• [deleted]

This sounds exactly like what Antony Valentini has been working on for some time now, where current quantum field theory, is only describing a "non-equilibrium" state which has relaxed out of a more fundamental theoretic, *non-local* description.

The only difference is that Valentini is approaching it from a pilot wave (Bohmian mechanics) basis. Valentini has proposed that there could be signatures of non-equilibrium states in the CMB and/or detectable n-e relic particles. As is the case in the article, this more fundamental state of non-locality would exist only in the very early universe or perhaps black holes.

I wonder if Steve is familiar with Valentini's work?

Dear Roy,

Thank you for highlighting that parallel between Giddings and Valentini's work -- I hadn't thought of it! It might also be worth revisiting what Valentini has been doing, on the site.

  • [deleted]

Hi Zeeya,

What would be great is if we could get Valentini to provide a summary and update of his current work on the site! Any chance do you think?

I personally think that all research into pilot wave or so called "hidden variable" theories has been criminally under resourced and misrepresented! This is why it is still under-developed compared to traditional, "acceptable" research in quantum theory. Then, because of this, people dismiss these ontological theories as being too incomplete, not able to fully describe the world. Well, no wonder!! People also tend to criticise them from the standpoint of the tradition interpretation of QT, not on their own terms!

Pilot wave type theories really aren't just another interpratation of QT, they are a different class of theories.

Love to hear from Mr V. !!

    • [deleted]

    General relativity is a geometric theory of spacetime, which means that quantum gravity means quantizing spacetime itself. It is not entirely clear what this means. A number of questions have to answered, and currently there are obstacles in our current theories which do not permit us to address these issues well. Standard quantum field theory is local, but the fundamental physical observables of quantum gravity, which means diffeomorphism-invariant, are necessarily nonlocal. Quantum mechanics is nonlocal, but the wave function is defined by the action of field operators that act on a Fock space so as to define an amplitude locally. In a related manner quantum field theory takes causality as a fundamental postulate, but in quantum gravity the spacetime geometry, and thus the light cones and the causal structure, are subject to quantum fluctuations. This has the curious meaning that a quantum field is propagating on spacetime, but where spacetime is the quantum field.

    LC

    • [deleted]

    "If faster-than-light signaling was allowed, you could, for instance, send a message back in time to persuade someone to kill your own grandfather before he met your grandmother - and thus you wouldn't exist to send the signal."

    This is both wrong and a red herring. Neither Einstein's relativity nor Newton's emission theory predicts anything like that. If faster-than-light signaling is possible, the conclusion is quite simple: special relativity should be abandoned and Newton's emission theory of light resurrected. In the example below, if the deactivation device is sending a superluminal signal to the bomb, then special relativity ends up in a contradiction: the bomb explodes in the frame of the train but does not in the frame of the tunnel:

    David Morin, pp. 41-42: 11.6. Train in a tunnel: "A train and a tunnel both have proper lengths L. The train moves toward the tunnel at speed v. A bomb is located at the front of the train. The bomb is designed to explode when the front of the train passes the far end of the tunnel. A deactivation sensor is located at the back of the train. When the back of the train passes the near end of the tunnel, the sensor tells the bomb to disarm itself. Does the bomb explode?"

    The solution to the train in a tunnel problem is on p. 57 in David Morin's text:

    p. 57: "Yes, the bomb explodes. This is clear in the frame of the train... (...) We can, however, also look at things in the frame of the tunnel... (...) Therefore, the deactivation device gets triggered before the front of the train passes the far end of the tunnel, so you might think that the bomb does not explode. We appear to have a paradox. The resolution to this paradox is that the deactivation device cannot instantaneously tell the bomb to deactivate itself. It takes a finite time for the signal to travel the length of the train from the sensor to the bomb. And it turns out that this transmission time makes it impossible for the deactivation signal to get to the bomb before the bomb gets to the far end of the tunnel, no matter how fast the train is moving. Let's show this. The signal has the best chance of winning this "race" if it has speed c, so let's assume this is the case..."

    Pentcho Valev

    • [deleted]

    Hi Roy, It's a good idea -- I'll ask him.

    10 days later
    • [deleted]

    By considering the correspondence between Hawking radiation and black holes quasi-normal odes, I found a time-dependent Schrodinger equation for the evolution of the black hole evaporation. Information comes out as the states of the correspondent Schroedinger wave-function can be written in terms of a unitary pure evolution matrix instead of a density matrix. Thus, they result to be pure states instead of mixed ones, see http://arxiv.org/abs/1304.1899.

