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

Whether there is a need for consciousness in the universe is something for future generations to determine. It might well be that the universe exists because it gives rise to sufficient complexity so that aspects of the universe can observe and think about itself. At that point we might find consciousness is somehow at the ultimate foundations of the unvierse. Of course the problem is that we really don't know what consciousness is exactly and further if consciousness becomes the foundations of cosmology it might be the end of physical foundations.

Cheers LC

The connection between black holes and the cosmological constant is the duality of D1-branes and D5-branes. This is a bit of a complicated business to show but in involves something called T-duality. You might also think of it this way. Consider the D1-brane (string) as an open string which results from a closed string. The closed string (like a loop) is distended into an open string, think of stretching a rubber band so without breaking it is a double stranded appearing filament. The graviton modes wrap around the string clockwise and counter clockwise. Yet these two modes on the now open string merge into states corresponding to tachyon condensates. These tachyons are "glued" at their ends to the M2-brane like flies on fly paper.

With the big bang consider an elementary model of two three dimensional balls where surface points are identified with each other in a multiply connected manner. Now these two three dimensional balls form the interiors of two wormhole openings, and the surfaces of the two balls are horizons which connect the two. The two 3-d balls then form a three dimensional sphere. Now in this elementary model suppose a point in the 3-d sphere is removed and "sent to infinity." The result is the 3-dim sphere is converted to a three dimensional flat space. This spatial surface is the observable universe that evolves to foliate spacetime according to the cosmological constant, or equivalently inflation. The boundary point removed almost instantly are tachyons again. Now in this toy-ish model the tachyons are removed to infinity and removed from the RR sector, which is what the M2-brane does by gluing tachyons in a condensate state in black hole interiors. These two aspects of physics have a duality of sorts.

Cheers LC

The quantum version of this is that entropy is S = -k Tr[ρ log(ρ)], for ρ the density matrix. Entropy amounts to some loss of knowledge about some submatrix of the density operator. Tracing over the unknown elements results in the loss of such data. So if we have a density matrix ρ = ρ'otimes ρ_{BH}, the black hole part of this might reflect the loss of information due to a quantum system being gravitationally coupled to the black hole. In the language of the Susskind stretched horizon, quantum bits are being stuck to the event horizon and lost.

To make this work with black hole complementarity the asymptotic observer and the infalling observer are observing string physics differently. The holographic string theory is a generalization of the S-matrix theory. The S-matrix requires a causal domain without any barriers. This is why the physics from the exterior is considered with tortoise coordinates where r* = r 2m ln(|r - 2m|), and the horizon is at infinity. So this domain is different from the domain the infalling observer witness string physics as r - -> 0. The gravitational Doppler shifting of a string observed by the exterior observer as it approaches a BH as seen at a distance amounts to a choice of how one measures the eigenstates of the string. If this were a classical system approaching the BH there would be no such issue in particular. The observation by the asymptotic observer then adjusts the system according to a commensurate set of observables (complete set of commuting observables etc), which are the complement of another set of observables an observer would select for measurements further into the BH. The horizon then prevents any communication of there being simultaneous observations of the two sets of observables. If the BH is quantum mechanical, IOW the horizon is a distribution of null congruences determined by a quantum measure over a state space, the two sets of observables occur in a superposition. The distant or asymptotic observer who detects photons from the horizon is then measuring states which are in an entanglement with the BH.

If you are watching a particle or string approach a black hole you do so with auxiliary quanta that it is entangled with. Your partner decides to follow the particle or string all the way into the black hole. You both share an EPR pair that is entangled with the string. The loss of that entanglement is a measure of the entropy which is measured on the black hole. The entanglement phase across the horizon is lost, except for the case when the black hole is quantum mechanical. The entanglement phase becomes "tangled up," if you will, similar to a blob of cream on the top of coffee that is stirred up and merged with the rest of the coffee. We lack the ability to track the fine grained details in how this happens, and so the entanglement phase is lost or buried away. This then counts into the entropy of the black hole.

Cheers LC

Jayakar,

I am almost done with the fuller version of this, which I can attach here. The matter involves a number of subtle connections. A crucial one is between S-duality of D1 & D5 branes and quantum homotopy or K-theoretic with something called Bott periodicity. This is where the M2-brane comes in. As I indicated the D1 branes can be tachyons which emerge from looped or closed strings by tidal forces into open string near the singularity. The tachyons are bunched together so they are no longer imaginary valued and faster than light, but are glued to this M2-brane. The M2-brane has physics simialr to what is found with graphene --- quantum critical phase transitions, quantum Hall effects and so forth. This is then dual to the D5-brane which is the outer portion we on the outside would call the black hole.

