" The Lorentzian length of a timelike curve (Lorentzian geometry and causality theory) connecting both endpoints of a classical computation is a function of the path taken through Minkowski spacetime.
Minkowski spacetime is a combination of three-dimensional Euclidean space and time (Euclidean space is the fundamental space of classical geometry) into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded.
The associated runtime difference is due to time-dilation: the phenomenon whereby an observer finds that another’s physically identical ideal clock has ticked at a different rate than their own clock. Using ideas appearing in the framework of computational complexity theory, time-dilation is quantified as an algorithmic resource by relating relativistic energy to an nth order polynomial time reduction at the completion of an observer’s journey. These results enable a comparison between the optimal quadratic Grover speedup from quantum computing and an n = 2 speedup using classical computers and relativistic effects. The goal is not to propose a practical model of computation, but to probe the ultimate limits physics places on computation."
Recently, in a significant breakthrough, scientists at JILA, a joint operation between the National Institute of Standards and Technology and the University of Colorado Boulder, have measured time dilation at the most miniature smallest scale ever using the most accurate clocks in the world. The team showed that clocks located just a millimeter apart—about the width of a pencil tip—showed slightly different times due to the influence of Earth’s gravity.
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CONCLUSION
This study is part of the research effort aimed at understanding what class of computations are made possible or ruled out by the laws of physics8,17,18,19. We have shown that finite nth root polynomial reductions in algorithmic run-time are made possible by relativistic effects. The runtime improvement is predicted by Einstein’s theory of relativity and the connection to computation was explored by considering polynomial reductions inside the black-box model. In the present study, the observer changes her own life history by taking an accelerated spacetime path to obtain a computational time efficiency improvement. The method would be more practical if instead an inertial observer sent the computer on an alternative spacetime path.
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