Why is the observed local region of space flat?

[PainterGuy]

Hi,

Also, I read this article, What Do You Mean, The Universe Is Flat? (Part I), on Scientific American; URL: https://blogs.scientificamerican.co...what-do-you-mean-the-universe-is-flat-part-i/

I have few questions about some points made in the article. I'd appreciate it if you could help me to understand it properly at basic level.

The following quote is from the Scientific American article mentioned above.

In the last decade—you may have read this news countless times—cosmologists have found what they say is rather convincing evidence that the universe (meaning 3-D space) is flat, or at least very close to being flat.
...
What I do want to talk about here is what it is that is supposed to be flat.

When cosmologists say that the universe is flat they are referring to space—the nowverse and its parallel siblings of time past. Spacetime is not flat. It can’t be: Einstein’s general theory of relativity says that matter and energy curve spacetime, and there are enough matter and energy lying around to provide for curvature. Besides, if spacetime were flat I wouldn’t be sitting here because there would be no gravity to keep me on the chair. To put it succinctly: space can be flat even if spacetime isn't.
...
Moreover, when they talk about the flatness of space cosmologists are referring to the large-scale appearance of the universe. When you “zoom in” and look at something of less-than-cosmic scale, such as the solar system, space—not just spacetime—is definitely not flat.
...
On a cosmic scale, the curvature created in space by the countless stars, black holes, dust clouds, galaxies, and so on constitutes just a bunch of little bumps on a space that is, overall, boringly flat.

Thus the seeming contradiction:

Matter curves spacetime. The universe is flat.

is easily explained, too: spacetime is curved, and so is space; but on a large scale, space is overall flat.

Question 1:
In the quote above, it says in green, "there are enough matter and energy lying around to provide for curvature".

If there is no matter in the universe, would there still be spacetime? In other words, if there is no matter, would time still exist?

My own attempt to find the answer is given below.

Can space exist by itself without matter or energy around?
No. Experiments continue to show that there is no 'space' that stands apart from space-time itself...no arena in which matter, energy and gravity operate which is not affected by matter, energy and gravity. General relativity tells us that what we call space is just another feature of the gravitational field of the universe, so space and space-time can and do not exist apart from the matter and energy that creates the gravitational field. This is not speculation, but sound observation.

Source: https://einstein.stanford.edu/content/relativity/a11332.html

I think it should have been "can NOT and do not". Could you please confirm this?

It was formerly believed that if all material things disappeared out of the universe, time and space would be left. According to relativity theory, however, time and space disappear together with the things.

Source: Einstein, 1921; for more info: https://www.nature.com/articles/d41586-018-05004-4

This webpage, https://physics.stackexchange.com/questions/552306/does-time-require-matter-to-exist , has an interesting discussion in the comment section.

I'd say that, informally speaking, matter, space, and time (or, matter and spacetime) are intertwined and exist together.

Is my basic understanding okay?Question 2:
In the quote above, it says in orange, "On a cosmic scale, the curvature created in space by the countless stars, black holes, dust clouds, galaxies, and so on constitutes just a bunch of little bumps on a space that is, overall, boringly flat".

Isn't there a possibility that on a very, very large cosmic scale (which we can't observe so far and perhaps will never be able to), the space is not really flat, it's rather curved.
The following quote is also taken from Scientific American article.

Finite or Infinite?

If everything in the nowverse has an x, a y and a z, it would be natural to assume that we can push these coordinates to take any value, no matter how large. A spaceship flying off “along the x axis” could then go on forever. After all, what could stop her? Space would need to have some kind of boundary; most cosmologists don’t think it does.

The fact that you can go on forever however does not mean that space is infinite. Think of the two-dimensional sphere on which we live, the surface of the Earth. If you board an airplane and fly over the equator, you can just keep flying—you’ll never run into the “end of the Earth.” But after a while (assuming you have enough fuel) you would come back to the same place. Something similar could, in principle, happen in our universe: a spaceship that flew off in one direction could, after a long time, reappear from the opposite direction.

Or perhaps it wouldn’t. Cosmologists seem to believe that the universe goes on forever without coming back—and in particular, that space has infinite extension. But when pressed, most cosmologists would also admit that, in fact, they have no clue whether it's finite or infinite.

In principle, the universe could be finite and without a boundary—just like the surface of the Earth, but in three dimensions. In fact, when Einstein formulated his cosmological vision, based on his theory of gravitation, he postulated that the universe was finite. Einstein’s Weltanschauung was rooted in his deep, almost mystical sense of aesthetics; the most symmetric, aesthetically perfect three-dimensional shape is that of a three-dimensional sphere.

