Weaving and sewing the fabric of space-time-matter the proper way

 Are Quantum Black Holes at(t)oms like any others?

A line of research has been set up by the author, with the aim of obtaining further insights in the texture of space, time and matter, by making one simple basic assumption: whatever happens during gravitational collapse, should be naturally incorporated in our theories of particles and forces. If we try to describe something like black holes, their behavior should be understood in the same language as the one we use for other particles; black holes should be treated just like atoms, molecules and nano particles, apart perhaps from being just a bit more exotic. If we find it useful to describe particles in terms of quantum wave equations, virtual particle states, and other such notions, we should expect that these should also apply to black holes; there will be virtual black holes, and so on.

...

In short, our present approach does not come with wild sceneries, new math languages, wild speculations, but only with small tangible facts. We do produce new results this way, notably about the topology in space-time and its non-commutative nature. Both are not what most people think; they are more interesting than that.

...

Our present aim is to precisely formulate the behaviour of the black hole microstates. First, consider the classical (i.e., non quantum mechanical) problem. It consists of a straightforward application of General Relativity without anything else added.

 ...

When we map the [Kruskal-Szekeres coordinates x and y] onto the finite line segments [−1,1], 

and we regard u⁺ + u⁻ as a space-like coordinate, while u⁺ − u⁻ is time-like, we get the well-known Penrose diagram, where the lines xy = −1 , x → ±∞ and y → ±∞ are straight lines, while light-like geodesics with dΩ = 0 run under 45◦ , see Figure 1...

Without quantum mechanics, General Relativity clearly states what observers will see. The region labelled I in this figure represents the visible universe. It is bounded by the lines x → ∞, y → ∞ and the future and past event horizons. All particles and fields in this universe are represented by the appropriately transformed fields in this region I . The other regions are mathematical extensions of the metric, but physically, they mean rather little. Region III is seen as where an unfortunate traveller who entered the black hole may think he is going, but nothing about his adventures can be followed by observers in the outside world. Regions labelled II and IV are often considered to be even more unphysical; these are actually not there, if we may assume that the black hole was formed by collapsing matter in the (recent or distant) past. We do not need to be bothered by any of this as long as we do classical physics, and limit ourselves to epochs well after the event(s) that produced the black hole. 

Figure 1: Penrose diagram for Schwarzschild black hole, showing regions I − IV . Equal-time lines are shown. In the standard picture, only region I corresponds to the visible universe. In this work, region II will represent the antipodal region of the same black hole. In the classical theory, this makes no difference. Arrows show the “firewalls”, caused by the very early in-going particles (in) and the very late out-going ones (out).

Also, at this point, if someone would suggest that region II corresponds to the antipodal region of the same black hole, this would not make any difference; to check such a statement, signals would have to be sent going faster than the local speed of light, and therefore, at this stage, such a statement is empty. It is not even obviously wrong... 

Now, switch on quantum mechanics. One important thing changes. Region III now definitely becomes important [or, at least, a small region of size ε around the centre of the diagram].

{to be continued in ...}

//added on August 30, 2016
Before the reader proceeds further, I choose to summarize the 't Hooft argument that is going to come with an extract from another article

In 3-space, the regions I and III form a wormhole connecting antipodal points on the horizon

//added on August 24, 2016

I have chosen to add the following digest to this fascinating paper just as an exercise for me to try to extract its very meat. This addendum makes the post longer but I hope it makes the reader convince that 't Hooft exposition below is not a pensum but the most illuminating description of the possible quantum gravity physics of Hawking particles at the horizon of a black hole, offering a tentative Einstein-Rosen bridge quantum crossing:

Indeed, we do need the existence of region III , smoothly seamed onto region I , in order to describe the vacuum state experienced by {an} observer {going in}. Performing the by now standard calculations by Hawking, one finds that the vacuum state as seen by the in-going observer, the Hartle-Hawking vacuum [8], is seen by the distant observer as a state containing particles, roaming in a different vacuum, the Boulware vacuum [9]. In terms of the complete set of quantum states in the Boulware vacuum, labelled as |E, ni, where E stands for the energy and n for any other kinds of quantum numbers, the Hartle Hawking vacuum can be written as 

|∅ >_HH = C ∑_{n, E} |E, n>_I |E, n>_II  e^{−1/2 β_HE} , (2.5) 

where ‘HH’ stands for ‘Hartle-Hawking’, β_H=1/kT_H=8πGM/ℏc³ is the inverse Hawking temperature, and C stands for a constant needed to normalize the state to one...

