Dear Ralph,

Thanks for your kind words and also for appreciating my Essay. Do not worry, your question is indeed very reasonable and it permits me to give more explanations. The analogy that you cited is very profound although we must be very careful in discussing it. I try to explain this issue. It is a general conviction that black holes should result in highly excited states representing both the "hydrogen atom" and the "quasi-thermal emission" in quantum gravity. At the present time, we do not yet have a full theory of quantum gravity, thus, we have to be content with the semi-classical approximation. In fact, as for large n Bohr's correspondence principle should hold, such a semi-classical description should be adequate. In this framework, my black hole model is somewhat similar to the semi-classical Bohr's model of the structure of a hydrogen atom. In fact, during a quantum jump a discrete amount of energy is radiated and for large values of the principal quantum number n the analysis becomes independent from the other quantum numbers. In a certain sense, QNMs represent the "electron" which jumps from a level to another one and the absolute value of the QNMs frequencies represent the energy "shells". In Bohr's model, electrons can only gain and lose energy by jumping from one allowed energy shell to another, absorbing or emitting radiation with an energy difference of the levels according to the Planck relation E=hf, where h is the Planck constant and f the transition frequency. In my black hole model, QNMs can only gain and lose energy by jumping from one allowed energy shell to another, absorbing or emitting radiation (emitted radiation is given by Hawking quanta) with an energy difference of the levels according to eq. (15) in my Essay. On the other hand, Bohr model is an approximated model of the hydrogen atom with respect to the valence shell atom model of full quantum mechanics. In the same way, my model should be an approximated model of the emitting black hole with respect to the definitive, but at the present time unknown, model of full quantum gravity theory. In any case, this analogy looks intriguing. Let me know if my replies are OK or if you need more details.

Thanks again for your important question.

Cheers,

Ch.

Dear Prof. Corda,

Its my privilege to comment on your essay.

Regarding your arguments on BH information restorations, I like to add some thing here which may support yours, and equally I invite you in my essay http://fqxi.org/community/forum/topic/1855)to have some comments.

It is well known that basic problem for not receiving any information from the opposite end of an event horizon of BH is due to its superluminal range of escape velocity. Special Relativity restricts the superluminal speeds of particles to emerge out of that event horizon. If in any way we can get the superluminal speeds to overtake that range of high escape velocities around the BH, the information sharing issues with the BH by penetrating the event horizon to the outside world could be instantly resolved.

That is why I sincerely like to invite you in my essay, particularly I request in Eqs.(28)-(31)and paragraph no.5.2 there, where we can get such superluminal speeds of particles in the same frames of special relativity (which is now we have respect to c).

With my regards

Dipak

    Dear Amos,

    Thanks for your kind comments.

    Actually, we do not know if the final result of the black hole evaporation is a puddle (having dimensions of order of the Planck scale) or if, instead, the black hole completely evaporates. In fact, avoiding evaporation is usually claimed when one invokes the Generalized Uncertainly Principle which should stop evaporation at the Planck scale. In any case, in quantum physics the complete information in a system is encoded in the wave function representing the quantum state of the system. i.e. in eq. (36) of my Essay. The wave function (36) represents a pure final state rather than a mixed final state. Thus, information is surely preserved because the evaporation process rigorously obeys the laws of quantum mechanics.

    In general, it is possible to apply method of analysis concerning black holes to the whole universe, but one has to be very careful because, although they have similar features, a black hole is different from the whole universe. In General Relativity the analogy depends on the issue that in both universe and black holes the ratio between mass and radius is of order 1 in natural units. I did not worked on information leaving a universe, thus I do not know if the analogy works also in this case.

    I do not think that in General Relativity, although gravity is king at long distances, everything loops back on itself, if you will. In fact, for example, particles can be attracted by others gravitational fields. We see the light of the sun and of distant stars on Earth. Thus, those photons do not loop back on them-self. Instead, they arrive to us. Then, one does not need quantum mechanics in order to have light rays going indefinitely outwards.