    Cheers,

    Ch.

    I find interesting:

    "One proposal being studied describes black holes and their environments as a network of Hilbert spaces (a Hilbert space for any quantum mechanical system defines all possible states that the system can be in). In the conventional picture of a black hole, locality prevents the Hilbert space of the interior of the black hole from influencing the Hilbert space of the exterior of the black hole."

    For good reason. The quantum configuration space according to Bell's result cannot be mapped onto physical space without a nonlocal model. Therefore, quantum configuration space in the Hilbert space behind the event horizon is assumed simply connected and local, while the Hilbert space outside the event horizon has to be assumed multiply connected and nonlocal. If it seems paradoxical that simply connected space adjoining multiply connected space is not itself multiply connected -- that's because it *is* a paradox. Either the interior quantum configuration space maps nonlocally to the exterior space -- or the exterior space outside the event horizon is simply connected, not multiply conected, and all the space inside and outside the black hole horizon is simply connected.

    Giddings chooses nonlocality to resolve the paradox. Does he have to? If one accepts the Hilbert space model of quantum configuration space, there is no way out; the choice is singular and correct.

    In the continuous functions of quantum field theory, though, locality is everywhere. The Hilbert space doesn't work to preserve locality except as Giddings has described it: constrained by the boundary of a black hole event horizon. Giddings thinks he has found a way out. The article continues:

    "In Giddings' model, however, the Hilbert spaces can exchange information. This allows a black hole to slowly evaporate, but not before it has dumped the information contained within into the environment."

    Problem is, this does not resolve the paradox -- "the environment," the space outside the event horizon, is nonlocal. So the mathematical model is accurate; the quantum configuration space of the black hole interior maps onto physical space with a nonlocal model, consistent with Bell's result. There's a catch:

    A physical observer sufficiently far from the event horizon has no concept of "fast" or "slow" information leakage. All she sees is on the 2-dimension surface of the event horizon, and those events are suspended in time, not dynamic, with no exchange of information between events inside and outside the horizon.

    The article continues:

    "The idea is that local quantum field theory can be derived as an approximation of this more fundamental underlying structure, in the same way that non-relativistic Newtonian physics can be derived from relativistic Einsteinian physics."

    Except that quantum field theory is everywhere local, not bounded by anything except the speed of light, and nothing physical. It's also inaccurate that non-relativistic Newtonian physics derives from Einstein relativity -- Einstein's relativity is an extension of Newton's physics as Newton is an extension of Galilean relativity. There's no discontinuity, no gap where relativity doesn't apply.

    Giddings seems aware of the conundrum, and is willing to eject spacetime from the physics canon: "Spacetime is doomed." If it is, Giddings' model doesn't do the job. The paradox created by the discontinuous dumping of information from the assumed local, simply connected quantum-configured black hole into the classical simply connected space outside the horizon, tells us at least three things:

    1. There is no boundary between the quantum configuration space of the black hole, and "the environment." (All the space is simply connected.)

    2. If the quantum configuration space cannot map onto the physical space without a nonlocal model, as Bell's result avers, the quantum configuration space behind the black hole event horizon is not local and not simply connected.

    3. Nonlocality -- not spacetime -- is doomed.

    Spacetime and quantum field theory nicely coexist with a continuum of Euclidean space, generalized to n-dimension topology. We simply don't need assumptions of Hilbert spaces and nonlocality.

    Tom

    15 days later
    • [deleted]

    I just noticed that the first sentence in my first comment post is incorrect. It should read that current quantum field theory describes an equilibrium state which has evolved out of a non equilibrium state.

    Zeeya, any luck with Antony Valentini yet?

    • [deleted]

    Quantum mechanics and gravitational field theory has already been unified and tested proving that space-time and wave functions are the same phenomena.

