Universe size

denism

thank you very much for your kind answer George. In fact, I was not particularly afraid by recession velocities higher than c given that this limit does not apply to space expansion, but I just wondered (for other reasons) if Vrec > c were really observed because H has been proven to be lower than H0 in the past (expansion accelerates)
I have an other question with respect to your answer: I wonder why SR is not sufficient for studying the geometry of expanding universe models, supposed to be isotropic and homogeneous as empty. General relativity deals with gravity?

Chronos

SR does not take gravity into account. That was the whole point of GR.

denism

Yes. But precisely, all what I read about universe expansion and horizons, begins with a redefinition of the line element of SR (ds²=(cdt)²+a(t)²dl²)

George Jones

I am not sure what you mean. How is this necessarily a redefinition of the line element of SR?

denism

Indeed I was not clear. I just wanted to say that authors talking about light connection rates in the expanding universe never make use of GR, but only derive their conclusions from the simple relationship cdt=a(t)*dl (note there is a typing error of sign in my previous equation). SR seems to be sufficient. Gravity and GR are generally not involved in the universe model used in these studies.

Furthermore, even SR seems to be not observed: calculations using speed substractions such as c-Vrec, rather resemble to classical mechanics ... even if I understood that Vrec is not a genuine speed.

qraal

There's no evidence - yet - that there's anything far beyond the present horizon. An infinite flat space-time is just the simplest topology to assume. Alternative theories exist for a range of different space-time structures which are consistent with the data, some of which are finite and some are actually smaller than the current apparent horizon. The problem with assuming the limits of what we currently see are the actual physical limits is the implication we exist in a privileged moment of history - a possibility that needs a theoretical explanation that'd convince other cosmologists or they'll stick to their current assumptions.

The limits of what we can see are about x3 times the current Hubble light-travel-time. A good place to learn about all the different cosmological distances is Professor Ned Wright's Cosmology Tutorial...
Ned Wright's Cosmology Tutorial
...which features a handy Java calculator of all sorts of cosmic parameters. So handy that people have referenced it in their papers.

denism

Many thanks for these informations


does "infinite flat" means euclidean 3D?

bapowell

"Flat" means Euclidean 3D. It need not be infinite.

denism

But how do you conceive a finite euclidean 3D space? with a peripheral boundary? looks strange

qraal

That requires some tricky topology I'm guessing :-)

bapowell

A torus.

denism

4D torus?

bapowell

You can think of it that way, yes. The best way to visualize it, though, is as a cube with opposite faces identified (by identified, I mean topologically connected -- for example, a circle is a line segment with the end points identified.)

denism

I have a naive perception of topology: the number of dimensions of any shape is that of the minimal euclidean space capable of embedding it. Your torus is 4D to me, even for something confined in its surface

the circle is a good example: unidimensional if envisioned from the interior, but in fact genuinely bidimensionnal

George Jones

[itex]ds^2 = 0[/itex] for a lightlike worldline in both special and general relativity.
This is just plain wrong. In order to use the equation in your post above, the dependence of [itex]a\left(t\right)[/itex] on [itex]t[/itex] is needed. This is given by the solution of the differential equation

[tex]
\left( \frac{da}{dt} \left(t\right) \right)^2 = H_0^2 \left( \Omega_{m0} a\left(t\right)^{-1} + \Omega_{r0} a\left(t\right)^{-2} + \Omega_{\Lambda 0} a\left(t\right)^2 + 1 - \Omega_{m0} - \Omega_{r0} - \Omega_{\Lambda 0} \right),
[/tex]
where the constants [itex]\Omega_{m0}[/itex], [itex]\Omega_{r0}[/itex], [itex]\Omega_{\Lambda 0}[/itex] are the current densities (relative to critical density) of matter, radiation, and dark energy, respectively. This equation comes from Einstein's equation of general relativity, i.e., it come form Einstein's theory of gravity.
With appropriate definitions of time and distance, c - V_rec is true in special relativity, and in the FRW cosmological models of general relativity

denism

sure, SR no way contradicts GR


I disagree with you, the scale factor has first been naturally postulated because of the observation of Hubble. You describe one of the multiple a-posteriori attempts to calculate the expansion rate(s) from the universe constituents: (matter/energy and now the more exotic dark energy). These attempts are very interesting from a physical viewpoint but please do not inverse the string. a(t) did not emerge from matter/energy density calculations but was just postulated a-priori. To my knowledge its time-dependence has not been firmly established yet and it is likely to be underlain by different successive functions in the course of cosmic time

you are certainly right but this typically looks a Newtonian approach in Galilean coordinates: you know the celebrated thought experiment of Einstein, this approach would lead to the absurd conclusion that the speed of light emitted by a lamp in a train, depends on the speed of the train. Ironically, this is erroneous for the train but true for galaxies