On Feb 14, 1:38 am, Eric Gisse wrote, in part:

There are NO interior solutions to Kerr in 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!

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

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 that Kerr describes 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 am tempted to claim that the exterior vacuum Kerr solution, 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.

The Kerr solution 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 a Kerr hole,
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.

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.
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.

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

I'm willing to bet money that the exact exterior solution for a
Weyssenhoff source will either be the full Kerr solution or will have
the full Kerr solution 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 a Kerr metric, we must look outside vanilla
GR. That's interesting, because the exterior Kerr solution is by all
accounts a perfectly physical looking GR vacuum solution. If the
Gisse hypothesis 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.

But even if all this were true I'm not sure it gets Kerr off 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 the Kerr solution 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.

I'm willing to bet money also (maybe $5), that at least one of the
following is true: (1) The Kerr metric 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.

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