iampete wrote:If one takes a projection of a 3-D sphere onto a 2-D surface, one can determine the center of the 2-D projection. Why would that analogy not work when trying to project a 4-D sphere into the 3-D space that we observe?
What works for the 3- and 2-D case
will work for the 4- and 3-D case, as well as for the simple 2- and 1-D case which simply projects the circumference of a circle onto a line segment.
But I don't think of the expanding 4-D supersphere (with time as its radius and space dimensions as its 3-D supersurface) as the
shape of the universe. I think of it as a
geometrical convenience that sets the time dimension perpendicular to space in the present day, merges a model of four dimensions with a model of expanding space, and places the big bang at a singular point in the 4-D center. If you want to know how far this analogy goes, you need only ask where we look for the parts of 3-D space closest to this singular point in the center. The answer is that we look as far away as we can in any direction. So now the center is all around and we're in the center of
it. The universe is inside out.
So I'm not sure what exactly your proposed projection corresponds to in the physical universe, and I have the same old problem with it anyway: If I am to believe in Einsteinian spacetime (even though "it just don't seem natural"), I must reject any line of thinking that depends on talking about an absolute position in 3-D space, and I think you need one to choose the orientation for your proposed projector and screen, so to speak. Anyway, your projection would tell you what orientation you chose, not what you want to know about the space (which I think is unknowable because it's not there).
The more I wing it, the less I know. I think I'll stop here.
iampete wrote:If you have further suggestions for layman type reading material beyond what you provided earlier, I'd be interested to see them.
Special Relativity, General Relativity, Cosmology ...
Special Relativity only took Einstein nine pages to describe completely. My textbook was a paperback less than a half inch thick (
Introduction to Special Relativity by Robert Resnick) and we did it in a half semester of a three-credit course. The book made it easy with illustrations, exercises, and I believe problems with answers. Used ones are cheaper than new ones, and the ones at the library are cheaper yet. After that book, the translated original Einstein was clear when I read it. For less technical, there's TV documentaries galore on Special Relativity. For more technical, some of us get state university classes on a free cable TV station. Any layman with an open mind and a little math can grasp Special Relativity completely.
There is no simple nor intuitive way to describe the blood and guts of General Relativity, and I'm sorry to have nothing to recommend. Even if they describe it well, it comes out lengthy (Hawking) or over my head in its brevity and heavy math (Einstein). Some TV documentaries about black holes, etc. may be worth a look.
Cosmology articles, not too technical nor non-technical, appeared regularly in
Sky & Telescope during the years I had a subscription. I don't know how it is today.
These days I mostly use google to find stuff on the internet. You need to pick through the results for something good, though.
[quote="iampete"]If one takes a projection of a 3-D sphere onto a 2-D surface, one can determine the center of the 2-D projection. Why would that analogy not work when trying to project a 4-D sphere into the 3-D space that we observe?[/quote]
What works for the 3- and 2-D case [i]will[/i] work for the 4- and 3-D case, as well as for the simple 2- and 1-D case which simply projects the circumference of a circle onto a line segment.
But I don't think of the expanding 4-D supersphere (with time as its radius and space dimensions as its 3-D supersurface) as the [i]shape of the universe[/i]. I think of it as a [i]geometrical convenience[/i] that sets the time dimension perpendicular to space in the present day, merges a model of four dimensions with a model of expanding space, and places the big bang at a singular point in the 4-D center. If you want to know how far this analogy goes, you need only ask where we look for the parts of 3-D space closest to this singular point in the center. The answer is that we look as far away as we can in any direction. So now the center is all around and we're in the center of [i]it[/i]. The universe is inside out.
So I'm not sure what exactly your proposed projection corresponds to in the physical universe, and I have the same old problem with it anyway: If I am to believe in Einsteinian spacetime (even though "it just don't seem natural"), I must reject any line of thinking that depends on talking about an absolute position in 3-D space, and I think you need one to choose the orientation for your proposed projector and screen, so to speak. Anyway, your projection would tell you what orientation you chose, not what you want to know about the space (which I think is unknowable because it's not there).
The more I wing it, the less I know. I think I'll stop here.
[quote="iampete"]If you have further suggestions for layman type reading material beyond what you provided earlier, I'd be interested to see them.[/quote]
Special Relativity, General Relativity, Cosmology ...
Special Relativity only took Einstein nine pages to describe completely. My textbook was a paperback less than a half inch thick ([i]Introduction to Special Relativity[/i] by Robert Resnick) and we did it in a half semester of a three-credit course. The book made it easy with illustrations, exercises, and I believe problems with answers. Used ones are cheaper than new ones, and the ones at the library are cheaper yet. After that book, the translated original Einstein was clear when I read it. For less technical, there's TV documentaries galore on Special Relativity. For more technical, some of us get state university classes on a free cable TV station. Any layman with an open mind and a little math can grasp Special Relativity completely.
There is no simple nor intuitive way to describe the blood and guts of General Relativity, and I'm sorry to have nothing to recommend. Even if they describe it well, it comes out lengthy (Hawking) or over my head in its brevity and heavy math (Einstein). Some TV documentaries about black holes, etc. may be worth a look.
Cosmology articles, not too technical nor non-technical, appeared regularly in [i]Sky & Telescope[/i] during the years I had a subscription. I don't know how it is today.
These days I mostly use google to find stuff on the internet. You need to pick through the results for something good, though.