    The definition of singularity in science is not simple. From an intuitive point of view, a singularity is visualized as a point at which a particular mathematical object is not defined. For example, the function 1/z is not defined in z=0. A rigorous definition of singularity in the gravitational collapse can be obtained following B. G. Schmidt, Gen. Rel. Grav. 1, 269-280 (1971). For example, in standard Schwarzschild coordinates one tells that in the internal geometry all time-like radial geodesics of the collapsing star terminate after a lapse of finite proper time in the termination point r = 0 and it is impossible to extend the internal space-time manifold beyond that termination point. I suggest you to search further details in the book C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W. H. Feeman and Company (1973).

    Although in General Relativity the speed of light can be different for different coordinates, an event horizon is defined as "the point of no return", i.e. photons emitted from beyond the horizon can never reach an outside observer.

    What do you mean with "weird 1/4 area to volume rule"? The famous formula of Bekenstein-Hawking entropy claims that the entropy of a black hole is 1/4 of its area in Planck units, but this is not connected with the information paradox.

    I regret for your statements claiming that "For all the math here, it seems a little one-sided, the other side maybe not even existing!" and that "I read more abstract math here than pictorial verbiage, which vaguely results because some maths just don't condense to easy imaging". In my Essay I used mathematics and physics on the same level of university studies on quantum mechanics. I read that you are currently an undergraduate physics student. Maybe, you have not yet completed your studies on quantum mechanics. On the other hand, rules of FQXi request verbatim that the Essay must be "Accessible to a diverse, well-educated but non-specialist audience, aiming in the range between the level of Scientific American and a review article in Science or Nature." I think that readers of Scientific American and review articles in Science or Nature should know mathematics and physics on the same level of university studies on quantum mechanics. Of course, this is not a criticism for you! In fact, I invite you to read again my Essay when you will end your graduate studies. I am sure that you will completely understand it.

    Finally, black holes evaporate because of quantum effects near the event horizon. Hawking provided a theoretical argument for this effect improving ideas by Parker, Zeldovich and Starobinski. In Hawking's original computation black hole radiation was a perfect black body radiation (purely thermal) which has a specific spectrum and intensity that depends only on the temperature of the body. By using arguments of energy conservation, Parikh and Wilczek have instead shown that the black hole emission is not exactly the one of a perfect black body. Hawking claimed that because of the strict thermality, information should be loss in black hole evaporation. As the radiation is, instead, not exactly the one of a perfect black body, Hawking's claim breaks down.

    I hope to have partially clarified your mix up.

    Cheers,

    Ch.

    Thanks Don.

    As I told you in your page Essay, I agree with Siri that the logic is OK. It should be OK for "Russell and Whitehead" too. Definitively, our viewpoints are the same!

    Cheers,

    Ch.

    Dear Branko,

    Thanks for your kind words. Yes, I agree that paradoxes are sometimes solved in wrong way. I will be pleasured to read, comment and rate your Essay in next days. Do not worry, I will not attach importance to the two errors in typing in formulas.

    Thanks again!

    Cheers,

    Ch.

    Dear Professor Corda

    Richard Feynman in his Nobel Acceptance Speech (http://www.nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html)

    said: "It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but with a little mathematical fiddling you can show the relationship. And example of this is the Schrodinger equation and the Heisenberg formulation of quantum mechanics. I don't know why that is - it remains a mystery, but it was something I learned from experience. There is always another way to say the same thing that doesn't look at all like the way you said it before. I don't know what the reason for this is. I think it is somehow a representation of the simplicity of nature."

    I too believe in the simplicity of nature, and I am glad that Richard Feynman, a Nobel-winning famous physicist, also believe in the same thing I do, but I had come to my belief long before I knew about that particular statement.

    The belief that "Nature is simple" is however being expressed differently in my essay "Analogical Engine" linked to http://fqxi.org/community/forum/topic/1865 .

    Specifically though, I said "Planck constant is the Mother of All Dualities" and I put it schematically as: wave-particle ~ quantum-classical ~ gene-protein ~ analogy- reasoning ~ linear-nonlinear ~ connected-notconnected ~ computable-notcomputable ~ mind-body ~ Bit-It ~ variation-selection ~ freedom-determinism ... and so on.

    Taken two at a time, it can be read as "what quantum is to classical" is similar to (~) "what wave is to particle." You can choose any two from among the multitudes that can be found in our discourses.

    I could have put Schrodinger wave ontology-Heisenberg particle ontology duality in the list had it comes to my mind!

    Since "Nature is Analogical", we are free to probe nature in so many different ways. And you have touched some corners of it.