There is a lot here with holographic principles and so forth. This is an extension of the Susskind holographic principle, done in part to lend firmer support for a number of things such as AdS/CFT correspondence.

Cheers LC

Dray and Manogue's work is essentially the same. The "O" in my matrix refers to elements from the octonions. It is any set of elements of the form

O = 1 Σ_io_ie_i

where the sum is over 7 octonionic basis elements. I read one of Dray's papers the other day. He works out how to get eigenvalues from the Jordan matrix. The scalars "z," noted differently in dray's papers, enter into polynomial equations from which the eigenvalues and vectors are determined. In these analyses the three octonionic elements are different, and we of course might imagine assigning a "color index" to these which are a QCD-like element.

Cheers LC

Ray,

This involves IMO some matters involving quantum critical points and nonlnear behavior of Luttinger fluids. I attach an interesting in "Perspectives" article which introduces a main article published near the first of this year. The nonlinear interactions are of the sort not identified by a linear Feynman graph.

Cheers LCAttachment #1: when_infinity_does_not_count.pdf

I have finished this extension of my essay paper in a rough form. This is still a bit of an outline of things. As I push into this business I find it encompasses considerable depth. This extended version is attached here.

In reference to the three color problem, I attach a file with diagrams lifted from Lisi's paper, since you made mention of it, which illustrates how three colors interact. This is a QCD matter, and even in Lisi's paper he works out how this fits in E_8 as QCD with g_2 = su(3) + 3 + 3-bar = gluons plus quarks. The G_2 group sits inside E_8 and the F_4 is stable or invariant under G_2 action. So for F_4 is the octahedral net of the 24-cell, which is dual to the D_4 subgroup, or 3 sets of 8 rectified tetrahedral cells. This is a source of the F_4 triality. The D_4 group is of course SO(8), which in a non-Euclidean form SO(7,1) decomposes into SO(3,1) and SO(4). Lisi at this point identifies SO(4) with the Pati-Salam weak interactions, but I will not make such an identification at this time. We could just as well identify this with the loop variable gauge-like curvature F^i_{ab} where the upper index is SO(4). The 24-cell is the minimal sphere packing arrangement for a four dimensional spacetime.

What is of course interesting is that F_4 and G_2 are the central groups (centralizers) in E_8 so that the action of one group leaves elements of the other group alone. The G_2 group is also interesting in its own right. For one thing if we have the Jordan exceptional algebra J^3(O) as I outline in my paper, the diagonal elements are scalars, which are the "ones" from each of the octonions. This gives the 3x8 + 3x1 = 27 dimensional exceptional algebra. Now suppose I decide to construct a similar matrix but with the "imaginary" or components of i, j, k, ii, jj, kk, l basis. This will be a 3x8 + 3x7 = 45 dimensional anti-Hermitian matrix. We now do our restrictions on the Hermitian J^3(O) to get 26 dimensions, and we do the same for the anti-Hermitian *J^3(O) with 38 dimensions. The tensor product of this element with the G_2 in 14 dimensions ( *J^3(O)xG_2 ) gives 38 + 14 = 52 dimensions, which is the dimension of the F_4 group. A further tensor product with the J^3(O) extends this to 52 + 26 = 78 dimensions, which is the size of the E_6 group. A similar trick once again with G_2 gives us 7x26 + 52 + 14 = 248. In this way we have that J^3(O) are the real octonions, *J^3(O)xG_2 the complex octonions, *J^3(O)xG_2xJ^3(O), the quaternionic octonions and eventually up to the octonionic octonions.

The exceptional group G2 is the automorphism on O, or equivalently that F_4 x G_2 defines a centralizer on E8. The fibration G_2 - -> S^7 is completed with SO(8), where the three O's satisfy the triality condition in SO(8). So the three octonions in the exceptional J^3(O) exhibit a triality condition inherited from SO(8) by the fibration of G_2. BTW, the triality condition of SO(8) can be seen with the Dynkin diagram with a central o surrounded by three other oriented 2π/3 from each other. So we might consider the role of G_2, which has as it maximal subgroup SU(3) and thus has QCD-like physics, as a source for the three colors. Indeed G_2 was considered as a possible candidate for hadronic gauge theory.

G_2 is the isotropy group of three forms in 7 dimensions. This definition and early structure was advanced by Fredrich Engel and his student Walter Reichel in the early 20 th century. G_2 acts on seven dimensions, the S^7 sphere with 3^2 - 1 generator for a total of 14 dimensions. How this comes about is the following. Consider the complex space C^5 and then for every point a ∈ C5, let the 2-plane π_a exist in the tangent space T_aC^5 t as the zero set of the Pfaffian system

Dx_3 = x_1 dx_2 − x_2 dx_1,

Dx_4 = x_2 dx_3 − x_3 dx_2,

Dx_5 = x_3 dx_1 − x_1 dx_3.

