# Universe's Flatness: Mass, Gravity, Dark Energy Explained

• B
• robbierobb
In summary, GR predicts that the universe is flat, but this is contradicted by the fact that mass curves space. This contradiction is resolved by taking into account the influence of mass on the curvature of space. This is why the universe is accelerating - the more mass there is, the more space it curves and the more dark energy is needed to counteract this curvature.f

#### robbierobb

So, this is a bit of an odd question, but something I was curious about. I recently learned in layman's terms that the universe is considered flat because of how we measure triangles in 3D space, in particular the CMB. I've also learned that the cosmological constant that was derived from GR predicts how much dark energy there is driving the anti gravity force that helps keep the universe flat.

But GR also tells us one important point, that space can be curved by mass. So on one hand GR is telling us that the universe is flat, but on the other hand it is also telling us that mass curves space. So this doesn't make sense, to me anyway, because how can you measure a perfectly flat triangle in a positively or negatively curved region of space?

So is that somehow related? Is the curvature of space somehow related to the flatness of the universe? And is the influence that mass has on the fabric of space correlated to the amount of dark energy required to keep the universe flat? Is this why the universe is accelerating? Because mass is forever causing more and more curvature that needs correcting with dark energy?

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But GR also tells us one important point, that space can be curved by mass. So on one hand GR is telling us that the universe is flat, but on the other hand it is also telling us that mass curves space. So this doesn't make sense, to me anyway.
When we say that the universe is spatially flat, that's on a large scale after we've averaged out the tiny and widely separated local divots around concentrations of mass like galaxies. It's like saying that the surface of a large body of water is flat, even though there are a few bubbles sticking up here and there.

Note also that mass curves spacetime, not just space.

robbierobb
Can you elaborate a little on what you mean by averaging out the pockets and divots of mass? I understand the concept with your bowl of water analogy. Does that mean you ignore them in the calculations? Or have you had to take them into account in some form?

Can you elaborate a little on what you mean by averaging out the pockets and divots of mass? I understand the concept with your bowl of water analogy. Does that mean you ignore them in the calculations? Or have you had to take them into account in some form?
They aren't important on the large scale. You may regard your lawn as flat. An ant trying to cross it will not agree. The ant isn't wrong to say that the lawn has a lot of surface roughness - it's just not important on the scale humans operate at.

Similarly, when we're thinking of gravity in our solar system or our galaxy, or our local group, then we're thinking like the ant. When we think about cosmology we're thinking like the human. We've just (mentally) stepped out to a scale where individual galaxies just aren't noticeable.

Imager, PeroK and robbierobb
They aren't important on the large scale. You may regard your lawn as flat. An ant trying to cross it will not agree. The ant isn't wrong to say that the lawn has a lot of surface roughness - it's just not important on the scale humans operate at.

Similarly, when we're thinking of gravity in our solar system or our galaxy, or our local group, then we're thinking like the ant. When we think about cosmology we're thinking like the human. We've just (mentally) stepped out to a scale where individual galaxies just aren't noticeable.

OK I get you. So when we say that pockets of mass aren't important, does that also include dark matter and dark energy? We're speaking specifically about measuring the flatness of the universe here, correct? Sorry if this is getting a little confusing, just trying to get a better understanding of my poorly understood initial picture. What we're saying is that as far as concerning the flatness of the universe goes, mass isn't important on a grander scale. So where does that involve incorporating anti gravity and dark energy into the picture in order to maintain stability? Is that on a grand scale as well? Or is that local like mass is?

What we're saying is that as far as concerning the flatness of the universe goes, mass isn't important on a grander scale. So where does that involve incorporating anti gravity and dark energy into the picture in order to maintain stability?
You may have missed a subtlety when we said that the universe is spatially flat. The universe is a four-dimensional spacetime, and that spacetime is curved by all the mass (and energy) in it, just as you're expecting. What's flat is space, a three-dimensional subset of that four-dimensional space. And even then, space is only big-picture flat, the way you and I might consider a plot of smooth and level ground to be flat while @Ibix's ant disagrees.

(Be aware that I commit a massive oversimplification, tolerable only in a B-level thread, when I speak of "a three-dimensional subset of that four-dimensional space").