Question 3:
In the quote above, I find the parts in red a little confusing.

If the universe is infinite as it is assumed, it means the space is also infinite. But if the space is curved, say spherical, on a very, very large cosmic scale then a laser beam would come back to the point of its origin after an indefinite long period of time. Since the space is spherically curved or has spherical symmetry, the laser beam would follow a path like a plane around the Earth's equator.

If the universe is infinite then the space has to be infinite. In other words, if the space is finite then the universe is finite as well.

Do you think what I'm saying above is correct from a layman's point of view?
Note to self:
1: The density of Milky Way Galaxy is about 1 kg for every 5 billion cubic km. Source: https://astronomy.stackexchange.com...s-the-theoretical-maximum-density-of-a-galaxy
2: The distance between Milky Way Galaxy and Andromeda Galaxy is 2.5 million light years, i.e. 2.4 x 10^19 km.
3: The Earth's curvature could be roughly approximated as shown here https://earthcurvature.com
4: The radius of Milky Way Galaxy is 5 x 10^17 km.
5: https://www.physicsforums.com/threads/can-energy-exist-by-itself-without-time-and-matter.941887/
6: https://map.gsfc.nasa.gov/media/030639/index.html
7: photos.app.goo.gl/sis37gYzA7nDYdtq7, How did the Universe Begin which is part of the series Secrets of the Universe by Curiosity.

 

[Drakkith]

If there is no matter in the universe, would there still be spacetime? In other words, if there is no matter, would time still exist?

Our theories are all derived from observing a universe full of matter, so it is impossible to do anything but speculate as to what would the universe would be like without any matter. In other words, we don't know and cannot know. All we can do is guess and give answers that are dependent on underlying assumptions that may or may not be true in a matter-less universe.

Isn't there a possibility that on a very, very large cosmic scale (which we can't observe so far and perhaps will never be able to), the space is not really flat, it's rather curved.

Of course.

If the universe is infinite then the space has to be infinite. In other words, if the space is finite then the universe is finite as well.

Do you think what I'm saying above is correct from a layman's point of view?

Yes.

 

[Ibix]

If there is no matter in the universe, would there still be spacetime? In other words, if there is no matter, would time still exist?

Can't be answered. GR can certainly describe empty universes, but whether such a thing can come into existence or not is not known. That's the two sentence version of the StackExchange debate.

Isn't there a possibility that on a very, very large cosmic scale (which we can't observe so far and perhaps will never be able to), the space is not really flat, it's rather curved.

Yes. But it has to be really large scale because it appears indistinguishable from flat.

But if the space is curved, say spherical, on a very, very large cosmic scale then a laser beam would come back to the point of its origin after an indefinite long period of time.

Unfortunately, a closed universe has a finite lifetime, which I believe is equal to the time needed to circumnavigate it at the speed of light. So, although you can imagine drawing a straight line that circles back on itself, you do not have time to follow the path.

If the universe is infinite then the space has to be infinite. In other words, if the space is finite then the universe is finite as well.

Both of these sentences seem to me to be tautologies, so true. I don't know if you intended something else.

 

[Orodruin]

Unfortunately, a closed universe has a finite lifetime

This is only true for a matter or radiation dominated universe. A universe dominated by, eg, a cosmological constant, could be closed yet expand forever.

 

[Ibix]

This is only true for a matter or radiation dominated universe. A universe dominated by, eg, a cosmological constant, could be closed yet expand forever.

I have a feeling you've told me that before. Any idea how rapidly the circumference grows? Can you orbit such a spacetime?

 

[timmdeeg]

I have a feeling you've told me that before.

I share this feeling with you.

Here is the background, see Figure 3.5. https://ned.ipac.caltech.edu/level5/Peacock/Peacock3_2.html

 

[Orodruin]

I have a feeling you've told me that before. Any idea how rapidly the circumference grows? Can you orbit such a spacetime?

With the right mix of matter and cosmological constant perhaps. I have not looked at that question specifically.

Intuitively I would expect it is going to be a fine line between expanding too fast to go around and collapsing too fast to go around. I couldn’t state for sure what exactly will occur in the transition between the regimes without looking at things more carefully.

 

[PeterDonis]

Intuitively I would expect it is going to be a fine line between expanding too fast to go around and collapsing too fast to go around.

Well, we know of one case that's exactly in between those two: the Einstein static universe. The question would be whether that represents a "set of measure zero" that can be circumnavigated, or if there is a finite range of parameters around that one case where the universe can still be circumnavigated.

