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

This paper is actually the stripped down version of a more technical paper which outlines the details more fully. I will be giving reference to that here before long. I will also write some very generic posts which outline in qualitative fashions some of the physics here.

It is hard to discuss p-branes and the like without things getting a bit technical.

Cheers LC

The Newtonian gravitational equation of importance is ∆φ = 4πρ, ∆ = ∂_i ∂^i, where φ is the gravitational potential and the gravitational force is ∂_iφ = -F_i, in the ith direction. The equation means that on a spherical surface that the divergence of the gravity field ∂_iF^I = -∆φ is given by the amount of mass energy in that sphere. In the case of the Earth that spherical surface would be defined by the radius of the Earth. If you go into a deep mine that spherical surface is reduced some. If you were at the center of the Earth (somehow without being burned up) that spherical surface is reduced to nothing. There is no gravity force. Alternatively you might think of how the even distribution of matter around you is pulling equally in all directions leading to a vector sum of forces = 0.

The black hole singularity is an exceptional case in general relativity. It is a region where the Weyl curvature diverges to infinity and spacetime simply comes to an end. Einstein was rather horrified by these prospects and refused to admit their reality. The black hole remained a mathematical solution, the Schwarzschild solution, to the Einstein field equation until 1939 when Oppenheimer and Snyder worked out what matter did in the Schwarzschild spacetime. They found that beyond r = 2GM/c^2 there was a breakdown in predictability of the material dynamics, but the spacetime seemed well behaved. Later Kruskal demonstrated the horizon at r = 2GM/c^2 was not a singular point and eventually Wheeler coined the term black hole. Not long after in the early 1970s evidence for BH's began to appear. So the singularity is something we have to deal with. Susskind worked out some interesting result with the event horizon and how quantum fields are preserved there as seen by a distant observer. However, for the infalling observer there are completely different results. The tidal forces (Weyl curvature) becomes utterly extreme and all constituents of anything are distended and causally separated as the curvature diverges.

In what I work out the classical singularity (classically infinite and nasty) is replaced near the string scale or Hagedorn temperature by M2-branes. The graviton mode of string is lost and replaced by tachyon modes. This is a quantum realization of the "end of spacetime," if you will. This then has all sorts of interesting properties, which correlated with a quantum phase transition. This also has implications for cosmology. If we think of the universe in a toy "email post" model as due to the interior of two wormholes, this defines a three dimensional sphere by the multiple connectedness of the two three balls inside. The quantum fluctuation for the occurrence of the universe is then a puncturing of the three sphere so that topologically S^3 - -> R^3 pt. The boundary where the point is removed is then shot off to infinity, and done so by the 2 vacuum modes of the 26 dimensional string. The tachyons in effect leave the universe everywhere, except for the projective point at ∞. That projective point at ∞ is the projective ray "end" for light cones or null congruences. This in part is how the physics of the M2-brane is involved with renormalizing the cosmological constant. The Fermi-like fields (internal symmetry or a veilbein index term) pack on a Fermi-like surface and at low energy exhibit Landau-like behavior. The bare cosmological constant is like the massive electron (spinon or quasiparticle) at the breakdown, and the physics of the quantum transition on M2 indicates a manner by which the cosmological constant is small, instead of enormous ~ L_p^4.

Cheers

Lawrence B. Crowell

My take on consciousness issues is that we should try to avoid them as long as possible. It is possible that consciousness is ultimately the foundations of reality. I don't think that is the case, but I also don't know that with certainty. This is one problem with the anthropic principle, particularly the hard AP. It is an argument which borders on a type of theology.

We really don't know what consciousness is. This is certainly a big problem with invoking consciousness as a foundation to cosmology. At this time I think it preferrable to develop quantum gravity and cosmology with no such reference. Of course this might turn out to be an approximation, and ultimately consciousness is crucial. However, if we do come down to that phase it could be the end to the foundations of physics. It could just as well be argued that we are "lost." Ernst Fischer worte an essay which borders on this, which does suggest at some time we may end up with a lot of unanswerable questions.

