Re: Why only gravity is "geometrical"?


"Nathan Urban" <nurban@xxxxxxxxxxxxxxxxxx> wrote in message news:ep8ae5$n5k$1@xxxxxxxxxxxxxxxxxxxxx
In article <ep84ud$v2k$1@xxxxxxxxxxxx>, sirix@xxxxxxxxxxxx wrote:

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?

Such a theory exists: it's a 5-dimensional spacetime theory called
Kaluza-Klein theory, unifying gravity and electromagnetism. However,
quantizing it is as difficult as quantizing gravity. Furthermore, we
already know that electromagnetism is unified with the weak nuclear
force in electroweak theory, part of the Standard Model. Witten
showed that it is not possible to obtain the Standard Model in a
Kalzua-Klein framework, at least in the context of compactifying on a
spatial manifold. There are various ways to dodge this, and I don't
know if anyone has looked at the orbifold compactifications that are
popular in string theory. But most people working in the "unified
field theories in extra dimensions" moved on to string theory.

Note that K-K theory is a unified theory of gravity and
electromagnetism, and it needs an extra spacetime dimension. It has
to be that way: electromagnetism itself can't be regarded as pure
(spacetime) geometry like gravity can, because, as another poster
pointed out, EM doesn't obey the equivalence principle. Thus, the
trajectory of a charged particle depends on internal properties of the
particle (its charge to mass ratio), not just on spacetime geometry.

The way K-K theory evades this consequence is by adding an extra
dimension in such a way that all of the charge-to-mass information
winds up being encoded in the particle's spacetime trajectory
(specifically in its 5th-dimensional momentum). Then a particle's
path is determined solely by 5D spacetime geometry, although its
apparent 4D spacetime path does depend on its internal properties.

*Very* nice intro :-)

Dirk Vdm



On the other hand, you can regard ordinary Maxwell electromagnetic
theory as a geometric theory in the sense of Yang-Mills theory
described on a fiber bundle. However, spacetime is not truly
dynamical in this theory as it is in GR or K-K theory, and it involves
geometry in something other than spacetime (namely, a fiber bundle
over spacetime), so you might regard that as "cheating".

I have discussed this topic in some more detail here:

http://groups.google.com/groups?as_umsgid=6hoi5o$ko7$1@xxxxxxxxxxxxxxxxxxxxxxxxxx
http://groups.google.com/groups?as_umsgid=i2h82$95q$1@xxxxxxxxxxxxxxxxxxxxxxxxxx
http://groups.google.com/groups?as_umsgid=6i520i$biu$1@xxxxxxxxxxxxxxxxxxxxxxxxxx
.

Relevant Pages

  • Re: Why only gravity is "geometrical"?
    ... electromagnetism, that would imply that electromagnetism is not a real ... geometric picture" for gravity and electromagnetism? ... it's a 5-dimensional spacetime theory called ... particle, not just on spacetime geometry. ...
    (sci.physics)
  • Re: Why only gravity is "geometrical"?
    ... electromagnetism, that would imply that electromagnetism is not a real ... geometric picture" for gravity and electromagnetism? ... described on a fiber bundle. ... In my differential geometry course lecturer ...
    (sci.physics)
  • Re: the basis of relativity
    ... > With regard to the theory of gravity as I described it. ... > one should say gravity is indistinguishable from geometry. ... spacetime being an imaginary frame for solving physics problems, ...
    (sci.physics.relativity)
  • Re: Questions (Space)
    ... All quantum theories of forces are based on a background spacetime, ... normally flat. ... conceptual difficulty about treating gravity as an effective field ... As I said in the previous post, it's the simplest possible geometry. ...
    (rec.arts.sf.composition)
  • Re: GR ?
    ... Mass distorts spacetime, causing it to curve. ... Gravity ... |> | is described by Euclidean or plane geometry. ...
    (sci.physics.relativity)
Loading