Recently, part of this argument has been put in doubt [7]. Arguments addressing possible entanglement between particles going in and out, in combination with no-cloning theorems, are used to claim that observers entering black holes with a long lifetime, will notice the presence of Hawking particles, and in fact be killed by the firewall formed by them. Even though we now claim that no firewall will arise, the tendency of very late (and very early) particles to form firewalls will play an important role in what follows, and pointing out the threat coming from these firewalls was justified. The model we bring forward, is a natural application of General Relativity, and based on the possibility to transform away the firewalls. The fact that, in our final results, the firewalls are to be removed, is an absolutely essential ingredient of General Relativity, and indeed we must show how this cure of the problem comes about, as we shall...

Today, many researchers assume that the states |E, n>_II (or at least the information they contain) will eventually escape from the black hole somewhere far in region III . This evidently false assumption lead to the black hole information problem and the firewall assumption;...

Gravitational back reaction 

The point is that there is more. Instead of analysing the situation further – to establish a third difficulty – most authors now turn on their marvellous fantasy to produce one crazy model after the next. We are not ready for that yet. First, we need to contemplate the ... difficulty ... that one cannot ignore gravitational back reaction.

Again, consider the classical theory, but now take into consideration that in-going and out-going particles all carry gravitational fields. The effects of these fields are far more bizarre than any other interactions they are involved in. The field of an in-going particle, and that of out-going ones, can be calculated precisely. All one has to do is consider the gravitational field of a particle at rest, and the way it affects the metric of the surrounding space-time, and boost that to very high velocities. The result of such a calculation is well-known: there is a dragging effect: when an out-going particle passes in ingoing one, both particles are shifted a bit... {on the black hole horizon parametrized with the solid angles Ω≡(θ,ϕ) and Ω′ one get expressions relating position u and momenta p according to   

u⁻_out=8πG/R² f(Ω, Ω′)p⁻_{in ,}  (2.8) 

with f(Ω, Ω′)} a Green function obeying (1−∆Ω)f(Ω)=δ²(Ω, Ω′) where R=2GM is the radius of the horizon}.

The most peculiar consequence of this dragging effect is, that it drags particles to and fro, across the horizons from region I and II and back, in the Penrose diagram. As soon as we try to take this effect into account, we have to abandon hopes that region I and region II can be handled independently...

Quantum dragging 

Carrying this dragging effect over to our quantum models is relatively simple. We were accused of doing a ‘semiclassical’ calculation, but this is not true. What we shall do is about as semiclassical as a calculation of the spectrum of a hydrogen atom using the classical electric potential fields between protons and electrons. One always takes this potential to be classical, because operators V (x) are commuting operators; regarding them as creation and annihilation operators for photons in unnecessary, at least in a first attempt to calculate atomic spectra reasonably accurately. Similarly here, the Green functions f(Ω, Ω′) in the previous section represent commuting operators in the Hilbert space of the gravitons. 

Thus, we relate the positions u⁻(Ω) of the out-going particles to the momenta p⁻(Ω) of the in-going ones, and vice versa. Note that we can easily take the momenta of the in-going particles all to commute with one another, as well as either all momenta, or all positions of the out-going ones as being commuting, but of course, momenta do not commute with positions of the same particles; they are controlled by the familiar commutation rules. {From Eq. (2.8) one gets the commutator algebra (2.9) that appears simple but tricky because one needs to interpret properly the sign switch in ↔ out}. 

... the dragging effects described by these equations are linear in the momenta and the positions. This begs for an expansion in spherical harmonics. A huge advantage of spherical wave expansions is that, expressed in terms of the spherical harmonics Yℓm(Ω), the different partial waves decouple. The angular operator 1−∆Ω becomes the C-number ℓ²+ℓ+1. This means that, as in the hydrogen atom, we are left with ordinary differential equations, and furthermore, near the horizon(s), we can also expand the in-going and out-going particles in terms of plane in-going and out-going waves. The resulting equations could not be simpler, and are now completely transparent. They tell us exactly what happens, for everyone to see...

