# How can the universe be flat?

Okay, so I just want to clarify a few things.

According to what I have read, we have measured the universe to be flat, and the shape of the universe is directly related to the mass-energy density.

So I have a few questions about this which may stem from misunderstanding, I'm just a researcher really.

Firstly, back in the day when the 'big bang' was occurring, wouldn't the mass-energy density be more than great enough to result in a curved universe, if so, how would the universe transition from being curved and finite into being flat and infinite?

Secondly, to my understanding GR states that gravity is the result of the curvature of the 3D surface of a four dimensional shape, OK, but it is a shape, so conceptually how can the surface of the shape be infinite in length, width, and depth if to be a shape, the edges/points must meet up and cannot technically be infinitely large, or is this just a conceptual problem of envisioning four dimensional shapes... Or could this problem be solved by having the multi-verse be the underlying four dimensional shape?

I'd love to hear answers because this is quite confusing to me.

• Your first problem is the famous flatness problem in cosmology. It is solved by the inflationary model of the universe which shows if inflation is true, then no matter how generic the early universe was, it relaxes eventually to a flat spacetime. Commented Jun 14, 2015 at 12:53
• For your second point, GR does not say that gravity is NOT a result of curvature of a 3D surface. Gravity is a result of curvature of the full 4D space-time. I suspect that this problematic view of GR has arisen from the rubber sheet analogy which is quite misleading. Commented Jun 14, 2015 at 12:55
• There is a conceptual difficulty with standard cosmology: only the visible universe seems to be flat. We don't know what the shape of the entire universe is. This also means that all attempts to explain the flatness of our portion of space might be a wild goose chase. Commented Jun 14, 2015 at 16:44

According to what I have read, we have measured the universe to be flat

More or less. I'm happy enough with the WMAP results that indicate that the universe is flat. To be blunt I never thought it could be anything other than flat.

the shape of the universe is directly related to the mass-energy density.

That's what they say. But IMHO two out of three options were always going to be wrong. I always thought it was going to turn out to be flat regardless of the energy density. Which means inflation is superfluous, but that's one for another day.

Firstly, back in the day when the 'big bang' was occurring, wouldn't the mass-energy density be more than great enough to result in a curved universe, if so, how would the universe transition from being curved and finite into being flat and infinite?

Good question. The answer is that curvature does not depend on energy-density. It depends on the delta energy-density. If the energy-density is the same at every point in space, light goes straight.

Secondly, to my understanding GR states that gravity is the result of the curvature of the 3D surface of a four dimensional shape

It isn't. Gravity is the result of a concentration of energy usually in the guise of a star "conditioning" the surrounding space and so altering its metrical properties, this affect diminishing with distance. I don't agree with Prahar that the rubber-sheet analogy causes problems, I think the problem comes from a modern-day confusion between curved space and curved spacetime, where "space is neither homogeneous nor isotropic". See Baez: "Similarly, in general relativity gravity is not really a 'force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial". Curved spacetime is a "curved metric", and a metric is to do with measurement. For example, you place optical clocks throughout an equatorial slice through the Earth and the surrounding space, then plot the clock rates. You depict lower slower clocks as lower down in a 3D image, and higher faster clock rates higher up. What your plot looks like, is this:

CCASA image by Johnstone, see Wikipedia

That's a picture from the Wikipedia Riemann curvature tensor article. It's the rubber-sheet depiction of curved spacetime. And because it's derived from optical clock rates, it's a plot of the "coordinate" speed of light. Your plot of measurements is curved, space isn't. Instead space is inhomogeneous, and because of this light curves and matter falls down. Note that you need the curvature to get the plot off the flat and level - you need the curvature to get the slope, but the curvature relates to tidal force while the slope relates to the force of gravity which relates to the degree of inhomogeneity.

OK, but it is a shape, so conceptually how can the surface of the shape be infinite in length, width, and depth if to be a shape, the edges/points must meet up and cannot technically be infinitely large, or is this just a conceptual problem of envisioning four dimensional shapes...

Those articles about "the shape of the universe" are nothing of the sort. The conceptual problem is with the universe having an edge. By the way, I don't agree with the assertion that a flat universe must be infinite. IMHO it's a non-sequitur, and it's at odds with big bang cosmology. I don't accept the claim that the universe was already infinite when the big bang occurred.

Or could this problem be solved by having the multi-verse be the underlying four dimensional shape?

No it can't. The multiverse solves nothing. It's pseudoscience, not science.

• You clearly put a lot of effort into this answer, but there seems to be an overabundance of personal opinion and speculation. One of the challenges when working in cosmology is remembering that every physicist and his/her dog has their own opinions and ideas about how the universe actually is. One needs to be able to communicate the verifiable and properly demonstrated science without mixing one's own ideas and theories into it. This answer is decent, but it needs to lose the personal opinions and the like.
– Jim
Commented Jun 15, 2015 at 14:48
• @Jims Bond : Apologies. But please note that what wasn't my opinion was my reference to Einstein describing a gravitational field as space that's "neither homogeneous nor isotropic". Combine that with the FLRW metric which "starts with the assumption of homogeneity and isotropy of space", and there's no overall gravitational field. Also note that there's no verifiable evidence for inflation, an infinite universe, a toroidal universe, a multiverse, or any breach of conservation of energy. Commented Jun 15, 2015 at 18:30
• I'm not saying the whole answer is opinion or poor science. There is a decent answer amid opinion/speculation. Furthermore, while there is no hard evidence for inflation or the topologies of the universe, I did say that the answer should be verifiable science. That means inflation fits the bill because the science behind it, even if lacking evidence, is proper, verifiable, and following in the scientific method. Personal beliefs and opinions often are not arrived at through following proper methodology, which is just one of the reasons to omit those ideas from writings such as this.
– Jim
Commented Jun 15, 2015 at 18:37
• @Jims Bond : all points noted, but I must challenge your assertion that inflation is verifiable science. It's a hypothesis for which we have no evidence. One that's been around for so long that the people who have grown up with it accept despite the lack of evidence. See this Steinhardt interview. As for BICEP2, apart from dust there's an issue in that the CMB dates from ~350,000 years after the big bang. But we're getting off topic, perhaps you could ask a separate question on inflation? Commented Jun 15, 2015 at 20:35
• Indeed we are getting off topic. I was merely trying to point out that you have a good answer surrounded by "I believe..."s and "I don't accept..."s, etc. If those were taken out, this would be a great answer. And I'd be the first to upvote it after the edits are made.
– Jim
Commented Jun 18, 2015 at 13:39

Just some minor remarks:

(i) An expanding universe filled with matter and a zero spatial curvature are not in contradiction. Please read up about the Friedmann equations and the Robertson-Walker metric. (Sorry, but I think that's the way to understand cosmology properly.)

(ii) Flat geometries are not necessarily infinite (non-compact). Take for example a torus, which is flat (i.e. can be obtained from a "flat" sheet of paper) and compact. So, our universe could for instance be a torus. However, this is so far speculation. The topology of the space-time geometry does not enter the Friedmann equations; only the geometry does.

psm