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

[Steve Crow]

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 or density is equal to critical. That strikes me as strange. Matter appears to flatten an expanding space rather than curve it. Does someone have an intuitive explanation of this phenomenon?

 

[Orodruin]

If you look at the vacuum case when the stress-energy tensor is identically zero, you will obtain a universe with negative ##k## which is linearly expanding (the Friedmann equation turns into ##\dot a^2 = -k##. Adding matter will increase ##k## and therefore eventually bring it to become positive once the critical density is reached. At one point, the critical density, the universe will be flat. Thus, it is a matter of matter bending the space in one direction and the vacuum solution having the opposite curvature.

 

[Steve Crow]

Yes, and well stated. Yet the field equations support a flat universe that is not expanding, or at least I think they do. Why does expansion create curvature as implied by your simplified Friedmann equation?

And how do you write such a nice equation in this environment? :)

 

[Orodruin]

Yet the field equations support a flat universe that is not expanding, or at least I think they do.

Indeed they do. This is just the case of ##\dot a^2 = -k = 0## and corresponds to the Minkowski space we all know and love from special relativity.

Why does expansion create curvature as implied by your simplified Friedmann equation?

The crass answer to this would be: "Because otherwise the space-time does not satisfy the Einstein field equations." In other words, according to what we know about how gravity works (i.e., GR) an empty expanding space-time must have a negative ##k##. If you are looking for a deeper reason than that, you are rather bordering on philosophy rather than physics.

And how do you write such a nice equation in this environment? :)

LaTeX Primer

 

[PAllen]

The empty FLRW universe with no cosmological constant is Minkowski spacetime in funny coordintates. These coordinates foliate Minkowski spacetime with hypersurfaces of negative spatial curvature - but the spacetime is still flat, thus identical to Minkowski space (conventional coordinates can be reached by coordinate transform). At the critical mass, the spactime has curvature, but the foliation displaying isotropic, homogeneous expansion has spatially flat hypersurfaces.

 

[Steve Crow]

I think Orodruin's answer is elegant, not crass. An equally elegant answer would be: "Because otherwise the Einstein field equations would be wrong as applied to cosmology." That answer would spare us dark matter and dark energy but would entail a lot of thought. PALLen's comment is interesting, and I shall give it a lot of thought. I assume the CMB triangulations pertain to space rather than to space-time.

Orodruin, thanks for the tip on LaTeX. I'll give it a shot when I write another equation.

 

[PAllen]

I thought of a motivation for why the empty cosmology must work the way it does (per SR + GR). An empty universe, per GR, must be flat spacetime, i.e. Minkowski space. A cosmology wants a congruence (family) of (massless and without energy or momentum, in this case, so can follow timelike geodesics) objects whose motion allows isotropy and homogeneity. In Minkowski spacetime, there are two possibilities - a standard Minkowski inertial frame with all static positions projected in time; or the family of all inertial world lines emanating from a single event (pseudo-big bang). In the latter case, to display isotropy and homogeneity, you must connect events on each world line with the same proper time from the 'beginning'. If you draw this on a 1+1 spacetime diagram, it is easily seen to be a hyperbola. Extending to a 3-surface, you have a hyperbolic space (negative curvature). Only by introducing spacetime curvature can you have such a congruence where the surface of common proper time from the beginning is spatially flat.

 

[Orodruin]

Just to be specific here, the coordinate transform is given by:
##t = \tau \cosh(\rho), r = \tau \sinh(\rho)##
in spherical coordinates based on some frame where the pseudo big bang occurs at the origin.

It is also worth pointing out that this is not a global coordinate chart but only covers the future light cone of the pseudo big bang for ##\tau > 0##.

The corresponding exercise in the region of space like separation from the pseudo big bang just gives the Rindler coordinates.

 

[Steve Crow]

That is a compelling argument. The Big Bang itself induces curvature. "Only by introducing space-time curvature can you have such a congruence where the surface of common proper time from the beginning is spatially flat". Bravo! Of course I still have to think a lot about it, but I shall not pull the plug on dark matter and dark energy just yet :)

 

[pervect]

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

yes

and could care less about density.

I'm not sure why you say this - or what you mean by it.

I think one thing that may be confusing the discussion is the difference between flat spatial slices, and flat space-time. If you have no matter and no cosmological constant, space-time will be flat, the Riemann tensor will be zero indicative of no space-time curvature.

However, you can describe the same flat space-time with a flat, non-expanding Minkowskii metric (such as ##dr^2 + r^2 d\Omega^2##, r being the radius and ##\Omega## being a shorthand for the solid angle terms in the metric involving theta and phi). You can also describe the same flat space-time with the Milne metric, see https://en.wikipedia.org/w/index.php?title=Milne_model&oldid=628558886

However, it turns out that the spatial slices in the Milne model are curved, not flat, even though the space-time curvature is identically zero in the Milne metric (as it is in the Minkowskii metric).

