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Florin and Ray,

I wrote here a rather long post in an attempt to address your questions.

It would seem that one should not be concerned with the interior of a black hole. And for a classical black hole in some sense that is right, unless you are the observer who wants to fall into the black hole. The issue does become of some relevance when the black hole is quantum mechanical. ...

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Florin and Ray,

I wrote here a rather long post in an attempt to address your questions.

It would seem that one should not be concerned with the interior of a black hole. And for a classical black hole in some sense that is right, unless you are the observer who wants to fall into the black hole. The issue does become of some relevance when the black hole is quantum mechanical. The observer who remains at a fixed distance from the black hole observes the string to approach this stretched horizon and an observer who falls in with the string observes it approach the so called singularity. For the observer who is removed from the black hole, what Susskind calls the FIFO, that is on the order of 10^2 to 10^4 Planck units of mass (or smaller of course) the event horizon is not the sharp boundary we have in general relativity. What is a null, timelike and spacelike region becomes uncertain. There is a geometric quantization at work which renders these regions uncertain. The observer is using a “probe” of sorts, but the spacetime geometry the probe detects is uncertain. So the FIFO observer will detect amplitudes the FREFO (freely falling) observer observes. The problem is that if you push the physics far enough you are inevitably forced to ask questions about what happens in the interior, for this is relevant with black holes which are themselves quantum systems.

The quantization of gravity can't requires some removal of obstructions in either QM or GR which prevents this from happening. The holographic principle by Susskind and black hole complementarity is the removal of such an obstruction. In this case it is relativity which has a problem. Lorentz transforms act only along the longitudinal direction a body is traveling. Yet we know from QFT that a high energy particle can have an increased cross section at high energy. Similarly with a string near a black hole the two light cone directions of a string X^+ = t + z, X^- = t - z and the transverse directions x, y, dual to transverse momenta p^┴, set us a similar cross section increase. The X^{+/-} are orthogonal, X^+ is considered to be “space,” and X^- is considered to be “time.” These are conjugate to a momentum P and a Hamiltonian H (we can think of them as just different momentum directions, but this language makes things easier) so that



H = [(p┴)^2 + m^2]/P.



We might think of the product HP = (p┴)^2 + m^2 as the E^2 in the standard relativistic momentum-energy interval. The bounds on the longitudinal momentum, say for some energy bound ε = 1/η the longitudinal momenta may be founded by P < m^2η, which may pertain to the string length (Hagedorn temperature). Hence for larger relativistic boosts the particle “multiplies” as with the old theory of partons. When translated to stringy language the string expands in the transverse direction around a black hole and winds around the horizon on what is called the stretched horizon.



This sets up the matter of black hole complementarity, where the external observer observes something entirely different from an infalling observer commoving with the string detects. In this case nothing in particular happens until the string reaches the singularity, which is a region of geodesic incompleteness. Here a closed string under huge tidal forces is distended into an open string with different quantum modes. This is something I explored in my essay last fall. The string observed interacting with a quantum mechanical black hole is some complementary state with the interior and the horizon, or the holographically projected exterior. So the stretched horizon states of a string are in some entanglement with the string states of the interior. If the quantum black hole is a pure state, and black holes are ultimately stringy-membrane objects this should be the case, then the total entropy is zero. The entropy of the exterior S(e) and the interior S(i) are equal if their underlying quantum states are in an entanglement. For the black hole as a pure state then the total entropy is

S = S(e) + S(i) – S(e|i)

The joint entropy S(e|i) = S(i|e) contains the mysterious nature of the black hole complementarity.

The stretch horizon has a noncommutative geometry with respect to transverse and longitudinal coordinate directions. Transverse modes of a string on the stretched horizon with a probe size Δx,

E_ Δx ~ ħc/ Δx )

For a black hole of mass E_ Δx = 1, distances probed include regions on the other side of the horizon with a radius 2GM = 2E_ Δx. The energy of a quanta has a spread E_ Δx ~ ħc/R with an upper bound at E_p. This spacetime uncertainty, or underlying noncommutative geometry, for the propagation distance to the screen Δy ~ cΔT is

Δx Δy = Δx cΔT ~ 2GM/ħ ~ 2L_p^2

The result for the distance to the screen is Δy = TΔS/F = ħδ/2πMkc. So the Holographic principle already implies noncommutative geometry.