Cheers LC

I think physics should pursue things as far as we can without invoking the anthropic principle. It would be nice to have a first order theory of quantum gravity and cosmology which does not require anthropic considerations. We might of course eventually find we are bereft of anything which serves as functional axioms and are forced into this tautological situation of the anthropic principle. I would just assume that situation is postponed as long as possible.

Cheers LC

If one is watching a mass fall into a black hole the proper distance for dt = 0 is

∫dR = ∫dr/sqrt(1 - 2m/r) =

r*sqrt((r - 2m)/r) m ln[r - m r*sqrt((r - 2m)/r)].

This is evaluated from some large R' >> 2m to r = 2m which is

R =~ R' m*ln(R') - {m*(m)} = R' m*ln(R'/m),

which is finite. So the ruler distance to the event horizon is finite. Now let us compute the radar distance. This is the distance as measured by EM waves or light from one point to another. In this case ds = 0 for the EM wave and we compute

∫dR = ∫dr(1 - 2m/r)

which results in the tortoise coordinates

r* = r 2m*ln(R/2m - 1),

which for time t = r/c and as r --> 2m the frequency ν approaches zero. This is the redshifting of radiation.

If one observes a string approach the BH its modes are observed to slow down and the string, using analogy with musical instruments, becomes distended along in order to give rise to these "low notes." A string is observed at a distance to have transverse modes slow down. The string action is determined by the so called string world sheet ~ T∫dσdτ sqrt{g}∂_aX^ν∂ ^aX_ν, where the Latin indices are string variables σ, τ and the Greek spacetime. The T is the string tension. The total action is conserved, but the observed quantum frequencies are reduced, as observed by the asymptotic observer. So as the action per unit of the string world sheet decreases, the string world sheet must increase in area. So the string ends up covering the BH horizon densely. Eventually the string approaches within a Planck unit of ruler distance from the BH which means in the tortoise coordinate R = 2m L_p, there is a log of L_p/2m which is hugely negative and the frequency is extremely small as observed by a far removed observer. That is the argument in a nutshell.

The stretched horizon is then a set of string modes with ν ~ 1/T_p sitting on Planck distance above the event horizon. I the BH absorbs some unit of mass δm so M --> M δm and the state of the BH adjusts accordingly. We can suppose there is a scattering matrix S = 1 2πiT which is involved with this transition. The S matrix is well understood for regions which are causally connected. So the S-matrix makes formal sense only for the tortoise coordinates, and the S-matrix as a flat space description of the holographic principle is formally extended properly. In this system the S-matrix has an appropriate domain, and it extends into elements of M-theory.

S-matrix theory is a flat spacetime case of the holographic prinicple, which in general is the AdS/CFT theory. S-matrix theory requires a complete causal domain to be applicable. This is one reason the tortoise coordinates are used, for there is no boundary. For this complementarity to work the S-matrix must apply in a dual manner for an observer on the commoving frame with the in falling string. However, there is a a singularity with geodesic incompleteness. Thus some care is required to understand this region, which I think is composed of tachyon condensates on M2-branes.

The tachyons are not a get out of jail free card. As soon as graviton modes vanish and the tachyon mode take their place they form condestate states. In effect they stick to the M2-brane and are like flies on fly paper. As open strings on the brane they form a quantum gas in a condesate state and do not fly off to infinity at arbitary speed v > c.

If as a euphemistic use of notation there exist states |ψ>_h and |ψ>s, for h and s standing for horizon and singularity, we might write a superposed state

|ψ> = |ψ>_h |ψ>_s,

where normalization is assumed. There is then the standard state reduction where a measurement of fields on the quantum black hole reduces the observation to one or the other. We can extend this further to an entanglement of states of the form

|ψ> = |ψ>_h|φ>_s |ψ>_s|φ>_h,

again normalization assumed. In this case the states |ψ>_i|φ>_j are mutually exclusive (orthogonal subsets of a Hilbert space). This would correspond to an EPR pair across the event horizon of the BH. To detect entanglement physics one needs to couple this to an ancillary state in some superposition. This could be used in a nondemolition measurement of the BH state where the ancillary state is read and then with a π/2 pulse the entanglement is restored. So we could determine the interior and horizon states are in superpositions or entanglements, but we of course could not measure both at the same time. So this is not a real cloning of states, but an entanglement of EPR states inside and outside the BH.

