Jonathan Scott

Gold Member

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## Main Question or Discussion Point

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?

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?