On Feb 14, 6:11 pm, Edward Green wrote:
On Feb 14, 1:38 am, wrote, in part:

There are NO interior solutions toKerrin general relativity to my
knowledge. That raises a red flag - I don't believe it is proven they
don't exist, but it's suggestive to me.

Yes, to me too.

I had intended to ask whether "interior solution" means interior to
the event horizon or interior to a surface within which the vacuum is
replaced with a mass density. It turns out, on some investigation,
both!

I should have been more specific.

There exists a coordinate chart for Kerr that extends past the
horizon. There does not exist, to my knowledge, a solution to the
field equations with some non-vacuum source that has Kerr as the
exterior solution. That's what I meant by "interior".


The vacuum solution inside the event horizon is said to be "unstable",
whatever that means, or else "unphysical" (that part of a respectable
theory which in lesser theories might be described as "wrong"). I also
learned, as you say, that there is no known solution including mass
density which can be smoothly joined to the outerKerrsolution

A Kerr black hole is unstable in the sense that it isn't the "ground
state" of a black hole. The Penrose process for extracting energy is
one reason, and super-radiant instability [nee black hole bomb] is
another. A Kerr hole is very ... obvious in its' environment. It can
impart energy to stuff, and drags stuff along its' direction of
rotation. I personally believe Kerr holes are behind quasars and the
more intense class of gamma ray bursts.

http://arxiv.org/abs/hep-th/0404096

(although some treatment of a spinning dust disk reduces toKerr
solution in a limiting case).

It isn't a GR solution though - its a solution to linearized Einstein-
Cartan theory. Unless you found something specific to GR, which I'd
encourage you to share.

I think Einstein-Cartan theory is very cute in that it not only
sidesteps a lot of the singularity and existence theorems that depend
on the strong energy condition [T_uv U^u U^v 0 iirc] because spin
counteracts positive stress-energy. Plus it has the same overall
structure as classical GR since you can cast the field equations in a
form that has everything packed into an effective stress-energy
tensor.


The evidence is firmly ambiguous: if we could show no non-vacuum
extension of the solution were possible inward, then I think we could
safely say that the hope thatKerrdescribes the exterior field of a
massive spinning body is doomed; however, the language is "not known"
-- and there is the tantalizing datum of a single known solution is a
single special case.

I have no idea how this would be done or even if it could be done.

I think the idea of the Kerr solution [rotating black hole] is very
beautiful and important, it's just that its' derivation is ugly as
hell. All sorts of boundary conditions and assumptions about algebraic
structure. Its' a mess.


I am tempted to claim that the exterior vacuumKerrsolution, while a
valid solution of the field equations, cannot represent the external
field of a spinning body. As I mentioned, a rotating massive object
seems to transmit no intelligence of its sense of rotation to the GR
source term. If we are in fact able to tell from purely gravitational
observation which way a spheroid is spinning, then there must be some
additional factor which breaks the symmetry.

Thus, Einstein-Cartan theory.


TheKerrsolution still retains a very definable sense of rotation
through its' behavior at infinity [ala Schwarzschild & the Komar
integral definitions of mass] and its' behavior in the strong field as
its' effect on photons and massive particles.

Except I don't know how to write the angular momentum of aKerrhole,
and I don't think you can - at all. The four dimensional & /covariant/
way to write angular momentum of a particle is the generalization of r
x p : L^uv = x^u /\ p^v - p^u /\ x^v. Except both x^u and p^v are both
singular as hell at the point [ok, annulus] and everything is vacuum
except at the singularity.

In the Wikipedia article, I see a parameter "J" in the metric? If
that's the angular momentum, it's built in at the outset.

Where...?




Since you might appreciate this, I'll tell you a little more that I
have found in my researching of Einstein-Cartan theory.

Basically Einstein-Cartan theory is what happens when you allow a
direct coupling with angular momentum that happens by allowing
torsion. I'm butchering this slightly, but in effect you have the same
field equations as general relativity except there is a source term
that couples to spin/angular momentum.

