The Shape of the Universe, and General Relativity

In summary: I'm not sure what you're asking. The FRW metric describes 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...)?The FRW metric describes 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...).
  • #1
Radiohannah
49
0
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
 
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  • #2
Radiohannah said:
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.

Radiohannah said:
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

Radiohannah said:
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
 
  • #3
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
 
  • #4
Radiohannah said:
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.
 
  • #5
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.
 
  • #6
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)?
 
  • #7
WannabeNewton said:
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.
WannabeNewton said:
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.
 

1. What is the shape of the universe according to General Relativity?

The shape of the universe according to General Relativity is believed to be flat. This means that the universe is infinite and has no curvature, similar to a sheet of paper. However, this remains a topic of ongoing research and there are other theories that suggest a curved or closed universe.

2. How does General Relativity explain the expansion of the universe?

General Relativity explains the expansion of the universe through the theory of cosmic inflation. This states that the universe underwent a rapid period of expansion in the first fractions of a second after the Big Bang, causing it to expand at an accelerating rate. This is supported by observational evidence such as the redshift of distant galaxies.

3. What is the relationship between gravity and the shape of the universe?

In General Relativity, gravity is not seen as a force but rather as a curvature of spacetime caused by the presence of mass and energy. This means that the distribution of matter and energy in the universe affects the shape of the universe and can even determine whether it is flat, curved, or closed.

4. Can General Relativity explain the origin of the universe?

General Relativity alone cannot explain the origin of the universe. It can only describe the behavior of the universe after the Big Bang. To understand the origin of the universe, scientists are still exploring theories such as the Big Bang theory and inflation.

5. How does General Relativity relate to other theories of gravity?

General Relativity is the most well-known and widely accepted theory of gravity, but it is not the only one. Other theories, such as Quantum Gravity, attempt to combine General Relativity with quantum mechanics to provide a more complete understanding of gravity. However, General Relativity remains the most accurate and reliable theory of gravity for most situations.

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