Re: Why only gravity is "geometrical"?



On Jan 24, 6:09 pm, mme...@xxxxxxxxxxxxxxxxxx wrote:
In article <ep84ud$v2...@xxxxxxxxxxxx>, sirix <s...@xxxxxxxxxxxxxx> writes:
hey ho!
What is the obstruction to developing a theory of - for example -
electromagnetism, that would imply that electromagnetism is not a real
force (like gravity in general relativity)?

Or maybe such a theory exists? If so, than why one can't find a "common
geometric picture" for gravity and electromagnetism?Well, if you take the lagrangian formalism (I know from your other
posts that you're quite familiar with it) with any fields and use it
to derive trajectory equations (instead of the usual equations of motion)
you get results which formally looks just like geodesics equations
(that's covered in Goldstein, as I recall). So you could say that
motion within any field can be represented as following geodesics in
appropriate geometry. Only (and this is a big only) the metric of
said geometry depends on the mass so you end up not with a single
geometry but with a potential infinity of such (different geometry for
each mass) which rather disagrees with our notions of what geometry
is, and is of little use besides. The exception is when the force
itself (or the potential, if you wish) is proportional to the mass of
the particle. In this case the mass cancels out in the metric and
we're left with a single geometry. And, we've forces like this. In
classical mechanics the ones that fit the bill are the inertial forces
(centrifugal and coriolis), where the geometric transformation is
trivial, and, of course gravity, where you've to abandon euclidean
geometry but other than this no problem. When the force is not
proportional to mass, though, such simple "geometrization of the
field" doesn't work.

A more physical way to express the above is: you can geometrize the
field when two different particles starting with the same initial
conditions will follow same trajectory. Gravity is like this, EM
isn't (just assume two particles, one charged and the other neutral,
and you see that it doesn't work).

But, all is not lost. One can say "well, these two have same initial
conditions in the dimensions we track, but perhaps there are some
additional dimensions involved which we're not aware of, and a
differnce in the initial conditions in those dimensions is what causes
the differnece in the trajectories. In other words, a field which
cannot be geometrized in our 4D spacetime, may be "geometrizable" in a
larger dimensional space containing the 4D spacetime. Attempts along
these lines were made, you may check on the Kaluza-Klein theory. As
far as I know (and here I'm rapidly approaching the limits of what I
can say without a refresher course), such attempts run into various
difficulties but the topic is still open.

Mati Meron | "When you argue with a fool,
m...@xxxxxxxxxxxxxxxxx | chances are he is doing just the same"

Stitch together Mati's and Nathan's responses, and you've got yourself
a nice little package. Well done!

PD

.

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