    With regards

    Than Tin

      Christian,

      Can I offer a completely different scenario?

      Energy manifests information. Information defines energy. Medium/message. Since energy is conserved, in order to create new information, old information is erased. This is the "arrow of time." It is not a vector from past to future, but the dynamic process in which future becomes past. Potential becomes actual. Tomorrow becomes yesterday. The cat lives or dies based on what physically happens. Clocks run at different rates because they are individual processes.

      To the extent black holes actually physically exist, they do eject out enormous amounts of radiation. Either the quasar jets out the poles of galaxies, or binary stars going supernova. The energy is conserved, in a cosmic convection cycle of expanding radiation, contracting mass and the structure is consumed, radiated back out.

      Information can only do what energy allows it to do. Reductionism is intellectually essential, but much is lost/radiated and when you finally reach the point of all message and no medium, it is delusional illusion. Not a black hole of infinite density, but simply the eye of the storm. There is no gravity at the center, only pressure. There is no platonic realm. Unmanifest structure is a multiple of zero. A dimensionless point is no more real than a dimensionless apple.

      The discipline of physics seems all encompassing and unstoppable, but it is only a matter of which of its many fudges and patches proves to be the Achilles heel.

      Rant over. Pardon my impropriety, but I just had to get that out.

      Regards,

      John

      Dear Dipak,

      Thanks for your kind comments. Actually, the quantum tunnelling framework which supports the emission of Hawking quanta can overtake the range of high escape velocities around the BH. This is the argument that, not only myself, but various other authors usually invoke. In any case, it will be my pleasure to read, comment and rate your Essay in next days.

      Cheers,

      Ch.

      Ch,

      Thanks for the detailed response, the length of which is quite impressive! I'll get that Gravitation book.

      Amos.

      My pleasure dear Amos. If you think to insert gravitation in your future academic studies be free to contact me for a potential collaboration. Also, I am going to read your Essay in next days.

      Cheers,

      Ch.

      Thanks for your kind comments, dear Than. I am honoured that you think that I have touched some corners of Nature. Like you, I have a strong admiration for Richard Feynman, and I completely agree with your and his point of view on the simplicity of nature. It will be my pleasure to read, comment and score your Essay in next days.

      Cheers,

      Ch.

      Dear John,

      Thank for offering your completely different scenario. You could be interested on a framework in which all the mass-energy has been lost/radiated during the gravitational collapse developed by the Indian physicist Abhas Mitra. Dr. Mitra published such a scenario in various important mainstream peer reviewed journals. On the other hand, John Baez claimed that "Mitra's work is based on some serious misunderstandings of this subject, and is full of mistakes."

      Cheers,

      Ch.

        Dear professor Christian Corda:

        The only things about of which I wrote, are the positive ones, my find is of not use for me, practically its only use are for theoretical physicists. I make it really short, about the subject can be written thousands pages, but this is the nut of it, this are the things were everybody get confuse and confuse everybody else. You know people think that because for two thousand years the problem of "time" was not solve, can't be solve, this is not true as you can see here.

        Anthropologist tell us men measured "time" since 30.000 years ago basically recording celestial bodies "constant" "motion" which was used as a system of measurement by people, for practical daily uses, as agriculture, hunting seasons, commerce.

        Civilizations learn from primitive men its use, but with it, acquired the incognita of what they were measuring?, that now days we call the experimental meaning of "time", since their beginnings they ask themselves for that and still do, but in that epoch, they were especially interested in better and more precise ways of measurement.

        Long 20.000 years after, science began a very slow beginning and as part of natural science was born a primitive physics no more than 2000 years ago, since the beginning physics included "time" a basic part, a foundational one, during those millenniums there was curiosity about its meaning but not a serious need to know the experimental one. The real need for physics was to improve precision of measurement.

        All the above is the most probable.

        Now on there is no hypothesis it is only an explanation using only proved facts.

        Centuries old proved facts, like earth rotation and its consequence the day, also a prove of an earth complete rotation of "constant" "motion", which last from one sunrise to the next one.

        This period of "constant" "motion" the day, it was divided in equal parts in 24 equal hours by Egyptians, the hour in 60 equal minutes and the minute in 60 equal seconds by Sumerians.

        If "motion" does not have those characteristics, still could be a variable but can't be divided in equal parts, so can't be use to measure any other "motion".

