Subject: categories: Re: cracks and pots

From: "Robert J. MacG. Dawson" <[email-masked]>
Date: Thu, 16 Mar 2006 14:41:35 -0400

Eduardo Dubuc wrote:
> Well Robert,
>
> 1)
>
>
>>Well, Einstein was not "trying to"; he was using it, and presented this
>>use as an accomplished fact.

...

> General relativity was born with differential geometry; it has no meaning
> without differential geometry. String theory was already there when a
> category theory approach began.

	Sorry, Eduardo! That's a little oversimplified.   See, for instance,
section 17.7 of Misner, Thorne, and Wheeler's "Gravitation", among other
references.

	 General relativity (though of course not in its modern form) goes back
to Einstein's formulation of the equivalence principle in 1907 (only two
years after special relativity), and the prediction of the gravitational
red shift. In 1911 Einstein also predicted the bending of light by
massive bodies; this too is intrinsically part of GR.

	But it was only in 1912 that he realized that Euclidean geometry awas
not compatible with this, and (encouraged by Grossmann and Levi-Civita)
started looking at differential geometry as a way to handle
non-Euclidean spacetime.  Einstein and Grossmann's 1913 attempt at a
general relativity theory was wrong; it did not transform correctly.
Some time after this,  Planck specifically warned him that the
differential geometry approach would not work and would not be believed
if it did.

	In  November 1915 Einstein submitted two papers.  The first of these
explained some observations such as the precession of the perihelion of
Mercury, but in other ways made wildly nonphysical predictions
(essentially ignoring many of the effects of mass -though  this
"linearized theory" does have some uses as an approximation) He
corrected this soon with a second paper in which he finally got it
right. Sort of.

	In 1917 Einstein introduced a cosmological constant into his field
equations to account for the "fact" that the universe wasn't expanding.
In the 1920's he took it out again when it turned out that the universe
*was* expanding.  Now astronomers think there ought to be one, but with
a value very different from what Einstein originally put forward.

	So GR got by without differential geometry for five years; and it was
another decade or so before it was a mature theory with enough of  the
bugs out to do what was expected of it.  And, as you know, there are
still scales, almost a century later,  on which its predictions are
unsatisfactory.

	-Robert