 

[Orodruin]

Well, we know of one case that's exactly in between those two: the Einstein static universe. The question would be whether that represents a "set of measure zero" that can be circumnavigated, or if there is a finite range of parameters around that one case where the universe can still be circumnavigated.

Sure, I agree. The Einstein static universe is however known to be unstable which is why I don’t want to make any statements about the surrounding solutions without looking at it proper.

 

[PainterGuy]

Thank you, everyone, for the help!

Unfortunately, a closed universe has a finite lifetime, which I believe is equal to the time needed to circumnavigate it at the speed of light. So, although you can imagine drawing a straight line that circles back on itself, you do not have time to follow the path.

What do you mean by ""closed universe"? Is it the universe with spherical symmetry? Could you please clarify?

This is only true for a matter or radiation dominated universe. A universe dominated by, eg, a cosmological constant, could be closed yet expand forever.

I think you're saying that in a closed universe (assuming it's the one with spherical symmetry) where the dark energy (which I take equivalent to cosmological constant) is dominant and remains dominant in the future, the universe won't collapse. In other words, the gravity won't be able to pull everything back into a big crunch. Is my interpretation okay?

 

[Orodruin]

I think you're saying that in a closed universe (assuming it's the one with spherical symmetry) where the dark energy (which I take equivalent to cosmological constant) is dominant and remains dominant in the future, the universe won't collapse. In other words, the gravity won't be able to pull everything back into a big crunch. Is my interpretation okay?

Yes, but the cosmological constant does not need to be dominant at present. It is sufficient that there is enough cosmological constant to eventually counteract the attractive effect of the matter. Then the cosmological constant will eventually come to dominate.

 

[Ibix]

What do you mean by ""closed universe"? Is it the universe with spherical symmetry? Could you please clarify?

It's the one usually illustrated with a sphere, which is what I think you are talking about. All three types of FLRW spacetime actually have spherical symmetry, although this may not come through in such diagrams.

 

[Orodruin]

It's the one usually illustrated with a sphere, which is what I think you are talking about.

… which is not only illustrated by a sphere. It is a three-dimensional sphere.

The main thing to note being that ”spherical symmetry” typically refers to a two-dimensional sphere. All 3+1-dimensional FLRW geometries have that symmetry.

 

[Ibix]

… which is not only illustrated by a sphere. It is a three-dimensional sphere.

Ok. It's normally illustrated by a 2-sphere, but is actually a 3-sphere. The others are normally illustrated with a flat plane and a "saddle".

The main thing to note being that ”spherical symmetry” typically refers to a two-dimensional sphere. All 3+1-dimensional FLRW geometries have that symmetry.

Yes - that was the point I was trying to make.

 

[PainterGuy]

All three types of FLRW spacetime actually have spherical symmetry

Are you talking about three different possible geometries of the universe: flat, saddle, spherical?
More info:https://map.gsfc.nasa.gov/media/030639/index.html

 

[Ibix]

Are you talking about three different possible geometries of the universe: flat, saddle, spherical?
More info:https://map.gsfc.nasa.gov/media/030639/index.html

Yes.

 

[timmdeeg]

I think you're saying that in a closed universe (assuming it's the one with spherical symmetry) where the dark energy (which I take equivalent to cosmological constant) is dominant and remains dominant in the future, the universe won't collapse.

Figure 3.5. mentioned in post #6 shows the tiny amount of the Cosmological Constant necessary to keep a closed FLRW-Universe expanding forever.

 

[PainterGuy]

Unfortunately, a closed universe has a finite lifetime, which I believe is equal to the time needed to circumnavigate it at the speed of light. So, although you can imagine drawing a straight line that circles back on itself, you do not have time to follow the path.

Sorry. In post #10 I made a mistake. I intended to use the word "geometry" instead of "symmetry".

What do you mean by ""closed universe"? Is it the universe with spherical symmetry geometry? Could you please clarify?

 

[kimbyd]

Can't be answered. GR can certainly describe empty universes, but whether such a thing can come into existence or not is not known. That's the two sentence version of the StackExchange debate.

I don't think the question is nearly so hopeless as that. In fact, based upon the current main cosmological model, an empty universe is exactly the far-future state of our own universe. And that model still has space and time, though the expansion is no longer meaningful.

Now, is this model correct? We don't know. But we do know that an empty space-time is a perfectly-valid solution to Einstein's equations, and we know of a way to produce such a state (wait a long time). We can't say for certain that nothing unexpected happens when that state is reached, but we can say the empty universe model makes sense given what we do know.

