Can we see into a black hole? by Lawrence B Crowell

Jonathan,

Thanks for your questions and in reading my paper. It is not entirely complete, and there are a few open questions with it at this time. The main point is this. If you have a string fall onto a black hole there are two ways one can choose to observe this. The smart observer reamins far removed from the black hole and watches the string fall in. The dumb observer sets themselves on a comoving frame that falls in with the string. L. Susskind has analyzed the situation for the smart observer. The string is observed to time dilate and spread out over the horizon of the black hole. The slowing of transverse oscillations of the string is seen in the lengthening of the string as it wraps around the event horizon. The dumb observer sees of course none of this. The string is in a local inertial frame as it falls through the horizon and it appears that nothing really changes. That is until you approach the singularity where tidal forces distend the string into a long thread. If the string is a closed string it ends up being distended into a single filament and is transformed into an open string. Graviton modes are in the 26 dimensional bosonic string elements of the SO(24) gauge group and the operator on the vacuum produces symmetric and antisymmetric modes from the vacuum. The symmetric parts are the graviton modes, which have clockwise and counterclockwise motion on the string. The antisymmetric modes are gauge-like fields. The transformation of the open string to a closed string destroys the graviton modes and transforms them into tachyon condensate states. These open strings which such modes are attached to D2 or M2-branes which are effectively the quantized singularity of the black hole.

The rest of the paper takes off from there. The M2-brane is similar to the physics of graphene, with anyonic statistics and Chern-Simons Lagrangian dynamics for fields on it. The S-duality, which has been established by Polchinski and others, with the M2-brane and the NS sector D5-brane leads to the extension of this system to the Jordan exceptional algebra. Unfortunately due to length limitations I did not break this out in full detail, but the D1-brane (open strings) attached to M2-branes leads to a physics dual to a D7-brane, with G_2 holonomy. The G_2 is the automorphism of the J^3(O) exceptional group.

This extended system then leads to further arguments about the cosmological constant. The dynamics of fermionic states on the 2-dimensional space (that part of the anyonic field that is fermionic) exhibits quantum criticality, or the onset of a quantum phase transition with the so called breakdown of the quasi-particle as its mass diverges. The cosmological constant is then proportional to the fermionic mass and the divergent mass m* is the bare cosmological constant Λ_b ~ m_p^4 while the small value is the physical cosmological constant. I will then leave the rest for you to read at this point. The arguments are somewhat heuristic as for the actual set point, involving a rapidity function that is yet to be explicitly determined, instead of the sketchy estimate I make in my paper.

Back to the issue of the string falling into the black hole, Susskind argues from the tortoise coordinate of the black hole. String theory is really a generalization of the S-matrix theory. S-matrix theory requires an infinite causal domain, and so the tortoise coordinates provide that. This provides a set of variables or basis elements for the S-matrix, or a basis of operators. The infalling (dumb) observer detects the string on a completely different causal domain for the S-matrix. This means the two observers perform measurements of the string according to different sets of observables. In standard quantum theory this would be as if the two observers orient Stern-Gerlach apparatus along different directions. For a quantum black hole, one where the event horizon has a quantum uncertainty to it, this means the string is really in a superposition of different field configurations --- one for the outside and one for the inside. This is a form of black hole complementarity.

I will try to get to your paper before the closing of the contest. There are a lot of them here, so it takes time to read them all. AGain thanks for your interest in my essay.

Cheers LC

Yes that is the case. The open string under the tidal forces or shearing is like a rubber band stretched out from opposite points so it becomes effectively a single rubber thread.

LC

PS: I have tried to make the arguments at the early stage and later stage of the essay as physical as possible. There is quite a bit here about K-theory and Fermi surfaces. Much of this is to advance the theory of how the cosmological constant is due to a breakdown of Landau-fermi fluids.

LC

Take your time. And if you have criticisms I would prefer to read them directly than to get a silent 1 or 2. Real feedback is important.

