Gustafson, S

My post, “An accurate age of the universe” is in the spirit of the well-known speculations of Nobel Prize winning physicist Paul Dirac. Essentially, it sets the electron mass energy divided by the quantum energy of the universe (Planck’s constant divided by the universe age) equal to the electron electrostatic energy divided by the electron gravitational energy. The result, after rearranging and using a factor of 2 and two factors of two times pi, is a value for the age of the universe that agrees with the value obtained from recent measurements of the cosmic background microwave radiation (13.7 billion years) to well within 1%. My questions are: Is this result interesting, significant, worthy of further study, etc.?

The age of the universe has recently been determined with unprecedented accuracy, by the Wilkinson Microwave Anisotropy Probe or WMAP, to be 13.7 billion years to within 0.9 %. Even if this value is in error by a percentage that is an order of magnitude larger (as may be the case if certain cosmological corrections are applied), it is sufficiently accurate for renewed consideration of “numerology” in the sense of the well-known Dirac large number hypothesis. In particular, the age of the universe to within 0.5 % of the WMAP value is given by A = [h/(2π)][e^2/(4πε_0)]/[2πc^2G(4πm)^3], where h is Planck’s constant, e^2/(4πε_0) = q^2 is the squared electron charge, G is the gravitational constant, c is the speed of light, and m is the electron mass. This expression has the following intriguing interpretation. Over an arbitrary distance r, the ratio of the electron mass force to the universe quantization force is (mc^2/r) / [(hA^-1)/r]. Over the same distance the ratio of the electron electrostatic force to the electron gravitational force is [(q^2/2)/ (2πr)^2] / [(Gm^2)/(r/2π)^2], where q^2 is divided by 2 because the electrostatic force is “signed”, r is multiplied by 2π for the electrostatic force because it is “circumferential”, and r is divided by 2π for the gravitational force because it is “anti-circumferential”. The terms “signed”, “circumferential”, and “anti-circumferential” obviously require further interpretation and justification. Nevertheless, setting the two ratios equal yields the above expression, which predicts an age of the universe that is 0.5% larger than the current WMAP value and which is well within its 0.9% error.

To "Insane alien": I agree that abstract math concepts such as pi may seem to have little to do with measureable quantities such as the radius of the universe. However the Platonic view of mathematics (perhaps the most widley accepted view among mathematicians) is that mathematical results are discovered---not invented or formally synthesized---in the same way that results are discovered in the physical sciences. To "John Cuthber": If space is curved, as is generally accepted, then the curvature may be said to correspond to a radius of curvature, which may be identified with the radius of the universe. Also, the value of pi is constant because, as you indicate, it is a mathematical construct, but some measues of the randomness of its digits may change with location of the digits. Finaly, it is generally accepted that the universe is expanding, but over times that are much smaller than its 13.7 billion year age we may regard it as stationary.

This post describes a computer experiment that could be carried out and that could produce a more accurate value for the age (or radius) of the universe than the currently estimated 13.7 billion years (or 13.7 billion light years). It refers to the post of 14 Feb 07 in the Speculations Forum, but the current post considers only factual material and realizable experiments. 1. Computer experiments which used an approximate entropy metric for randomness have shown that the digits of √2 are more random than the digits of e. This result is “mainstream” and has been reported in a leading peer-reviewed journal: S. Pincus and R. E. Kalman, “Not all (possibly) ‘random’ sequences are created equal”, Proc. Nat. Acad. Sci., vol. 94, pp. 3513 – 3518, April, 1997. 2. There is reason to believe that the randomness of pi expressed in any base (i.e., 3.14159…in base 10) changes near a precision of one part in 10 to the 121st power and that the precision at which this change (if any) occurs could be determined accurately using the following computer experiment. Find the randomness of the sequence of the digits of pi from digit 1 to digit n and from digit n + 1 to digit 2n using approximate entropy, find the fractional change in randomness, repeat for n = 2, …, 242, and plot the fractional change in randomness versus n. Examine the plot for changes in level, slope, curvature, etc., near n = 121. Generate plots using other measures of randomness and using pi expressed in other bases to characterize the changes (if any). 3. If changes in randomness are found for many measures of randomness and for many bases and if they can be interpolated to the same fractional digit location (e.g., n = 121.327), then a more accurate value for the radius of the universe (currently estimated at 13.7 billion light years) could be proposed. The proposed value would be such that the precision at which the change in randomness occurs equals three times the square of the ratio of the Planck radius (4.05 10E-35 meters) to the radius of the universe. 4. Someone fluent in Matlab, Mathematica, etc., could carry out this computer experiment. Is anyone interested?