 

[PeterDonis]

based upon the current main cosmological model, an empty universe is exactly the far-future state of our own universe.

Only if we consider de Sitter spacetime, with a positive cosmological constant but (to a good enough approximation) no other stress-energy present, to be "empty". But that's not the usual usage; see below.

we do know that an empty space-time is a perfectly-valid solution to Einstein's equations

When questions are raised about whether an "empty" spacetime is physically realistic, "empty" usually means Minkowski spacetime, not de Sitter spacetime. Minkowski spacetime is not the end point of any known process of "cosmic evolution".

 

[PeterDonis]

When questions are raised about whether an "empty" spacetime is physically realistic, "empty" usually means Minkowski spacetime

Sometimes similar questions are raised about the maximal analytic extensions of vacuum solutions, such as the black hole solutions. But those also have zero cosmological constant. The simplest black hole solution with a nonzero cosmological constant is Schwarzschild-de Sitter, and the Schwarzschild part, if it does not arise from the collapse of a massive object (i.e., if it corresponds to the maximal analytic extension of the vacuum Schwarzschild solution), raises exactly the same issues of physical realism as Minkowski spacetime or the ordinary Schwarzschild solution.

 

[kimbyd]

Only if we consider de Sitter spacetime, with a positive cosmological constant but (to a good enough approximation) no other stress-energy present, to be "empty". But that's not the usual usage; see below.When questions are raised about whether an "empty" spacetime is physically realistic, "empty" usually means Minkowski spacetime, not de Sitter spacetime. Minkowski spacetime is not the end point of any known process of "cosmic evolution".

I suppose if you reject de Sitter spacetime as being empty, this is mostly true. The one quibble is that Minkowski spacetime cannot be the end result. It definitely can be if "dark energy" is not the cosmological constant, but some quintessence field which slowly decays. Provided that decay is slow enough that each Hubble volume is emptied of all matter prior to the quintessence field reaching zero, Minkowski spacetime would result (edit: maybe not...the behavior at infinity would get complicated...will have to think about this).

That's a little ad-hoc, but no more so than any other non-cosmological-constant solution to the dark energy problem. Either way, Minkowski spacetime is a valid solution to Einstein's equations, so we have no reason to believe it's unphysical. Whether physical processes can ever produce a Minkowski spacetime is a different question, one that is very much uncertain. That may prove to be impossible because it's impossible to have zero cosmological constant, for instance.

 

[PeterDonis]

It definitely can be if "dark energy" is not the cosmological constant, but some quintessence field which slowly decays. Provided that decay is slow enough that each Hubble volume is emptied of all matter prior to the quintessence field reaching zero, Minkowski spacetime would result.

I don't see how such a solution would satisfy the Bianchi identities. Do you have an actual reference for such a solution?

 

[PeterDonis]

Minkowski spacetime is a valid solution to Einstein's equations, so we have no reason to believe it's unphysical.

As you state it, this seems to be way too permissive a condition. There are many valid solutions of the EFE which are widely believed to be "unphysical", such as, for example, Godel spacetime. Even if we add additional constraints, such as that the solution must be globally hyperbolic (which Minkowski spacetime of course is), there are still plenty of solutions, such as the maximal analytic extensions of the black hole spacetimes, which are widely believed to be "unphysical" despite being valid solutions.

 

[kimbyd]

As you state it, this seems to be way too permissive a condition. There are many valid solutions of the EFE which are widely believed to be "unphysical", such as, for example, Godel spacetime. Even if we add additional constraints, such as that the solution must be globally hyperbolic (which Minkowski spacetime of course is), there are still plenty of solutions, such as the maximal analytic extensions of the black hole spacetimes, which are widely believed to be "unphysical" despite being valid solutions.

Well, generally they're considered unphysical because of some aspect of the stress-energy tensor that seems unphysical. A stress energy tensor that is zero everywhere is generally not considered unphysical.

 

[PeterDonis]

generally they're considered unphysical because of some aspect of the stress-energy tensor that seems unphysical.

Not in the cases I mentioned. For Godel spacetime, it's the presence of closed timelike curves. For the maximal extensions of the black hole solutions, it's the presence of the "white hole" and second exterior regions. The latter are vacuum solutions so obviously the stress-energy tensor is not the problem.

A stress energy tensor that is zero everywhere is generally not considered unphysical.

Sure, but that doesn't mean a vacuum solution can't be considered unphysical on other grounds.