Motion-dependent levels of order in a relativistic universe

Abstract

Consider a generally closed system of continuous three-space coordinates x with a differentiable amplitude function ψ(x). What is its level of order R? Define R by the property that it decreases (or stays constant) after the system is coarse grained. Then R turns out to obey R=8(-1)L(2)I,where quantity I=4∫dx[nabla]ψ(*)·[nabla]ψ is the classical Fisher information in the system and L is the longest chord that can connect two points on the system surface. In general, order R is (i) unitless, and (ii) invariant to uniform stretch or compression of the system. On this basis, the order R in the Universe was previously found to be invariant in time despite its Hubble expansion, and with value R=26.0×10(60) for flat space. By comparison, here we model the Universe as a string-based "holostar," with amplitude function ψ(x)[proportionality]1/r over radial interval r=(r(0),r(H)). Here r(0) is of order the Planck length and r(H) is the radial extension of the holostar, estimated as the known value of the Hubble radius. Curvature of space and relative motion of the observer must now be taken into account. It results that a stationary observer observes a level of order R=(8/9)(r(H)/r(0))(3/2)=0.42×10(90); while for a free-falling observer R=2(-1)(r(H)/r(0))(2)=0.85×10(120). Both order values greatly exceed the above flat-space value. Interestingly, they are purely geometric measures, depending solely upon ratio r(H)/r(0). Remarkably, the free-fall value ~10(120) of R approximates the negentropy of a universe modeled as discrete. This might mean that the Universe contains about equal amounts of continuous and discrete structure.