Dear Shawn,

This is a very interesting idea you are proposing. I have two questions.

1. Have you thought about inflation in the early universe in this context? It immediately comes to mind after reading your essay as something your ideas could possibly explain. Some physicists have proposed an as-of-yet unidentified "inflaton field" that drove inflation but seems to be absent today. From the perspective of your idea, one could hypothesize that the decrease in energy density (and hence gravitational interaction) "turned off" or "damped out" the inflaton field after the initial expansion.

2. There is a theoretical energy limit (called the GZK limit) for cosmic rays from distant sources, based on the hypothesized interaction of the particles with background radiation along their trajectories. However, cosmic rays have been observed with energies above this limit. This seems to be another piece of data your idea could possibly explain: along most of their trajectories, cosmic rays from distant sources would be in regions of low gravitational interaction, and hence would not interact with the background radiation. Thus, they would preserve more of their energy than conventionally predicted. Have you thought about this possibility?

If you like, take a look at my essay On the Foundational Assumptions of Modern Physics. It has a completely different viewpoint, but it's possible you may get some interesting ideas from it as I did from yours. Take care,

Ben Dribus

    Hi Ben,

    It's really hard to say for me whether the model is just a toy or would have a desirable effect on the real phyiscs. I have tried to think of how things would have been like near the Big Bang in the context of the model, but I haven't any crystal clear thoughts on the matter.

    It was actually the GZK limit that got me started on this. It started out as a numerology (centred around the energy scale 10^19 eV) and then came the creation and annihilation idea a while later. I have thought about how this would affect the propagation of cosmic rays, but again, nothing crystal clear.

    To be honest, it's been a month now since I thought about the whole thing for more than a couple of minutes at a time. I'm bored with it.

    At worst, it is a toy model that can make for a possibly useful video game idea.

    - Shawn

    Oh yeah... do notice how the data bits x per message is always an integer, and that y and r are not necessarily so unless the number of messages is a power of two and all messages are equiprobable.

    Notice that when you analyze the classical binary messages that the mean radial distance increases, but the standard deviation decreases.

    It does kind of seem, at first glance, like a "spherization" of the positions in the message (state) space.

    The C++ code is attached.

    In the following list, the "max message size" is the number of bits per message. So the first in the list analyzes just the two 1-bit messages, the second in the list analyzes the four 2-bit messages, etc, etc. I have to redo the code so that it does not store the radii in an array, since it inevitably runs out of memory on this system when the "max message size" gets to about 25. I can just analyze the radii twice; the first time to get the mean, the second time to get the standard deviation.

    The n-bit messages of course "live" in a discrete n-dimensional space that has only two positions (0, 1) for each dimension.