Modelling the black hole microstates 

Using only legitimate transformations in General Relativity, can we now categorize the black hole microstates? And how do these evolve in time?

... in the Schwarzschild metric, let us consider any quantum state of the Standard Model (or whatever theory is used near the Planck scale, as long as it allows for perturbative calculations). There will be particles travelling in, and particles going out, but we cannot allow particles whose momenta – in terms of the Kruskal-Szekeres coordinates x and y – are so high that they cause a significant amount of gravitational dragging. These would contribute to firewalls, and thus make our calculations invalid. At first sight, this demand seems to force us to abandon the condition that our states are stationary, but wait and see. 

We consider the most general state of light (i.e. non energetic) particles in the Penrose diagram, with this important limit on their momenta. These are the states that will generate the microstates we want...

Thus we assume that, exactly when positions in the transverse coordinates (the angles on the horizon) reach Planckian values, other branches of physics will have to be addressed. That happens at ℓ≈M in Planck units; all partial waves with considerably smaller values of ℓ will be allowed, which is a sizeable fraction of all partial waves that may exist. Our description of the microstates will be limited to these. 

Now comes the good part. Consider the time evolution of the microstates we have now under consideration, and remember that every partial wave can be considered independently of all others. The in-going particles will gather alongside the past event horizon; their position operators u⁺ will shrink exponentially. Their momenta p⁻ will grow exponentially, and sooner or later, we will have to consider the gravitational dragging effect caused by them.

Since we are looking at a spherical wave, the calculation of the effect this gravitational dragging has on the out-going particles is easy. Everything happens at one value for ℓ and m only. Their position operators u⁻_ℓ,m will grow, together with the  p⁻_ℓ,m  of the in-going particles. The relation between u⁻_out and p⁻_in stays the same, both grow exponentially. It is given by a 2×2 matrix, whose entries refer to whether the particles (more precisely, the waves) sit in region I of region II . As soon as u⁻ and p⁻ become too large, we can consider the in-going particle as having become invisible (it is stuck against the past horizon), while the out-going particle has left the system. Our information retrieval process has done its job. The out-going particle carries the information of the in-going one back to the outside world. It’s gone. We remove them both...

We do conclude one important thing: the momenta of in- and out-going articles grow when they come close to a horizon. As soon as this happens, we replace them by the effect they have on the other articles, which are being shifted. This way, we can always limit ourselves to low momentum particles, even as time evolves: there is no firewall, either on the future or on the past horizon – in-going particles threatening to produce a firewall are simply removed, while putting the information in their energy-momentum tensor in the new coordinates of the shifted out-going particles. Thus, as time evolves, we can keep using the eternal Penrose diagram of Figure 1.  

...The antipodal identification

If we would end our story here, there would still be a problem: what does region II stand for? Our problem is that the drag, described in the algebra (2.9), carries particles (more precisely, their spherical waves) from region I to II and back. The resulting scattering matrix is a two-by-two matrix [15]. Miraculously, we have two asymptotic regions with in-going and out-going particles, instead of one. This does not seem to describe the universe outside the black hole properly. Until shortly, most investigators (including this author) have been thinking that regions III and IV refer to the “inside” of the black hole.

General Relativity tells us that coordinate transformations can be used to describe regions of space and time differently. However, this only makes sense if every point that is described by one set of coordinates in one frame, should also be described by just one set in the other frame. Not two, as we have now (Note: the Schwarzschild coordinates r and t in region II are the same r and t as in region I , so, unless we do something about this, we have here a one-to-two mapping, or: the Penrose diagram seems to describe two black holes rather than one).  

So what does the Einstein-Rosen bridge (the central region of the Penrose diagram) do? Some authors suspect that region II refers to an other black hole, somewhere else in this (or another) universe. This would completely violate the most basic concept of locality, and the present author sees no way to make sense out of such suggestions. A more elegant resolution had to be searched for.

