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.

 

[hedgehug]

In the Milne metric spatial components are scaled by $t$ (their squares are scaled by $t^2$) and that's where linear expansion comes from. What a funny coordinates indeed. But now there are two different coordinate systems used in the same plot - the one I've pasted.

 

[Ibix]

But now there are two different coordinate systems used in the same plot - the one I've pasted.

I don't understand what you mean by this. The diagram you pasted in #1 is a plot of $a$ for different spacetimes and the same scheme for assigning coordinates is used in each case. So one could either argue that the same coordinate system was used on each spacetime or (the purist view) that comparing coordinate systems on different manifolds isn't possible without first defining a map between the manifolds.

 

[hedgehug]

If the empty universe scale factor's curve line was plotted in the FLRW coordinates like the rest of the curves, it would be a horizontal line with $a(t)=\text{const}$.

 

[Ibix]

If the empty universe scale factor's curve was plotted in the FLRW coordinates like the rest of the curves, it would be a horizontal line with $a(t)=\text{const}$.

No - they're computing the scale factor for Milne coordinates.

 

[hedgehug]

Please, read again what I wrote, and try not to deny it again.

If the empty universe scale factor's curve line was plotted in the FLRW coordinates like the rest of the curves, it would be a horizontal line with $a(t)=\text{const}$.

 

[renormalize]

If the empty universe scale factor's curve was plotted in the FLRW coordinates like the rest of the curves, it would be a horizontal line with $a(t)=\text{const}$.

@hedgehug, you are simply wrong.
Below is an excerpt from the recent paper:
Cosmological redshift from translations in an Einstein-Milne universe relativity with bounded position and velocity
that derives the Milne metric in FLRW-coordinates by starting from the usual flat Minkowski metric:

Clearly, in FLRW-coordinates the scale factor $a\left(\tau\right)$ of the empty Milne universe, when plotted as a function of time $\tau\,$, yields a straight line of slope $c$ and not a horizontal line of slope zero. In other words, the plot in your post #1 is exactly correct for an empty universe.
And, to quote you:
"Please, read again what I wrote, and try not to deny it again."

 

[hedgehug]

@hedgehug
Clearly, in FLRW-coordinates the scale factor $a\left(\tau\right)$ of the empty Milne universe, when plotted as a function of time $\tau\,$, yields a straight line of slope $c$ and not a horizontal line of slope zero. In other words, the plot in your post #1 is exactly correct for an empty universe.
And, to quote you:
"Please, read again what I wrote, and try not to deny it again."

@renormalize If $\tau$ was equal to $t$, then it would be the same FLRW coordinates. Are they equal?

 

[renormalize]

If $\tau$ was equal to $t$, then it would be the same coordinates. Are they equal?

Note that there is no "$t$" that appears in the derivation shown in post #16, just the Minkowski-space time $T$ and the Milne/FLRW-time $\tau$, which means physically that $\tau$ is the cosmic time-coordinate used in any standard discussion of the FLRW metric. You're free to relabel that coordinate using the symbol $t$ if that's what you prefer.

 

[hedgehug]

So you're basically saying that Milne's $a(t)\propto t$ uses the same time coordinate as $a(t)$ from the FLRW metric?

 

[renormalize]

So you're basically saying that Milne's $a(t)\propto t$ is the same time coordinate as in $a(t)$ from the FLRW metric?

Yes! The derivation in #16 proves it.

 

[hedgehug]

Apart from Milne, why isn't Minkowski's $T$ equal to $\tau$ from the FLRW metric in this derivation?

 

[renormalize]

Apart from Milne, why isn't Minkowski's $T$ NOT equal to $\tau$ from the FLRW metric in this derivation?

It's because the transformation from Minkowski-coordinates to FLRW-coordinates in eq.(4) of the derivation mixes time- and radial-coordinates. You can summarize the total transformation from Minkowski coordinates $T,R$ to FLRW-coordinates $\tau,\bar{R}$ (i.e., cosmic time and co-moving radial distance) by the equations $T=\tau\sqrt{1+\bar{R}^{2}},\:R=c\tau\bar{R}\,$. Only at the spatial origin $\bar{R}=0$ does $T=\tau$.
But bear in mind that this analysis only holds for the expanding Milne universe, where $a\left(\tau\right)=c\tau$. A general FLRW spacetime with $a\left(\tau\right)$ some non-linear function of cosmic time is curved and thus it cannot be a coordinate-transformation of flat Minkowski spacetime.

 

[hedgehug]

A general FLRW spacetime with $a\left(\tau\right)$ some non-linear function of cosmic time is curved and thus it cannot be a coordinate-transformation of flat Minkowski spacetime.