There then 14 vector fields on C^5 whose local flows map the planes π_a to each other satisfy commutator relations of the Lie algebra g_2. Engel derived it from the first by a contact transformation, while Cartan identified g_2 as the symmetries of the solution space of the system of second order partial differential equations5 (f = f (x, y))

f_{xx} = 4/3(f_{yy})^3, f_{xy} = (f_{yy})^2.

The complex Lie group G2 has two non-conjugate 9-dimensional parabolic subgroups P_1 and P_2, and G_2 acts on the two compact homogeneous spaces M5i := G2/Pi , i = 1, 2. The lattice inside the maximal linearly independent, pairwise commuting elements of G_2 are spanned by the 12 roots of g_2, the root lattice, with 2 additional roots at the origin. The roots define a hexagonal planar lattice. The root diagram is with the attached gif file. The 9-dimensional parabolic groups P_1 and P_2 subgroups of G_2 with Lie algebras p_1 and p_2. By general results, the space G_2/P_i is a compact homogeneous variety, and a projectivization of the representation space V_i with highest weight ω_i as the G_2 orbit of some distinguished vector v_i ; but ω_i generates the 7-dimensional representation, here spanned by the six short roots and zero with multiplicity one. ω_2 is the highest weight of the adjoint representation that is spanned by all roots and zero with multiplicity two.. This means that

M^5_1 = G2/P1 = G2 • [v1] ⊂ P(C^7) = CP^6,

M^5_2 = G2/P2 = G2 • [v2] ⊂ P(g_2) = CP^13.

The first space M^5_1 is thus a quadric in CP^6.

There are two real 9-dimensional subgroups P_ i inside the noncompact real form G_2 corresponding to the complex parabolic groups Pi ⊂ G2; but they have no counterparts in the compact Lie group G_2 (roughly speaking, g_2 ⊂ so(7) consists of skew symmetric matrices, while parabolics are always upper triangular): a maximal subgroup of G_2 is isomorphic to SU(3) and thus 8-dimensional. M^5_1 is a projective variety over generic three forms in 7 dimensions, where the three forms have dimension

dimG = dimGL(7,C) − dimΛ^3(C^7) = 14.

The second algebraic varieity M^5_2 = G2/P2 ⊂ CP^13 is a more complicated projective algebraic variety: it has degree 18 and its complete intersection with three hyperplanes is a K3 surface of genus 10. Geometrically G2/P2 in terms of ω is seen in the 21-dimensional representation Λ^2C^7 splits G_2 into g_2 ⊕ C_7 Consequently G2/P2 is a subvariety of P(Λ^2C^7) as well. By the Plücker embedding, the 14-dimensional Grassmann variety G(2, 7) of 2-planes in C^7 lies in P(Λ^2C^7). This then leads to a GL(7,C) group structure on 2-planes, which is a gauge-like fibration.

So G_2 can then act as an internal gauge structure which has a triality on the octonions. BTW in Conway and Sloane on page 559 there is discussion of the Mathieu group automorphism on the Fischer Greiss group with a high order triality condition. As time is getting a bit late I will have to continue this later. Yet I think that this could how the gauge structure of this might look, and it would be comparatively elementary. This three color system would then be an internal gauge structure on the octonions which results in three-way products with nonassociative structure.

Cheers LCAttachment #1: jordan.pdfAttachment #2: 3color.GIF

The basic triality is due to SO(8), where three of the roots in the Dtnkin diagram are at an angle of 2π/3 oridented around another root. However, triality can in general refer to any system of roots or elements which are related to each other by some operator where the cube of that operator is unity. There are a number of circomstances where this does happen.

I will have to look at the Lisi paper again on the triality matter. I seem to remember this came about from SO(8) or SO(7,1) in E_8, or a decomposition of E_8. There are probably two subtle issues which occur though. I am not sure if the triality exists in the full E_8. Further, a problem with Lisi's argument is that he frames internal and external symmetries together, which runs into trouble with the Coleman-Mandula theorem.

With the three O's in the Jordan matrix there is a triality which exists. The octoinions are related to each other by a discrete symmetry with T^3 = 1. It is of course tempting to say this is a signature of some larger symmetry which involves three E_8's, such as the Mathieu sporadic group or Leech lattice.