I hope this clarifies a few things. Unfortunately I have for the last few days been pretty sick and my efforts at posting an extended version of this has stalled.

Cheers LC

Before I answer these question explicitly I first will lay down a little bit of background. I do this in hopes this will give some idea of the arguments I advance. The conservation of quantum information with BH's is argued for by putting the black hole in an AdS_4 spacetime. This is the BTZ black hole, which due to the odd structure of AdS is perfectly confined there and does not interact with the QFT information at the boundary. The boundary is a sort of repeller, as geodesics leave the boundary. Hence the conformal field theoretic information on the system holographically projected in the AdS spacetime AdS_{d+1} ~ CFT_d is preserved. It is for this reason that the black hole is found to preserve quantum information. In that sense quantum physics is ultimately preserved, as laid down by Susskind, Vafa, Banados, Teitelboim, Zanelli .(BTZ), Maldacena and others.

A theoretical understanding of how black holes or quantum gravity in general processes quantum information means that there is some nonlocality involved with QFT field inside and outside the black hole. The quantum information which is spread out over the event horizon is thermalized (encrypted if you will) by the Hawking radiation which emerges from the horizon. The black hole is then a type of processor, which converts quantum information with string that attach to the horizon into quantum information that is emitted quantum mechanically by the BH. This is an aspect of black hole complementarity, where if you observe quantum fields of a string wrapped on the stretched horizon and the quantum fields emitted by the black hole the two sets of states do not defne complete sets of commuting observables. In other words if I could observe a string and its quantum information on the horizon and measure it in the Hawking radiation emitted I have then performed a quantum cloning. So the thermalization of radiation emitted by a black hole is a manifestation of QM, and not some violation of QM. It is curiously related to the measurement problem, which Penrose has made some light of.

The string world sheet increasing in size as it approaches the event horizon harkens back to Fermat's argument of principle of least time and how light can't be focused to temperatures higher than the source. Getting a phase space volume preservation at the foundation of physics, so |J(ρ')| = |J(ρ)|, where J is a Jacobian, and ρ' is a symplectomorphism map of ρ, is critical to having sane physics IMO. With black holes this has always been a bit of a problem, and one reason Einstein did not like the black hole idea at all. The BH focuses geodesics into infinite caustics which throws all of this into the trashcan, or so it appears. In effect we have been battling this problem for decades now. With string theory we might be getting the start of some resolution to this problem.

The physics I invoke involves the M2-NS5-brane duality. The NS5-brane (black brane) has a duality with the AdS_4, as an AdS-4 black hole. So the interior of a black hole has states which are dual to the stretched field theoretic configuration For an AdS_4 black hole metric

ds^2 = {1/z^2}[α^2(-f(z)dt^2 + dx^2 + dy^2) + dz^2}/f(z)^2],

for f(z) = 1 + (h^2 + q^2)z^4 - (1 + h^2 + q^2)z^3. The boundary on the AdS and the black hole, z = 0, 1 resectively there are mirror field theoretic configurations with a potential one-form A = {1/2}(hα^2xdy - hα^2ydx) + qα zdt and the electric and magnetic fields satisfy as S-duality for a dyonic black hole. This structure carries over in a T-duality from the NS5-brane to the M2. The dyonic BH then has two sets of charges that are e and g so that eg = nħ, which is the Bohr-Somemrfeld quantization condition. In this sense I think the field theoretic information from the horizon projected outwards (holography) and projected inwards are quantum mechanically related.

The closed string contains gravitational modes that upon the collapse of the closed string into an open string transition into tachyonic modes, but such modes are in a condensate. A tachyon condensate is a sort of glue of tachyons on a M2-brane. Now if there were no quantum physics here I can't say what would happen. The singularity of a black hole must quantum mechanically transformed form an infinite region to something "rational." This rational dynamics is I think based on duality principles of M-theory.