    2 months later
    • [deleted]

    I reckon that blackholes aint that black.these are points in spacetime that spacetime fabric uses as synthensis points of matter into energy.or energy into matter-These are reconversion proccess,probably leading to a cyclic process.The blackholes can be regarded as machines.Machines that compute,they are computers in that they regulate stars in the galaxies.stars that can no longer orbit the Super massive blackholes fall into them,resynthensised and later pop out into existence in another part of the galaxy . their region of operation lies within the shwarzchild radius,and they have a fixed entropy bound by the berkenstein hawking radiation limit.Informatiom in a singularity is likely to obey the principle of conservation of energy-Energy/mass cannot be created but can only be converted from one form to another.May a photon may pop into a blackhole and emerge as a Graviton.Chan paton factors in string /M-theory also kind OF IN A SENSE,WHEn applied to loop quantum gravity display,the garviton to be the unseen superpartner to the photon.Maybe the photon is the Fermio-bosonic bearer of both EM force,and Gravity.instantenously

    7 days later

    Title -

    DIGITAL STRING THEORY DELETES QUARK STARS - EXPLAINS BLACK HOLES

    Author - Rodney Bartlett

    Abstract -

    "Eminent Princeton physicist Ed Witten famously conjectured that the true ground state of matter (in the sense of the lowest energy per particle) consists of a mixture of roughly equal numbers of up, down, and strange quarks, with enough electrons thrown in to ensure that this soup is electrically neutral." ("Is it possible for a Quark Star to exist?" by Victoria Kaspi - Astronomy magazine, June 2013) Scientists have never demonstrated this conjecture to be true, and don't have evidence that stars made of such matter ("quark stars") exist. This reminds me of something Stephen Hawking and Leonard Mlodinow wrote on p.49 of "The Grand Design" (Bantam Press, 2010) - "It is certainly possible that some alien beings with seventeen arms, infrared eyes and a habit of blowing clotted cream out their ears would make the same experimental observations that we do (regarding the existence of quarks), but describe them without quarks." In a similar way, we non-aliens with two arms, eyes that respond to visible light and no clotted cream in our ears could conceive of a quark-electron mixture forming Quark Stars only to find that it actually describes a different mixture (of binary digits) forming black holes instead of quark stars.

    E=mc^2 is referred to in the description of the conversion from gravitational energy to the "coherent, organized energy" that is matter; and also in the description of gravitational energy producing mass in black holes. We should remember that E=mc^2 appears to only be partly correct because the highest speed possible is Lightspeed. Physically speaking, it cannot be multiplied. Einstein himself proved this. The equation E=mc^2 can be considered a degenerate form of the mass-energy-momentum relation for vanishing momentum. Einstein was very well aware of this, and in later papers repetitively stressed that his mass-energy equation is strictly limited to observers co-moving with the object under study. The version of the equation applicable here may be E=m/c^2*c^2.

    Referring to the paragraph which states "hidden variables called binary digits could ... allow time travel into the past by warping a 5D hyperspace" - With a single extra dimension of astronomical size, gravity is expected to cause the solar system to collapse ("The hierarchy problem and new dimensions at a millimetre" by N. Arkani-Hamed, S. Dimopoulos, G. Dvali - Physics Letters B - Volume 429, Issues 3-4, 18 June 1998, Pages 263-272, and "Gravity in large extra dimensions" by U.S. Department of Energy - http://www.eurekalert.org/features/doe/2001-10/dbnl-gil053102.php

    However, collapse never occurs if gravity accounts for repulsion as well as attraction. It does this not only on astronomical scales but on the subatomic, too. It accounts for dark energy and familiar concepts of gravity, as well as repelling aspects of the electroweak force [such as placing two like magnetic poles together] and attracting electroweak aspects like the strong force. "Electroweak" and "strong" force can be united in that sentence because, as we'll see, gravitation and space-time are united with both the (electro)weak and strong nuclear forces.

    Content -

    "Eminent Princeton physicist Ed Witten famously conjectured that the true ground state of matter (in the sense of the lowest energy per particle) consists of a mixture of roughly equal numbers of up, down, and strange quarks, with enough electrons thrown in to ensure that this soup is electrically neutral." ("Is it possible for a Quark Star to exist?" by Victoria Kaspi - Astronomy magazine, June 2013) Scientists have never demonstrated this conjecture to be true, and don't have evidence that stars made of such matter ("quark stars") exist. And they never will demonstrate it to be true, nor will they find Quark Stars. Why? Because the binary digits of 1 and 0 are the ground state or lowest possible energy level - not only of matter but also of space-time.*

    * Suppose Albert Einstein was correct when he said gravitation plays a role in the constitution of elementary particles (in "Do Gravitational Fields Play An Essential Part In The Structure of the Elementary Particles?" - a 1919 submission to the Prussian Academy of Sciences). And suppose he was also correct when he said gravitation is the warping of space-time. Then it is logical that 1) gravitation would play a role in constitution of elementary particles, and their mass, and also in the constitution of the nuclear forces associated with those particles, and 2) the warping of space-time that produces gravity means space-time itself plays a role in the constitution of elementary particles, their mass, and the nuclear forces. Electromagnetism is not separate from space but is waves within it - the frequencies and wavelengths of these waves identify them as electromagnetic. Matter can be thought of as "coherent space" that is bound by forces0. Gravity, being united with EM** and the nuclear forces, is the ultimate physical source of all repelling and attracting (from the attractive strong nuclear force to the repulsive dark energy).

