Calculating the Size of the Universe

[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]

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).

ds^2 = 0 for a lightlike worldline in both special and general relativity.

SR seems to be sufficient. Gravity and GR are generally not involved in the universe model used in these studies.

This is just plain wrong. In order to use the equation in your post above, the dependence of a\left(t\right) on t is needed. This is given by the solution of the differential equation

<br /> \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),<br />
where the constants \Omega_{m0}, \Omega_{r0}, \Omega_{\Lambda 0} 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.

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.

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]

ds^2 = 0 for a lightlike worldline in both special and general relativity.

sure, SR no way contradicts GR

This is just plain wrong. In order to use the equation in your post above, the dependence of a\left(t\right) on t is needed. This is given by the solution of the differential equation

<br /> \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),<br />
where the constants \Omega_{m0}, \Omega_{r0}, \Omega_{\Lambda 0} 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.

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

With appropriate definitions of time and distance, c - V_rec is true in special relativity, and in the FRW cosmological models of general relativity

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