From: mcirvin@husc8.harvard.edu (Mcirvin)
Newsgroups: sci.physics
Subject: Re: Is Spacetime Absolute?
Date: 6 Sep 92 17:12:58 GMT
Article-I.D.: husc8.mcirvin.715799578

prim!dave@germany.eu.net (Dave Griffiths) writes:

>Just to muddy the waters, here is a quote from Einstein's Relativity:

>"There is no such thing as an empty space, i.e. a space without field.
>Space-time does not claim existence on its own, but only as a structural
>quality of the [gravitional] field."

Hmmm... that's an interesting way of thinking about it. 
If you define "gravitational field" in the right way, I suppose it
can be thought of as true; if you call the metric the gravitational 
field (in which case it exists even in situations where we'd normally 
say there's "no gravity,") it exists everywhere that there is spacetime,
and (with the exception of statements about boundary conditions) the
only way the properties of spacetime enter into physics is through the
metric, at least in torsion-free general relativity. 

In *quantum* field theory, it's absolutely true that there's no space
without a field, but I don't think that's what Einstein was talking about.

I prefer a point of view in which "space-time" and "field" are 
interchanged in the second sentence, personally. 

>Part of my confusion results from trying to imagine 4D spacetime. I imagine
>first of all Euclidean space with the Universe as a big 3D cube, then
>extend that to a 4D hypercube where it is still possible to assign
>coordinates (x, y, z, t) to every event and where this 4D spacetime still
>has an objective existence.

There's no problem with that, except that the spacetime is anything but
Euclidean, and that any given coordinate system you assign might
fail in spots because of curvature, and that there's nothing special about
how you assign the coordinates.  Physics ought to be expressible in such
a way that it doesn't matter what coordinate system you use to write down
a statement; this is actually applicable to everything, but in GR it
became a pressing requirement, since in a theory in which variations
in geometry play a central role, you don't want effects of the coordinate
system to obscure things that actually have to do with the geometry. 

Whether the spacetime has an objective existence or not is more of a
philosophical question.  I tend to think of it as a very real, physical
thing.  

>(Not to mention the problem of, "if spacetime is curved, in what dimension
>is it curving?" :-)

A central idea of GR, Riemannian geometry, and, for that matter, all of
modern differential geometry is that it doesn't matter.  There's nothing
that makes Euclidean geometry so special that you have to embed everything
in a higher-dimensional Euclidean (or, for that matter, Minkowskian) space.
It's easier to see what things like Gaussian curvature mean by looking
at surfaces embedded in a Euclidean space, but the Euclidean space is
completely nonessential.  

What I think about when I visualize spacetime is somewhat vague.  I suppose
I try to think of some higher-dimensional analogue of a floppy surface
with lots of lumps.  One way to get around imagining this as embedded
in Euclidean space (which isn't always possible, for a pseudo-Riemannian
thing like spacetime: how do you embed things so that lightlike geodesics
have zero length?) is to instead imagine the definition of distance
changing as you move around, which is, after all, rather close to the
way the mathematics describes it.  As the meter sticks tilt toward the
light cone they become really "long," so that apparent immensities barely
hit the one-millimeter mark.  Straight lines look "curved," though
of course they really aren't; if you take those morphing meter
sticks and measure distances, you find that they are extrema of
distance.  When the intervals become timelike this breaks down again,
since the meter sticks would then get imaginary lengths; so you replace
them with traveling clocks that time-dilate. 
 
-- 
Matt McIrvin, Cambridge, Massachusetts, USA 
Newsgroups: sci.physics