Thanks to all of you for your comments on my “A CONJECTURE: PI CHANGES WITH TIME”. I completely agree with comments to the effect that pi is a mathematical construct and not a measured quantity. Some less philosophical observations are as follows. 1. The conjecture is not idle---it can be verified by computer experiments. These experiments would be similar to experiments which used an approximate entropy metric for randomness to verify that the digits of √2 are more random than the digits of e (reference: S. Pincus and R. E. Kalman, “Not all (possibly) ‘random’ sequences are created equal”, Proc. Nat. Acad. Sci., vol. 94, pp. 3513 – 3518, April, 1997). However, the conjecture-verifying experiments should be conducted by persons more familiar with Matlab, Mathematica, etc., than I am. 2. Here is an initial recipe for the computer experiments. Find the randomness of the sequence of the digits of pi from digit 1 to digit n and from digit n + 1 to digit 2n using approximate entropy, find the fractional change in randomness, repeat for n = 2, …, 242, and plot the fractional change in randomness versus n. Examine the plot for changes in level, slope, curvature, etc., near n = 121. Generate plots using other measures of randomness and using pi expressed in other bases to characterize the changes (if any). Note that if changes in randomness are found for many measures of randomness and for many bases and if they can be interpolated to the same fractional digit location (e.g., n = 121.327), then a more accurate value for the radius of the universe (currently estimated at 13.7 billion light years) might be proposed. 3. Is anyone out there interested in performing the conjecture-verifying experiments?

The third paragraph is intended to be entirely rational. It states, AS A CONJECTURE, that digits before a precision of about one part in 10 to the 121st power in a listing of calcualted digits of pi will be found to have a different degree of randomness than the following digits. It also states that this conjecture can be tested using various metrics for the degree of randomness (a particular entrpy-based metric is suggesed as an example), and it indicates that pi can be expressed in any base, where base 10 or base 2 would be common choices. In base 10 a precision of one part in 10 to the 121st power corresponds to the 121st decimal place, whereas in base 2 (binary) this precision corresponds to the 404 binary digit (i.e., more binary than decimal digits are needed to express a given precision).

A CONJECTURE: PI CHANGES WITH TIME Steven C. Gustafson, 14 February 2007 The following is conjectured: The measured value of pi changes with time. At the big bang it was 2, but now it is very slightly smaller than the calculated value of 3.14159... . The current measured value of pi, if it could be measured accurately enough, would be found to be smaller than the calculated value by about one part in 10 to the 121st power. The difference in the measured and calculated values of pi will be indicated by a significant change in the randomness of the digits of the calculated value of pi before and after about the 121st decimal place. Thus determining randomness will constitute measurement, and the significant change will be robust in that it will be found using digits generated in various bases and using various measures of randomness. A Justification is as follows As indicated in reference 1 and its citations, at the big bang the universe can be modeled on the surface of a sphere of approximately the Planck radius r = (hG/cE3)E1/2 = 4.05 10E-35 meters (where r is obtained by eliminating m from mcE2 = GmE2/r and mcr = h). The smallest circle that could have been drawn on the surface of this sphere would have had a radius of r, and since the diameter of this circle must have been measured along the surface of the sphere, the measured diameter would have been half of the measured circumference, and thus the measured value of pi would have been 2. Any method of measuring pi (e.g., summing the angles of a triangle drawn on the surface of the sphere) would have produced this value of pi. Currently the universe can be modeled on the surface of a sphere of radius R = 13.7 10E9 light years, but the smallest circle that can be drawn on the surface of a sphere of this radius still has the Plank radius (see reference 1 and its citations). Thus the diameter of this circle, which again must be measured along the surface of the sphere, is very slightly curved and is therefore longer than it would be in a flat or Euclidian space (i. e., a space in which the radius of the universe is infinite). Accordingly, the measured value of pi, which is the ratio of measured circumference to measured diameter, is very slightly less than the calculated value, which is currently known to more than a trillion decimal places. Again, any method of measuring pi, including summing the angles of a triangle, using Buffon’s needle, etc., would yield a value of pi slightly less than 3.14159… . As is easily shown (see reference 3) this line of reasoning leads to a measured value of pi that is smaller than the calculated value by about one part in 10 to the 121st power, which is three times the square of the ratio of R to r. The calculated value of pi found using any of many possible formulas is of course accurate to the calculated number of decimal digits, binary bits, etc. However, the calculation may be considered to be a measurement of pi in that the sequence of digits or bits obtained has a degree of randomness which is not predetermined. Thus it is conjectured that the degree of randomness of about the first 121 decimal digits (or about the first 404 binary bits) is significantly different than the degree of randomness of the following digits (or bits). The degree of randomness might be quantified using various metrics, including the approximate entropy considered in reference 2, where this metric was used to determine, for example, that the digits of √2 are more random than the digits of e. If the change in the degree of randomness is sufficiently sharp, then the digit or bit (or digit or bit region) at which the change occurs might be used to specify a more accurate value for the radius or age of the universe. 1. A. Ashtekar, T. Pawlowski, P. Singh, “Quantum Nature of the Big Bang”, Physical Review Letters, vol. 96, pp. 1413011- 1413014, 14 April 2006. 2. S. Pincus and R. E. Kalman, “Not all (possibly) “random” sequences are created equal”, Proc. Nat. Acad. Sci., vol. 94, pp. 3513 – 3518, April 1997. 3. S. C. Gustafson, “The Planck radius, the randomness of pi, and the age of the universe”, http://www.scienceforums.net, 28 May 2006.

