Questions About the Flat Universe: Answers Inside

[Levi Porter]

Sorry for posting this in the wrong spot.

Thanks for moving it.

If there is a consensus throughout mainstream science that the Universe is flat, I have a few questions that I have not found the answers for.What types of explosions are flat?

What forces are required to keep the universe flat?

Is the Universe considered disc shaped?

 

[Chris Hillman]

"Flat explosion"?! Where did you see that term?

If there is a consensus throughout mainstream science that the Universe is flat

The current consensus is that the large scale structure of the observable universe is well approximated by an FRW dust (with nonzero Lambda) in which the spatial hyperslices orthogonal to the world lines of the dust particles are locally flat, i.e. have vanishing three-dimensional Riemann tensor. Cornish and Weeks (and others) are looking for indications that these flat hyperslices are actually quotient manifolds of E^3. (For example: take a "solid" three-cube [0,1]^3 and identify opposite faces; this gives a compact, globally nonisotropic Riemannian three-manifold which is locally flat--- and thus locally isotropic).

However, the Riemann tensor of the four-dimensional Lorentzian manifold which is the setting ("spacetime") for physics in this FRW model (or quotient of FRW model, as the case may be), is nonzero, as is required by Einstein's equation! OTH, the Weyl tensor of this simple spacetime vanishes (which implies that in this approximation we are neglecting gravitational radiation, and we also neglect inhomogeneities in matter density and so on).

What types of explosions are flat?

Where did you see a description of a "flat explosion"? Right now I can't think of any way to make sense of the phrase "flat explosion" in gtr, since even a simple model of a spherically pulsating isolated spherically symmetric star would have nonzero Weyl curvature in the exterior.

What forces are required to keep the universe flat?

What theory of gravitation do you have in mind? In the context of cosmology, the current default is gtr. If that is the theory you have in mind, gtr does not treat gravitation as a "force law" but rather as spacetime curvature; the content of the field equation of the theory, the Einstein field equation, is captured approximately by Wheeler's slogan, "spacetime curvature and material stress tell bits of matter how to move; the density and momentum of bits of matter and nongravitational fields tells spacetime how to curve".

Is the Universe considered disc shaped?

Sounds like you are confusing our galaxy with the entire universe! Are you perhaps reading a book which dates to about 1850 or something? Seriously, its been recognized for a long time that our galaxy is only one of many. (Many many!)

 

[JesseM]

Levi, have you read the book Flatland? If not, I recommend it...either way, if you can imagine a 2-dimensional universe, a "flat" 2D universe would just correspond to an infinite plane, while a positively-curved "closed" 2D universe would be like the surface of a sphere, and a negatively-curved "open" 2D universe would be something like the surface of a saddle (see the images at the top of this page). Now increase the number of dimensions by one--we can't actually visualize what the 3D surface of a 4-dimensional "hypersphere" would look like, but that's what it would mean for our 3D space to have positive curvature, for example. A "flat" universe just means the ordinary 3D euclidean geometry that we're all used to, it doesn't mean anything is squashed into a 2D plane or disc (our universe would look 'flat' from the perspective of an imaginary 4-dimensional being, just as a 2D plane looks 'flat' from our perspective).

 

[marcus]

... A "flat" universe just means the ordinary 3D euclidean geometry that we're all used to, it doesn't mean anything is squashed ...

Jesse is right, Levi.

when cosmologists say the universe is "spatially flat", or NEARLY spatially flat, they mean something special by the word flat.
basically what it means is that if you construct larger and larger triangles (even if it means finding points outside the solar system to set up your surveying equipment) the sum of the interior angles that you measure is always exactly 180 degrees.

It is hard to imagine technically how one would survey large triangles with interstellar or intergalactic distances because of the communication problem. But that is the simplest way to imagine how one could check for perfect flatness or "zero curvature" if you like that term better.

if the universe is NEARLY flat, spatially, but has a slight positive curvature, then if you checked with a big enough triangle you might notice that the sum of the angles was not exactly 180 but was just barely over---a tiny bit more than 180.

Fortunately there are other ways of checking for curvature besides setting up big triangles and it comes down to the same thing---and a lot of checking has been done!

There is no agreement that it is PERFECTLY flat but pretty wide agreement that it is either perfectly flat or very very close----flat or very nearly so.

