A view to a Kill (a farewell to continuous spacetime?)

Light rays to illuminate (the geometrodynamics behind) the entangled dark body radiation of black holes  

This post will be the penultimate of my August-series with titles constantly borrowed from the James Bond films list. 

More seriously it is also the follow up of the last one as announced in its ending picture caption. I try to convey once more the physical insight of the 't Hooft vision on black holes with his own worlds but using texts I have not quoted yet and which are more synthetic. 

“what happens to the quantum information carried by particles that enter the black hole, soon to be absorbed by the central singularity? Do they wiggle their way out again, or at least the quantum information carried along with them? When you take the existing theory and equations literally, there is no chance that they can wiggle out. Should we ignore this?  

First of all, we claim that this problem is a much more elementary and important one than what the literature suggests, and that most of the suggested cures are not well enough thought through. What should be done, is to stretch as much as is possible what one can do using standard, mainstream physics. Then, finally, impose the demand that black holes should behave, as much as possible, as ordinary forms of matter. When doing this properly, this leads to amazing observations.

Only few people noted that the application of existing physical knowledge can bring us much further than any wild speculation. Only recently we discovered a way to calculate things that could have been applied decades earlier. Our observation was [22] that we can use a partial wave expansion to describe energy and momentum entering the black hole, and that, applying Einstein’s equations, the information carried in by each partial wave is carried out again, in a partial wave with the same values of the quantum numbers ℓ and m.  

This calculation seems to be as elementary as the calculation of the spectrum of a hydrogen atom using quantum mechanics. As in the hydrogen atom, the boundary conditions are of crucial importance, but in a black hole, these boundary conditions are quite counter intuitive.  

What this calculation does, is to expose our problem more clearly than ever before, and now we can see what the answers must be. The general coordinate transformation relating the “inside” of a black hole with what an outside observer sees, must be a topologically nontrivial one. [23,24,25] Only then do the equations make sense.  

This does remarkable things with the fabric of space and time itself, not noticed before. For one thing, space and time must be discrete.

Imagining the future, or how the Standard Model may survive the attacks

Gerard ’t Hooft (10 June 2016)

For the quantization of gravity, it is crucial to understand the role of (real as well as virtual) black holes. The region in the immediate vicinity of the horizon, is characterised by the fact that time translations are substituted by Lorentz transformations. This means that observables near the horizon are undergoing unlimited processes of Lorentz contraction. 

Light rays are essential for defining the exact location of a black hole horizon: it is the boundary that separates regions from which escape to the outside universe by light rays is still possible, from the domain where all light rays are trapped. We shall now show how to use light rays to describe the backreaction of Hawking particles upon the presence of matter entering the black hole. 

The density matrix, that is, the probability distribution, of Hawking particles basically only depends on mass, charge and angular momentum of the black hole, but the actual configuration of the out-going particles characterises the microstate of the black hole, of which we have a large number (≈ e^4πM2 in natural units) of distinct elements [The number 4πM², corresponding to the black hole entropy, also represents the total number of Hawking particles emitted by a black hole in its life time [14]]. Whenever a particle enters the hole from outside, transitions to different microstates take place. This happens because a particle entering a black hole interacts with what comes out. The most disruptive interaction is the gravitational one, in spite of its apparent weakness... 

The gravity field surrounding a particle causes a very slight Shapiro effect: a passing light beam is slowed down and dragged towards the particle [15]. For a particle at rest, this effect is very small, but when the particle is Lorentz boosted, its gravity field increases in strength. At the black hole horizon, this Lorentz boost enhances its energy exponentially in time. 

Let p^µ_in(Ω) be the momentum distribution of the in-going particles at the spot Ω=(θ,φ) on the horizon, and δx^µ_out(Ω) the Shapiro displacement for the Hawking particles going out. An elementary calculation shows that 

δx^µ_out(Ω)=8πG ∫d²Ω′f(Ω,Ω′)p^µ_in(Ω'),     (4.1)

where f(Ω, Ω′) is a Green function, defined by ∆_S f(Ω,Ω′)=−δ²(Ω,Ω′) , and here, the angular operator ∆S is defined by

∆_S=∆_Ω−1 = −ℓ(ℓ+1)−1 .                          (4.2) 