LC

I will try to answer these question as best I can:

Frank Martin DiMeglio:

1) Why is it NOT your position that it is plain and simple common sense that the known mathematical unification of Einstein's theory of gravity (general relativity) with Maxwell's theory of light (electromagnetism) that is achieved by the addition of a fourth dimension of space to Einstein's theory must be plainly and significantly obvious in our direct experience? This, obviously, bears heavily upon the predictive effectiveness of mathematics as well. How do you see this mathematical unification in relation to black holes? GR is superceeded and advanced by said mathematical union, is it not? The key is to identify the physical/actual/real basis for said unification, correct?

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Which it comes to #1 the Kaluza-Klein theory imposes a fifth spatial dimension into general relativity. You then impose the condition that spacetime (4-dim) quantitites are independent of the fifth coordinate, which leads to 5-th coordinate connection terms which are Maxwell's equation. This was first done around 1921. There are a number of odd things though. One is that we do not observe a 5-th dimension, gauge theories differ from gravitation as internal symmetries, or principle bundle fibrations, so the idea then was to compactify the 5-th dimension. So it is curled up into a little circle. That has one very significant departure from a straight fiber bundle. If we write the wave equation for the EM field as

[Δ - ∂^2/∂t^2 ∂^2/∂ρ^2](E, B) = 0,

and we write the fifth coordinate partial solution as exp(iρ/R}, where R is the radius of compactification the wave equation appears as

[Δ - ∂^2/∂t^2](E, B) = (1/R)^2(E, B),

where the radius of compactification acts as a mass term. It is also not hard to see that if the radius of compactification is Planck scale then this is a hefty mass! Clearly electromagnetism is not a Proca wave equation of this sort.

So to work around this much has been done with renormalization and S-duality or Montenen-Olive duality similar to a Bohr-Sommerfeld quantization condition. Of course things get more difficult because there are the nuclear and weak interactions as well.

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Frank Martin DiMeglio:

2) Do you agree that the fundamental union of gravity and electromagnetism/light necessarily/ideally involves balancing scale by making gravity repulsive and attractive as electromagnetic energy/light?

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I guess I don't quite understand this. I would say that gravity has a duality in its strength to other fields, in particular QCD. Yet gravity is simply attractive because it has a hyperbolic group structure. There is only one physical root corresponding to a single source of the field, which is mass-energy >= 0.

As for the rest of your questions, I tend not to consider consciousness much. I base this on the fact we don't know what consciousness is particularly, and further attempts to infuse consciousness into physics leads to a muddle. I tend to be an objectivist of sorts. If we measure something in the world, and this observation is accessible to others or is repeated and so forth, then the phenomenon is relatively observer independent. Even quantum mechanics which involves observer induced state reductions is stochastic, so there is not predictability. So on balance I prefer not to invoke ideas of observer dependencies, or whether the exterior world is in ways dependent on the mental state of the observer.

Cheers LC

I am anonymous above! Forgot to log in. Here are more thoughts on where this is going, which I have posted elsewhere on FQXI

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I am working on a problem involving octonionic black holes, or a way to use the heavenly phere structure of the Jordan exceptional algebra as local systems of 26 dimensional Lorentz spaces. The identification between the three octonions by the triality operation and the light cone structure of the diagonal elements reduces this to 10 and 11 dimensions. The J^3(O) is locally diagonalized by the F_4 group. The F_4 group is the Hurwitz quaternion "1154" representation of the 24-cell. The 24-cell has B_4, D_4 and F_4 representations and the quotient F_4/B_4 determines a short exact sequence between spin(9) and the Moufang plane OP^2. This local Lorentzian system is then given by connection coefficients on this system.