 

[SpaceTiger]

If there is a consensus throughout mainstream science that the Universe is flat

To the precision we're currently able to measure, yes. Inflation, which is a very popular theory in physics and astrophysics, predicts a deviation from perfect flatness within the observable universe. If I remember correctly, it's a fractional deviation of order 10^-5, a few orders of magnitude more precise than the current measurements.

 

[Wallace]

The current consensus is that the large scale structure of the observable universe is well approximated by an FRW dust (with nonzero Lambda) in which the spatial hyperslices orthogonal to the world lines of the dust particles are locally flat, i.e. have vanishing three-dimensional Riemann tensor. Cornish and Weeks (and others) are looking for indications that these flat hyperslices are actually quotient manifolds of E^3. (For example: take a "solid" three-cube [0,1]^3 and identify opposite faces; this gives a compact, globally nonisotropic Riemannian three-manifold which is locally flat--- and thus locally isotropic).

However, the Riemann tensor of the four-dimensional Lorentzian manifold which is the setting ("spacetime") for physics in this FRW model (or quotient of FRW model, as the case may be), is nonzero, as is required by Einstein's equation! OTH, the Weyl tensor of this simple spacetime vanishes (which implies that in this approximation we are neglecting gravitational radiation, and we also neglect inhomogeneities in matter density and so on).

I'm curious. Do you think that if the OP could have comprehended any of that reply that they would have needed to ask the question they did?

 

[Levi Porter]

Thanks for the great responses

Where did you see a description of a "flat explosion"? Right now I can't think of any wayto make sense of the phrase "flat explosion" in gtr, since even a simple model of a spherically pulsating isolated spherically symmetric star would have nonzero Weyl curvature in the exterior.

I was just visualizing the Big Bang and thinking about the external forces required to create a flat universe. With my better understanding of the scientific meaning of the word "flat", I want to suggest or ask if it is appropriate to consider the Big Bang as an example of a "flat explosion."

What theory of gravitation do you have in mind? In the context of cosmology, the current default is gtr. If that is the theory you have in mind, gtr does not treat gravitation as a "force law" but rather as spacetime curvature; the content of the field equation of the theory, the Einstein field equation, is captured approximately by Wheeler's slogan, "spacetime curvature and material stress tell bits of matter how to move; the density and momentum of bits of matter and nongravitational fields tells spacetime how to curve".

I wasn't referring to gravity alone as a singular force that is applied for the containment of the universe. I was trying to find out what other forces are at work to contain the 3 dimensional universe and space/time without having it expand exponentially into infinite dimensions.

Sounds like you are confusing our galaxy with the entire universe! Are you perhaps reading a book which dates to about 1850 or something? Seriously, its been recognized for a long time that our galaxy is only one of many. (Many many!)

No I wasn't. They do seem to have similarities. I was asking if there was a consensus to a general shape of the universe; i.e. flat sphere, disc, cube...?

I wonder if it's appropriate to say that the Earth is flat once again?

 

[JesseM]

I was trying to find out what other forces are at work to contain the 3 dimensional universe and space/time without having it expand exponentially into infinite dimensions.

What reason is there to think there are additional spatial dimensions for it to expand into? There are some "braneworld" models coming out of superstring theory that postulate that our universe is a kind of membrane in a higher-dimensional space, see here for some more info, but there's no need to make such an assumption, and in general relativity curved space is not modeled as sitting in any sort of higher-dimensional space.

No I wasn't. They do seem to have similarities. I was asking if there was a consensus to a general shape of the universe; i.e. flat sphere, disc, cube...?

usually it's assumed that matter is pretty much evenly distributed throughout all of 3D space on large scales, when people talk about the shape of the universe they're talking about the curvature of space itself, not an idea of the matter being confined to a certain shape (say, a disc or a cube) within a larger empty space.

 

[Feonix12]

The universe as we know it is not flat, but as far as all the dimentions interlacing with each other, the second dimention does have something to do with the universe, as does all the others. How many there supposedly were i forgot.

 

[Chris Hillman]

Thanks for the great responses

You are very welcome

I was just visualizing the Big Bang and thinking about the external forces required to create a flat universe. With my better understanding of the scientific meaning of the word "flat", I want to suggest or ask if it is appropriate to consider the Big Bang as an example of a "flat explosion."

