Dear Biswajoy,

It is a reasonable question, since angular momentum relates to the only "degrees of freedom" a "black hole" might exhibit under excitation from a Schwarzschild black hole. Of course there is the generalization to the Kerr metric, the Penrose cosmic sensorship etc.

A black hole rotor might exhibit many features in analogy with molecular rotations and their transitions, symmetric-, spherical-, etc. rotors with fundamental degeneracies influencing associated properties etc., described entirely in terms of rotational parameters without reference to coordinate metrics, which do not exist in a black hole singularity.

Biswajoy> L= r x p is a 3 dimensional relation. In 4 dimensions r and p are 4 vectors and L is a 4x4 matrix.

But the natural extension of L (conserved angular) to 4 dimensions is not a 4-dimensional cross product, but two 3-vectors L and K. (Well, these _can_ be collected in an antisymmetric 4x4 matrix...)

In special relativity the conservation of K means that the center-of-energy moves with constant velocity. That is in general different from the conserved quantity resulting from translation invariance (consider the motion of a charged particle in a translation invariant magnetic field).

I believe one must be able to use the Lorentz group to transform black hole solutions to a moving frame, and through this get an approximate description of two well-separated black holes, moving with respect to each other. 

Or simpler, a description of how a fast-moving black hole looks from a stationary observer.

In 3-D  Euclidean space if a sphere contracts under it's own gravity, conservation of angular momentum will increase it's angular velocity. However, in 4 space-time dimensions it need not be the case by the following reason. As a black hole collapses it distorts space-time around it. The metric changes. Because conservation of angular momentum is a consequence of rotational invariance (in general non-zero angular momentum breaks rotational invariance), and because we do not know whether rotational invariance is maintained by the space-time surrounding a collapsing black hole, we cannot say whether angular momentum is conserved or not.

Internal symmetries however do not have such a problem. As a black-hole collapses, local U(1)_em gauge invariance is also maintained and electric charge still remains conserved. Electric charge then remains a valid number to label a black hole.

For the Kerr (-Newman) black holes this is given by the Komar integral belonging to the Killing field, associated to the SO(2) isometry group. For details, see e.g. my GR- book (Springer 2013), eq. (8.187).

Classically, SO(3)-symmetry leads to conservation of three components of angular momentum. If you introduce a (classical) spinning electron into space, its presence reduces the symmetry to SO(2). This is important when describing objects interacting with the electron, but that does not imply that the overall SO(3)-symmetry has been broken, when you rotate _both_ electron and objects.

The same must be true for (something that can be treated like) a single black hole in the universe. But, all of whatever is collapsing for a collapsing black hole must of course be included in the accounting (and nothing radiated away).

Ik like the interpretation of the Komar Mass/Angular momentum essentially from http://www.math.ru.nl/~waltervs/Einstein.pdf to which I will add a bit of standard standard topological lore to interpret the Komar Mass angular momentum as a topological invariant of a Killing vectorfield and a choice of "infinity" (which must be well known by the experts).

The gist is as follows.

Suppose (M, g) is a space time of signature (+,- - -)  (things go through essentially unchanged in higher dimensions except for an occasional minus sign that is irrelevant for the argument) , Let K \subset  M be such that (M,g) is Ricciflat outside of K (i.e. in the relativity context, the stress energy tensor T  has support on K, in practice it is also good if in a loose sense, K contains all singularities. This is loose because the singularities are not actually part of M but you want to choose K such that all incomplete timelike geodesics hit K first, i.e. K is typically contains everything inside the event horizon)  

Assume that (M,g) admits a Killing vector k = k^a\partial_a with a dual one form

\xi = \xi_a dx^a = g_{ab}k^a dx^b.

Since k is a Killing vector we have

d\xi = \nabla_[a \xi_b] dx^a dx^b = \nabla_a \xi_b dx^a dx^b = \nabla \xi 

and using the Hodge star *

*d* \xi = (+/-) g^ab \nabla_a \xi_b = 0

It then follows from the Weitzenböck formula that

d*d\xi = *i_k Ric = 2Ric(k, --)

          = 2*R_{ab}k^a dx^b

          = 2 \sqrt{-g}k^a\epsilon_{abcd}dx^b dx^c dx^d

Therefore, for any 3 cycle \Omega \subset \M with boundary \partial \Omega we  have

\int_{\Omega} *Ric(k, --) = (1/2) \int_ {\partial \Omega} *d\xi

or if the Einstein equations (with \Lambda = 0) hold:

\int_Omega (*T(k, --)  - 1/2 T *\xi) = (1/2) \int_{\partial \Omega} *d\xi.   (1)

In particular since  *d\xi is a closed 2 form on M - K, the Kumar invariant "inside" a closed cycle \Sigma \subset defined as 

M(\xi, \Sigma) = \int_\Sigma *d\xi           (2)

only depends on the homology class of \Sigma  in H_2(M - K)  and so we can think of the Kumar invariant M(\xi) for all \Sigma's (of which in practice there is only one homology class) as the cohomology class  

[*d\xi] \in H^2_{de Rham}(M-K) = H^2(M-K; R).

The space H^2(M-K; R)  is a topological invariant of M - K. Typically M-K retracts to a sphere and so integration over that sphere (or any homologous one) in H^2(M-K; R) gives a canonical isomorphism of H^2(M-K; R)  with the real numbers. 

To interpret the Kumar  invariant we must choose particular cycles and space times.

In the document it is explained that for a space time foliated in slices orthogonal to a timelike Killing vector and in the Newtonian approximation for a cycle Omega in a space slice, the lefthandside of (1) is the total mass-energy in Omega. Therefore it is natural to interpret (2) as a mass for such killing vectors, even if the cycle \Sigma cannot be interpreted as a boundary in M or not as a boundary in a slice. 

Likewise for a Killing vector with periodic orbits (like the Kerr black hole) and space like foliations and a cycle \Sigma tangent to \xi contained in the spacelike foliation, the lefthand side of (1) can be interpreted as  an angular momentum. Again we stick to this interpretation even if the cycle \Sigma is not a boundary in M.

Actually, before looking at black hole and momentum it is useful to think further about the meaning of motion itself. Even in a classic 3 dimensional universe, it is assumed that material bodies (even at the level of a material point) exhibit a feature : there is an orthonormal frame attached to it, which rotates. So "motion" must be understood as the combination of a transversal motion and a rotation. It is represented in classical physics as a displacement, that is the semi direct product of a rotation and a translation, but this is wrong : the origin of the frame of the observer has nothing to do in the matter. The only mathematically correct way to deal with that is to use fiber bundles : then the rotation is just an element of a principal bundle (a rotation located at a point). It is then easy to extend this representation to GR. And if we look further we see that the right way to represent motion (both translation and rotation) is in Clifford algebra.

The next step is to go from motion to momentum. This is not as straightforward as is seems, even in Newtoninan Mechanics. The basic assumption is that, for a material body, there is a fixed relation between the motion and the forces which are exercised on the body : the usual forces and torques. And it is not easy to represent torques...Motion and forces (or momenta) are not the same physical quantities : they are related but measured by different protocols. Indeed one can measure a change of motion without measuring the motion itself (take your smartphone). So it is bit more complicated, and the right representation of momenta is byspinors, which are one of the components of the QTF particles. But it works for any material body, even deformable material bodies in GR. However there is no short cut to get there.

The natural framework for discussing conservation laws related to angular momentum in 4 space-time dimensions is O(4,2) dynamical symmetry group which provides an its analogy with electromagnetic antisymmetric tensor field and its unification with gravitational symmetric metric tensor (required for introducing the black hole).