The effect of Planck scale space time fluctuations on Lorentz invariance at extreme speeds

Abstract

The starting point of this work is the axiomatic existence of a smallest measurable interval, viz. the Planck time $t_P$, set by quantum fluctuations in the vacuum metric tensor. By the Relativity Principle, the same limit must then apply to the accuracy of all clocks which register time of events in their own frames. Further, it implies that the ordinary meaning of distance also ceases in the same manner beyond a scale l_P = c t_P. We demonstrate that quantum space-time, if real, may be made manifest by observing very energetic collisions, defined as interactions which occurred with the center-of-mass frame $\Sigma'$ (of the participating bodies) moving at a high speed $v$ relative to our laboratory frame $\Sigma$. In such situations, the initial conditions of the interaction are determined from direct measurements of the ultra-energetic particles or photons by instruments aboard $\Sigma$: they gather a raw dataset ${\cal S}$ which are subject to the limiting uncertainties \~ (t_P, l_P). Yet a meaningful (i.e. experimentally verified) version of the interaction is one where the interaction is viewed from frame $\Sigma'$. Since no instruments aboard $\Sigma'$ have taken any data, the way we proceeded is by a Lorentz transformation of ${\cal S}$ from $\Sigma$ to $\Sigma'$. Beware however that the resulting ${\cal S'}$ no longer consists of raw data, i.e. Lorentz distortions of probability distributions render the uncertainties non-Planckian - in fact, it will be shown that as v -> c they are >> (t_P, l_P), and are no longer negligible. Examples will given to indicate how a proper interpretation of the most current high energy cosmic and gamma-ray data necessitates incorporation of the said effect.