        For practical reasons men copy the "constant" "uniform" celestial bodies "motion" designing clocks. When the hour hand rotate twice around the clock dial, it represents an earth complete "constant" "uniform" rotation "motion".

        Looking at the clock dial we will know at what hour of the day we are which is the same that knowing, on what part of the earth complete "constant" rotation "motion" we are.

        With the clock dial hands "constant" "motion", representing earth rotation motion, comparatively we measure all "no constant" "motions" which are part of every change or transformation occurring to us, or everything around us.

        The real definition of Duration: Is the period of change or transformation allowed by "motion" and limited by men.

        Change, transformation and motion, none of the concepts can exist without the other two.

        Above it is proved that what theoretical physicists needed for the last 50 years, the experimental meaning of "time" it is "motion".

        That with a "constant" "uniform", "regular" "motion" we measure all the rest of "motions" without those characteristics.

        Also it is proved that "motion" is a quality or property of every physical existing thing, and as such can relate to every physical existing thing.

        Within those "motion" can be affected and affect, gravity, inertia, mass

        Etc. etc..............

        Héctor

        Christian,

        Thank you for the response and the links. I have to say my thoughts on black holes are arrived at tangentially, as a consequence of the conclusion that gravity already balances expansion in a concurrent, convective-type cycle. (Sort of as if the rubber sheet analogy were placed over water and wherever the ball is not, the sheet is pushed up by an equal amount.)

        Rather than take up your thread with the entire argument, I will point out what I see as a conceptual fallacy incorporated into the cosmological model; 1)According to Einstein, "Space is what you measure with a ruler." 2) Space expands. 3) This will eventually result on distant galaxies disappearing, as the light takes ever longer to reach us.

        So in this description of expanding space as measured in terms of lightspeed, which is the denominator? Presumably it is lightspeed, yet that would mean space as measured by the ruler of C is not expanding. But if the expanding space were the denominator, what metric would provide and sustain a stable speed of light? If C is the denominator, then it would be an expansion IN space, not OF space and that would mean we are at the center of the universe.

        Of course this perception would be quite reasonable if redshift is due to an optical effect, but that would mean light does not travel as a point particle, but is only absorbed as one.

        So; How does intergalactic space expand, yet our most basic metric of it remain constant?

        Dear John,

        Sorry, but I do not see your conceptual fallacy incorporated into the cosmological model. Yes, distant galaxies disappears as the light takes ever longer to reach us, but this merely implies that the universe has an horizon. Where is the problem? Stable speed of light enters in the well known Friedmann - Lemaître - Robertson - Walker metric which evolution is governed by Einstein Field Equation. This does not imply that we are at the center of the universe. It is exactly the opposite which works. Friedmann - Lemaître - Robertson - Walker metric is founded on the Cosmological Principle which states that "Viewed on a sufficiently large scale, the properties of the Universe are the same for all observers". When intergalactic space expands, our most basic metric of it does not remain constant. It is only the curvature which remains constant, the universe scale factor increases.

        Cheers,

        Ch.

        Dear Héctor,

        Thanks for your historical reconstruction on "the concept of "time". Notice that, when you claim that "men measured "time" since 30.000 years ago" you are using "time", i.e. the 30.000 years. I agree with your positivism to write only positive things. You can easily convert "time" in "motion" by using the speed of light c. In fact, in natural units "time" and space ("motion" is space covered) have the same unit. I have no doubts that "motion" is a quality or property of every physical existing thing, and as such can relate to every physical existing thing. Uncertainty Principle is a proof of these statements.

        Cheers,

        Ch.

        Christian,

        If the speed of light is not our most basic metric, why is it used as the denominator?

        Consider normal doppler effect; The train moving away doesn't stretch the tracks, or the spacing of the telegraph poles, or the length of the train. Only the distance between it and the person hearing the whistle changes. So say the train goes from 1 telegraph pole away, to 10. That would be 1/1 to 10/1.

        Now consider galaxies moving apart, as the universe expands; Two galaxies go from x lightyears apart, to 2x lightyears apart. x is the denominator, so the distance goes from 1/x to 2/x. Like the length between telegraph poles, a lightyear doesn't get longer. Like the train moving away, these galaxies grow further apart in terms of lightyears. How is it that we can have this constant measure as a denominator, when the very fabric of space is expanding?

