Volume of Dimension reduced due to gravitation

[benk99nenm312]

I've had this on my mind for a while. I'm pretty confident that I have the right idea, but I can't find a formula that I could use to prove it to someone. So, here it is.

Imagine a sphere of space-time with absolutely nothing inside of it. It is a large sphere, sun-size large. Now, take that sphere and insert something like the sun into it, so that the mass of the sun causes curvature on the surrounding space. Because the sun is in this sphere of space-time, the curvature of the sun will compress the entire sphere inward, and reduce the volume of the sphere.

This reminds me of Einstein's universe without the cosmological constant.

Is there any equation that shows this? If so, then what is it? My guess is that it would be one of Einstein's, but I can't find it anywhere.

 

[confinement]

Because the sun is in this sphere of space-time, the curvature of the sun will compress the entire sphere inward, and reduce the volume of the sphere.

I am not convinced of this statement. I'm not saying it is wrong, as I have not worked through the details, but it does appear to me as not to follow from your considerations up to that point.

What are the conditions for the metric on the boundary of the space time region? I ask because you make it sound as if the "shrinking sphere" is part of a larger spacetime. If not, if the sphere is "everything", then what you have described is similar to the assumptions that go into the positive-curvature (spherical curvature) solutions of the Friedman equations, which is the basis for the study of cosmology.

It would help if you explained what you mean by:

Einstein's universe without the cosmological constant.

because I don't know any cosmological models that are named after Einstein, although his field equations of General Relativity can come with or without a cosmological constant. But these are just equations that describe, mathematically, how the curvature of spacetime responds to the presence of matter at each point. It is rather a different occupation to create a model of the universe out of a solution to Einstein's equations (with or without a cosmological constant).

Here is a technical article at the undergraduate level which describes how to turn your thoughts into mathematical assumptions which then lead to model that is simple enough so that the Einstein equations can be solved exactly, but still rich enough to make non-trivial predictions.

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

 

[benk99nenm312]

What I implied was that the sphere would be "all there is", like us with the universe. I said that I wanted no cosmological constant because I wanted to see how the sphere would shrink without complications.

So, am I trying to describe the Friedmann equations pictorially?

 

[Naty1]

Are you asking if the presence of gravity via a mass shrinks surrounding space as well as causing curvature in spacetime?

 

[benk99nenm312]

Yes. From what I understand, the presence of matter in the universe would, before the observations and the cosmologoical constant, shrink the universe and keep it from being static. This would mean that the volume of an imaginary sphere surrounding an object should be shrunk due to the curvature. I just want to find an equation to show this.

 

[Naty1]

Are you asking if the presence of gravity via a mass shrinks surrounding space as well as causing curvature in spacetime?

Ok so that's the basic problem statement...

(Edit: after posting I now wonder if the following misinterprets conditions applicable to the Einstein field equations...I'd remove this entire post, but maybe it will give somebody an idea..or a chance to howl in dismay ..)

I'm trying to figure out what various solutions to the Einstein field equations mean and how they are related. So take the following perhaps as some clues rather than an immediate simplified answer

Wikipedia: http://en.wikipedia.org/wiki/Friedman_equation#Assumptions

The Friedman equations start with the simplifying assumption that the universe is spatially homogeneous and isotropic; empirically, this is justified on scales larger than 100 Mpc.

Einstein's equations now relate the evolution of this scale factor to the pressure and energy of the matter in the universe.

Wikipedia: http://en.wikipedia.org/wiki/Einstein_field_equations#Vacuum_field_equations

The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution.

None of these conditions appears to match your hypothetical question. Further, as far as I know a "cosmological constant" caused inflation during a brief transition phase when energy density remained constant until a more stable lower energy universe evolved. Since then we have had a much smaller, perhaps variable, cosmological "constant" which is now causing acceleration of the universe (experimentally observed via Hubble observations).

Once again, these conditions do NOT appear to match your hypothetical...

My guess is that you need to see if you can eliminate (simplify) the basic Einstein Field Equations shown here, http://en.wikipedia.org/wiki/Einstein_field_equations#Mathematical_form

and see if the resulting equations can be solved... or even apply...and I think that is your posted dilemma..

If there is no cosmological constant, no negative pressure/density,there is no vacuum energy, so there is no gravitational attraction...Does space curve but not contract?? It's such an unrealistic set of conditions I have no idea...

Maybe your question boils down to this: If space had no vacuum energy, would gravity cause curvature of space time?? That might require knowing how the various components of the Einstein stress energy tensor interact.

 

[Naty1]

another idea...I don't have the math skills to know whether it really makes things simpler or not: Since you are dealing with a hypothetical situation, maybe an infinite plane of mass which would result in a uniform gravitational field would offer simplified insights to your question...
There are people here with a lot more math training and experience than I; I'm surprised they have not posted ...

 

[Al68]

I've had this on my mind for a while. I'm pretty confident that I have the right idea, but I can't find a formula that I could use to prove it to someone. So, here it is.

Imagine a sphere of space-time with absolutely nothing inside of it. It is a large sphere, sun-size large. Now, take that sphere and insert something like the sun into it, so that the mass of the sun causes curvature on the surrounding space. Because the sun is in this sphere of space-time, the curvature of the sun will compress the entire sphere inward, and reduce the volume of the sphere.

This reminds me of Einstein's universe without the cosmological constant.

Is there any equation that shows this? If so, then what is it? My guess is that it would be one of Einstein's, but I can't find it anywhere.

How is your imaginary sphere defined? If it's defined by a specified radius, then it's volume would remain constant by definition.

 

[benk99nenm312]

My original sphere, my imaginary sphere, with nothing inside of it, has a specific radius. After this, If we insert an object into the imaginary sphere, the object would curve space and reduce it's volume due to the mass of the object.

This seems to be a lot more complicated than I thought. (smile)

 

[Mentz114]

My original sphere, my imaginary sphere, with nothing inside of it, has a specific radius. After this, If we insert an object into the imaginary sphere, the object would curve space and reduce it's volume due to the mass of the object.

This seems to be a lot more complicated than I thought. (smile)

Does the object fill the designated sphere ?
If not you can model both situations with the exterior Schwarzschild metric, one case with a very small M and the other with a significant M.

It is not correct to think of space-time as shrinking and expanding, you should think about how observers will measure the radius of your sphere for the two situations. If two observers with identical rulers travel to the two spheres and measure a radius of 20 units, say, would a third observer who stayed at home see the spheres as equal ?

And that probably is more complicated that either of us thought.

 

[benk99nenm312]

The first sphere is empty.

The object fills up a certain fraction of the second sphere, but not all of it. The object is inside the sphere, pulling inwards. I would like to consider the frame of reference to be from someone on the outside looking at the spheres.

Also, another key question is this.

The sphere is empty. There is nothing inside to bend the space and pull it inwards. It has a specified radius and volume. Then, someone (for the sake of theoretical science) appears out of nowhere inside the sphere. They bend the space inside the sphere, and pull it inwards. Would they, from their reference frame, measure the sphere to have essentially contracted?