The Jordan algebra of a vector space V, which can be O, is ~ V\oplus R according to the mixed product

(u, α)*(v, β) = (αv + βu, (u,v) + αβ), (u,v) = \langle u, v\rangle

but the inner product gives

(u, α)*(v, β) = (u,v) - αβ,

which is the Lorentz metric. Thus the diagonal entries of the J^3(O) set the three copies of the octonions in a Minkowski geometry. The diagonalization of the Jordan matrix is the defined locally with a "gauged F_4," which defines both the Minkowsi structure and the quantum algebra locally and constructs connection terms between local charts (local inertial frames).

There is for the Cl_8 group 256 dimensions with 9-grading for spins in a cycle of -2, -3/2, -1, -1/2, 0, 1/2, 1. 3/2, 2, which appears as

1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 1 + 8 + 28 + 56 + (35+35) + 56 + 28 + 8 + 1,

which predict eight Rarita-Schwinger fields and a single graviton. E_8 has a 7-grading 8 + 28 + 56 + 64 + 56 + 28 + 8, where the graviton sector and the scalar and pseudo-scalar fields are removed. The extra-octonionic graded structure 1,0,0,0,3+3,0,0,0,1 indicate that the graviton and the scalars (Higgs & dilaton) are derived elsewhere. Yet in this way the Clifford-8 can embed the exceptional E_8. That the graviton and scalars are not independent in E_8 is why there is the extended CL_{16) system, which embeds E_8xE_8, where now the graviton is due to two copies (in string theory interpreted as due to the handedness of the fields on a closed string) .

In particular, in the J^3(O) the two E_8s are related to each other by the "point" corresponding to one octonion, are dual to lines in OP^2, which are themselves OP^1 --- 7 dimensional spaces. The holonomy for the 7-sphere is the G_2 group, which is centralized with respect to F_4 in E_8. The duality here then describes a graded system on O^2 with dual 8_vx8_s representations. This is then one reason why octonionic gravitation involves the F_4 transformation of the octonionic group which diagonalizes the Jordan exceptional algebra. The action of the F_4 and G_2 groups I indicate in the attached jpg file.

BTW, this is one problem with the Lisi program. The graviton was assumed to be framed with the electroweak interactions. Yet the gauge interactions are internal symmetries, while gravitation is an external symmetry. The intertwining of them is through supersymmetric transformations and the graded structure over Lie algebras.

Lawrence B. CrowellAttachment #1: 1_octonionic_curved_space.JPG

The absense of gravity in free space is related to why a freely falling frame has not gravity as well. This is the falling elevator issue. A person on a falling elevator is not able to determine locally if they are in free space or falling in a gravity field. This is the essense of the Einstein equivalence principle. Now of course the freely falling frame may have some spatial extent and so the radial dependency of a gravity field will result in small deviations in motion between distributed masses. This is the Weyl curvature or tidal accelerations.

Cheers LC

I am not terribly educated in psychology, and so I can't comment on dreams too much. It is my understanding that dreams are ways the brain processes information at a time when glycogen stores are restored by oligoastrocytes.

One might say in some sense that models are mental ideations similar to dreams. I have a certain speculation that beneath what I think is the ultimate theory of everything, based on the monster group, there might at the Planck scale exist self-referential states which are similar to Liebniz's monads, or the Indra net of the Uppanishads. In that loose sense the universe might ultimately be something similar to the eternal dream of Vishnu.

The universe might then be some ultimate self-referential wholeness, and what we consider to be physical principles are various accidents, spontanous emergences or (fill in the blank) which come about. Human consciousness might then be some sort of recherche or mirror of this. It might then possibly be said that our capacity to abstract the nature of the universe is some mirror or similar process by which it came about.