Apart from Milne, are you also saying that $dt$ in the Minkowski's $ds^2=dt^2-dr^2$ and $dt$ in the FLRW metric $ds^2=dt^2-a(t)^2dr^2$ are NOT the same $dt$?

But bear in mind that this analysis only holds for the expanding Milne universe, where $a\left(\tau\right)=c\tau$.

There's the rub.

 

[PeroK]

There's the rub.

For in that metric tensor,
What curvature may come,
When we have shuffled off comoving time,
Must give us pause.
And rather bear the time we have,
Than transform to others that we know not of.

 

[hedgehug]

Didn't you all (except @renormalize) make a distinction between Milne and FLRW coordinates?

Is Milne's scale factor $a(t)\propto t$ independent from both Milne's and FLRW's spatial coordinates?

 

[hedgehug]

No - they're computing the scale factor for Milne coordinates.

@Ibix Since you upvoted all the comments of @renormalize, do you also claim that Milne and FLRW are the same coordinates?

Is Milne's scale factor $a(t)\propto t$ independent from both Milne's and FLRW's spatial coordinates?

 

[PeterDonis]

do you also claim that Milne and FLRW are the same coordinates?

@hedgehug, the coordinate transformation between standard Minkowski coordinates and Milne coordinates is explicitly given to you in post #16. That post also shows what the metric looks like in Milne coordinates; it is obvious by inspection that that metric is in the FLRW form (and the quoted passage notes this explicitly). So it is pointless for you to be asking the question you ask in the quote above; the answer is already here in the thread.

Is Milne's scale factor $a(t)\propto t$ independent from both Milne's and FLRW's spatial coordinates?

It's $a(\tau)$, not $a(t)$, per post #16. That post makes it obvious that the two time coordinates are not the same. And that post also makes it obvious that the answer to the question you ask in the quote above is yes.

There's the rub.

There is no "rub" here anywhere. The Milne model is perfectly consistent mathematically, and describes a flat spacetime, which, as has already been noted, is to be expected with a vanishing stress-energy tensor and zero cosmological constant. There is no issue here at all.

Is there any actual substantive question you have about the Milne model that has not been answered at this point?

 

[hedgehug]

It's because the transformation from Minkowski-coordinates to FLRW-coordinates in eq.(4) of the derivation mixes time- and radial-coordinates. You can summarize the total transformation from Minkowski coordinates $T,R$ to FLRW-coordinates $\tau,\bar{R}$ (i.e., cosmic time and co-moving radial distance) by the equations $T=\tau\sqrt{1+\bar{R}^{2}},\:R=c\tau\bar{R}\,$. Only at the spatial origin $\bar{R}=0$ does $T=\tau$.
But bear in mind that this analysis only holds for the expanding Milne universe, where $a\left(\tau\right)=c\tau$. A general FLRW spacetime with $a\left(\tau\right)$ some non-linear function of cosmic time is curved and thus it cannot be a coordinate-transformation of flat Minkowski spacetime.

@renormalize Since $dt$ in Minkowski's $ds^2=dt^2-dr^2$ is THE SAME $dt$ as in the FLRW's $ds^2=dt^2-a(t)^2dr^2$, therefore Minkowski's $t$ is the same $t$ as in FLRW, so making a distinction between them by introducing a coordinate transformation between Minkowski's $T$ and FLRW-Milne's $\tau$ (remembering that Milne IS FLRW in your dictionary), that is $T=\tau\sqrt{1+R^2/(c\tau)^2}$, introduces quite a mess.

From this transformation I can tell that $\tau$ depends on both $T$ and $R$, because $T$ depends on both $\tau$ and $R$. Since in your dictionary $\tau$ is both Milne's and FLRW's time coordinate, and in mine its only Milne's, I say that it can't be the same time as $t$ in $a(t)$ in the FLRW metric, because this $t$ doesn't depend on any spatial coordinate, and $\tau$ depends on $R$.

So if there are Milne's $a(\tau)$ and FLRW's $a(t)$ in the same plot that I've pasted, then there are two different coordinate systems used in the same plot, because $\tau\neq t$.

 

[hedgehug]

@PeterDonis please, read my previous comment and tell me again there is no problem.

 

[PeterDonis]

Since $dt$ in Minkowski's $ds^2=dt^2-dr^2$ is THE SAME $dt$ as in the FLRW's $ds^2=dt^2-a(t)^2dr^2$

No, it isn't. That's made clear in post #16, as I've already pointed out.

please, read my previous comment and tell me again there is no problem.

I did. It's wrong. See above.

There is no problem.