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Ok, I had another look at the links.
I am still not completely comfortable with the idea of a patchwork universe with all the patches (effectively?) being the same patch.
I am however comfortable with the idea of the universe being "compact", with no sharp edges or discontinuities.
It is entirely possible that you will not like what I am about to suggest. That's ok, since I am not totally comfortable with it either.
You wanted to know what mapping regime I had in mind. As I have pointed out I didn't concern myself with that initially, but now I have thought it through and cannot justify the projection of a plane onto the surface of a sphere or a volume to the hypersurface of a hyperspere. But I can justify the projection of a plane onto the surface of a hemisphere or a volume to the hypersurface of a hyperhemisphere (hemihypersphere?)
This unfortunately, from my perspective, then demands the sort of patchwork arrangement discussed in the links you sent so that anything moving past the border of the hemisphere (let's stick with 2+1 to make it simpler) would appear on the other side of the universe travelling along the same line (or arc).
Each one of us would perceive the universe as a plane stretching out tangentially from the surface of the sphere, effectively out to infinity. But that effective infinity is in terms of metres right now. What is infinity today won't necessarily be infinity tomorrow. (Yes, I don't like this either.)
Take a look at the diagram now. I will try to show what I mean graphically since words seem to fail me here.
Location A can be thought of as lying on the plane but that version of the location is in different time from the one we are "in". It's in the future. The version that is on our surface of simultaneity is closer and that is the one that really matters. Note that we cannot "see" either, since photons have to get to us.
The same applies to Location B. If you take a line like the one to Location B and increase the angle of it from the top of the hemicircle until it nears pi/2, then you can see that the plane effectively stretches out to infinity. But when that version of the location lies on the same surface of simultaneity as me, it won't be infinitely distant (admittedly though, it might be at an infinitely distant time).
Anyway, it is this plane (flat in 2d) that I want mapped onto the surface of simultaneity.
To the best of my knowledge the transformation would be something like:
(t*tan(theta),t*tan(phi)) > (t,theta,phi) ...or... (x,y) > (t,arctan(x/t),arctan(y/t))
I don't think this schema is bad locally, but I really would not want to be fiddling around at the edges.
I did say I wasn't totally comfortable, didn't I?
cheers,
neopolitan
I am still not completely comfortable with the idea of a patchwork universe with all the patches (effectively?) being the same patch.
I am however comfortable with the idea of the universe being "compact", with no sharp edges or discontinuities.
It is entirely possible that you will not like what I am about to suggest. That's ok, since I am not totally comfortable with it either.
You wanted to know what mapping regime I had in mind. As I have pointed out I didn't concern myself with that initially, but now I have thought it through and cannot justify the projection of a plane onto the surface of a sphere or a volume to the hypersurface of a hyperspere. But I can justify the projection of a plane onto the surface of a hemisphere or a volume to the hypersurface of a hyperhemisphere (hemihypersphere?)
This unfortunately, from my perspective, then demands the sort of patchwork arrangement discussed in the links you sent so that anything moving past the border of the hemisphere (let's stick with 2+1 to make it simpler) would appear on the other side of the universe travelling along the same line (or arc).
Each one of us would perceive the universe as a plane stretching out tangentially from the surface of the sphere, effectively out to infinity. But that effective infinity is in terms of metres right now. What is infinity today won't necessarily be infinity tomorrow. (Yes, I don't like this either.)
Take a look at the diagram now. I will try to show what I mean graphically since words seem to fail me here.
Location A can be thought of as lying on the plane but that version of the location is in different time from the one we are "in". It's in the future. The version that is on our surface of simultaneity is closer and that is the one that really matters. Note that we cannot "see" either, since photons have to get to us.
The same applies to Location B. If you take a line like the one to Location B and increase the angle of it from the top of the hemicircle until it nears pi/2, then you can see that the plane effectively stretches out to infinity. But when that version of the location lies on the same surface of simultaneity as me, it won't be infinitely distant (admittedly though, it might be at an infinitely distant time).
Anyway, it is this plane (flat in 2d) that I want mapped onto the surface of simultaneity.
To the best of my knowledge the transformation would be something like:
(t*tan(theta),t*tan(phi)) > (t,theta,phi) ...or... (x,y) > (t,arctan(x/t),arctan(y/t))
I don't think this schema is bad locally, but I really would not want to be fiddling around at the edges.
I did say I wasn't totally comfortable, didn't I?
cheers,
neopolitan
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