Hi,

I am happy that you liked my essay.

An observer in an accelerating motion will experience the so-called Rindler horizon, which has properties very similar to those of the black hole horizon. There is some evidence that whenever you take a finite part of a Rindler horizon, that part possesses entropy which, in natural units, is one-quarter of the area of that part. Such an assumption was used, in effect, by Ted Jacobson in a remarkable paper (Phys. Rev. Lett. 75 1260 (1995)), where he obtained Eintein's field equations by means of thermodynamical arguments. I think that to explain the entropic properties of the Rindler horizon, and thus the Unruh effect, one should construct the Rindler horizon, in the same way as the black hole horizon, out of discrete constituents.

Best,

Jarmo

Hi,

I guess that Newton thought at first of identifying the microscopic states with the unordered, instead of ordered strings, of the quantum numbers associated with the constituents of the event horizon of the black hole. That would have taken him to the partitions of integers. Indeed, such choice would have been justified on grounds of possible symmetries between the quantum states of the constituents of the horizon: If constituent 1 is in a quantum state identified by a quantum number n1, and constituent 2 in a state identified by a quantum number n2, the state should, according to this view, be the same as the one where the constituent 1 is in a state n2 and constituent 2 in a state n1. In other words, the constituents are, like bosons in quantum mechanics, indistinguishable.

However, it seems that there are no reasons for such symmetries. The symmetry properties of many-particle states in quantum mechanics follow from the spin-statistics theorem which, in turn, follows from the symmetries of flat spacetime. In the Planck scale of distances there are no such symmetries.

I think that there are some grounds to believe that Newton was, at least in his private relations, more congenial than it is usually thought. For instance, the well-known story that Newton laughed only once in his life, was known already during his life time. However, one of his contemporaries, when he was told this story, said that he had seen Newton laughing several times, and it was easy to make him smile. (See Gleick's book)

One of the best reasons for thinking that Newton was far from an inhuman monster is that his niece Catherine Barton, who was one of the most admired women of the London of her time, and known both for her beauty and wit, worked as Newton's house-keeper for a pretty long time. Several years later, after getting married, Ms. Barton, her husband and her little daughter lived together with Newton. I do not think such arrangements would have been possible, if Newton had been an entirely unbearable person.

Best,

Jarmo

Hi,

Sloppy reasoning is certainly not on Newton's side. In the discussion Newton said: "At macroscopic scales one may reduce the concept of distance to the concept of area." When talking about macroscopic scales he was talking about ordinary, classical spacetime at everyday (say 1 meter) scales. At such scales you may consider spacetime inside the four-simplex as a flat Minkowski spacetime, and to calculate the edge lengths and the triangle areas (in an appropriate system of coordinates) by means of the flat Minkowski metric. Thus you obtain a one-to-one relationship between the edge lengths and the triangle areas.

For a more technical account on how to reduce the concept of distance to the concept of area in four-dimensional Riemannian manifolds you may have a look at my recent pre-print in arXiv:1011.2052. More generally, one finds that equations equivalent to Eintein's field equations (the so-called Regge-Einstein equations) may be expressed in terms of the concept of area (see J. Mäkelä, Class. Quant. Grav. 17 4991 (2000) and J. Mäkelä, together with J. Mäkelä and R. Williams, Class. Quant. Grav. 18 L43 (2001)).

Best,

Jarmo

  • [deleted]

  • Post #14

The partition of integers is important in counting the number of states on a black hole horizon. The area of a black hole is composed of little quanta of areas given by a sum of integers n_i >= 0,

A = 4 π a(n_1 n_2 ... n_m)

where this total number N = n_1 n_2 ... n_m can be written according to the integer partition. Another way of thinking about this is that the string modes can exist in a distribution which is an integer parition. This is the holographic principle in action, where the event horizon or stretched horizon is composed of a "gas" of strings.

The density of states for a string is tr(w^N) , which for N = \sum_nα_{-n}α_n the string number operator. Given there are 24 string operator the computation of this generating function is tr(w^N) = f(w)^{-24} for

F(w) = prod_{n=1}^∞(1 - w^n)

This is a form of the Dedekind η-function and the remaining calculation leads to a form of the Hardy-Ramanujan approximation for the integer partitions.

The black hole in the holographic setting has a stretched horizon which is a gas of strings. If we consider the string to be the bosonic string in d = 26 then 24 correspond to the SO(24) group for the graviton plus dilaton and a gauge field. So the Newtonian insight here seems to be pointing in this direction.