So it's important to make a clear distinction between the space curvature and the space-time curvature. While the spatial slices in the Milne metric are curved, the space-time is still flat. I can only offer the general observation that that the spatial slices are curved as a consequence of the particular method used to separate space-time into space + time, I don't have any explanation other than "that's the way the math works out".

 

[Orodruin]

I can only offer the general observation that that the spatial slices are curved as a consequence of the particular method used to separate space-time into space + time, I don't have any explanation other than "that's the way the math works out".

I think it is worth pointing out that the selection of what spatial slices constitute "now" (or more generally anything that defines events to be simultaneous) is present already in special relativity and is nothing else than the relativity of simultaneity. The only thing there is that we select our coordinate system in such a way that the spatial slices become flat. The only generalisation here is to allow splitting space and time in a manner such that the geodesics on the spatial surface are not geodesics of the space-time.

 

[Steve Crow]

All these comments have been most helpful. Here is a summary of my current understanding.

1. An expanding empty space-time is flat.
2. Because of diverging world lines, the 3-space component of an expanding empty space-time is curved.
3. The curvature parameter of the three-space component is k = -adot^2, where adot is the expansion rate of the scale factor.
4. Addition of matter and energy can raise k to 0 ...
5. ... in conformity with observations of the CMB.

As pervect said, the key is to distinguish space from space-time. Please correct anything that is wrong.

 

[Orodruin]

2. Because of diverging world lines, the 3-space component of an expanding empty space-time is curved.
3. The curvature parameter of the three-space component is k = -adot^2, where adot is the expansion rate of the scale factor.

2. I would say it is because of the geometry of Minkowski space. If you did something similar in a Euclidean space, you would get a sphere rather than a hyperboloid, so what you get is ultimately dependent on the underlying geometry.

3. Note that this is with this very peculiar definition of what "space" is. Since it is Minkowski space, you could pick a definition of space such that it is flat as well, but it would not be expanding.

Another note of interest is that the "pseudo big bang" as PAllen dubbed it would be just that, a coordinate singularity and not an actual singularity of the space-time.

 

[PeterDonis]

1. An expanding empty space-time is flat.

You're stating this backwards. It should be: empty spacetime is flat; and we can choose a (peculiar, as Orodruin says) way of splitting up a portion of this spacetime into space and time so that it appears to be "expanding". (It's also important to note the "a portion" part; as Orodruin pointed out in post #8, this way of splitting into space and time only covers the future light cone of the "pseudo Big Bang" at the origin. So this "expanding" slicing is incomplete--there are regions of spacetime that it doesn't even cover.)

4. Addition of matter and energy can raise k to 0

But it also changes the geometry of the spacetime. If you're trying to imagine a solution with ##k = 0## and matter and energy present (at the critical density) as just an "adjusted" version of empty Minkowski spacetime, I don't think that's going to be a fruitful strategy. (For one thing, the "expanding" slicing of the critical density spacetime does cover the entire spacetime, unlike the "empty expanding" case, as noted above.)

 

[PAllen]

You're stating this backwards. It should be: empty spacetime is flat; and we can choose a (peculiar, as Orodruin says) way of splitting up a portion of this spacetime into space and time so that it appears to be "expanding". (It's also important to note the "a portion" part; as Orodruin pointed out in post #8, this way of splitting into space and time only covers the future light cone of the "pseudo Big Bang" at the origin. So this "expanding" slicing is incomplete--there are regions of spacetime that it doesn't even cover.)

Peculiar as it is, if falls out of the FLRW metric ansatz for density approaching zero. Thus it 'belongs' as the limit of this family of solutions. I guess, on the other side, the limit of the Schwarzschild exterior metric as M->0 is also Minkowski space with a polar coordinate singularity at the origin replacing the horizon coordinate singularity. In more complete coordinates, the singularity vanishes in the limit. In both cases, a limit which seems 'smooth' as to local geometry produces a radical topology change. But maybe not strange - the limit of 2 spheres as curvature approaches zero is a plane. This is locally smooth - and really is for a patch - but for the 2-sphere manifold as a whole it is a radical topology change.

But it also changes the geometry of the spacetime. If you're trying to imagine a solution with ##k = 0## and matter and energy present (at the critical density) as just an "adjusted" version of empty Minkowski spacetime, I don't think that's going to be a fruitful strategy. (For one thing, the "expanding" slicing of the critical density spacetime does cover the entire spacetime, unlike the "empty expanding" case, as noted above.)

On the other hand, looking at local geometry, this is exactly what happens as noted in Orodruin's #2. The increasing curvature of spactime allows spatial hyper-surfaces of constant proper time (for the Hubble flow) to evolve from maximally negative (hyperbolic) curvature to flat and then to positive curvature.