Often, we ignore the −1. It was further assumed that the momentum p^µ_in is dominated by the radial component p⁻_in, so that, also, its derivatives w.r.t. θ and φ are small. Thus, whatever the positions x^µ_out(Ω) of the Hawking particles in the original microstate were, they are now replaced by the displaced positions, and since the particles emerge with a trajectory that separates itself from the horizon exponentially in Schwarzschild time, the significance of this displacement also increases exponentially in time. This is how information regarding ingoing particles is imprinted in the out-going ones. In Ref [17], a formal expression for the ensuing unitary evolution matrix is derived, to be summarised in the next section. One important feature is seen to emerge: the effect is purely geometrical. Only the momentum distribution p^µ_in(Ω) (mainly the minus component) enters. Consequently, unitarity of this evolution (scattering) matrix implies that these in-going particles are to be entirely characterised by their momenta...  

The question is now, how to deduce the black hole microstates from this evolution matrix. Let us simplify the situation a bit by replacing the Schwarzschild black hole metric by Rindler space. This implies that the angular coordinates Ω=(θ,φ) are replaced by two transverse, flat coordinates x=(x,y).... The Green function f(Ω,Ω′) becomes 

f(x,x′) = −1/2π log(|x−x′|).                    (5.1) 

Reserving the symbol z⁻ for the radial component of the positions, we write the displacement (4.1) due to the Shapiro shift as 

δz⁻_out(x) = −4G δp⁻(x′) log(|x−x′|/C).   (5.2) 

The constant C will soon drop out. Assume now that a black hole produced by one given initial state |in₀⟩, upon its final explosion leads to a given final state |out₀⟩. We then calculate the final state when a slight modification is brought about to the state |in⟩. Let the modification consist of adding one light particle with momentum δp⁻ entering the Rindler horizon at the transverse position x. All particles at the transverse position x' in the final state |out⟩ are then dragged along such that their out-coordinate z ^- is modified by an amount given by Eq. (5.2). 

We write this modification as a property of the black hole scattering matrix:  

S |in₀⟩ = |out₀⟩ → S |in₀ + δp⁻(x)⟩ = exp(−i ∫d²x p⁺_out(x')δz⁻) |out₀⟩,     (5.3)

... now we can reach any other initial state |in⟩, when described by the distribution of the total momentum going in, p⁻_tot(x) (as compared to the original initial state |in₀⟩), to find the new final state |out⟩ as a displacement of the original finite state |out₀⟩: 

⟨out|S| in⟩ = ⟨out₀|S|in₀⟩ exp[4iG ∫d²x′ log(|x'−x/C) p⁺_out(x') p⁻_in(x)].        (5.4) 

Note that, here, the operators p⁻_in(x) and p⁺_out(x') both describe the total momenta of all in- and out going particles as distributions on the Rindler horizon. The important step to be taken now is to postulate that the entire Hilbert space of the in-particles is spanned by the function p⁻_in(x), and the black hole scattering matrix maps that Hilbert space onto the space of all particles going out, spanned by the function p⁺_out(x'). We arrive at the unitary scattering matrix S : 

⟨p⁺_out|S|p⁻_in⟩ = N exp[ 4iG ∫d²x′ log(|x'−x/C) p⁺_out(x') p⁻_in(x)].              (5.5) 

This procedure leads to one common factor N , which now can be fixed (apart from an insignificant overall phase) by demanding unitarity... From Eq. (5.3) and the expression (5.5) for the scattering matrix, we can now deduce the relations and commutation rules between the operators p⁻_in(x), p⁺_out(x'), z⁻_in(x) and z⁺_out(x'), where the latter can be regarded as the coordinates of in- and out-going particles relative to the Rindler horizon... The algebra ... is quite different from the usual Fock space algebra for the elementary particles. In fact, it resembles a bit more the algebra of excited states of a closed string theory, but even that is not the same...

{As the operators z^± and  p^±  form a linear set one can decompose these new physical degrees of freedom into eigenmodes through decoupled k-waves in Rindler space. This diagonalisation provides in particular a pair of equations that can be seen as boundary conditions on the horizon implying the waves moving in are transformed into waves going out (these waves are not to be interpreted as particles as they may consist of many ones). This can be interpreted} as a real bounce against the horizon. The information is passed on from the in-going to the out-going particles. We do emphasise that in- and out-going particles were not assumed to affect the metric of the horizon, which is fine as long as they do not pass by one another at distances comparable to the Planck length or shorter; in that case, the gravitational effect of the transverse momenta must be taken into account. For the rest, no other assumptions have been made than that the longitudinal components of the gravitational fields of in- and out-going particles should not be ignored. This must be accurate as long as we keep the transverse distances on the horizon large compared to the Planck length. 

