Doubt about our spacetime manifold

In summary, the data from WMAP7 suggests that the universe has a slight negative spatial curvature. The curvature is not constant in time, but is constant on spatial slices. This curvature is related to measurable dynamical quantities in a Friedmann-Robertson-Walker universe.
  • #1
TrickyDicky
3,507
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I understand that accordingt to GR mass curves the spacetime (I'm not referring to spatial curvature k), so that the universe globally considered is a manifold with constant curvature, is this right?
If so, is this curvature positive or negative in the current cosmological model?
 
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  • #2
  • #3
The latest data from WMAP7 is that the universe is flat to within a few percent.
 
  • #4
TrickyDicky said:
I understand that accordingt to GR mass curves the spacetime (I'm not referring to spatial curvature k), so that the universe globally considered is a manifold with constant curvature, is this right?
If so, is this curvature positive or negative in the current cosmological model?

The spatial manifold (say M) has constant curvature. The spacetime manifold (RxM) has some non-trivial time-dependent curvature. Current data favours a universe with a slight negative spatial curvature, though the other scenarios are by no means ruled out.
 
  • #5
nicksauce said:
Current data favours a universe with a slight negative spatial curvature, though the other scenarios are by no means ruled out.
Hi Nick,

Could you provide a source for this? In reading through WMAP's most recent findings (arXiv:1001.4538), they report:

WMAP+BAO+SN (95% CL): [tex]-0.0178 < \Omega_k < 0.0063[/tex]
WMAP+BAO+H (95% CL): [tex]-0.0133 < \Omega_k < 0.0084[/tex]
 
  • #6
bapowell said:
Hi Nick,

Could you provide a source for this? In reading through WMAP's most recent findings (arXiv:1001.4538), they report:

WMAP+BAO+SN (95% CL): [tex]-0.0178 < \Omega_k < 0.0063[/tex]
WMAP+BAO+H (95% CL): [tex]-0.0133 < \Omega_k < 0.0084[/tex]

Yes, you're right. I should have said, slight positive curvature, with slight negative curvature not ruled out.
 
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  • #7
bapowell said:
The latest data from WMAP7 is that the universe is flat to within a few percent.

But this refers to (lack of) curvature of spatial slices, not to (lack of) spacetime curvature, and the original poster wrote
TrickyDicky said:
I'm not referring to spatial curvature k




TrickyDicky said:
so that the universe globally considered is a manifold with constant curvature, is this right?
If so, is this curvature positive or negative in the current cosmological model?

The spacetime curvature is non-zero, is constant on spatial slices, but is not constant in time. If curvature were not dynamical, einstein's equation wouldn't lead to a dynnaical Friedmann equation. For the components of the spacetime curvature tensor written in terms of the scale factor, see page 271 from

http://books.google.com/books?id=IyJhCHAryuUC&printsec=frontcover&dq=gron&cd=3#v=onepage&q&f=false.
 
  • #8
George Jones said:
But this refers to (lack of) curvature of spatial slices, not to (lack of) spacetime curvature, and the original poster wrote
Indeed. Thanks for pointing this out.
 
  • #9
Thanks for the answers.
I was thinking in terms of curvature R, as in this models from cosmoogy books: a de Sitter spacetime, and an Einstein spacetime have R>0, Anti de Sitter spacetime has R<0 , Minkowski spacetime has R=0. But of course all of these models are of static universes, I didn't realize that in our dynamical (expanding) universe the curvature is not so straightforward as is it is dynamical and I guess it can vary (noncostant and nonzero) as GeorgeJones pointed out.
Am I on the right track?
 
  • #10
Yes, you are on the right track. The scalar curvature, R, cannot be measured directly, but can be related to measurable dynamical quantities in a Friedmann-Robertson-Walker universe:

[tex]R \propto \dot{H} + 2H^2 + \frac{k}{a^2}[/tex]

where H is the Hubble parameter, k the curvature of spatial slices (the thing that WMAP constrains to be close to zero), and 'a' the scale factor. Using the Friedmann equations, this can be recast in terms of the energy content of the universe:

[tex]R \propto \frac{1}{3}\rho - p = \rho\left(\frac{1}{3} - w\right)[/tex]

where [tex]\rho[/tex] is the energy density of the universe and [tex]p[/tex] the pressure. The final equality is written in terms of observable parameters that are actively being constrained by current observations.
 