Cheers LC

The AdS/CFT result indicates that quantum information is not really destroyed. The AdS spacetime turns out to be a bottle that can hold a black hole, since the hyperbolic curvature (Gaussian curvature ~ -1) means the boundary of the spacetime is repelling. Geodesics are great arcs, where a two dimensional version is the Poincare disk, or the Escher disks tessellated by stylized fish or bats vs birds, angels vs devils etc, which leave the boundary at high energy and return at near zero energy. A black hole placed in the spacetime is guaranteed to not hit the boundary. The equivalency between conformal fields and the boundary of the spacetime, and further the holographic principle on field theoretic information on the black hole horizon and the spacetime illustrates that information is preserved.

Information is then never really destroyed, where here we are referring to quantum information. Hence the entropy S = -k*ρln(ρ) is constant and the entanglements between EPR pairs or n-tuples is never destroyed. Such information might be hidden, such as if one particle in an EPR pair approaches a black hole, while the other particle remains at asymptopia, but the decoherence means entanglement phase has been lost. That entanglement phase is not destroyed, but in a manner similar to chaotic dynamics in classical physics, it has become distorted and folded into the environment in ways which are hard to dynamically predict. The domains of causality of the S-matrix for the two entangled states are not equivalent, and this results in the apparent loss of entanglement between the two states. It is also why Susskind says you will see an object, or its string-field theoretic content, suspended on the stretched horizon while at the same time the black hole will emit radiation from that string-field theoretic information (Hawking radiation) which burns up the fields on the stretched horizon. Working this problem out is one of the purposes of the sketch I wrote in my FQXI paper. The two domains of S-matrix causality for the two entangled states are related to each other by modular structure which ultimately preserves quantum information. This is also one basis for quantum error correction codes.

Cheers LC

The SO(24) comes from the three representations on the algebraic level 8 () 8 () 8 of the Jordan exceptional algebra, here () means "oplus.". One of these gives the vector terms in the J^2(O) 2x2 matrix and the other are for the supersymmetric pairs θ θ-bar. So for elements in the vector portion there is the framing

Φ = v ξ-bar θ ξθ-bar ξξ-bar F

of a superfield, for ξ the anticommuting Grassmannian.

This exercise is meant to build up to the Leech lattice which has F_4 automorphism as does the J^3(O). Then the Leech system can be decomposed into a pure N = 8 supergravity multiplet of 32 supersymmetries with a Bloch lattice like system as [E_8xE_8] x[H_4]^2. The H_4 stands for the 120/600 cells with Coxeter group H_4, [3,3,5] and Coxeer-Dynkin diagram *-5-*--*--* . SO(12) is a D_6 group which has the Dynkin diagram

*\

~~*--*--*--* Here the ~ means "space" since the editor does not like them.

*/

So how might we pull SO(12) from H_4? The H_4 emerge from E_8 in the Weyl group diag[H_4,H_4], where there are additional cyclic permutations of 8 letters. This is unless we get creative and do something like

*\

~~*--*--*--* *-5-* =

*/

*\

~* --* *-5-*--*--*

*/

which is almost a group theory "chemical reaction" on the algebraic level

so(12) () h_2 = so(8) () h_4

where h_2 is the algebra for the pentagonal group *-5-*.

In order to do this it requires a complete E_8 to construct this. So for this to work, where we have the above decomposition, I think we need to work in the domain of octo-octonions, or E_8-octonions. There is a whole lot of work to do before we get their. Yet this is not entirely impossible.

Cheers LC

E_8 is of course interesting, as the most complex of the ADE groups. There are of course the sporadic groups which stretch from the Mathieu _{24} group to the monster. There things get really crazy.

F_4 is the automorphism group of the Jordan matrix. G_2 is the automorphism of the octonions. The J^2(O) is the spin factor J^2(O) = O()R [() = oplus] and the J^3(O) is the extended spin factor

J^3(O) = J^2(O)()R = ([O()R]()R.

There is then an interplay here between G_2 and F_4. The spin factor define light cone structure in 11 dimensions, and we can certainly consider physics where this is a local frame. A curved space analogue of this can be examined according to the transformation roles of F_4 and G_2, which reduces the complexity of the problem.

Cheers LC