S'il vous plaît M. 't Hooft... chante nous un cℏoeur quantique d'espace-temps

(Dreaming of) A portrait of a black hole horizon (as a quantum cℏoir of spacetime)...
... reading 't Hooft last preprint on his proposed new constraints on the topology and the boundary conditions of general coordinate transformations to solve both the firewall and information problems in black hole physics :

In this paper, we shall primarily make use of a partial answer that we claim to have arrived at recently [10]: the necessity of revising the boundary conditions for Nature’s degrees of freedom at the horizon of a black hole. Since our analysis started out with our desire for consistent descriptions of stationary (or approximately stationary) black holes, it was not immediately clear how the revised boundary conditions should have been enforced during the formation of a black hole, but, in a somewhat formal fashion, one may well argue that, during black hole formation, the horizon starts out stretching over an infinitesimally tiny region; it opens up at a single point [Upon close inspection, one might conclude that the horizon first forms on a fractal subspace of spacetime, but since the scale at which this fractal extends may end up to be small even in Planck units, we ignore this complication in this paper.] in space and time. At that single point, it now appears to be necessary to revise the structure of this infinitesimal horizon to obey the new boundary condition, but since all this should happen at Planckian dimensions, the revision needed in our laws of Nature here can easily be argued to have escaped our notice up to today. 

After the horizon opens up, a black hole can grow quite big; the black hole horizon area grows rapidly towards macroscopic sizes during collapse, and as our modified boundary condition keeps track, it turns space and time into a non-trivial topological manifold. As our new boundary condition must end up as being indestructible, its implications will be sizeable. We emphasise that, nevertheless, our modified boundary condition will not affect the visible properties of a black hole in the classical limit. Also, we shall ensure that the modified boundary condition is of a kind that is not directly observable for a local observer. 

The boundary condition that we shall arrive at is characterised as an antipodal identification. In short, what it means is that the region of space-time inside the horizon is removed completely, as if by surgery, after which the edges are glued together by identifying the antipodes. This is continued throughout the lifetime of the black hole [Do keep in mind that, strictly speaking, the horizon is entirely timeless.] It is important, subsequently to insist that, locally, space and time remain smooth across the seams, while particles, including the information they carry, can cross. The seams must be locally invisible—only global observers notice this boundary condition. We argue that the antipodal mapping is the only way to attach the edges together such that strict geometrical conditions are obeyed. 

It boils down to a single “new physics” ingredient in black hole physics as soon as quantum effects are being considered: 

• from now on, when quantised particles and fields are considered, only those general coordinate transformations are permitted that map space and time continuously, and they must be one-to-one,

a condition not obeyed by the standard, classical Schwarzschild metric: every space-time point in the physically observable part of the universe is mapped onto two points in the Kruskal-Szekeres coordinates. 

As will be demonstrated (subsection 3.1), mapping the Schwarzschild metric onto the space-time metric of a local observer forces us to glue together regions in such a way that time-inversion takes place. Inverting the time direction is associated with an interchange of creation operators and annihilation operators. This implies that a region almost devoid of particles for one observer, is mapped onto a region almost filled with particles for the other observer. At first sight, this seems to be an unfamiliar and unwanted complication, but it is unavoidable...

At first sight, it may seem that our way of handling space and time near a black hole, will make a decent quantum field theoretic description of the elementary particles in there, hopelessly inadequate but the contrary is true. [... in earlier reports, the author had expressed his opinion that the causal order of events has to be respected by the coordinate transformations used; we now found compelling reasons to abandon that demand]. The apparently drastic rearrangement of the space-time continuum is exactly what is needed to arrive at pure quantum states for the black hole, and to obtain a unitary scattering matrix, so as to eradicate both the black hole information problem and the firewall problem, while accurately respecting the laws of general relativity.

There is a number of points that we should keep in mind. One is that wild guesses concerning possible answers, such as ‘novel uncertainty relations’, will be almost fruitless, as history shows*. The best thing to do is to split our problems into small pieces, and try to address each of these small fragments of questions in turn. Every now and then, such fragmented questions will lead to surprises. It helps enormously if we can convince ourselves of the correctness of our partial answers, and it is these that we should be able to use as new starting points for our next steps.  

On the other hand, we do not claim that all mysteries are resolved now. A systematic procedure must be found for a one-to-one mapping of the states generated by the spherical waves of momentum distributions and positions, onto states of the Fock space of a quantum field theory (some grand unified version of the standard model, relevant in the vicinity of the Planck scale, simply referred to as “standard model” elsewhere in this paper). It is here that the machinery of string theory might be of much help. 

Another point where our theory becomes vague is where the Planckian dimensions are reached. Usually it is assumed that string theory will provide all the answers, but string theory did not tell us about gravitational back reaction or antipodal identification, so we respectfully conclude that string theory is not fool-proof. 

The firewall transformation for black holes and some of its implications

Gerard 't Hooft  (Submitted on 27 Dec 2016) 

* Personal comment or wishful(l) thinking : more recent history might prove this 't Hooft statement is not exactly right...

Credit: X-ray: NASA/CXC/Penn State/B.Luo et al.
Press Image and Caption

This is the deepest X-ray image ever obtained... collected in eleven and a half weeks, of Chandra spatial telescope observing time. The image comes from what is known as the Chandra Deep Field-South. The central region of the image contains the highest concentration of supermassive black holes ever seen, equivalent to about 5,000 objects that would fit into the area on the sky covered by the full Moon and about a billion over the entire sky...

About 70% of the objects in the ... image are supermassive black holes, which may range in mass from about 100,000 to ten billion times the mass of the Sun. Gas falling towards these black holes becomes much hotter as it approaches the event horizon, or point of no return, producing bright X-ray emission.

Deepest X-ray Image Ever Reveals Black Hole Treasure Trove

For Release: January 5, 2017

This post is dedicated to a hiker friend, Emmanuel ;-)