Something occurred to me today. This involves the whole matter of how one defines the black hole complementarity according to conformal symmetry. My FQXI paper(s) indicate how the Weyl curvature is involved with the transformation of the closed string to an open one. This seems to lead to a generalization. Black hole complementarity involves the causal domain of the S-matrix. An exterior observer may remain outside a black hole and observe a string approach a black hole. It will redshift, or equivalently the time on the frame of the string slows down. The string will appear to slow down and its high frequency vibrations appear slower. The conservation of action and the observation of such slow quantum oscillations means that the string is lengthened and covers the horizon of the black hole. L. Susskind uses tortoise coordinates r* = r - 2m ln(|r - 2m|) to describe this physics. In effect the S-matrix for the string is extended to an infinite causal set this way. The infalling observer of course observes something very different. The string does none of this, it remains a nice closed string until it approaches very close to the singularity where the tidal forces (Weyl curvature) becomes large enough to distend the string into an open string. This is where the graviton modes disappear, tachyons in a condensate state appear and there is the M2-brane.

Now let us consider a metric g_{ab} related to a bare metric ĝ_{ab} by g_{ab} = Ω^2 ĝ_{ab}. The light cones in the spacetime are determined by the base metric ĝ_{ab} and the conformal factor Ω determines the scales for meter sticks and clocks. The bare metric has unit determinant, and the conformal factor is (-det(g_{ab})^{1/8}. The conformal factor is in general a function of position Ω = Ω(x), and the coordinates of space may transform by local Lorentz transformations according to y^a = y^a(x)

g_{ab}(x) - -> g'_{ab}(y) = (∂x^a/∂y^c)(∂x^b/∂y^d) g_{cd}(x),

and the conformal factor transforms as

Ω(x) - -> Ω'(y) = Ω(x)det(∂x/∂y)^{1/4},

Which means that

ĝ_{ab}(x) - -> ĝ'_{ab}(y) = (∂x^a/∂y^c)(∂x^b/∂y^d) ĝ_{cd}(x)det(∂x/∂y)^{-1/2}

This conformal structure is set up because this is what determines scale. A universe with just the bare metric would have no length scale at all, just a metric which determines null directions. The point of my essay is that the quantum critical point is what breaks the perfect scale invariance at that critical point and determines a fermionic-like Landau fluid which breaks the symmetry of the scale invariance. I think from a particle physics perspective this has something to do with zitterbewgung and the Higgs field which determines mass. In greater generality scales are determined by a conformal breaking which imposes a scale on nature, or a cutoff.

Hawking-Unruh radiation is due to the existence of an event horizon which partitions spacetime into two regions. The two regions have hyperbolic curves of constant proper distance from the horizon with

Region 1: Ψ(x) = ψ_1(R,t), z = Rcosh(t)

Region 2: Ψ(x) = ψ_2(R,t), z = -Rcosh(t)

For ψ_1(x) and ψ_2(x) the field configurations available observers the two regions. The ingoing observer then has available the field ψ(x) with the vacuum configuration from the spil states

|0> = Σ_n|n>_1|n>_2 exp(-πE_n)

Where the Boltzmann-like factor exp(-πE_n) means the vacuum in this frame appears with a thermal distribution of particles.

The bare metric perspective of this can be seen as light cones for ingoing and outgoing radiation. In effect the black hole with no sense of distance or scale is really just a point where equivalent light cones in the past and future are joined together. This is a very sterile view of the physics, so the conformal factor is brought into the picture. It is not hard to compute for curvature for the conformal and bare metrics with

R_{ab} = Ř_{ab} Ω^{-2}(4∂_aΩ∂_bΩ~-~ ĝ_{ab} ĝ^{cd}(∂_cΩ∂_dΩ) - Ω^{-1}(2D_a∂_bΩ - ĝ_{ab} ĝ^{cd}D_c∂_dΩ)

With the Ricci scalar

R = Ω^{-2}Ř - 6Ωĝ^{ab}D_a∂_bΩ

The Ricci curvatures are R_{ab} = R_{ab}(g) = R_{ab}(Ω^2ĝ) and Ř_{ab} = R_{ab}(ĝ). The system requires the imposition of a gauge, and with the above equations the simplest gauge choice is R_{ab}(Ω^2ĝ) = 0, which leads to the equation

ĝ^{ab}D_a∂_bΩ = (1/6)ŘΩ.

This little equation is then what becomes of interest. The differential of the conformal factor ∂_bΩ is related to the connection coefficient of the spacetime. Since g_{ab} = (1/4)γ_aγ_b it is possible to express the equation as a Dirac-like equation. Consequently there is a fermionic connection which may be established. This equation for the case of a black hole will then exhibit AdS-fermionic properties. The conformal factor, which is a scalar, may then represent a quantum field which exhibits statistical properties related to scaling. This then ties this physics into quantum criticality.

More later, cheers LC