- #26

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

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- #26

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

- #27

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That requires some tricky topology I'm guessing :-)But how do you conceive a finite euclidean 3D space? with a peripheral boundary? looks strange

- #28

bapowell

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

- #29

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4D torus?

- #30

bapowell

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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.)4D torus?

- #31

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

- #32

George Jones

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[itex]ds^2 = 0[/itex] for a lightlike worldline in both special and general relativity.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).

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

[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 relativityFurthermore, 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.

- #33

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sure, SR no way contradicts GR[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.

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.

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

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