Time dilation and length contraction in hyperbolically-curved spacetime

[kmarinas86]

For the universe to be totally flat, and not just asymptotically closer to being flat as the universe expands, wouldn't there have to be pockets of hyperbolically-curved spacetime?

In either case, I still would like to know how one expresses time dilation and length contraction in hyperbolic spacetime. Is it possible in hyperbolic space that the gravitational time dilation factor can be less than one while matter length expands rather contracts due to spacetime curvature?

 

[Naty1]

You might find this of interest:

http://en.wikipedia.org/wiki/Rindler_coordinates#The_Rindler_observers


As a general comment, I'd expect all sorts of locally curved spacetimes given the variety of energy and mass distributions in the universer...supernovas, stars, black holes, pulsars,etc.

Why pick hyperbolically shaped??

 

[kmarinas86]

You might find this of interest:

http://en.wikipedia.org/wiki/Rindler_coordinates#The_Rindler_observersAs a general comment, I'd expect all sorts of locally curved spacetimes given the variety of energy and mass distributions in the universer...supernovas, stars, black holes, pulsars,etc.

Why pick hyperbolically shaped??

Because I already know the answer for elliptical and flat geometries. The Rindler coordinates are for locally flat spacetime. It's not even sufficient in mapping the global curvature of space due to mass. Your response does not answer my question, "Is it possible in hyperbolic space that the gravitational time dilation factor can be less than one while matter length expands rather contracts due to spacetime curvature?" This is a GR question, not an SR one.

http://en.wikipedia.org/wiki/Rindler_coordinates

In relativistic physics, the Rindler coordinate chart is an important and useful coordinate chart representing part of flat spacetime, also called the Minkowski vacuum.

http://en.wikipedia.org/wiki/Minkowski_spacetime

Strictly speaking, the use of the Minkowski space to describe physical systems over finite distances applies only in the Newtonian limit of systems without significant gravitation. In the case of significant gravitation, spacetime becomes curved and one must abandon special relativity in favor of the full theory of general relativity.

So I am definitely not talking about Minkowski spacetime.

 

[Bill_K]

So I am definitely not talking about Minkowski spacetime.

Well what are you talking about?? If you want an answer that's on target you need to express yourself clearly. The first thing you said was,

For the universe to be totally flat,

and the only universe which is totally flat is Minkowski space. You then talk about "hyperbolic spacetime", along with elliptical and flat geometries. My only guess is that you're referring to the three varieties of Friedmann cosmologies for k=0, ±1. Beyond that, I have no idea what you're looking for. Are you confusing hyperbolic curvature with negative mass?

 

[Naty1]

kmarinas: pssssst!
If no one answers your question, try Roger Penroses THE ROAD TO REALITY.. he has a number of Hyperbolic geometry discussions...I skimmed them but the math later in his book. where the details reside, is too advanced for me.

 

[kmarinas86]

Well what are you talking about?? If you want an answer that's on target you need to express yourself clearly. The first thing you said was,

and the only universe which is totally flat is Minkowski space. You then talk about "hyperbolic spacetime", along with elliptical and flat geometries. My only guess is that you're referring to the three varieties of Friedmann cosmologies for k=0, ±1. Beyond that, I have no idea what you're looking for. Are you confusing hyperbolic curvature with negative mass?

I'm talking about cosmologically "flat" space with imperfections. I would think that one needs more than dimples near matter (positively-curved geometry) to explain this flatness. It would seem that the void would have to be filled with saddles (negative-curved geometry) to balance out the curvature.

No one of fame has seemed to appreciate very much the implications such might have on what the universe looks like. Hyperbolic spacetime would have the effect of making stars seem farther apart than they really are. This would make observed masses of galaxies look like they are incapable of holding themselves together, when in fact they are.

Such hyperbolic space-time could be a way to explain observations currently attributed to "microlensing" without postulating some hidden masses.

http://www.google.com/search?q="saddle+points"+microlensing

Also, in showing the universe to be flat, one might also suggest that there is some conservation principle in which the total potential associated with the creation of positive curvature and that associated with the creation of negative curvature in spacetime sum to zero.

 

[Bill_K]

OKay, we are talking about the Friedmann cosmologies, in which the space sections can either be spherical, hyperbolic or flat. The source for the Friedmann universe is a smoothed-out mass density, and the thing that determines the geometry of the space sections is how the mass density compares to a critical value. If Ω = 8πGρ/H² where H is the Hubble constant, then Ω > 1 is spherical, Ω < 1 is hyperbolic, and Ω = 1 is flat. One could imagine replacing the uniform mass density with a collection of Schwarzschild masses that cause 'dimples' in the spacetime, and the same arguments would apply.

But in the first place, it's only the curvature of the space sections that can be hyperbolic, not the entire spacetime. (You can easily cut Minkowski space into hyperbolic slices!)

Secondly, for the flat case (which is what we appear to live in) there's no requirement that regions of positive curvature must be balanced by regions of negative curvature, only that Ω = 1. (What would cause a negative curvature?? Again, I think you're trying to argue for the presence of negative masses.)

Thirdly, the cosmological curvature is much too small to have the effects you're talking about on individual stars or even individual galaxies. It would cause the number count of galaxies to be greater with increasing distance, but even this is too small to observe. The evidence that the universe is 'flat' comes from counts of Ia supernovas, plus variations in the cosmic microwave background.

 

[Naty1]

BillK:

Is there any possibility dark energy (instead of negative mass) could be a source of negative curvature? If it's anything like the 70% of total mass-energy currently believed, seems like it should curve something somewhere.

I don't necessarily mean to the extent that kmarinas likely means and I don't know if the currently perceived distribution of dark energy would even remotely match any spacetime curvature anyway.

 

[kmarinas86]

Thirdly, the cosmological curvature is much too small to have the effects you're talking about on individual stars or even individual galaxies. It would cause the number count of galaxies to be greater with increasing distance, but even this is too small to observe.

Distance as we know it wouldn't be the same in a hyperbolically curved space. Things would appear farther away due to this curvature - the opposite of a "gravitational lens". So even if the number of galaxies would increase with distance at a rate exceeding that which would be expected from a cosmological principle, you would not even notice it by just looking at a photograph because they would look smaller and thus farther away, compensating for their greater presence. The idea would testable though if it were occur on an interstellar scale. All one would have to do is send two telescopes in opposite directions from our solar system into deep interstellar space to detect any anomalies in the observed parallax of stars.