Does GR say anything about solid angle deficit?

[Jonathan Scott]

We describe mass as creating "curvature" in the universe. Taking a simplified lower dimensional analogy, the universe at a point in time could be seen on average as being like the surface of a ball, but at a more detailed level the surface could consist of shallow cones (made of locally flat material) around rounded locations corresponding to masses. With this model, the proportion of the total mass enclosed within any loop would be equal to the total angular deficit around that loop as a proportion of 4 pi.

When GR is used to derive the shape of space around a central object, as in the Schwarzschild solution, the assumption is made that space at a sufficient distance from the central mass is flat. However, I feel that since the universe is finite, some property similar to the angular deficit should probably apply, so the exact limit would not be flat but rather something like a 3D cone, representing a fraction of a finite universe. (That is, the space would be locally flat, like a paper cone, but would be missing a finite angle around a central object).

For example, one hypothesis by analogy with the ball model might be that there would be a solid angle deficit equal to 8 pi times the fraction of the total mass of the universe enclosed by a surface.

I'm only familiar with GR being used for cosmological solutions and for local central solutions, and not anything in between. Is there any known GR result which might relate to this "solid angle deficit" or any similar way of describing the shape of space at a boundary which encloses a large amount of mass but is at a large distance from the central mass?

 

[Mentz114]

Wow, I have trouble getting my head around that. Are you perhaps talking about 'parallel transport' or holonomy ? If you're not familiar with those terms, try doing a look-up.

If you think the universe is finite, have a look at the Robertson-Walker space-time with k = 0. At any given time, the spatial part has constant curvature.

 

[yuiop]

Wow, I have trouble getting my head around that. Are you perhaps talking about 'parallel transport' or holonomy ? If you're not familiar with those terms, try doing a look-up.

If you think the universe is finite, have a look at the Robertson-Walker space-time with epsilon=0. I think it's conformally flat.

This FAQ http://www.astro.ucla.edu/~wright/cosmo_constant.html

suggests the bulk of the recent evidence points towards a universe with positive curvature and therefore a closed and finite universe.

 

[Mentz114]

Hi Kev:

that's an interesting link. I don't have a view on whether the universe is bounded/unbounded, I just thought the OP could investigate his problem in a GR cosmological context. It would be interesting work out whether a vector is parallel transported around a circle radius r in a FLRW space-time.

M

 

[Jonathan Scott]

OK, I've looked up parallel transport and holonomy, and yes, it's related to that. However, this is obviously very closely related to the concept of "angular deficit" (as in Descarte's law of closure deficit), sometimes called "angular defect", yet there seem to be very few references which link the two.

I've now found that Greg Egan has a web page called "General Relativity in 2+1 dimensions" which does mention all of these terms and uses a very similar model to the one I described in the first post in this thread. However, I've not been able to find anything about this in the usual space-time dimensions.