The posting describes calculations carried out for a collapsing spherical mass shell, whose radius approaches its own Scwartschild radius. The metric outside the shell with radius larger than rS is assumed to be Schwartschild metric. In the interior of the shell the metric would be Minkowski metric. The system considered is second quantized massless scalar field. One can calculate the Hamiltonian of the radiation field in terms of eigenmodes of the kinetic and potential parts and by canonical quantization the Schrödinger equation for the eigenmodes reduces to that for a harmonic oscillator with time dependent frequency. Solutions can be developed in terms of solutions of time-independent harmonic oscillator. The average value of the photon number turns out to approach to that associated with a thermal distribution irrespective of initial values at the limit when the of the shell approaches its blackhole radius. The temperature is Hawking temperature. This is of course highly interesting result and should reflect the fact that Minkowski vacuum looks from the point of view of an accelerated system to be in thermal equilibrium. Manifest unitary is just what one expects.
The authors assign a density matrix to the state in the harmonic oscillator basis. Since the state is pure, the density matrix is just a projector to the quantum state since the components of the density matrix are products of the coefficients characterizing the state in the oscillator basis (there are a couple of typos in the formulas, reader certainly notices them). In Hawking's original argument the non-diagonal cross terms are neglected and one obtains a non-pure density matrix. The approach of authors is of course correct since they consider only the situation before the formation of horizon. Hawking consider the situation after the formation of horizon and assumes some un-specified process taking the non-diagonal components of the density matrix to zero. This decoherence hypothesis is one of the strange figments of insane theoretical imagination which plagues recent day theoretical physics.
Authors mention as a criterion for purity of the state the condition that the square of the density matrix has trace equal to one. This states that the density matrix is N-dimensional projector. The criterion alone does not however guarantee the purity of the state for N> 1. This is clear from the fact that the entropy is in this case non-vanishing and equal to log(N). I notice this because negentropic entanglement in TGD framework corresponds to the situation in entanglement matrix is proportional to unit matrix (that is projector). For this kind of states number theoretic counterpart of Shannon entropy makes sense and gives negative entropy meaning that entanglement carries information. Note that unitary 2-body entanglement gives rise to negentropic entanglement.
Authors inform that Hawkins used Bogoliubov transformations between initial Minkowski vacuum and final Schwartschild vacum at the end of collapse which looks like thermal distribution with Hawking temperature in terms from Minkowski space point of view. I think that here comes an essential physical point. The question is about the relationship between two observers - one might call them the observer falling into blackhole and the observer far away approximating space-time with Minkowski space. If the latter observer traces out the degrees of freedom associated with the region below horizon, the outcome is genuine density matrix and information loss. This point is not discussed in the article and authors inform that their next project is to look at the situation after the spherical shell has reached Schwartschild radius and horizon is born. One might say that all that is done concerns the system before the formation of blackhole (if it is formed at all!).
Several poorly defined notions arise when one tries to interpret the results of the calculation.
- What do we mean with observer? What do we mean with information? For instance, authors define information as difference between maximum entropy and real entropy. Is this definition just an ad hoc manner to get sum well-defined number christened as an information? Can we really reduce the notion of information to thermodynamics? Shouldn't we be very careful in distinguishing between thermodynamical entropy and entanglement entropy? A sub-system possessing entanglement entropy with its complement can be purified by seeing it as a part of the entire system. This entropy relates to pair of systems. Thermal entropy can be naturally assigned to an average representative of ensemble and is single particle observable.
- Second list of questions relates to quantum gravitation. Is blackhole really a relevant notion or just a singular outcome of a theory exceeding its limits? Does something deserving to be called blackhole collapse really occur? Is quantum theory in its recent form enough to describe what happens in this process or its analog? Do we really understand the quantal description of gravitational binding?
What TGD can say about blackholes?