It is also important to emphasise that, even though we describe modes of in-going matter that “bounce back against the horizon”, these bounces only refer to the information our particles are carrying, while the particles will continue their way falling inwards as seen by a co-moving observer. In accordance with the notion of Black Hole Complementarity [20], an observer falling in only sees matter going in all the way, and nothing of the Hawking matter being re-emitted, since that is seen as pure vacuum by this observer. Rather than stating that this would violate nocloning theorems, we believe that this situation is asking for a more delicate quantum formalism...

One could try to compute the black hole entropy from the contributions of these reflecting modes. For each mode, the result is finite... Using the thermodynamical equations ... one can derive the contribution of each mode with transverse wave number k to the total entropy... The expression we obtained must now be summed over the values k. If we take these to describe a finite part of the black hole horizon area, we see that the summed expression will be proportional to the area, as expected, but the sum diverges quadratically for large k... The explanation for this divergence is that... our expressions are inaccurate at very large k, where transverse gravitational forces should be taken into account.  

It is not easy to correct for this shortcoming, but we can guess how one ought to proceed. It was remarked already in Refs. [18], that the algebraic expressions we obtain on the 2-dimensional horizon, take the form of functional integrals very much resembling those of string theory. We did treat the transverse position variables x and wave number variables k very differently from the longitudinal variables z^± and  p^± , but it is clear that we are dealing with the full expressions of an S₂ sphere. This sphere should be given two arbitrary coordinates σ=(σ¹,σ²), after which these should be fixed by a gauge condition relating them to the transverse coordinates x. We took σ=x, but apparently this fails when the longitudinal variables fluctuate too wildly. As long as a more precise procedure has not been found, we can simply insert a cut-off for the transverse wave numbers k, or equivalently, the angular momentum quantum numbers ℓ for a finite-size black hole. The divergence is quadratic. The expected expression, Hawking’s entropy S=4πM², comes about of we introduce a sharp cut-off at |k|≈M_Planck. Such wave number cut-offs impy that the conjugate variables, x, or equivalently, Ω=(θ,φ), form a discrete lattice [21]... 

Note, that we did not apply second quantization, such as in Ref. [17], since now we are not dealing with a quantum field theory. At every value of k, there are exactly two wave functions Ψ...(one at each side of the horizon, which mix). Second quantization would fail here, since the microstates appear to be determined by a single function such as p⁻...

It was observed that the in-going and out-going particles with which we started, produce vertex insertions as in a string world sheet, as if all particles considered should be regarded as closed string loops. It all takes the form of a string theory. Strings were not put in, however, rather, they come out as inevitable objects! But beware, these are not “ordinary” strings. The black hole horizon is the string world sheet [18]. If ordinary strings were to be Wick rotated to form space-like string world sheets, all factors i would disappear from the action, whereas our expressions are still in the complex plane, as if the string slope parameter α′ should have the purely imaginary value 4Gi. In most string treatments of black holes, the string world sheets are assumed to be in the longitudinal direction, that is, the world sheets are taken to be orthogonal or dual to the horizon.  

Our analysis appears to be closely related to ideas using the BMS approach [22], although there, the emphasis seems to be specially on the in- and out-going gravitational waves, while we focus on all particle types entering or leaving the black hole. Secondly, ... we attribute the black hole properties to the immediate surroundings of the future and past event horizon. Also, one may note that both approaches now focus on light-like geodesics, which justifies attempts to employ conformally invariant (or covariant) descriptions of quantum gravity. 

How quantization of gravity leads to a discrete space-time

Gerard 't Hooft 2016

What is the required Nature's Book keeping system?

A (last) mid-summer night's dream (or day's homework) could (have) be(en) to (start to) harness its potential for the construction of the proper geometric framework for quantum gravity. 

