🚀 go-pugleaf

mcirvin@husc8.harvard.edu (Matt McIrvin) writes:

: I get the feeling sometimes that something like this is what people
: want when they express a fondness for an embedded universe: you have
: a four-dimensional spacetime, and you explain curvature by saying 
: that it "curves through some still higher dimensions."  You can do this
: if the bigger manifold is Minkowskian, of course, but it takes away
: some of the elegance of the idea.

Why so?
Indefinite geometry is much more elegant than definite
(for instance, two _indefinite_ real quadratic forms with
the same zero sets must be scalar multiples of each other),
but that's not the point at issue here.
The question is whether intrinsic curvature
(Gaussian scalar curvature in 2D,
the Rieman tensor in general)
can always be realized by a suitably curved embedding
in a flat space --- an affine space with a translation-invariant metric.
The answer is Yes for a pseudo-Riemannian manifold,
though around black holes --- where the metric goes kerfloop ---
I'm not up to date on whether embeddability is known.
To make even a local embedding mentally useful in the case of spacetime,
you do need to beef up your intuitive sense of (e.g.) how
geodesics work in such indefinite embeddings;
I drew some relevant pictures in
Dodson & Poston, Tensor Geometry, Springer.

The fact of this possibility does _not_ mean that spacetime
`is' isometrically embedded in some flat space.
A very nice example is in the other direction: PacMan's universe.
That is a flat square with opposite edges identified,
topologically therefore a torus.
But to get a _flat_ intrinsic geometry for the torus
by an embedding, you need more than the usual 3D surrounding space;
you can't do it in less than four flat dimensions
(where it's easy, as the product of a circle in the (x0,x1)-plane
with a circle in the (x2,x3)-plane ).
Thus that manifold needs more dimensions to be `flat around'
than to be `curved around', and nothing about PacMan's 2D universe
has any stronger flavour of 4D than our own has of the many
flat dimensions we would need (even locally) to embed that.
The simplest local embedding of a 2D (time,radius) slice around the Sun,
(which does help in understanding the way that geodesics go,
i.e., how planets move) is periodic in time
(Tensor Geometry, Fig.XII.3.2);
this does _not_ mean that time is looped for the Solar System.
Embeddability and embeddedness are very different questions.

Tim Poston


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