Steven C. Gustafson, 28 May 2006 Conjectures: (1) The measured value of pi changes with time. At the big bang it was 2; now it is smaller than the calculated value of π = 3.14159... starting at about the 121st decimal place. (2) The difference in the measured and calculated values of pi is evident in significant differences in the randomness of the digits of the calculated value of pi before and after about the 121st decimal place. Justification: Consider a circle of circumference 2πr on the surface of a universe of radius R, where all measurements are on the surface. The diameter of this circle measured along the surface is 2θR, where θ = arcsin(r/R). Therefore the measured value of pi (circumference divided by measured diameter) for the circle is p = πr/[R arcsin(r/R)]. As indicated in reference 1 and its citations, at the big bang R and r were equal and were approximately the Planck radius r0 = (hG/cE3)E1/2 = 4.05 10E-35 meters (where r0 is obtained by eliminating m from mcE2 = GmE2/r0 and mcr0 = h). Thus the measured value of pi at the time of the big bang was 2, and any value of pi measured at a later time is larger. At infinite time any measured value of pi is the calculated value, which is known to more than a billion decimal places. Currently R is Rc = 13.7 10E9 light years, and thus p is only slightly less than π for circles that are not light years in size. Specifically, the above expression for p yields (π – p)/p = (r/R)E2 /3 for r much less than R, and since the smallest r is the Planck radius and the current R is Rc, the current value of (π – p)/p is at least (r0/Rc)E2/3 = 10E-121 = 2E-404. The calculated value of pi found using any of many possible formulas is of course accurate to the calculated number of decimal digits, binary bits, etc. However, the calculation may be considered to be a measurement of pi in that the sequence of digits or bits obtained has a degree of randomness which is not predetermined. Thus it is conjectured that the degree of randomness of about the first 121 decimal digits (or about the first 404 binary bits) is significantly different than the degree of randomness of the following digits (or bits). The degree of randomness might be quantified using approximate entropy as indicated in reference 2, where this metric was used to determine, for example, that the digits of √2 are more random than the digits of e. Verification and application: The above conjectures are certainly “out of the mainstream”, and to restate a famous saying by Carl Sagan, “extraordinary claims require extraordinary evidence”. In particular, suppose someone finds a significant difference in the randomness of the digits of the calculated value of pi before and after about the 121st decimal place using some randomness criterion. This finding would require verification by many others in many ways, including different randomness criteria, different number bases, and different definitions of “significant”, before the “extraordinary claim” might be said to have the support of “extraordinary evidence”. However, if verification is achieved, then the digit or bit at which a change in randomness occurs might be used to specify a more accurate value for the radius or age of the universe. 1. A. Ashtekar, T. Pawlowski, P. Singh, “Quantum Nature of the Big Bang”, Physical Review Letters, vol. 96, pp. 1413011- 1413014, 14 April 2006. 2. S. Pincus and R. E. Kalman, “Not all (possibly) “random” sequences are created equal”, Proc. Nat. Acad. Sci., vol. 94, pp. 3513 – 3518, April 1997.