The Planck radius, the randomness of pi, and the age of the universe

[Gustafson, S]

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.

[arivero]

The guy is right, just it is using a different definition of PI, which is a very stupid thing to do. This "varying Pi" has a name, it is a measurement of the curvature of space.

 

Note that there are other flat definitions for pi; for instance the sum of angles of a triangle, and I am not sure all will vary the same way. Better stick with the well defined, century old, concepts of curvature.

[kenshin]

Umn...changing value of Pi.....The defination of Pi is given for the euclidian circles ie. the space is considered flat.So,the Pi has to be constant and is given as the ratio of circumfrance to the diameter in a flat space.(we are considering the euclidian geometry).But,there is an intresting point,although the Pi is a constant,but,its value has to be determined by experiment.Now,we should know that experiment is being performed on a curved space(as ours),so,the experimentally callculated value will change with the change in curvature.Hence,although the Pi is a constant,it's(calculated) value changes because of the experimental limitations.Well,at least this is what I think.

[gcol]

Does this mean that pi will always be a fixed number relative to the circle as we have done and always will know it, but to an observer from a different universe, our circle and his may be different?

[DV8 2XL]

Pi is a numerical value, an exact quantity. An exact quantity leaves no room for randomness. Mathematics is in no way random.

 

Actually it is transcendental - the term exact value is not quite right in this context. However it is a constant and any theory that claims otherwise must add other entities to support that contention drawing a charge of violating Occam's Razor.