But GR also tells us one important point, that space can be curved by mass. So on one hand GR is telling us that the universe is flat, but on the other hand it is also telling us that mass curves space.
GR tells us that spacetime can be curved by the presence of energy, momentum, and stress. When cosmologists are talking about flat space they are only talking about the 3-dimensional space-like subspaces of spacetime that are homogeneous and isotropic and represent the same cosmological time. The spacetime describing the universe is certainly curved in cosmology.

So when we say that pockets of mass aren't important, does that also include dark matter and dark energy?
Dark matter is just the same as normal matter as far as cosmology is concerned. It's a kind of stuff that's difficult to detect and it doesn't clump so much as normal matter for various reasons, but its gravitational effect is the same as normal matter.

Dark energy is (more or less) the modern name for the cosmological constant. As the older name implies there are no "pockets" of it - it's the same everywhere. Or, at least, that's our current model. Part of the reason for not calling it the cosmological constant is not to get too attached to "it's the same everywhere" as dogma when we don't really know much about it yet.
What we're saying is that as far as concerning the flatness of the universe goes, mass isn't important on a grander scale
You missed my point. Mass is important - but only the average density. Tiny little local variations in mass like galaxies don't matter on the cosmological scale. Like our human doesn't need to care that the ground is actually an aggregate of tiny particles of dirt. The ground is important (you'd fall if it weren't there), but the detailed structure of the dirt isn't of any real concern to us.
so where does that involve incorporating anti gravity and dark energy into the picture in order to maintain stability?
It doesn't. Trying to use the cosmological constant to balance expansion and model a non-expanding universe was what Einstein called his greatest mistake.

The universe is spatially flat because of the average density of normal and dark matter. This turns out to mean that the universe will expand forever. But the expansion we see can't quite be modeled by any matter density. We need to add a tiny amount of cosmological constant (which is very different in behaviour to normal matter, as noted above) to make our models match our observations. We don't really know much about it beyond that.

You may have missed a subtlety when we said that the universe is spatially flat. The universe is a four-dimensional spacetime, and that spacetime is curved by all the mass (and energy) in it, just as you're expecting. What's flat is space, a three-dimensional subset of that four-dimensional space. And even then, space is only big-picture flat, the way you and I might consider a plot of smooth and level ground to be flat while @Ibix's ant disagrees.

(Be aware that I commit a massive oversimplification, tolerable only in a B-level thread, when I speak of "a three-dimensional subset of that four-dimensional space").

No it's cool I understand the 4D nature of sapce-time, SR is super intereating and sapce-time diagrams are particularly cool, I just didn't factor in time as a GR component of my OP which I probably should have done. As much as I understand time in SR, and how gravity effects time in GR, my understanding of it in cosmology in the grand scheme of things is pretty limited. I guess I lucked out in that respect!

Dark energy is (more or less) the modern name for the cosmological constant. As the older name implies there are no "pockets" of it - it's the same everywhere. Or, at least, that's our current model. Part of the reason for not calling it the cosmological constant is not to get too attached to "it's the same everywhere" as dogma when we don't really know much about it yet.

Trying to use the cosmological constant to balance expansion and model a non-expanding universe was what Einstein called his greatest mistake.

Sorry, by stable I didn't mean static, I just meant stable as in not changing. But yeah, I've read a few things about it's use in quantum field theory and other areas of quantum mechanics where it constantly has to be adjusted. And thus, the comolgical constant problem?

GR tells us that spacetime can be curved by the presence of energy, momentum, and stress. When cosmologists are talking about flat space they are only talking about the 3-dimensional space-like subspaces of spacetime that are homogeneous and isotropic and represent the same cosmological time. The spacetime describing the universe is certainly curved in cosmology.

I guess this comes full circle back to my original point, so thanks for the answers. Although this is where my initial thoughts came from, I thought that perhaps if we measured the rate of acceleration it might correlate to the amount of curvature. But I really have no idea what that would mean so it was best just to try and understand each individual component, I don't think that statement allows for much explanation.

^
I thought that perhaps if we measured the rate of acceleration it might correlate to the amount of curvature.

A bit like how you sit on a cushion, and the more you sit on the cushion the more it sags, and in order to restore it you have to put energy into it to plump it up and make it nice again and comfortable to sit on. Like mass is constantly sitting on the cushion and dark energy is constantly plumling it up.