        (A lightyear is approx. a trillion miles, so the distance in miles increases as well.)

        Regards,

        John

        Hello Dr. Corda -

        On the subject of abstractions, I think we are essentially in agreement. My meaning is that we need to rigorously question our assumptions - know which ones we are employing, and justify their inclusion in our emerging paradigm; only then can we attempt to proceed logically and empirically.

        In short, before we can answer ultimate questions, we must define the foundational assumptions.

        I think my reason for saying this will become clearer once you've had a chance to look over my essay - something I am very much looking forward to!

        Best regards,

        John.

        Wow ... I found myself engrossed in that vitriolic exchange of 2004. Sounds exactly like the mess we got into with Joy Christian's result, and for the same reasons. I mean, it appears that when someone -- anyone, it seems -- proposes a purely classical framework (which both Mitra's and Christian's are), it's like waving a red cape in the face of a bull.

        I respect John Baez's knowledge and skill in mathematical physics; however, I think he may have been a bit unfair to say things like, "Starting from the solution which describes a black hole of mass m, he attempts by a calculation to show that m = 0. It's a bit like taking an arbitrary prime number and proving that it must equal 37."

        However, Mitra didn't assume a black hole of mass M (it should be M in this context). So Baez's criticism amounts to saying that Mitra made an erroneous assumption, not that the calculation is wrong. Nevertheless, Baez then uses what he regards as the erroneous assumption to show that Mitra makes the elementary mathematical error of dividing by zero.

        Baez quotes Mitra: "For the benefit of the serious readers, I give below the essence of my proof: In Lemaitre coordinates, the radial geodesic (angular part=0), the metric of a test particle around a BH is ds^2 = dt^2 - g_rr dr^2 (1)"

        To which Baez replies, " ... the phrase 'the metric of a test particle' makes no sense. The metric is something on spacetime, and it applies to all particles moving in spacetime, so one never speaks of the metric 'of a test particle'."

        While this is exactly true, I think it's ungenerous and a bit condescending. I get the impression that Mitra is using the word metric to mean "trajectory." This would have to be so, in order to fix time coordinates for the endpoints of a geodesic. That is, a particle trajectory on the event horizon traces a metric; it doesn't define a metric. My suspicions is confirmed later on when John says:

        "Somehow M = 0 has popped out. It's popped out because in equation (10) he gets ds^2 = 0 at R = 2M, 'following the radial geodesic'. He's not very clear about (what) that means ...

        (Well, it's quite clear to me that Mitra means the metric trace of a massless particle, for which t = 0 both at the event horizon *and* for a distant observer.)

        " ... but interpreting him generously I'd say he's concluding the change in proper time vanishes for a test particle freely falling into a black hole as it crosses the horizon."

        (Mitra's test particle never crosses the horizon.)

        "This would indeed be a contradition since general relativity (quoting Mitra here) 'demands that the geodesic must remain timelike there and we should have had ds^2 > 0.'

        "So, his mistake may lie in his derivation of ds^2 = 0. Where does this come from? He says it comes from ds^2 = dt^2 - dr^2 at R=R_g (6) and (dr/dt)^2 = 1 at R = R_g (9) I've already said I see no flaw in (6) so probably the flaw is in (9). And indeed, (9) is false for a test particle freely falling into the black hole: in LeMaitre coordinates, r is constant for such a particle, so dr = 0 contradicting (9).

        "The rest is a mopping-up operation ..."

        Only with John's assumptions. And those assumptions make his criticism look suspiciously like a straw man argument to me, another common characteristic of the Joy Christian controversy. Ah, well.

        Professor Corda, if you find this diatribe irrelevant to your forum discussion, please just pull the plug on it.

        All best,

        Tom

        Christian,

        "the curvature which remains constant, the universe scale factor increases."

        Yes, the curvature remains constant, much as a circle is always 360 degrees, but you have this scale that allows you to say it is larger, or smaller and that scale is the speed of light.

        If the universe was the scale and "the light takes ever longer to reach us," wouldn't it be necessary to say it must be the light that is slowing down, if space is the universe?

        The scale is the stable base line, not what is changing. If you need to say the speed of light is constant and the universe expands relative to this constant, it just seems to me that you have already defined space in terms of the speed of light.