The usual objection of string theory hegemony is that there are no competing scenarios so that superstring is the only "known" interesting approach to quantum gravitation (knowing in academic sense is not at all the same thing as knowing in the naive layman sense and involves a lot of sociological factors transforming actual knowing to sociological unknowing: in some situations these sociological factors can make a scientist practically blind, deaf, and as it looks - brainless!) . I dare however claim that TGD represents an approach, which leads to a new vision challenging a long list of cherished notions assigned with blackholes.
To my view blackhole science crystallizes huge amount of conceptual sloppiness. People can calculate but are not so good in concetualizing. Therefore one must start the conceptual cleaning from fundamental notions such as information, notions of time (experienced and geometric), observer, etc... In attempt to develop TGD from a bundle of ideas to a real theory I have been forced to carry out this kind of distillation and the following tries to summarize the outcome.
- TGD provides a fundamental description for the notions of observer and information. Observer is replaced with "self" identified in ZEO by a sequences of quantum jumps occurring at same boundary of CD and leaving it and the part of the zero energy state at it fixed whereas the second boundary of CD is delocalized and superposition for which the average distance between the tips of CDs involve increases: this gives to the experience flow of time and its correlation with the flow of geometric time. The average size of CDs simply increases and this means that the experiences geometric time increases. Self "dies" as the first state function reduction to the opposite boundary takes place and new self assignable it is born.
- Negentropy Maximizaton Principle favors the generation of entanglement negentropy. For states with projection operator as density matrix the number theoretic negentropy is possible for primes dividing the dimension of the projection and is maximum for the largest power of prime factor of N. Second law is replaced with its opposite but for negentropy which is two-particle observable rather than single particle observable as thermodynamical entropy. Second law follows at ensemble level from the non-determinism of the state function reduction alone.
The notions related to blackhole are also in need of profound reconsideration.
- Blackhole disappears in TGD framework as a fundamental object and is replaced by a space-time region having Euclidian signature of the induced metric identifiable as wormhole contact, and defining a line of generalized Feynman diagram (here "Feynmann" could be replaced with " twistor" or "Yangian" something even more appropriate). Blackhole horizon is replaced the 3-D light-like region defining the orbit of wormhole throat having degenerate metric in 4-D sense with signature (0,-1,-1,-1). The orbits of wormhole throats are carries of various quantum numbers and the sizes of M4 projections are of order CP2 size in elementary particle scales. This is why I refer to these regions also as light-like parton orbits. The wormhole contacts involved connect to space-time sheets with Minkowskian signature and stability requires that the wormhole contacts carry monopole magnetic flux. This demands at least two wormhole contacts to get closed flux lines. Elementary particles are this kind of pairs but also multiples are possible and valence quarks in baryons could be one example.
- The connection with GRT picture could emerge as follows. The radial component of Schwartschild-Nordström metric associated with electric charge can be deformed slightly at horizon to transform horizon to light-like surface. In the deep interior CP2 would provide gravitational instanton solution to Maxwell-Einstein system with cosmological constant and having thus Euclidian metric. This is the nearest to TGD description that one can get within GRT framework obtained from TGD at asymptotic regions by replacing many-sheeted space-time with slightly deformed region of Minkowski space and summing the gravitational fields of sheets to get the the gravitational field of M4 region.
All physical systems have space-time sheets with Euclidian signature analogous to blackhole. The analog of blackhole horizon provides a very general definition of "elementary particle".
- Strong form of general coordinate invariance is central piece of TGD and implies strong form of holography stating that partonic 2-surfaces and their 4-D tangent space data should be enough to code for quantum physics. The magnetic flux tubes and fermionic strings assignable to them are however essential. The localization of induced spinor fields to string world sheets follows from the well-definedness of em charge and also from number theoretical arguments as well as generalization of twistorialization from D=4 to D=8.