    0 This makes sense to me because of Einstein's paper saying gravitation may be involved in the structure of particles. Gravity is only the warping of space, which means space (or space-time) itself may be involved in the structure of particles. To understand how this could happen, we need to remember that higher-frequency light (in the visible part of the electromagnetic spectrum) can be converted into lower-frequency infrared light (by being absorbed as visible light and re-emitted at infrared wavelengths by interstellar gas and dust). Similarly, the gravitational waves of space could have a frequency even greater than gamma rays. When converted into lower frequencies, they'd produce all the electromagnetic wavelengths. Einstein's paper suggests these interact with gravity to form particles of matter (E=mc^2 would probably describe the conversion from gravitational energy to the "coherent, organized energy" that is matter).

    Hidden variables is an interpretation of quantum mechanics which is based on belief that the theory is incomplete (Albert Einstein is the most famous proponent of hidden variables) and it says there is an underlying reality with additional information of the quantum world. Their identification would lead to problems having exact, instead of merely probabilistic, outcomes - and could also restore a reality that exists independently of observation ("Quantum" by Manjit Kumar - Icon Books 2008, p.379) I suggest this underlying reality is the binary digits generated in 5D hyperspace^. These allow time travel by making it possible to warp space, simultaneously adding precision and flexibility to the elimination of distances and the "fitting together" of subuniverses to form a continuous, analog superuniverse.

    ^ Maybe hidden variables called binary digits could permit time travel into the future by warping positive space-time. And maybe they'd allow time travel into the past by warping a 5D hyperspace that is translated 180 degrees to space-time, and could be labelled as negative or inverted. (The space-time we live in is described by ordinary [or "real"] numbers which, when multiplied by themselves, result in positive numbers e.g. 2x2=4, and

    -2x-2 also equals 4. Inverted positive space-time becomes negative hyperspace which is described by so-called imaginary numbers that give negative results when multiplied by themselves e.g. i multiplied by itself gives -1.) The past can never be changed from what occurred, and the future can never be altered from what it will be. Both are programmed by the 1's and 0's.

    "Empty" space (according to Einstein, gravitation is the warping of this - it is not empty but is filled with the energy of binary digits) seems to be made up of what is sometimes referred to as virtual particles by physicists since the concept of virtual particles is closely related to the idea of quantum fluctuations (a quantum fluctuation is the temporary change in the amount of energy at a point in space). The production of space by BITS (BInary digiTS) necessarily means there is a change in the amount of energy at a certain point, and the word "temporary" refers to what we know as motion or time. Vacuum energy is the zero-point energy (lowest possible energy that a system may have) of all the fields (e.g. electromagnetic) in space, and is an underlying background energy that exists in space even when the space is devoid of matter. Binary digits might be substituted for the terms zero-point energy (since BITS are the ground state or lowest possible energy level) and vacuum energy (because BITS are the underlying background energy of empty space). Relativistically, space can't be mentioned without also mentioning time - the measurement of particles' properties - which can therefore also be viewed as gravitation (since "dark matter" is invisible but has gravitational influence, its existence could be achieved by ordinary matter travelling through time).