However this does not give us any complementarity with respect to the interior. It does imply some very strange things with respect to the location of a field in spacetime, which suggests the geometric notion of an event standard in quantum field theory is not correct. This is where octonions enter into the picture. However, this is not at all clear physically at this point to me. The exterior observer observes scattering matrix physics (the S-matrix) according to the delay or tortoise coordinate --- easily derived or looked up in a GR text. These coordinates provide a causal domain for the S-matrix. The FREFO observer finds there is another domain for the S-matrix, where the causal domain of support extends to the singularity, which is I think a D3 brane or a lamination of M2-branes. The FREFO finds there is a completely different S-matrix and the two observers will record this fact as the FREFO approaches the horizon of a quantum black hole. The two S matrices define different quantum groups G and G’, or equivalently braid systems. However a map between them m:G - -> G’ has Bogoiubov coefficients, so there is no super-braid group between different braid groups here. This requires a nonassociative system of triplets of braid groups. So the universe locally has some quantum group g and in another region another g' with A:g --- >g', or Ag = g' or that g^{-1}Ag = g^{-1}g'. This is the "overlap" of states predicted by associator transformed quantum groups. The noncommutative geometry of quantum spacetime is a quantum group which pertains locally, and the vacuum of the universe connects these quantum groups by associators (e_ie_j)e_k - e_i(e_je_k) = T_{ijk}^le_l. There is a set of 7 such quantum groups which form a basis of possible quantum groups. What one observer will measure as the vacuum state given by a quantum group is different from what another observer will probe. The associator preserves quantum information, but in a sense encrypts it. If we coarse grain over the associators we end up with the Bogoliubov transformation, which is the thermalized transformation involved with black hole radiation.

So this is additional element with black hole complementarity, things are not just noncommutative, as determined by a braid group, but involve nonassociative structures within a set of braid groups. Now this may involve the S-matrix where one S matrix has the ordering of operators or fields (ab)c and the other S matrix has a(bc). The associator [a,b,c] is in multiplication tables usually brings a negative sign so (ab)c = -a(bc). This makes things somewhat convenient with respect to noncommutative systems, for it does provide a way of relating commutators in different associator related braid groups according to sign flips. So this does suggest a way of working this. The rub is that we can’t physically justify current in operator product expansions (OPEs) as being nonassociative. It might be one thing to have operators, which in a way are just formal mathematical objects as nonassociative, but physically it seems unlikely we can justify having nonassociative currents in OPEs. So there is still a physical unknown here, besides some formal mathematical issues.

So I hope this covers most of the questions posed here.

Cheers LC

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Florin,

This ended up being a bit on the long side.

The octonionic approach to quantum mechanics is one approach I take with these problems. To be honest I see no problem with a set of field operators in an S-matrix having nonassociativity, so the ordering of fields has this extra “freedom.” The problem comes with the current algebra of measured fields, where nonassociativity seems...

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Florin,

This ended up being a bit on the long side.

The octonionic approach to quantum mechanics is one approach I take with these problems. To be honest I see no problem with a set of field operators in an S-matrix having nonassociativity, so the ordering of fields has this extra “freedom.” The problem comes with the current algebra of measured fields, where nonassociativity seems problematic, and OPEs with currents would have funny interpretations of energy-momentum tensors. Different S-matrices, or classes of them, would then be determined according to associators of elements of braid groups (quantum groups), and the relationship between these S-matrices is nonassociative. The question that I don’t understand is how one can then get OPEs and current algebras which do not have nonassociativity. However, this is not the only way one might think about this matter. Below I will illustrate one plausible way of thinking about this.

The problem with BH violation of quantum information, which I prefer to separate from unitarity, is that this would mean the fields contained on the stretched horizon are not removed from the horizon in BH quantum evaporation in a proper manner. The quanta which tunnels out of the BH annihilate the degrees of freedom on the stretched horizon, and if the interior does not have degrees of freedom which match those on the stretched horizon, or S(int) = S(ext) then this means that from the perspective of the exterior observer where the S-matrix is extended to r = 2m to ∞ is not unitary. Quantum mechanics simply fails. Now, what you say about degrees of freedom is on tack here. The fields on the exterior or stretched horizon and those on the interior or singularity are in some general form of entanglement so the BH evaporates in a way which preserves quantum information.

Don’t worry about baryon violation as fundamental loss os quantum information. Susskind does talk about this on pages 90-92, where on the stretched horizon a proton may have an enhanced prospect of being trapped on the horizon in the state X + e^+ rather than in the state p. Further, if we had some ideal trap for a proton, say we could put it on a toroidal space that acts as a perfect mirror box, the decay products of the proton will wind around the torus and eventually recombine into the proton! So in the intermediate state the quantum information for the proton still exists. The same is the case for the random thermal S(therm) > 0 radiation which emerges from a black hole. On a fine grained level the information for everything which entered the black hole still exists. It is just not easy to obtain that information and reconstruct the state of matter-fields which constituted the black hole.

Instead of thinking about octonions let us think about modular systems. In fact if octonionic QM is to be justified physically it has to be built up from this basis. The anti de Sitter spacetime exhibits periodic time which is removed by considering a universal cover or a patch on the spacetime. The AdS spacetime on this patch is



which in the limit as x --> 0 defines a Minkowski metric



which is a Minkowski spacetime. This means that the evolute of AdS from a spatial surface is an entire spacetime. So there is a loss of causality here. What is then required is a conformal completion of AdS. In doing so the Cauchy data on the AdS is defined on a conformal set of metrics. The boundary space ∂AdS_{n+1} is a Minkowski spacetime, or a spacetime E_n that is simply connected that with the AdS is such that (AdS_{n+1})UE_n is the conformal completion of AdS_{n+1} which exhibits a conformal completion under the discrete action of a Klienian group. For the Lorentzian group SO(2,n) there exists the discrete group SO(2,n,Z) which is a Mobius group. For a discrete subgroup Γ subset SO(2,n,Z) that obeys certain regular properties for accumulation points in the discrete set AdS_{n+1}/Γ is a conformal action of Γ on the sphere S_n. This is then a map which constructs an AdS ~ CFT correspondence.



