Re: General relativity versus black holes

From: Kefka G (kefkag_at_aol.com)
Date: 08/16/04 

Date: 16 Aug 2004 13:55:41 -0400

Oz writes:

>KefkaG <kefkag@aol.com> writes:
>>Well, if you want to pull EM into GR, there's always Kaluza-Klein theor=
y,
>where
>>we add an extra periodic spatial dimension to 4-D GR and find that this
>exactly
>>gives us EM gauge invariance if we "factor it away" (Kaluza's cylinder
>>condition). It's not too satisfying, though, for reasons too detailed =
to go
>>into here (there's a paper at Living Reviews which analyzes a lot of th=
is
>stuff
>>in detail). Maybe you're thinking about something else...
>
>Probably I was, but I expect KK were closer.
>
>Why do we have to 'factor it away'? Is it because we don't see this
>extra dimension, but then given we are electrically neutral then I
>wouldn't expect us to see it directly. Perhaps I am missing something
>here.=20
>

Here are the basic assumptions of Kaluza-Klein theory in its original for=
m:

1) The laws of general relativity hold fast
2) We have five dimensions - one time, four space
3) The fourth spatial dimension is periodic, and somehow so small that "n=
othing
depends" on it - i.e. we drop all derivatives in this direction, and assu=
me
that we can't measure any distances in that direction.
4) Gravity and matter is all that we have

To start out, following Kaluza we take the line element to look like this=
:

ds^2 =3D g*_uv (x,y) dx*^u dx*^v
      =3D g_uv (x) dx^u dx^v + ( dy + k A_u (x) dx^u )^2

where the *'d variables refer to the bulk 5-d space, and the unstarred on=
es
refer to 4-d space (in which case the fourth spatial dimension is called =
y).=20
Notice that A_u does NOT depend on y here, so we've already essentially t=
aken
the Kaluza cylinder hypothesis 3). I'll let you work out that in order t=
o have
invariance of this line element under transformations y -> y' =3D f(x^u) =
we must
have the following:

A_=B5 -> A'_=B5 =3D A_u - (df/dx^=B5) / k where the d's should be parti=
als.

Note that f does not depend on y - convince yourself that this is necessa=
ry for
the line element to be preserved.

Also note that we can take this last equation as the gauge transformation=
 of
the scalar potential, so this gives us hope that the EFE will be equivale=
nt to
gravity coupled to e/m. I'll let you work out the curvatures and whatnot=
 -
it's not that tough if you use a non-orthonormal basis. Do remember to d=
rop
all y-derivatives by the cylinder condition! But in the end, we find a s=
imple
relation between the 5-d and 4-d curvatures:

R* =3D R - (k^2 / 4) F^uv F_uv

where F_uv is the usual Maxwell tensor. Note that this equation is invar=
iant
under choice of metric signature (being the scalar contractions of two
even-rank tensors), which is nice if you're like me and use different
conventions depending on your whim. Now just plug this into the Hilbert
action, integrate out the fifth coordinate (trivially since nothing depen=
ds on
it) and you'll get a 4-d action integral describing (for the appropriate =
choice
of k) 4-d gravity coupled to Maxwell's equations.

So in short, the reason we need to factor out the fifth dimension is that
otherwise, we're just doing relativity in 5-d. We'll have waves in that
direction, particles moving in that direction, etc. In order to get anyt=
hing
else, we need to pretend that we're in 4-d, and a good way to do this is =
to
pretend that the fifth dimension is just too small to see. Keep in mind t=
hat so
far, all of this is in empty (5-dimensional) space, too, so in 4-d, at mo=
st
we're covering e/m radiation coupled to gravity with no matter.

In particular, one thing that sucks about this theory is the following. =
In
five dimensions everything is sourceless, so the scalar R* =3D 0. Conseq=
uently,
thanks to the 5-d EFE, R*_ab =3D 0 for all a, b. I haven't shown this, b=
ut as it
turns out,

R*_44 =3D - (1/4) F_ab F^ab =3D 0

which in turn means that F_ab F^ab =3D 0 everywhere, so all of these equa=
tions
are trivial! As it turns out, we'd need to have included a scalar field =
in the
original metric hypothesis in order to avoid this problem. It's much mor=
e
complicated to get anything interesting out, so I'll leave it to you to f=
ind
papers where this is discussed - there are plenty. But I think this disc=
ussion
at least illustrates the basic idea quite nicely - if you tack on an extr=
a set
of dimensions invariant under some isometry group, then the coordinate
invariance of GR will automatically turn that isometry group into a group=
 of
gauge transformations in higher dimensional gravity, at least if we "fact=
or
them away," i.e. just don't pay attention to them. This creates problems=
 of
its own, however, such as the fact that the resulting equations may turn =
out
trivial if we're not careful (here the problem arises from assuming that =
g_44
is always equal to 1(indices from 0-4), which we're not free to do - if w=
e
allow this to "flap in the wind", we can obtain non-trivial equations whi=
ch are
no longer so simple as 4-d gravity + e/m).

It's an interesting exercise to try this out with linear fields as well -=
 in
particular, try this whole "factoring out a dimension" thing on a scalar =
field
(where as it turns out there are no observable consequences - a 5-d scala=
r
theory compactified to 4-d is exactly equivalent to a 4-d scalar theory) =
and on
the Maxwell field (using the obvious generalization of the Maxwell equati=
ons).=20
As it turns out, it is NOT true that a fifth dimension would be unobserva=
ble if
it were really small under Maxwell's equations - we end up with an extra =
scalar
field which has a non-trivial effect on 4-dimensional matter, i.e. 5-d Ma=
xwell
compactified to 4-d =3D 4-d Maxwell + 4-d scalar. Of course, to produce =
this
extra field requires current flowing in the extra dimension, which maybe =
we'd
prefer to "factor out" - in any case, these are interesting matters, and
there's a million fun things to play with if you're interested.

And that's not even starting to talk about what happens when we add in qu=
antum
mechanics, higher gauge groups, projective spaces, stability issues, sour=
ces,
black holes, Casimir forces, etc (much of which I don't understand very w=
ell
yet). Actually, all that I've talked about above was really Kaluza's the=
ory -
Klein made separate contributions which I haven't talked about yet. Anyh=
ow,
I'm hungry so I need to go out now. Sorry if all this was a little vague=
 -
e-mail me if you'd like a longer, more cogent explanation.

-Eric

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