“To get Hawking radiation we have to give up on the idea that spacetime always had 3 space dimensions and one time dimension to get a quantum theory of the big bang.” I’ve been studying relativity for some years now in the hopes of saying something intelligent to the author (Jim Hartle), if we’re both lucky enough to make it to our 60th college reunion in 2 years. Hartle majored in physics under John Wheeler who essentially revived relativity from obscurity during the years when quantum mechanics was all the rage. Jim worked with Hawking for years, spoke at his funeral and wrote this in an appreciation of Hawking’s work [ Proc.Natl. Acad. Sci. vol. 115 pp. 5309 – 5310 ’18 ].

I find the above incomprehensible. Could anyone out there enlighten me? Just write a comment. I’m not going to bother Hartle

Addendum 25 May

From a retired math professor friend —

”

I’ve never studied this stuff, but here is one way to get more actual dimensions without increasing the number of apparent dimensions:

Start with a 1-dimensional line, R^1 and now consider a 2-dimensional cylinder S^1 x R^1. (S^1 is the circle, of course.) If the radius of the circle is small, then the cylinder looks like a narrow tube. Make the radius even smaller–lsay, ess than the radius of an atomic nucleus. Then the actual 2-dimensional cylinder appears to be a 1-dimensional line.

The next step is to rethink S^1 as a line interval with ends identified (but not actually glued together. Then S^1 x R^1 looks like a long ribbon with its two edges identified. If the width of the ribbon–the length of the line interval–is less, say, than the radius of an atom, the actual 2-dimensional “ribbon with edges identified” appears to be just a 1-dimensional line.

Okay, now we can carry all these notions to R^2. Take S^1 X R^2, and treat S^1 as a line interval with ends identified. Then S^1 x R^2 looks like a (3-dimensional) stack of planes with the top plane identified, point by point, with the bottom plane. (This is the analog of the ribbon.) If the length of the line interval is less, say, than the radius of an atom, then the actual 3-dimensional s! x R^2 appears to be a 2-dimensional plane.

That’s it. In general, the actual n+1-dimensional S^1 x R^n appears to be just n-space R^n when the radius of S^1 is sufficiently small.

All this can be done with a sphere S^2, S^3, … of any dimension, so that the actual k+n-dimensional manifold S^k x R^n appears to be just the n-space R^n when the radius of S^k is sufficiently small. Moreover, if M^k is any compact manifold whose physical size is sufficiently small, then the actual k+n-dimensional manifold M^k x R^n appears to be just the n-plane R^n.

That’s one way to get “hidden” dimensions, I think. “