One also ends up with the analog of AdS/CFT duality applying to the generalization of conformal invariance in TGD framework. This duality states that one can describe the physics in terms of Kähler action and related bosonic data or in terms of Kähler-Dirac action and related data. In particular, Kähler action is expressible as string world sheet area in effective metric defined by Kähler-Dirac gamma matrices. Furthermore, gravitational binding is describable by strings connecting partonic 2-surfaces. The hierarchy of Planck constants is absolutely essential for the description of gravitationally bound states in thems of gravitational quantum coherence in macroscopic scales. The proportionality of the string area in effective metric to 1/heff2, heff=n× h=hgr=GMm/v0 is absolutely essential for achieving this.
If the stringy action were the ordinary area of string world sheet as in string models, only gravitational bound states with size of order Planck length would be possible. Hence TGD forces to say that superstring models are at completely wrong track concerning the quantum description of gravitation. Even the standard quantum theory lacks something fundamental required by this goal. This something fundamental relates directly to the mathematics of extended super-conformal invariance: these algebras allow infinite number of fractal inclusion hierarchies in which algebras are isomorphic with each other. This allows to realize infinite hierarchies of quantum criticalities. As heff increases, some degrees are reduced from critical gauge degrees of freedom to genuine dynamical degrees of freedom but the system is still critical, albeit in longer scale.
- A naive model for the TGD analog of blackhole is as a macroscopic wormhole contact surrounded by particle wormhole contacts with throats connected to the large wormhole throats by flux tubes and strings to the large wormhole contact. The macroscopic wormhole contact would carry magnetic charge equal to the sum of those associated with elemenentary particle wormhole throats.
- What about black hole collapse and blackhole evaporation if blackholes are replaced with wormhole contacts with Euclidian signature of metric? Do they have any counterparts in TGD? Maybe! Any phase transition increasing heff=hgr would occur spontaneously as transitions to lower criticality and could be interpreted as analog of blackhole evaporation. The gravitationally bound object would just increase in size. I have proposed that this phase transition has happened for Earth (Cambrian explosion) and increases its radius by factor 2. This would explain the strange finding that the continents seem to fit nicely together if the radius of Earth is one half of the recent value. These phase transitions would be the quantum counterpart of smooth classical cosmic expansion.
The phase transition reducing heff would not occur spontaneusly and in living systems metabolic energy would be needed to drive them. Indeed, from the condition that heff=hgr= GMm/v0 increases as M and v0 change also gravitational Compton length Lgr=hgr/m= GM/v0 defining the size scale of the gravitational object increases so that the spontaneous increase of hgr means increase of size.
Does TGD predict any process resembling blackhole collapse? In Zero Energy Ontology (ZEO) state function reductions occurring at the same boundary of causal diamond (CD) define the notion of self possessing arrow of time. The first quantum state function reduction at opposite boundary is eventually forced by Negentropy Maximization Principle (NMP) and induces a reversal of geometric time. The expansion of object with a reversed arrow of geometric time with respect to observer looks like collapse. This is indeed what the geometry of causal diamond suggests.
- The role of strings (and magnetic flux tubes with which they are associated) in the description of gravitational binding (and possibly also other kinds of binding) is crucial in TGD framework. They are present in arbitrary long length scales since the value of gravitational Planck constant heff = hgr = GMm/v0, v0 (v0/c<1) has dimensions of velocity can have huge values as compared with those of ordinary Planck constant. This implies macroscopic quantum gravitational coherence and the fountain effect of superfluidity could be seen as an example of this.
The presence of flux tubes and strings serves as a correlate for quantum entanglement present in all scales is highly suggestive. This entanglement could be negentropic and by NMP and could be transferred but not destroyed. The information would be coded to the relationship between two gravitationally bound systems and instead of entropy one would have enormous negentropy resources. Whether this information can be made conscious is a fascinating problem. Could one generalize the interaction free quantum measurement so that it would give information about this entanglement? Or could the transfer of this information make it conscious?
Also super string camp has become aware about possibility of geometric and topological correlates of entanglement. The GRT based proposal relies on wormhole connections. Much older TGD based proposal applied systematically in quantum biology and TGD inspired theory of consciousness identifies magnetic flux tubes and associated fermionic string world sheets as correlates of negentropic entanglement.