# B Shape of the universe

1. Dec 29, 2017

### wolram

This may be a silly question, but how do we give a shape to the universe if it is The same from every point we look?

2. Dec 29, 2017

### Arman777

We dont give a shape.

We can make an argument that universe looks the same at larger scales (you can further search cosmological principle). This argument reduces the possible geomtries that universe can have.

1-Flat with zero curvature
2-Spherical geometry with positive curvature
3-Hyperbolic geometry with negative curvature

From observation we can conclude or derive the geometry of universe.

Recent data (Plank 2015 results or etc) shows that universe is flat and has zero curvature.

It may have another shape like a torus maybe but even in that case the size of torus must be huge cause obersvable universe seems flat over a large scale.

3. Dec 29, 2017

### PeroK

You could ask the same about the surface of the Earth, assuming it were ideally spherical. Its curvature can be defined by its being a 2D surface embedded in 3D space: i.e. it's the surface of a 3D sphere.

Or, its shape can be defined using differential geometry to define the infinitesimal distance in any direction. Using spherical polar coordinates this is:

$dS^2 = R^2(d\theta^2 + \sin^2 \theta d\phi^2)$, where $0 \le \phi < 2\pi$ and $0 \le \theta \le \pi$, and $R$ is some parameter, which equates to the radius of the 3D sphere above.

This "line element", in fact, encapsulates the shape of the sphere's surface without directly appealing to an embedding in a higher dimension.

Note that this is the distance along the surface, not taking any shortcuts through the body of the Earth!

4. Dec 29, 2017

### wolram

I get the analogy of the Earth, but as regards to the universe what does flat mean, surly due to gravity and dark matter the universe must be globular, but how can that be flat?

5. Dec 29, 2017

### PeroK

It's flat on a large scale. But, locally, where there are galaxies, black holes or simply where there are stars or planets you have locally curved spacetime.

6. Dec 29, 2017

### Arman777

Why it would be globular ? And how can it be globular and flat at the same time ?

There's something you should mention, are you talking about the universe or observable universe.

We cant now the geometry of the universe, (since we cant travel or observe the whole universe etc), but we can claim that observable universe is flat. Becasue that is what we observe (by experiment and measurement)

The universe can have a spherical geomtery or it can be just flat as ours, but If its sphere the R (radius of the sphere) must be so huge that, we observe, the observable universe as flat.

Mathematically it means that curvature is zero, In example a plane is a flat becasue, when you set a triangle, and you measure the angles it gives you $π$. But in spherical geometry it gives you more then $π$.

Of course in metric terms it would be more different

Try to think that you are a small ant, and you are travelling on a piece of paper. Wherever you walk or go it feels like you are on a flat surface and when you draw a triagle and measure the angles you get $π$. Observable universe is a piece of paper and you are an ant.

If we were bigger then ant, then we could have notice the curvature, but since we are not, we cannot know the real geometry of the universe.

(thats why our ancestors thought that earth is flat in the first place casue it was hard to observe that earth is spherical, in our case we cannot observe or dont have tools to see the real geometry of the universe, we are more like a bacteria respect to the universe)

Last edited: Dec 29, 2017
7. Dec 29, 2017

### phinds

But if the universe were a sphere, that would clearly imply a preferred direction and there is zero evidence of such a thing.

8. Dec 29, 2017

### Staff: Mentor

It means spatially flat: a spacelike slice of constant time for a comoving observer is Euclidean 3-space.

9. Dec 29, 2017

### Staff: Mentor

No, it woudn't; there is no preferred direction in a 3-sphere. A 3-sphere is spatially isotropic.

You might be confusing this with the case of a spherically symmetric, but not homogeneous, spacetime such as Schwarzschild spacetime. In this case, yes, the radial direction is different from the tangential directions. But that's due to the fact that there is a mass at the center, but not anywhere else; i.e., the spacetime is not homogeneous. In a closed FRW universe, which has the spatial geometry of a 3-sphere, that's not the case: the density is the same everywhere, and all directions are the same

10. Dec 29, 2017

### wolram

Thank you for your replies, I can see now that when we talk about the universe having shape we only mean the observable universe

11. Dec 29, 2017

### phinds

That's not very exciting since the OU is just a sphere centered on you.