As for the gamma rapidity factor, this is an S-duality (strong-weak) result from the dyonic AdS BH. I will confess I have only weak estimates on the value of this and at this time can only give what are approximate values. I agree this factor is crucial to the whole argument, and it hinges upon the whole duality system here.

This is a work in progress to a degree, so if this works out and is not completely insane, there are further developments to come. I do think that vacuum state can be though of as a fluid of sorts. To take the vacuum energy density as

∫kdk(k^2/2) = K^4_end/6 ~ (1/M_p)^4

evaluates the quantum fluid at its quantum critical point, where for a small singular perturbation away from that point the value is renormalized to a much smaller value at the Landau-Fermi fluid state.

BTW, I still feel pretty wretched, though I got better sleep last night.

cheers LC

The singularity at the center of a black hole is more than a point, and in the case of a Schwarzschild metric is actually a spatial hyperbolic 3-surface. It is a bit odd to think that this could "fit inside" a sphere, but the horizon is a boundary between two different spacetime regions. Near the event horizon the BH metric appears similar to the Rindler wedge for an accelerated observer held at some constant distance from the BH. This accelerated observer will watch an inertial geodesic cross the horizon and become frozen on the horizon and never see it actually cross. According to the accelerated observer the inertial observer crosses hyperbola of constant radius until crossing the Rindler horizon at a proper distance = 0. The metric switches signature and the accelerated observer's metric then predicts spatial hyperbola of constant time-like radius on the "other side" which the inertial observer will pass through until eventually reaching r = 0.

The QFT information the exterior observer watches pile up on the horizon, in the sense of a frozen star as the Russians called it, is observed to exist on a causal domain which differs from that of the infalling observer. Classically this might not make much difference, and we don't worry about what happens to accretion disk material that enter an astrophysical black hole. Yet from a quantum perspective this amounts to a coordinate frame change analogous to orienting a Stern-Gerlach apparatus by different SO(3) rotations. So the infalling observer and the distant observer will see quantum field theoretic information according to different observational domains which have complementary aspects to them.

Lenny Susskind has worked out what happens to string on the horizon of the BH. The red shifted QFT and the conservation of action means the string world volume must increase as the energy per length on a string decreases. So for the transverse modes to red shift enormously it is like making a string distend and cover the black hole horizon. What happens to the string according to a commoving observer who foolishly falls into the BH? The string appears unaffected until it approaches the singularity. The tidal force on the string distends it from a closed loop (I assume we start out with a closed string with gravity modes) into an open string as it elliptically distends into a single strand. This is the spaghettification effect. The gravity modes disappear and are transformed into tachyon modes in a condensate form. These strings are D1-branes which have a duality with NS5-branes. At this point gravity disappears, and in effect it is the end of spacetime from the stringy perspective as the now open string gets caught like a fly on sheets of M2-branes which compose what we classically call the singularity.

The complementarity then is invoked here where there is a complemtarity between the information the distant observer sees frozen on the event horizon and what occurs in the BH interior with QFT information on the M-2brane. The D1-branes caught like flies on the M2-branes are an SO(2,1) system which is dual to observing a string tied to a 2-dim horizon. The physics of the SO(2,1) system is then analogous to graphene with quantum phase criticality and quantum Hall physics.

I have been working on a more comprehensive write up to include here, though I have been rather ill this past week. I will try to get that up and a more general write up on this whole business.

Cheers, LC

Steve,

I must confess I have a hard time following your line of reasoning at times. I am not sure if you are trying to draw parallels with string or branes with other structures. There does appear to be some sort of self-similarity of structure. Strings encode information as does DNA, DNA binds onto higher dimensional structures such as ribosomes which result in peptides, and open superstrings attach to d-branes. Beneath this is what I think a solid state physics type of lattice system of Planck volumes or quantum-gravity-bits, which has parallels in crystals and viruses and so forth.

Cheers LC