Essay Abstract
A report of a discussion with Isaac Newton.
Author Bio
I received my PhD in theoretial physics from the University of Jyväskylä, Finland, in 1994, and did a post-doc in the Department of Applied Mathematics and Theoretical Physics of the University of Cambridge during the years 1995-1996. Since the year 2000 I have worked as a Senior Lecturer of mathematics and physics in the Vaasa University of Applied Sciences located in Vaasa, Finland.
Very enjoyable -- a most clever and creative way to approach this topic. I have a question: Toward the very end, Newton proposes "a still unknown law of nature," related to the 2nd law of thermodynamics, which recovers time and causality at macroscopic scales. This sounds a lot like decoherence. Is it?
Dear Jarmo,
Nice! I said first, but then I remembered about "Conversation with a mummy" by E.A. Poe.
Otherwise, there is clearly something about quantization of area, but to me the link with the distance and the 4 simplex is just numerology.
Dear Jarmo,
I enjoyed reading this beautiful essay. The story flows nicely and the explanations are clear - I particularly liked that of the black hole entropy. I agree that the BH entropy may be the major key to the unification between general relativity and quantum theory.
I do have one question. The Newton character said: "When reading Wald [11] I learned that an observer in a uniformly accelerating motion will detect thermal radiation of particles even when all inertial observers detect a vacuum". We know about the Unruh effect because is predicted by the standard (based on continuous spacetime) QFT, applied to an accelerated observer. Then the Newton character said "to me it seems likely that to explain effects like this one must assume a sort of atomic structure of spacetime". Why would Newton use an unobserved effect, which is predicted by a theory, as a supporting argument for a rival theory?
Best regards,
Cristi
Newton was convinced: God is more important than physics or mathematics. God winds up the big clock again and again. Time and space time are absolute.
Leibniz argued that this does restrict God too much. God is not obliged to wind up the big clock, i.e. the universe, which was created by himself. Samuel Clarke who spoke on behalf of Newton in their correspondence commented: This makes God redundant and puts Leibniz at the brink of atheism.
I reminded strong Newtons religiosity in order to stress that Newton would hardly have said "... it would be a strong violation of causality ...". To him causality was given by God.
Who knows any case where causality is violated except for phantasm of mediocre physicists? How to manage violation of causality? Plagiarism is not a violation of causality.
Eckard Blumschein
Jarmo, Thanks for a very enjoyable read. The technical discussions were too specialized for me, but the idea of Newton surveying modern physics is delicious. You could have made him say "I told you so" about General Relativity's conclusion that starlight bends slightly around the sun. You might have - in your dream - taken as a present a graphing calculator which I am sure he would have found extremely amazing and useful! You did not mention how Newton knew the meaning of the phrase "analog or digital" - An interesting discussion of how the words got their current meaning only in 1940 is here Cheers. Vladimir
Jarmo,
Your paper is interesting and it suggests a more complete form of this theory is to consider the set of partitions of the integers. G.H. Hardy and Ramanujan did some work along these lines.
Your depiction of Newton has him a bit more congenial than I would have imagined. He was a pretty disagreeable and barbed, and taken to unamicable relationships with people. In particular Robert Hooke became his main nemesis. He also ran the mint in a pretty draconian way, and was a supporter of the repression of Ireland after the Jacobites were defeated at the Boyne. That little episode killed off about 100,000 Irish. He was an odd man, for he was clearly one of the greatest geniuses our somewhat tawdry species has produced, yet morally he was also a bit on the order of one of the more craven characters from a Dicken's novel.
Cheers LC
Hi,
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.
Best,
Jarmo
Hi,
I have not read Poe's story before reading your post. The four-simplex really has as 10 edges and 10 triangles. You may try this by yourself: Write the numbers 1,2,3,4 and 5 on the paper and make a list of all different pairs and triples of numbers. You will find 10 pairs and 10 triples. The areas of the 10 triangles of the four-simplex are functions of the 10 edge lengths of the four-simplex. When you write the triangle areas in terms of the edge lengths, you will get a system of 10 equations for 10 unknowns (edge lengths), and you may solve edge lengths from your equations in terms of areas.
Best,
Jarmo
Really this is not a big deal. You mean euclidean area and length, so you are going to use this in the hypothesis, along with the "amazing" (5 choose 2) = (5 choose 3), for what? For proving that in fact in this very particular euclidean space there is a formula which expresses some lenghts in term of some areas.
The problem is that the formula you are going to use is true only in euclidean space of dimension 4 (maybe there is some extension of it to some configurations of points in Hilbert spaces). Therefore you already put R^4 (with a scalar product) in the hypotheses and you try to justify that (euclidean) area is somehow quantized.
Newton was a great mathematician, there are no reasons to put in his mouth such sloppy reasonings.
Best, Marius
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
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.
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.
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.