The Shape of the Universe, and General Relativity

[Radiohannah]

Hello!

I'm trying to get my head around general relativity at the moment...(!), and there's one aspect of it that's really causing me a lot of kerfuffle.

I understand that in an appropriately sized local inertial frame, the laws of special relativity occur. On those scales the curvature of space-time will not be noticeable, and you are in an inertial frame of reference.

How, if at all, does this relate to the shape of the Universe?

If the Universe is spherical in the k=+1 case, is the space-time curved in the same way in that it curves around mass? Does that then have an additional/the same effect on things like orbits? Or is it on far too large a scale?

Do the Einstein field equations have any relation to the Friedmann equation, I think, is what I am trying to ask?!

I hope that makes sense!

Cheers!
Hannah

 

[bcrowell]

If the Universe is spherical in the k=+1 case, is the space-time curved in the same way in that it curves around mass?

Yes, it's curved. No, the curvature isn't exactly the same type as, say, the curvature that exists around the sun. For example, you get a type of curvature that produces tidal forces around the sun, but there can't be that specific type of curvature in an FRW cosmological model. That's because the cosmological model is constructed to be isotropic and homogeneous, so by symmetry there is no preferred direction for a tidal effect.

Does that then have an additional/the same effect on things like orbits? Or is it on far too large a scale?

You mean like the Earth's orbit around the sun? No, it has no measurable effect on that: http://arxiv.org/abs/astro-ph/9803097v1

Do the Einstein field equations have any relation to the Friedmann equation, I think, is what I am trying to ask?!

The Friedmann equations are derived from (1) the Einstein field equations, plus (2) the assumption of homogeneity and isotropy.

-Ben

 

[Radiohannah]

Hello! Cheers!
I think I see what you mean.
Do the metrics take into account the shape of the Universe? Or only the curvature of space-time by mass (since the shape "has no measurable effect on things like orbits")? I think I am confused because in the metrics I have seen, there is a term for 'curvature', which may apply to spheres, flat surfaces and hyperbolic surfaces,...but these are curvatures to do with the mass alone rather than the shape of the Universe?
Hannah

 

[bcrowell]

Do the metrics take into account the shape of the Universe? Or only the curvature of space-time by mass (since the shape "has no measurable effect on things like orbits")? I think I am confused because in the metrics I have seen, there is a term for 'curvature', which may apply to spheres, flat surfaces and hyperbolic surfaces,...but these are curvatures to do with the mass alone rather than the shape of the Universe?

There is no real distinction between curvature due to mass and curvature not due to mass. For example, we have good evidence that gravitational waves exist. A gravitational wave is a traveling oscillation in the curvature of spacetime. Some gravitational waves may be due to mass (e.g., the famous Hulse-Taylor binary pulsar) or some may be left over from the Big Bang. They both act the same.

We all visualize curvature using two-dimensional surfaces embedded in three dimensions, because that's how our brains work. But there is no need for any such embedding in GR.

The main reason that the spacetime curvature due to the sun has a big effect on the Earth's orbit, while cosmological curvature doesn't, is that the curvature due to the sun is simply much stronger. However, after one year the Earth's orbit brings it back to the same point where it was before. But cosmological effects accumulate, which is why, for example, a quasar can be ten times farther away from us now than when it was first formed.

 

[JesseM]

It's also worth adding that the overall curvature of space does depend on mass/energy--if the average energy density (which includes energy due to mass) throughout the universe is equal to a certain critical value then space is flat, if it's above the critical value then space is positively-curved like the surface of a 4D sphere, if it's below the critical value then space has negative or "hyperbolic" curvature. This is discussed in part 3 of Ned Wright's cosmology tutorial, for example.

 

[WannabeNewton]

Just to clarify something for myself: when the mass density is above the critical density and k = +1 does the FRW metric describe a universe with the geometry of a 4 - sphere for each hypersurface of t = const. in the domain of t or a universe that evolves with the geometry of a 4 - sphere for the domain of t (as in it starts at a point then grows to a maximum and then recedes again)?

 

[JesseM]

Just to clarify something for myself: when the mass density is above the critical density and k = +1 does the FRW metric describe a universe with the geometry of a 4 - sphere for each hypersurface of t = const. in the domain of t

Yes, the three possible geometries refer to the geometry of space on surfaces of constant t in the FLRW coordinate system.

or a universe that evolves with the geometry of a 4 - sphere for the domain of t (as in it starts at a point then grows to a maximum and then recedes again)?

If there's no cosmological constant/dark energy then a universe with positive spatial curvature is also bound to collapse back into a big crunch eventually (while a universe with flat curvature is bound to expand forever but with the expansion rate approaching zero, and negative curvature means eternal expansion without the expansion rate approaching zero), but if there is CC/DE then this isn't necessarily true any more.