    max message size: 1

    min radius: 0

    max radius: 1

    mean radius: 0.5 -/+ 0.5

    max message size: 2

    min radius: 0

    max radius: 1.41421

    mean radius: 0.853553 -/+ 0.521005

    max message size: 3

    min radius: 0

    max radius: 1.73205

    mean radius: 1.12184 -/+ 0.491409

    max message size: 4

    min radius: 0

    max radius: 2

    mean radius: 1.33834 -/+ 0.456989

    max message size: 5

    min radius: 0

    max radius: 2.23607

    mean radius: 1.52183 -/+ 0.428974

    max message size: 6

    min radius: 0

    max radius: 2.44949

    mean radius: 1.68313 -/+ 0.408759

    max message size: 7

    min radius: 0

    max radius: 2.64575

    mean radius: 1.82867 -/+ 0.394921

    max message size: 8

    min radius: 0

    max radius: 2.82843

    mean radius: 1.96247 -/+ 0.385622

    max message size: 9

    min radius: 0

    max radius: 3

    mean radius: 2.08713 -/+ 0.379348

    max message size: 10

    min radius: 0

    max radius: 3.16228

    mean radius: 2.20439 -/+ 0.37503

    max message size: 11

    min radius: 0

    max radius: 3.31662

    mean radius: 2.31552 -/+ 0.371968

    max message size: 12

    min radius: 0

    max radius: 3.4641

    mean radius: 2.42143 -/+ 0.369719

    max message size: 13

    min radius: 0

    max radius: 3.60555

    mean radius: 2.52281 -/+ 0.368006

    max message size: 14

    min radius: 0

    max radius: 3.74166

    mean radius: 2.62022 -/+ 0.366657

    max message size: 15

    min radius: 0

    max radius: 3.87298

    mean radius: 2.7141 -/+ 0.365562

    max message size: 16

    min radius: 0

    max radius: 4

    mean radius: 2.80482 -/+ 0.364652

    max message size: 17

    min radius: 0

    max radius: 4.12311

    mean radius: 2.89268 -/+ 0.36388

    max message size: 18

    min radius: 0

    max radius: 4.24264

    mean radius: 2.97793 -/+ 0.363215

    max message size: 19

    min radius: 0

    max radius: 4.3589

    mean radius: 3.0608 -/+ 0.362634

    max message size: 20

    min radius: 0

    max radius: 4.47214

    mean radius: 3.14148 -/+ 0.362121

    max message size: 21

    min radius: 0

    max radius: 4.58258

    mean radius: 3.22012 -/+ 0.361665

    max message size: 22

    min radius: 0

    max radius: 4.69042

    mean radius: 3.29689 -/+ 0.361256

    max message size: 23

    min radius: 0

    max radius: 4.79583

    mean radius: 3.37191 -/+ 0.360886

    max message size: 24

    min radius: 0

    max radius: 4.89898

    mean radius: 3.44529 -/+ 0.360552

    max message size: 25

    min radius: 0

    max radius: 5

    mean radius: 3.51713 -/+ 0.360246Attachment #1: radius.txt

      Attached is an image of the radii for different spaces of n bits, where n = 10, 18, 26. The radii are normalized by sqrt(n), binned, and then drawn. The lighter coloured lines (bins) have more messages in them.

      The radii slowly creep together as n increases.Attachment #1: shell.jpg

      ... This is all equivalent to saying:

      The set of n-bit messages contains 2^n messages total. The bits in each message are Cartesian coordinates in the message space, and the radial distance squared of each message (the count of the bits with the value 1 in the message) can be any one of the integer values 0 through n.

      The number of n-bit messages with the radial distance squared x is

      f(n, x) = n! / [x! (n - x)!]

      which is a way of counting the combinations (see binomial coefficient).

      Summing the various f(n, x) for x from 0 through n, the grand total is 2^n.

      5 days later

      • [deleted]

      • Post #54

      Hello,

      Thanks for your comment. Have a good day.

      - Shawn

      After studying about 250 essays in this contest, I realize now, how can I assess the level of each submitted work. Accordingly, I rated some essays, including yours.

      Cood luck.

      Sergey Fedosin

        • [deleted]

        • Post #56

        Hi Sergey,

        Thanks for your rating, whatever it may have been. Good luck in the contest as well.

        - Shawn

        If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is [math]R_1 [/math] and [math]N_1 [/math] was the quantity of people which gave you ratings. Then you have [math]S_1=R_1 N_1 [/math] of points. After it anyone give you [math]dS [/math] of points so you have [math]S_2=S_1+ dS [/math] of points and [math]N_2=N_1+1 [/math] is the common quantity of the people which gave you ratings. At the same time you will have [math]S_2=R_2 N_2 [/math] of points. From here, if you want to be R2 > R1 there must be: [math]S_2/ N_2>S_1/ N_1 [/math] or [math] (S_1+ dS) / (N_1+1) >S_1/ N_1 [/math] or [math] dS >S_1/ N_1 =R_1[/math] In other words if you want to increase rating of anyone you must give him more points [math]dS [/math] then the participant`s rating [math]R_1 [/math] was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

        Sergey Fedosin

          • [deleted]

          • Post #58

          Okay!

          "Van der Pauw sheet resistance and the Schwarzschild black hole"

          The entropy of the Schwarzschild black hole is considered in terms of Shannon's

          mathematical theory of communication and van der Pauw's theory of sheet resistance.Attachment #1: 1210.0021v1.pdf

          • [deleted]

          • Post #61

          Could very well be "sheet conductance", if you want to think of a black hole with zero temperature as a superconductor... Which might be more reasonable than taking the backwards approach in the paper. Will add this in for v2.0, with references re: extremal black holes.

          5 days later
          5 days later

          Oh yeah, moved the mention of the factorial / Stirling's approximation to the section on binary discrete signals. ;)

          • [deleted]

          • Post #66

          Perhaps it is not obvious that getting the "root" signals (a set of points on an (n-1)D shell) from all of the signals (a set of points on an nD ball) is a matter of ignoring noise and phase distortion. Came up with a toy "wavelet" reconstruction of the continuous signal to show how this "dimensional reduction" is straightforward and natural (even for toy models). Will add it in soon. I got a job in construction! Should be indoors. Bonus.

          • [deleted]

          • Post #67

          Looking around at all the comments from the past two weeks... Like I kind of hinted at a month or two ago: if this particular essay contest was not a psychology experiment, it should have been.