they
necessarily detect the same photons. Drawing conclusions about the origin of
those photons (e.g. reflection off a clock face) is a very complicated
business. Our innate sense of distance, which is based on binocular vision
and atmospheric scattering and the known size of familiar objects and other
such cues, does not work well in the relativistic domain.
Ned Wright's page says there is a large excess of bright
galaxies in the "northern part of the sky" which I can only guess means
galactic north. This is the direction that I called "down" earlier.
What he says is, "Hubble [...] found approximately the same number of faint
galaxies in all directions, even though there is a large excess of bright
galaxies in the Northern part of the sky." What this means is that the
universe is anisotropic on a small scale, but isotropic on a large scale.
The local anisotropy around the Milky Way is typical of the local anisotropy
one would see from anywhere else.
I'm not sure what he means by "northern", but it may well be terrestrial
north rather than galactic north.
And YES, my theory says the distribution of matter is anisotropic in
the present era--at least the parts of it we can see. The dark areas,
I believe, are still isotropic--undisturbed from the original
explosion.
Imagine for the moment that our present worldline pointed straight back to
the big bang. Would the universe then appear isotropic to us at large
scales, in your model? This is a physically meaningful question, so it
doesn't depend on coordinates -- you're free to analyze it with respect to
SR inertial coordinates. Your first impression might be that it won't appear
isotropic if we're near the edge of the expanding sphere, but if I
understand your theory, a careful analysis will show that the universe will
appear isotropic no matter where we are. Our motion with respect to the CMBR
cannot change this -- see below about the 600km/sec boost.
I do see that gravitational lensing
actually happens, but I have my doubts that gravity can effect the
redshift of passing photons.
Considering those two different coordinate covers of flat space may help. In
one, the redshift is explained by the SR formula. In the other, it's
explained by the change in the scale factor between emission and absorption.
This equivalence is a mathematical fact which doesn't depend on any
additional physical hypothesis. Einstein made the additional physical
hypothesis that every gravitational effect can be understood in the same
way, and he seems to have been right.
(I shouldn't really say this, because there is a coordinate-independent
sense in which gravitational fields do exist.)
As my model does nothing to the scale factor of space, I would say that
distant galaxies should not appear larger than nearby ones.
Actually I've changed my mind: I'm pretty sure I was wrong, and your theory
does predict that distant galaxies appear larger. :-) This is easier to see
if you use the FLRW coordinates, but since it's a physically real
prediction, you can in principle analyze it from SR inertial coordinates as
well.
The simplest difference I know of is that I predict that a 600km/second
change in velocity would not significantly effect a measurement of the
CMBR dipole. This is very much at odds with the explanation for the
dipole given by NASA.
But that's not even consistent with SR, let alone GR or the big bang theory.
A 600km/sec boost leads to Doppler shift and aberration *of your visual
field* which is completely independent of where that light originally came
from. The effect of a 600km/sec boost on the CMBR dipole is independent of
any cosmological assumptions. It only depends on local Lorentz symmetry.
http://www.astro.ucla.edu/~wright/cosmo_02.htm
I lose him when he defines D_now as any event on the same hyperbola
instead of on the horizontal plane. That would be fine if he just said
"interesting idea" and moved on, but he appears to use it throughout
the rest of the tutorial as though it were the actual distance. Is he
correcting for this error in judgment when he introduces the scale
factor?
Crux of the matter again. :-) It's not an error in judgment.
-- Ben