    ** Gravity would be united with EM if gravitation and electromagnetism are both products of a mathematical foundation to the universe. Let's borrow a few ideas from string theory's ideas of everything being ultimately composed of tiny, one-dimensional strings that vibrate as clockwise, standing, and counterclockwise currents in a four-dimensional looped superstring. We can visualize tiny, one dimensional binary digits of 1 and 0 (base 2 mathematics) forming currents in a Mobius loop - or in 2 Mobius loops, clockwise currents in one loop combining with counterclockwise currents in the other to form a standing current. Combination of the 2 loops' currents requires connection of the two as a four-dimensional Klein bottle (I like to refer to these ideas combining binary digits, Mobius loops and Klein bottles as "digital string theory"). This connection can be made with the infinitely-long irrational and transcendental numbers. Such an infinite connection translates - via bosons being ultimately composed of 1's and 0's depicting pi, e, √2 etc.; and fermions being given mass by bosons interacting in matter particles' "wave packets" - into an infinite number of Figure-8 Klein bottles.[0.1 and 4] Slight imperfections in the way the Mobius loops fit together determine the precise nature of the binary-digit currents (the producers of gravitational waves, electromagnetic waves, the nuclear strong force and the nuclear weak force) and thus of exact mass, charge, quantum spin, and adherence to Pauli's exclusion principle. Referring to a Bose-Einstein condensate, the slightest change in the binary-digit flow (Mobius loop orientation) would alter the way gravitation and electromagnetism interact, and the BEC could become a gas (experiments confirm that it does).

    [0.1] A four-dimensional Klein bottle's construction from binary digits would make it malleable and flexible, deleting any gap and molding its border to perfectly fit surrounding subuniverses. This Klein bottle could possibly be a figure-8 Klein bottle because the latter's similarities to a doughnut's shape describes an idea suggested by mathematics' "Poincare conjecture". The conjecture has implications for the universe's shape and says you cannot transform a doughnut shape into a sphere without ripping it. One interpretation follows: This can be viewed as subuniverses shaped like Figure-8 Klein Bottles gaining rips called wormholes when extended into the spherical spacetime that goes on forever (forming one infinite superuniverse which is often called the multiverse when subuniverses - which share the same set of physics' laws - are incorrectly called parallel universes which are wrongly claimed to each possess different laws). Picture spacetime existing on the surface of this doughnut which has rips in it. These rips provide shortcuts between points in space and time - and belong in a 5th-dimensional hyperspace. If binary digits are strings, and if digits create rips in the space of a universe that obeys fractal geometry, Steven Weinberg would be correct to imagine strings as rips in space ("What String Theory Tells Us About the Universe" by Dr. Sten Odenwald (Astronomy - April 2013, p.35). The boundaries where subuniverses meet could be called Cosmic Strings (they'd be analogous to cracks that form when water freezes into ice i.e. cosmic strings would form as subuniverses cool from their respective Big Bangs).

    [4] Each one is a "subuniverse" composing the physically infinite and eternal space-time of the universe (our own subuniverse is 13.8 billion years old). We don't have to worry about accelerating cosmic expansion - the result of more space being continually produced by binary digits - leaving our galaxy alone in space. As "dark energy" causes known galaxies to depart from view, more energy and matter - also the product of binary digits - can replace them (since the universe obeys fractal geometry, gravity is the source of repelling and attracting not only on a quantum scale but on a cosmic scale, too i.e. it accounts for dark energy - it accounts for dark matter and Kepler's laws of planetary motion, too [but that's a long explanation best left in my article "Unified Field, Relativity and Quantum Mechanics Meet String Theory, Parallel Universes, the Mathematical Universe, and TOE" - http://vixra.org/abs/1303.0218]). The Law of Conservation says neither matter nor energy can be created or destroyed (though the quantity of each can change), so a better phrase might be "binary digits recycle spacetime" (when matter changes into energy or energy becomes matter, we commonly say matter or energy has been created). As well, other expanding subuniverses can collide with ours and their galaxies enter our space to keep our galaxy company. (see "Cosmic evolution in a cyclic universe" by Paul Steinhardt and Neil Turok - Phys. Rev. D 65, 126003 (2002) [20 pages] - also see "Will Our Universe Collide With a Neighboring One?" by Zeeya Merali: http://discovermagazine.com/2009/oct/04-will-our-universe-collide-with-neighboring-one#.UY3YTKL-Gbs (from the October 2009 issue of Discover). That speaks of the "axis of evil", an unexpected alignment of cold and hot [denser and less dense] spots in the cosmic microwave background; one of the possible explanations of this being collision with another universe [other proposals are that the universe's inflation wasn't perfectly symmetrical, and that the entire universe is rotating])

    So there's no such thing as a quark-electron mixture forming Quark Stars. But there is a mixture of 1's and 0's forming matter, energy, forces, and all space-time. The form binary digits assume that most resembles stars, or masses of perhaps billions of stars, would be that part of space-time called Black Holes. Black holes aren't composed of matter but do have mass because they are meeting-places and "sinks" for the gravitational currents flowing in and between galaxies (see end of 2nd paragraph), and all this gravitational energy produces much mass via E=mc^2 (to use another description, they're whirlpools of space-time). They possess charge because the universe's mathematical foundation unites gravity/spacetime with electricity/magnetism (see above). Since it has mass, a black hole can naturally possess the 3rd property of holes viz. spin.