Cheers LC

Hi,

thank you for your answer. From the Newton character I understand that:

(1) Unruh effect => "a sort of atomic structure of spacetime"

But how do we know about the Unruh effect? It is not an experimental fact in search of an explanation. We know about it because it follows from the principles of QFT. So we have:

(2) QFT => Unruh effect

From (1) and (2) we have

(3) QFT => "a sort of atomic structure of spacetime"

But QFT is based on continuous spacetime. So, we have something like

continuous spacetime (combined with other principles) => "a sort of atomic structure of spacetime"

This is what I don't understand.

Best regards,

Cristi

Hi Jarmo,

Actually you agree with me that you take as a hypothesis that the simplexes you use for discretization live in the "ordinary, classical spacetime", so you build upon a continuous background. I was under the impression that your Newton character claims that reality is discrete.

Just in case you think that "simplex in a riemannian space" is sufficiently flexible, let me give you a distance distribution with the property: there is no embedding of this finite metric space into any riemannian manifold. Take the simplex vertices to be O, A, B, C, D and distances between them as follows: OA=OB=OC=OD=1, AB=AC=AD=BC=BD=CD=2.

  • [deleted]

  • Post #17

The Question wether Reality is digital or analogue struck me as extremely unrealistic: those are just different ways humans can handle what they believe to be reality. Of course reality is neither and is beyond any human approach except by intuition. Therefore I think this question can only be put by extremely short-sighted people. It is not really aserious problem at all.

I know this is not the subject here, but I saw a possibility to make my point.

  • [deleted]

  • Post #18

Jarmo Matti Mäkelä,

I just printed off your essay. I will be looking for why you asked Newton about the 2nd law of thermodynamics instead of asking Clausius:

"I am not quite sure what Newton really had in mind. I think that he simply meant that at appropriate length scales the metric tensor, which depends on the quantum states of the constituents of spacetime, has a signature (-,+,+,+). The process, which brings this result may or may not have something to do with decoherence, but I am not sure about that."

I don't believe that either of them, even today, would state the quote above. I think that both of them would have first determined what thermodynamic entropy is before skipping past it to what I consider to be sidestepping the question. In other words, giving it later assigned meaning that I do not see applying to Clausius' definition.

James

    An excellent and entertaining entrance to your essay Jarmo, congratulations on your imagination and ingeniuty. I have a burning question which I've always wanted to ask Newton though, which is this:

    Q: Since he equated the ancient greek philosophy of the smallest irreducible particle, called an atom, with the motions of the planets as observed by Galileo Galilei, does he want to know what his very large unspoken logical assumption was, which has now meant that humanity has been led down the wrong scientific path?

    Ans: He assumed that the cores of the planets and sun are composed of the same everyday matter which is found on the external crust. (It's not necessarily the case and so invalidates the whole of Einstein's space-time concept imo and also invalidates the results of the Cavendish experiment to 'weigh' the Earth).

    Eckard, An excellent comment. I found Valdes-Marin's essay on 'Structure and Force' to define causality rather well.

    Jarmo, An interesting and enjoyable format.

    • [deleted]

    • Post #21

    Actually, the Hawking effect (black hole radiation) also follows from ordinary QFT, although applied in curved spacetime. Also, the derivations of the Hawking and the Unruh effects are petty similar. The point is that in both effects one may (at least in principle) measure some temperature for matter coming out of the vacuum. Since spacetime interacts with matter, one may consider the temperature of the matter as the temperature of spacetime from the point of view of an appropriate observer. If spacetime has temperature, it presumably has some internal structure, which produces that temperature. For more dwetails you may have a look at Refs. [13] and [14] in my essay.

    Best,

    Jarmo

    • [deleted]

    • Post #22

    Enormously clever! And I don't just mean the conceit of talking to Isaac Newton (with a nice paradox-saving ending). I mean the journey through algebra, combinatorics, geometry, to the 4-simplex whose edges correspond to the 10 non-redundant points of the Riemann metric tensor. That is literally where "the rubber meets the road," as they say -- where the discrete comes smack up against the continuous, where " ... the causal properties of spacetime ..." imply the existence of " ... a still unknown law of nature ..." that we hope quantizes spacetime.

    I'm betting that you're a very popular lecturer. Bravo.

    Tom

    • [deleted]

    • Post #23

    A temperature for spacetime is an interesting possibility, but one which might have some problems. To assign spacetime an internal structure with a metric means the entropy of spacetime can be very large. This is an enormous number of degrees of freedom. However, the temperature of spacetime might just be confined to horizons, which reduces the numbers of degrees of freedom, and fluctuations or dynamics outside an event horizon has a map or holographic realization on an event horizon.