What's really cool is this:

Look at the dust solution for GR. Let it be spherically symmetric n'
****, and the exterior solution is Schwarzschild.

Try the same thing but different in Einstein-Cartan theory. Use
something called the Wessenhoff fluid as a source - it is a fluid with
intrinsic angular momentum.

I've long been suspicious of "intrinsic angular momentum". I've
grappled with the idea several times, and I seem to recall concluding
it doesn't make sense. IIRC, my objection was that holding the volume
density of angular momentum constant and shrinking the elementary
spinning bits required the angular speed of the bits to blow up.

A fair objection, but one rooted in classical mechanics. You and I
know it isn't that simple.

Classical E&M contains a similar internal inconsistency regarding the
self-energy of an electron.

OTOH, I can see the idea might make sense for a system with
significant unpaired spins: macroscopically it might be simplest to
treat the spins as a smooth volume density of intrinsic angular
momentum.

But you may agree we ought to be able to model a spinning star or
planet without appealing to "intrinsic angular momentum": the body's
angular momentum is out there in the open.

When I say spin, I do not specifically refer to the quantized angular
momentum of a particle in quantum theory. I treat, in this context,
spin and angular momentum to be the exact same thing. They are as such
in quantum mechanics, but "spin" has a specific connotation that I
don't mean to imply.


The /approximate/ exterior solution to the
Weyssenhoff fluid is...get this...the linearizedKerrsolution.

I'm willing to bet money that the exact exterior solution for a
Weyssenhoff source will either be the fullKerrsolution or will have
the fullKerrsolution as a subset

Interesting guess: I take it Einstein Cartan theory is an extension to
plain Einstein theory. Your guess means that to make sense of an
extended source supporting aKerrmetric, we must look outside vanilla
GR. That's interesting, because the exteriorKerrsolution is by all
accounts a perfectly physical looking GR vacuum solution. If theGissehypothesis were true we could say that GR is incomplete, in the
sense that it allows self-consistent solutions which can only be fully
understood by admitting more terms into the theory.

I do believe GR is incomplete in the sense that the theory handles
angular momentum poorly. I am not saying I believe it to be wrong [I
have no evidence] or that it can't, it's just that /I don't see how/,
and I know enough to be able to differentiate between the two. To me,
Einstein-Cartan theory handles it /better/.

Look very carefully at what Roy Kerr had to do to obtain the Kerr
solution. He had to go digging very deep into the algebraic structure
of general relativity to find something that could be interpreted as a
spinning black hole.

In my superficial understanding of both theories, the structures you
can have are /exactly the same/. The only significant difference
between the two is that in GR you make the explicit choice to forbid
torsion and in Einstein-Cartan [EC-GR?] theory, torsion is allowed.

At the moment it is my personal belief that Kerr engineered a solution
that would be a natural solution in EC-GR by digging through the
overall structure that is common to both theories.


But even if all this were true I'm not sure it getsKerroff the hook
as being the vacuum metric of a spinning body: as appealing as
Weyssenhoff fluid is, it seems unlikely we need it to model spinning
balls of rock and gas. Maybe theKerrsolution describes the
gravitational field of a giant collapsed globule of electrons, whose
gravitation has overcome their electrostatic repulsion. :-) I guess
then we need the charged version, too.

Ooowww. Charged.


I'm willing to bet money also (maybe $5), that at least one of the
following is true: (1) TheKerrmetric is not the vacuum solution
associated with ordinary rotating matter, or (2) GR requires an
extension to handle ordinary rotating bodies, but not the one you
propose. A role will appear for the four momentum density -- not just
for the the stress tensor, whose terms are sensitive to orientation or
surfaces, but not to the choice of positive normals.

I think you'll lose #1, and that you stand a good chance of making
money with the first part of #2.

At the moment I am satisfied that the Kerr metric is an adequate
description of the exterior metric of a rotating body, Lense-
Thirring effect, geodetic precession, and the observation of near-
extremal Kerr black holes.


Since our bets are not opposite sides of the trade, I'm not sure whom
we're betting!

We are betting on angular momentum.