The quotient space AdS/ Γ is a Kleinian structure. The group SO(2,n) is a map from the unit ball B_{n+1}, with boundary ∂B_{n+1} = S_n, into R^{n+1}. The discrete group Γ acts as a conformal on the sphere S^n by the action of the Mobius transformation on S_n. The discrete set of maps on S^n has accumulation points on the limit sphere S^n_∞ are determined by the limit set g_i \in G for i --> ∞. This is denoted by Λ(G), G = O(2,n). The discontinuous set is then the complement of this or Ω(G) = S_n - Λ(G). The manifold Ω(G)/G is an orbifold. This means that the Mobius transformation on the limit sphere S^2_∞ is equivalent to the conformal transformation of N^{n+1} which is equivalent to the isometries of AdS_{n+1}. The Ω(Γ)∩E_n/Γ is then a Lorentzian manifold ∂AdS_{n+1), and a set of discrete points in E_n pertaining to spatial hyperbolids of equivalent data. In this way the data on any spatial surface of AdS_{n+1} is contained in this conformal completeness of AdS_{n+1}. This is equivalent to the discrete action of Γ on S_n.

For more mathematics on this you can go to C. Frances’s paper on this. The paper is mathematics of AdS spacetimes and their conformal boundaries as spacetime according to Klein group systems.



The discrete structure here is isomorphic to the discrete set for the Taub-NUT spacetime. This Taub-NUT spacetime is similar to the Schwarzschild spacetime, but where time serves the role radius does. This system The spacetime has a timelike region I that connects to a region which is spacelike II, which in turn is connected to a timelike region III with closed timelike curves. The manifold for TN is S^3xR, with S^3 reduce to S^1 modeled as a cylinder. The spacetime is a sort of time version of a black hole, where the time coordinate defines the condition for the BH instead of the radius. The manifold (M,g) has a discrete structure to it. An Euler angle in space wraps a geodesic around the tube in region I, and defines intervals s^2 = t^2 – x^2 that are equivalent. These discrete points define a discrete subgroup d = SL(2,Z) in g = SL(2,C). Each of these points defines a neighborhood such that the action of the discrete group on that neighborhood determines d(U)∩U = ø, or the null set. This is a Hausdorff condition. Now the region I of M and II of M (I,g) and II,g) are cases where the t > -x and then (I + II, g) is Hausdorff. Similarly (I,g) and (III,g) is the case where t > x and again (I + III,g) is Hausdorff. However, this can’t hold for (I + II + III, g), which is then not Hausdorff. This then leads to a funny situation where field on these discrete points for I+ II + III exist on a Hilbert space with some frequency determined at each discrete point, but this nonseparability of points means there exist eigen-bases |E_i> where we can define a time operator



so that the [T_{op}, H] may be evaluated on such a basis element so that (E_i|[T_{op}, H]|E_i) = iħ. This is due to the nonseparability condition with the non-Hausdorff property of the manifold. If there is such separability it is not hard to show that this matrix element must be zero. These regions I II and III are entirely analogous to the patches seen in the Kerr-Newman solution for a black hole. The region III with closed timelike curves is entirely responsible for this behavior, it is what permits the existence of a discrete time operator in this spacetime.

This is then a modular system which can preserve quantum information, but which is not unitary. This is a key feature of what I think is needed to begin to understand the dynamics of how quantum information is preserved.

The TN spacetime is then analogous to the Dirac monopole. The unit which plays the role of mass is the NUT parameter. It is similar to mass in the same way a magnetic monopole is similar to charge. In a pure S-duality theory both charges exist and play a role in a “braney physics.” The D3-brane of spacetime, which world brane volume V^4 ~ D4-brane, is dual to a compactified 7-brane. This 7-brane then contains the NUT charges. The 7-brane is compactified into ‘tubular spaces” which exhibit G_2 (7dimensional 3-form gauge-like group) dynamics and define the NUT charges associated with then as Misner-Dirac monopoles where they intersect (or terminate) on their dual D3-branes. Further, this Misner-Dirac monopole derives naturally N = 2 and N = 4 supersymmetry.

Now I want to return to the AdS~CFT correspondence according to the conformal completion. The one thing this lacks is any role of the Weyl curvature. This works for a 2+1 spacetime, where C. Frances does connect his work with the BTZ black hole , where for the extremal condition the BTZ black hole has zero temperature. For the 3 + 1 spacetime black hole the temperature is nonzero, and in four dimensions one gets the Weyl curvature. The Weyl curvature apparently permits additional degrees of freedom which give a finite temperature for the extremal black hole. This is important for the AdS spacetime, for a BTZ black hole in the AdS (the perfect black hole “box’) is used to understand the preservation of quantum information in black hole physics. For higher that 2+1 dimensions there is the inclusion of degrees of freedom which define a temperature, and this is I think a mechanism of the Weyl curvature in a quantum black hole.

Cheers LC

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