Impossible, linear expansion of "empty universe"

[hedgehug]

https://en.wikipedia.org/wiki/Age_of_the_universe#Cosmological_parameters

I have two problems problem with the "empty universe". If it's completely empty, it can't expand, but its scale factor function is linearly increasing, so it's $a(t)=t/t_0$ where $t_0$ is the age of the universe. The second problem with linear function of the scale factor is that the integral of its inverse is $$\int\frac{dt}{t/t_0}=t_0\ln{t}+C$$ and the proper distance equal to the observable universe radius is $$d(t_0)=a(t_0)\int_{0}^{t_0}\frac{cdt}{a(t)}$$ $$=\frac{t_0}{t_0}\int_{0}^{t_0}\frac{cdt}{t/t_0}=ct_0(\ln{t_0}-\ln{0})$$ $$=ct_0\ln{\frac{t_0}{0}}$$ It's undefined, because $\ln{0}=-\infty$.

What's the explanation or solution to these problems?

 

[renormalize]

I have two problems problem with the "empty universe". If it's completely empty, it can't expand ...

$\Omega=1$ is just flat empty Minkowski spacetime as described in peculiar coordinates that involve a scale-factor expanding linearly with the time coordinate. Look up the "Milne model" and see this PF discussion:

Graduate Thread 'What's wrong with a flat, empty, and expanding universe?'

Aug 16, 2015

I'm new to this impressive forum and have a question that may have been addressed a thousand times, but here goes.

A FLRW metric is happy with a time-varying scale factor a(t) and zero curvature parameter k and could care less about density. The combination of a FLRW metric and the Einstein field equations, however, requires that a flat universe either not expand or that it have a density equal to a critical density much larger than independent observations support. The constraint is imposed by the Friedmann equation,

k = Ho^2 (density ratio - 1).

For a flat space, either Ho is zero...

• Steve Crow
• Cosmology Empty Expanding Expanding universe Flat Universe
• Replies: 14
• Forum: Special and General Relativity

 

[hedgehug]

Eintein field equations for the empty universe tell me that $T_{\mu\nu}=0, \Lambda =0, R_{\mu\nu}=0, R=0$. Are you saying that I can still have a conformal metric tensor $g_{\mu\nu}$ with the changing scale factor?

 

[PeterDonis]

Eintein field equations for the empty universe tell me that $T_{\mu\nu}=0, \Lambda =0, R_{\mu\nu}=0, R=0$.

Yes.

Are you saying that I can still have a conformal metric tensor $g_{\mu\nu}$ with the changing scale factor?

With an appropriate choice of coordinates, yes. If you compute the Riemann curvature tensor for that metric tensor in those coordinates, you will find that it vanishes, indicating that the spacetime is flat. The funny coordinates just obfuscate that fact somewhat.

 

[PeterDonis]

If it's completely empty, it can't expand

What expands in the "empty universe" is the set of worldlines that are defined as "comoving"--more precisely, that congruence of worldlines has a positive expansion scalar. In ordinary inertial coordinates on flat Minkowski spacetime, this congruence of worldlines is the set of all timelike geodesics that emerge from the origin. The funny coordinates of the "empty universe" model use proper time along these worldlines as the time coordinate, and arc length along curves of constant proper time from the origin (which in ordinary inertial coordinates are hyperboloids with the future light cone of the origin as asymptotes) as the definition of "spatial distance" (i.e., those hyperboloids are the "surfaces of constant time" in the empty universe model).

 

[PeterDonis]

the proper distance equal to the observable universe radius

Is not really meaningful in this model. Mathematically, as you point out, it's undefined. Physically, the concept of "observable universe" doesn't really make sense in this model, because conformally it doesn't look like the other FRW models, since it's just flat Minkowski spacetime in funny coordinates, and conformally flat Minkowski spacetime looks very different from any FRW spacetime that has nonzero stress-energy in it.

 

[hedgehug]

Are you defining conformal coordinates in a way that static universe expands in these coordinates? Are you simply changing them in time to get the effect of expansion of static universe in these coordinates?

 

[Ibix]

Are you defining coordinates in a way that static universe expands in these coordinates? Are you simply changing them in time to get the effect of expansion of static universe in these coordinates?

Well, I think it was originally a serious proposal for a cosmology and it remains the zero-mass member of the FLRW family of spacetimes. But basically, yes. In the usual Minkowski/Einstein coordinates your "stationary points" are a set of parallel lines. In Milne coordinates they're the future half of the set of timelike lines passing through an arbitrarily chosen event.

It's an interesting piece of mathematics and can be interesting to study (and it's interesting how much of FLRW cosmology carries across), but it's not a very likely physical model.

 

[hedgehug]

Well, I think it was originally a serious proposal for a cosmology and it remains the zero-mass member of the FLRW family of spacetimes. But basically, yes. In the usual Minkowski/Einstein coordinates your "stationary points" are a set of parallel lines. In Milne coordinates they're the future half of the set of timelike lines passing through an arbitrarily chosen event.

So in Milne coordinates they are no longer parallel, since they have to cross at the event, right? Either past or future event.

 

[Ibix]

So in Milne coordinates they are no longer parallel, since they have to cross at the event, right? Either past or future event.

Yes. They expand out from that event, and would be the paths comoving galaxies followed if there were any mass to form galaxies. Milne coordinates only cover the interior of the future light cone of that event.