There is an elegant solution. It tells us that the transformation from coordinates of a distant observer, to local coordinates in the Penrose diagram, must be topologically non-trivial. Is there a problem with that? Not at all if the point mapping is one-to-one, and if it does not lead to singularities. This is what we get if we ordain that region II of the Penrose diagram exactly describes the domain of the universe extending from the antipodes of region I  We do not have to fear that this would mean that two halves of our universe would be identical. Precisely not: the two domains I and II of the Penrose diagram are distinct. Inside the black hole, one cannot travel from one to the other, since these regions are space-like separated. Also, the points that are to be identified are always far away from one another. The closest they get is at the horizon itself, but that has a finite radius R=2GM , so even there, we have no singularity. We conclude that this mapping is legitimate. Indeed, we claim that it is inevitable. 

It is also interesting. In Ref. [6], it is shown that there is a remarkable consequence for the Hawking particles. Eq. (2.5) now implies that both the states |E, n>_I and |E, n>_II describe visible particles outside the horizon. This means that our state no longer requires summation over the ‘unseen’ states to form a density matrix, but instead corresponds to a pure quantum state. If one sums over the other states because they are too far away for a local observer to observe (all the way to the antipodal area), then one would conclude these particles to have the usual Hawking temperature. However, the state entire is not thermal at all. There is a superb entanglement between the Hawking particles on one hemisphere of the black hole and the other...

As for the original ‘firewall problem’, we now see that it originates from an incorrect counting of states. If we hadn’t identified regions I and II with antipodal points in the black hole, every quantum state would require the existence of a clone somewhere inside the black hole (or elsewhere in the universe). Now, we have restored the one-toon nature of the general coordinate transformation. The gravitational drag phenomenon returns information carried by particles that were about to escape to regions III or IV , back into our universe, so that this quantum information is preserved... 

The Quantum Black Hole as a Hydrogen Atom: Microstates Without Strings Attached 

Gerard t Hooft (Submitted on 17 May 2016)

What " resemblance ... not more than superficial"?

Id.

Now could there be a link between the position and momentum operators from 't Hooft work with the Dirac operator and Feynman slash of real scalar fields emphasized in the following recent development in spectral noncommutative geometry or is it just wishful thinking? Time could tell...

In spectral geometry the metric dimension of the underlying space is defined by the growth of the eigenvalues of the Dirac operator. As shown in [2] for even n, equation (2) {below}, together with the hypothesis that the eigenvalues of D grow as in dimension n, imply that the volume, expressed as the leading term in the Weyl asymptotic formula for counting eigenvalues of the operator D, is “quantized” by being equal to the index pairing of the operator D with the K-theory class of A...

In this letter we shall take equation (2), and its two sided refinement ... using the real structure, as a geometric analogue of the Heisenberg commutation relations [p, q] = iℏ where D plays the role of p (momentum) and Y the role of q (coordinate) and use it as a starting point of quantization of geometry with quanta corresponding to irreducible representations of the operator relations. The above integrality result on the volume is a hint of quantization of geometry... (For details and proofs see [3, 4])...

It is natural from the point of view of differential geometry, to consider the two sets of Γ-matrices and then take the operators Y and Y′ as being the correct variables for a first shot at a theory of quantum gravity. Once we have the Y and Y ′ we can use them and get a map (Y, Y′): M→Sⁿ×Sⁿ from the manifold M to the product of two n-spheres. Given a compact n-dimensional manifold M one can find a map (Y, Y′): M→Sⁿ×Sⁿ which embeds M as a submanifold of Sⁿ×Sⁿ. This is a known result, the strong embedding theorem of Whitney, [7], which asserts that any smooth real n-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2n space. Of course R²ⁿ= Rⁿ×Rⁿ⊂ Sⁿ×Sⁿ so that one gets the required embedding. This result shows that there is no restriction by viewing the pair (Y, Y′) as the correct “coordinate” variables.

Quanta of Geometry: Noncommutative Aspects

Ali H. Chamseddine, Alain Connes, Viatcheslav Mukhanov

(Submitted on 8 Sep 2014 (v1), last revised 11 Feb 2015 (this version, v4))