Ah--- progress! Next step: there are actually several technical meanings of "flat" which might be relevant here.

The exact solutions of the Einstein field equation most often used to illustrate how Hubble expansion and the "Big Bang" can be modeled in gtr are the FRW dust models. These happen to be conformally flat but not Ricci flat.

In gtr, Ricci curvature is the kind which is proportional to the stress-energy tensor, which represents the density and momentum of any matter or nongravitational fields. Also, "dust" is short for pressure-free perfect fluid, so these are matter-filled models. Once you know these items of information, it is obvious that the FRW dusts are not Ricci flat. The other kind of curvature (of spacetime) is called Weyl curvature or conformal curvature and is measured by the Weyl curvature tensor. Together, the Ricci and Weyl tensors contain the same information as the Riemann curvature tensor. The terms "Ricci flat", "conformally flat" and "flat", when applied to a spacetime model, just mean that respectively the Ricci, Weyl and Riemann tensor of the model vanish.

And then there is the issue of spatial hyperslices orthogonal to the world lines of the dust particles in these FRW models, or speaking informally "spaces at a (global, coordinate) time". (Not every congruence of timelike curves, i.e. a family of non-intersecting world lines which fill up a spacetime model, admits a family of orthogonal hyperslices; the congruence must be irrotational.) These may or may not be "flat" in the sense of having vanishing Riemann tensor when treated as Riemannian three-manifolds. Neglecting \Lambda for simplicity the FRW dusts come in three types, according to whether or not these hyperslices are locally S^3, \, E^3, \, H^3; these happen to expand and recollapse (first case) or to expand forever (second and third cases). Then, we can say that only the second features flat spatial hyperslices orthogonal to the world lines of the dust.

A bit complicated, perhaps --- but it sounds like you know enough to expect complications in cosmology!

So, er, one, two, three, several distinct uses of "flat" might be relevant. You can ask others here to clarify some of the technical terms I used if you are curious (or you can look at some cosmology textbooks--- I always hope someone who drops by PF will be so intrigued by what they find that they pick up a good textbook! ...color me bookish, I guess).

I wasn't referring to gravity alone as a singular force that is applied for the containment of the universe. I was trying to find out what other forces are at work to contain the 3 dimensional universe and space/time without having it expand exponentially into infinite dimensions.

Sorry, you lost me there.

No I wasn't. They do seem to have similarities. I was asking if there was a consensus to a general shape of the universe; i.e. flat sphere, disc, cube...?

Someone mentioned a possible interpretation of your post which hadn't occurred to me, that you might have thought that "flat" meant something like "squashed like a bug". If so, any "squashing flat" would entail non-isotropic dynamics, whereas the FRW models are homogeneous and isotropic (all directions are equivalent). If you really did mean "squashing", you will probably want to learn a bit about the Kasner dust, another exact solution often used as a cosmological model (for pedagogical purposes).

Cube: imagine identifying opposite faces of a solid cube. Well, if you didn't grok that, first imagine identifying the endpoints of a line segment, to form a circle, then imagine identifying opposite edges of a square, to form a torus. Anyway, yet another complication: we can also consider quotient manifolds formed from the FRW models by doing something like this with all the spatial hyperslices. So for example we could have an FRW dust with "locally flat spatial hyperslices" which each have the topology of a three-dimensional torus, S^1 \times S^1 \times S^1. And the slice corresponding to our own epoch might be bigger than the "observable universe", which would mean that we couldn't tell the difference. Or it might be small enought that we could tell the difference, hence the proposal by Cornish and Weeks to look for evidence of "nontrivial topology" in the spatial hyperslices. As this discussion shows, it might be impossible in principle for cosmologists to nail down all the details, unless they are prepared to wait billions of years--- and maybe not even then. The universe is a really big place--- I think it's amazing that scientists sitting on this tiny blue pearl can say with justifiable confidence anything at all about it!

It often happens that a casual inquiry involves all kinds of subtle issues--- that certainly seems to have happened here! This can sometimes make it difficult to compose a suitable response to a newbie posting.

I wonder if it's appropriate to say that the Earth is flat once again?

Don't follow, but many others here can help you. I think I'll bow out here, since it seems that I might be trying to pull you up to too high a level too quickly, which unfortunately can induce narcosis (yeah, a mangled diving metaphor) And at this higher level you would find--- you guessed it, more complications!