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  • #11
bapowell, what's w stand for in the last equation?
Thanks
 
  • #12
TrickyDicky said:
bapowell, what's w stand for in the last equation?
Thanks
Sorry. It's just a parameter that relates the energy density of the fluid to the pressure:

[tex]p = w\rho[/tex]

so nothing new...just a retooling of the previous equation. I write it this way because you frequently see [tex]w[/tex] constrained in experiments -- it is referred to as the equation of state parameter, or simply the equation of state. For reference, [tex]w = -1[/tex] is de Sitter expansion, [tex]w = 0[/tex] is pressureless dust, and [tex]w = 1/3[/tex] is radiation. You'll notice that a universe that is filled with radiation (uniformly) has R = 0.
 
  • #13
nicksauce said:
Yes, you're right. I should have said, slight positive curvature, with slight negative curvature not ruled out.
Well, no, that's not a correct interpretation of the data. The fact that zero is within the one-sigma limit means that curvature being zero is fully consistent with the data. Due to simple random statistical noise, we expect the experimental result to, on average, be about one sigma away from the true value anyway, so we cannot interpret any deviation from zero within one sigma as being evidence of a true value different from zero.
 
  • #14
bapowell said:
[tex]p = w\rho[/tex]
so nothing new...just a retooling of the previous equation. I write it this way because you frequently see [tex]w[/tex] constrained in experiments -- it is referred to as the equation of state parameter, or simply the equation of state. For reference, [tex]w = -1[/tex] is de Sitter expansion, [tex]w = 0[/tex] is pressureless dust, and [tex]w = 1/3[/tex] is radiation. You'll notice that a universe that is filled with radiation (uniformly) has R = 0.

Just one thing, I don't understand the case w=1/3. How is this a radiation filled universe? It would seem it correspnds to a universe with one third as much dust as radiation pressure if we follow that w=P/rho . What am I missing?
Thanks
 
  • #15
TrickyDicky said:
Just one thing, I don't understand the case w=1/3. How is this a radiation filled universe? It would seem it correspnds to a universe with one third as much dust as radiation pressure if we follow that w=P/rho . What am I missing?
Thanks
The pressure of radiation is one third its energy density, hence w = 1/3 for pure radiation (dust has no pressure).
 
  • #16
bapowell said:
[tex]R \propto \frac{1}{3}\rho - p [/tex]

I tried to obtain this by myself but I didn't get the same, After adding the right terms of the Friedmann equations I got:

[tex]R \propto - p [/tex]

I know this is simple math but I'm not sure what I did wrong.
Thanks in advance.
 
  • #17
TrickyDicky said:
I tried to obtain this by myself but I didn't get the same, After adding the right terms of the Friedmann equations I got:

[tex]R \propto - p [/tex]

I know this is simple math but I'm not sure what I did wrong.
Thanks in advance.
Make sure you're using the second Friedmann equation that casts the time derivative of the Hubble parameter in terms of the density, as well as the first to simplify things.
 

1. What is a spacetime manifold?

A spacetime manifold is a mathematical concept used to describe the four-dimensional structure of the universe. It combines the three dimensions of space (length, width, and height) with the dimension of time to create a four-dimensional framework for understanding the physical world.

2. How does the concept of spacetime manifold relate to Einstein's theory of relativity?

Einstein's theory of relativity is based on the idea that the laws of physics are the same for all observers, regardless of their relative motion. The concept of a spacetime manifold allows us to mathematically describe this idea by representing space and time as a single entity.

3. Why is there doubt about our understanding of the spacetime manifold?

While the concept of a spacetime manifold has been widely accepted and used in physics, there are still unanswered questions and inconsistencies within our understanding of it. Some scientists believe that there may be alternative theories or modifications to the current understanding of the spacetime manifold that could better explain certain phenomena.

4. How does the concept of spacetime manifold impact our understanding of the universe?

The concept of a spacetime manifold has greatly influenced our understanding of the universe and its fundamental laws. It has allowed us to develop theories such as general relativity and explore concepts like time dilation and the curvature of space. Without the concept of a spacetime manifold, our understanding of the physical world would be limited.

5. Are there any ongoing research or experiments related to the spacetime manifold?

Yes, there are ongoing research and experiments exploring the concept of spacetime manifold. Some scientists are looking for evidence of spatial distortions or gravitational waves that could support our understanding of the manifold. Others are exploring alternative theories that could provide a better understanding of the fundamental nature of space and time.

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