    Cheers LC

    8 days later

    • [deleted]

    • Post #24

    Jarmo Matti Mäkelä,

    If you think that my point was silly and not worth your time to respond I understand. I don't agree, but I understand that my viewpoint is not the same as a professionally trained physicist. I thought it was important to point out that I think that no one knows what thermodynamic entropy is. I will move on to other essays. We have a very good quantity along with sufficient quality this year.

    James

    • [deleted]

    • Post #25

    I said that no one knows what thermodynamic entropy is. The point I am making is that Clausius' definition of entropy is the real definition of entropy. The transferrence of Boltzmann's constant onto Boltzmann's entropy was the beginning of losing sight, if we ever had it in sight, of the meaning of Boltzmann's constant. Boltmann's interpretation of entropy was the first step in ignoring Clausius' definition and finding some more simplistic substitute. I think that Newton would have first said: What is thermodynamic entropy? Perhaps there are others here who will clarify the meaning of thermodynamic entropy for me?

    James

    4 days later

    Hi Mäkelä,

    Your article is ingenious because you put in Newton autority mouth, your ideas.

    Guessing, I supose that Newton will agree more with an universal gravitation law modified than with Einstein TGR.

    If you replace Beta in the partition formula by one divided Boltzmann constant times Temperature, and as Hawking temperature in a Black Hole is proportional to the inverse of Energy the formula doesn´t diverge.

    Bests,

    JC

      • [deleted]

      • Post #27

      Hi,

      When you calculate the partition function, you must assume that the inverse temperature Beta is a constant, and therefore independent of the energy. Once after you have calculated the partition function, you will find the relationship between energy and temperature using the formula of Note 6.

      Best,

      jarmo

      • [deleted]

      • Post #28

      Your metric tensor, and your Newton analogy are very clever. I would like to understand how that contributes to the quantum limit as you propose.

      Could you elucidate this point more clearly ?

      Andy

        Dear Jarmo

        I am embarrassed to admit I seem to have a confession to make. Some time ago I heard someone fall down some stairs and went to help, he was in a bad way, mumbling is some alien language and delirious. I picked him up with his papers and bits and carried him off to hospital.

        I found afterwards I still had his papers and a strange little machine with HG Wells stamped on it.. but the poor fellow had mysteriously disappeared. I read the papers but they had no name or address. They were entitled "The Complete Quantum Theory of Gravitation". I really didn't know what to do with them. If I'd known they were yours I'd have put them in a time capsule...

        As it was I thought I must try to put them and the machine to good use. They both rather scared me! I was worried about causality. What I decided to do was pop forward the 300 years the machine was set to and check if all was ok and we'd managed to find the theory.

        Imagine my horror when I discovered it was my fault that we hadn't, ..and that science seemed to have gone backwards in time! I didn't have a clue if you were around or where to find you. The world was strange to me but I managed to survive by popping back and making the odd bet and investment, and have had fun directing some great blockbuster films about a land I visited using your... well never mind about that. I spent a few years checking the theory and it seemed spot on so I eventually decided it best to write it up in my own way and give it to the world. - the complete atomic structure of space time, unifying SR, GR and QM.

        But apparently the world doesn't understand it! As only you will I hope you'll read my essay, gently slipping in that the sub space-time medium condenses into an atomic structure to produce space-time itself. The quantum states are ions, re-polarised by black holes, and distributed as plasma in such a way that the every day notion of time and causality are recovered. Actually I think that wording is yours, I mean Sir Isaac's, not mine!

        It's been a difficult time, but a massive relief to have found it's owner. I did tentatively refer to it in last years contest, and touch on it quite lightly in my current essay languishing some way below yours (2020 Vision..) But it's better covered in a number of preprints I can pass to you. Here are some recent ones; http://vixra.org/abs/1001.0010 and http://vixra.org/abs/1102.0016

        I hope you're able to finally confirm I'm not totally insane. The only way I've managed to hide from the men in white coats so far is to be secretly the 3rd richest man in the world.

        I pass it all to you, in Isaac's name. - Do you think I should withdraw or just give you both credits and citations in the paper currently accepted for review!

        With massive relief

        Peter Jackson

        PS. You have a well earned 10 from me. But please do reply asap.

          • [deleted]

          • Post #30

          Dear Jarmo,

          I must agree with many of your other readers, in that you have given a very creative presentation. The introduction and conclusion are very well and clearly written and are enjoyable. Unfortunately, not being a physics specialist, I found the middle of the essay too technical to follow. Also as I have limited mathematical skill I can not judge if what you have said is technically correct and rigorously argued. I will have to leave that judgment to others.I can see some lively debate and much interest in your comments thread. It is certainly topical and foundational.I suspect it is far more interesting and goes far deeper than I can appreciate and understand.I wish you good luck, Georgina.