The Effect of Planck-Scale Spacetime Fluctuations on Lorentz Invariance at Extreme Speeds

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Abstract
Although our own timekeeping devices have finite precision, we often question whether this must mean the flow of time in nature is continuous at all scales. The starting point of this work is the axiomatic existence of an ultimate limit, or smallest measurable interval tP = (G/c5)1/2 (the Planck time), set by quantum fluctuations in the vacuum metric tensor. By the relativity principle, this same limit must then apply to the accuracy of all clocks that register time of events in their own frames. Furthermore, it implies that the ordinary meaning of distance also ceases in the same way beyond a scale lP = ctP. Owing to the smallness of tP and lP, their roles have hitherto been considered only in the context of the early universe. Here we demonstrate that quantum spacetime, if real, may be made manifest not only by probing very fine scales or large masses but also by observing very energetic collisions, defined as interactions that occurred with the center-of-mass frame Σ' (of the participating bodies) moving at a high speed v relative to our laboratory frame Σ. In such situations, the initial conditions of the interaction are determined from direct measurements of the ultraenergetic particles or photons by instruments aboard Σ: they gather a raw data set that is subject to the limiting uncertainties ~(tP, lP). Yet a meaningful (i.e., experimentally verified) version of the interaction is one where the interaction is viewed from frame Σ'. Since no instruments aboard Σ' have taken any data, the way we proceeded is by a Lorentz transformation of from Σ to Σ'. Beware, however, that the resulting ' 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 much greater than (tP, lP) and are no longer negligible. Examples are given to indicate how a proper understanding of existing high-energy cosmic- and gamma-ray data necessitates incorporation of the said effect.