I am amazed by the physical intuition of 't Hooft that shines through his prospective quantum modelisation of black hole dynamics. I must confess I am also intrigued by the vague but pushy analogy one might tentatively establish between the physicist computing scheme outlined in bold characters in the text above with the recent new degrees of freedom devised by spectral noncommutative geometers to describe tentative quanta of geometry. Since the quantum information of incoming matter is supposed to impinge on outgoing one only through a momentum operator while two position operators are required to describe in a unitary way outgoing matter, could it be that a set made of a Dirac operator D and two Feynman-slashed position operators Y and Y' as defined by noncommutative geometry is the Nature's book keeping system to track the entangled spacetime-matter dynamics of black hole horizons?

Laws of gravity, as they are known today, then suggest that all forms of matter will be geometric: the way they affect the curvature of space and time is the only form of information that is conserved [5]–[7]. We think we drew the important conclusion that observations of the sort mentioned here, will be the only way to reconcile finiteness of the degrees of freedom with an unbounded group of local Lorentz transformations. This is extremely important, if true. It means that Fock space will not be the appropriate language; rather, we get something that resembles a bit more string theory, which is also basically geometric. String theories known today, however, seem not yet to be based on very sound book keeping... 

The demand from string theory that space and time themselves must feature either 10 or 26 dimensions, however, seems to be too restrictive. If indeed, as we suspect, physical degrees of freedom form discrete sets, then dimensionality of space and time may be less well-defined notions, so that such ‘predictions’ from string theory may well be due to some mathematical idealisation having little to do with reality. All in all, we are badly in need of a more orderly listing of all conceivable configurations of physical variables in a small region of space and time... 

String theory was an interesting guess, but may well have been a too wild one. We are guessing the mathematical structures that are likely to play a role in the future, but we fall short on grasping their internal physical coherence and meaning. For this, more patience is needed.

Nature's Book Keeping System

Gerard t Hooft

(Submitted on 29 Apr 2016)

Comment: the scrutinizing reader will notice that in the first text of 't Hooft I quote today I choose not to select his commentary about noncommutative geometry. This is because I think it addresses specifically the noncommutative field theory program, or to say more geometrically it refers to the construction of noncommutative spaces via deformation of commutative algebra while I am interested in this post (and in this blog) to chronicle advances in the isospectral deformation of classical manifolds, what I use to call spectral noncommutative geometric program (see chap 19  in this review for details).

To (almost) conclude this 2016 August-series

There is real adventure to be had at a time in which pure mathematics, theoretical physics, astronomy, philosophy and experiment are all coming together in a manner unseen for almost a hundred years. You probably have to go back still further to the seventeenth-century Scientific Revolution to get a full sense of what I mean, to the era of ‘natural philosophy’, Copernicus, Galileo, Newton. I do sometimes hear some of my physicist colleagues lamenting the old days when physics was young, when there was not a vast mountain of theory or technique that a young researcher had to climb before even starting out... actually we are at the birth of a new such era right now.

On Space and Time, Shahn Majid, 2007

Now it seems that the empirical notions on which the metric determinations of Space are based, the concept of a solid body and that of a light ray, lose their validity in the infinitely small; it is therefore quite definitely conceivable that the metric relations of Space in the infinitely small do not conform to the hypotheses of geometry; and in fact one ought to assume this as soon as it permits a simpler way of explaining phenomena. The question of the validity of the hypotheses of geometry in the infinitely small is connected with the question of the basis for the metric relations of space. In connection with this question, which may indeed still be ranked as part of the study of Space, the above remark is applicable, that in a discrete manifold the principle of metric relations is already contained in the concept of the manifold, but in a continuous one it must come from something else. Therefore, either the reality underlying Space must form a discrete manifold, or the basis for the metric relations must be sought outside it, in binding forces acting upon it. An answer to these questions can be found only by starting from that conception of phenomena which has hitherto been approved by experience, for which Newton laid the foundation, and gradually modifying it under the compulsion of facts which cannot be explained by it. Investigations like the one just made, which begin from general concepts, can serve only to insure that this work is not hindered by too restricted concepts, and that progress in comprehending the connection of things is not obstructed by traditional prejudices. This leads us away into the domain of another science, the realm of physics, into which the nature of the present occasion does not allow us to enter”.

On the Hypotheses which Lie at the Foundation of Geometry

Bernhard Riemann (10th June 1853)