    ---------------------------------------------------------------------------------------------------------

    8 days later

    For a minute, let's focus on the article's references to gravitational waves - the following paragraph shows that the Newtonian theory of gravity may have things back-to-front (in physics, nothing is dogma and everything needs to be questioned - even the work of the great Isaac Newton).

    If the Sun's gravity can keep something as distant and massive as Jupiter in orbit and prevent it from flying off into space, why can't it stop a relatively puny Coronal Mass Ejection from going anywhere? Surely this means the gravity associated with the Sun doesn't emanate from its centre. It must be gravitational waves from deep space that both push against CMEs (but are too weakened - because of the energy they lose during mass formation - to stop their progress) and push them from behind (giving them a boost that's constantly moderated by the waves pushing against CMEs). The waves from deep space would both push Jupiter towards the Sun, and away from the Sun. They'd thus cancel and maintain the planet's orbit (in the short term). Over billions of years, Einstein's paper "Do Gravitational Fields Play An Essential Part In The Structure of the Elementary Particles?" - a 1919 submission to the Prussian Academy of Sciences - implies that planets gradually move farther away because gravity waves that first encounter the sun would help form the solar mass. They'd be diverted to the Sun's centre - during this journey, the increasing density would concentrate and magnify the gravitational waves. Therefore, they'd be more powerful when they emerge from the Sun's opposite side, and gradually push planets farther away. According to "Celestial Mechanics & Dynamical Astronomy", Volume 90, Issue 3-4, pp. 267-288 by Krasinsky, G. A. and Brumberg, V. A., the distance between Sun and Earth is growing by approx. 15 centimetres per century.

    See http://vixra.org/abs/1303.0218 for further explanation (in nomathematical, layman's language) regarding gravity's attraction, Kepler's laws of planetary motion, tides, orbits, dark matter.

    a month later
    • [deleted]

    The black hole information paradox is the object of my Essay for the 2013 FQXi ESSAY CONTEST "It from Bit or Bit from It", which recently won the community rating. I propose a natural solution by improving my analysis on the correspondence between Hawking radiation and black hole quasi-normal modes which obtained an honorable mention in the 2012 Gravity Research Foundation Essay Contest.

    Cheers,

    Ch.

    5 months later

    Thanks for the news, Eckard. I think Hawking is right -- no firewalls. Further though, I think that Hawking radiation is a compelling bit of evidence that we live (or at least, measure physical events) on the 3-sphere manifold, for this reason:

    That physical singularities are extinguished in finite time comports with the mathematical proof (Perelman) that Ricci flow with surgery continues the flow for infinite time on the half open interval [0,1). What this means, is that the singularity never reaches the endpoint in more than finite time, which implies that the manifold is simply connected.

    In physical terms, imagine that the observer applies "surgery" to the singularity by sending a test particle to the black hole event horizon; to the observer, the test particle appears frozen as it sends back information of its state, because the time intervals between pulses are longer and longer. The image dims, and never disappears. Information is conserved for infinite time because no information is lost in any finite time interval.

    Maybe Hawking is right that the event horizon is itself a superfluous concept. I hope so, because it is a step closer to reconciling general relativity with quantum rules. (See my post of 9 April 3013 in this article forum.)

    Tom

    Eckard,

    You've got a really good eye for important results -- Hawking's short paper, I predict, will be a game-changer.

    " ... black holes should be redefined as metastable bound states of the gravitational field. It will also mean that the CFT on the boundary of anti deSitter space will be dual to the whole anti deSitter space, and not merely the region outside the horizon."

    Same thing I said in my 9 April 2013 (not 3013! LOL) post:

    "If it seems paradoxical that simply connected space adjoining multiply connected space is not itself multiply connected -- that's because it *is* a paradox. Either the interior quantum configuration space maps nonlocally to the exterior space -- or the exterior space outside the event horizon is simply connected, not multiply conected, and all the space inside and outside the black hole horizon is simply connected."

    Tom

    I meant to write the half open interval [0, oo), not [0, 1). I was trying to express that the flow never reaches length 1.

    (Something I don't understand -- why is the "edit" function available only occasionally? Is it just my computer, or my settings?)