Register to reply 
Doubt about our spacetime manifold 
Share this thread: 
#1
Jun1110, 08:56 AM

P: 3,035

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? 


#2
Jun1110, 09:19 AM

P: 48

Good question, I'd actually just logged on to ask a similar one myself. As such I shall not directly answer you but I've just been reading this wiki page that seems pretty descriptive: http://en.wikipedia.org/wiki/Shape_of_the_Universe



#3
Jun1110, 09:50 AM

Sci Advisor
P: 1,682

The latest data from WMAP7 is that the universe is flat to within a few percent.



#4
Jun1110, 10:03 AM

Sci Advisor
HW Helper
P: 1,275

Doubt about our spacetime manifold



#5
Jun1110, 10:09 AM

Sci Advisor
P: 1,682

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
Jun1110, 10:18 AM

Sci Advisor
HW Helper
P: 1,275




#7
Jun1110, 12:57 PM

Mentor
P: 6,242

http://books.google.com/books?id=IyJ...page&q&f=false. 


#8
Jun1110, 01:00 PM

Sci Advisor
P: 1,682




#9
Jun1110, 01:46 PM

P: 3,035

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
Jun1110, 01:59 PM

Sci Advisor
P: 1,682

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 FriedmannRobertsonWalker 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. 


#11
Jun1110, 02:05 PM

P: 3,035

bapowell, what's w stand for in the last equation?
Thanks 


#12
Jun1110, 02:10 PM

Sci Advisor
P: 1,682

[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
Jun1210, 01:11 AM

Sci Advisor
P: 4,800




#14
Jun1310, 05:27 AM

P: 3,035

Thanks 


#15
Jun1310, 05:33 AM

Sci Advisor
P: 4,800




#16
Jun1310, 03:12 PM

P: 3,035

[tex]R \propto  p [/tex] I know this is simple math but I'm not sure what I did wrong. Thanks in advance. 


#17
Jun1310, 05:00 PM

Sci Advisor
P: 4,800




Register to reply 
Related Discussions  
Spacetime manifold: initial condition or result of GR?  Special & General Relativity  4  
Manifold ?!  Differential Equations  4  
Embedding curved spacetime in higherd flat spacetime  Special & General Relativity  9  
4 dimensional spacetime manifold question  Differential Geometry  7  
Why Minkowski spacetime and not Euclidean spacetime?  Special & General Relativity  10 