The ancient Greek philosopher Zeno of Elea, who lived between 495–430 BC, came up with the arrow paradox about motion and the continuity of time: “If everything when it occupies an equal space is at rest at that instant of time, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless at that instant of time and at the next instant of time but if both instants of time are taken as the same instant or continuous instant of time then it is in motion,” to which the ancient Greek philosopher Aristotle responded in his manuscript titled Physics written in 350 BC: “…a line cannot be composed of points, the line being continuous and the point indivisible… no continuous thing is divisible into things without parts… everything continuous is divisible into divisibles that are infinitely divisible… The same reasoning applies … to time… Zeno’s argument makes a false assumption in asserting that it is impossible for a thing to pass over or severally to come in contact with infinite things in a finite time.”
Is Aristotle’s assertion about the continuity of time correct?
Any physical variable, including time, obtains its meaning through a measurement process. According to the Nyquist-Shannon sampling theorem, a signal sampled at a rate f can be fully reconstructed if it contains only frequency components below half of that sampling frequency: f/2. To get infinite time resolution in accordance with the Aristotelian perspective, we must be able to increase the sampling rate of experimental clocks indefinitely. Is there an upper limit to the sampling frequency of time?
The best atomic clock in the world achieves a frequency stability noise of about 10^{18} Hertz over one second, where Hertz is the inverse of a second. Equating this sampling frequency to twice the Nyquist frequency implies that laboratory clocks can resolve time up to 5×10^{-19} seconds over one second. But technology gets better over time. Aristotle and Zeno might have wondered whether there is any limit to the highest sampling frequency of a clock?
Aristotle suggested that time is continuous and can be divided into arbitrarily small bins. The shortest measurable time bin would correspond to the highest sampling frequency of a clock. As I argue below, quantum gravity sets a fundamental limit to the highest possible sampling rate of any clock.
Imagine a hypothetical clock based on the interaction between two particles, each carrying an energy E. According to quantum mechanics, the particle energy can be expressed as E=hf, where h is Planck’s constant and f is the particle frequency. The hoop conjecture of Einstein’s General Relativity suggests that if an energy E is packed within a spherical region smaller than its Schwarzschild radius, 2GE/c⁴, where G is Newton’s constant and c is the speed of light, then the system would collapse into a black hole and no information will escape out of its event horizon. In the case of our hypothesized particles, the energy E is packed within a region equal to the particle wavelength, c/f. Requiring that the Schwarzschild radius be smaller than the particle wavelength, is equivalent to the inequality: (2GE/c⁴)<(c/f). Since E=hf, this inequality ends up being quadratic in f which sets an upper limit on the clock sampling frequency that allows timing to be recorded by an experimentalist outside a black hole.
The maximum sampling frequency of time is the square root of (c⁵/2Gh), a combination of fundamental constants of nature. By substituting the values of these constants, we get a maximum Nyquist frequency for measuring time of 5.4×10^{42} Hertz. Its inverse equals a minimum time bin of 1.9×10^{-43} seconds.
These considerations imply that if we wished to sample time over bins shorter than 1.9×10^{-43} seconds, then the measurement quanta used by our hypothetical clock would have collapsed to black holes without displaying the timing information we are seeking. In other words, it is not feasible to measure time with finer resolution.
Aristotle’s argument loses validity for time bins shorter than 1.9×10^{-43} seconds because it is not possible to design a clock that would display temporal information with finer resolution. This limit is based on our current understanding of quantum-mechanics and gravity.
Given this perspective, what happens on timescales shorter than this bin duration cannot be measured experimentally. The best data possible on Zeno’s arrow would display a sequence of snapshots separated by a time interval longer than 1.9×10^{-43} seconds.
Whether time exists on shorter intervals remains in the realm of philosophy rather than physics. Quantum-gravity limits experiments from resolving whether the Aristotelian conjecture of a continuous time is correct to an arbitrary level of precision. But this should not trigger despair. Where fundamental physics has its limits, metaphysics comes to the rescue.
The great privilege of being a physicist is the “carte blanche” to answer unresolved fundamental questions with the phrase: “we do not know” or “we will never know.” The latter option is most appropriate when dealing with the interface between quantum-mechanics and gravity on other questions like: “What is the ultimate fate of matter which falls into a black hole?” or “What happened before the Big Bang?”
ABOUT THE AUTHOR
Avi Loeb is the head of the Galileo Project, founding director of Harvard University’s — Black Hole Initiative, director of the Institute for Theory and Computation at the Harvard-Smithsonian Center for Astrophysics, and the former chair of the astronomy department at Harvard University (2011–2020). He is a former member of the President’s Council of Advisors on Science and Technology and a former chair of the Board on Physics and Astronomy of the National Academies. He is the bestselling author of “Extraterrestrial: The First Sign of Intelligent Life Beyond Earth” and a co-author of the textbook “Life in the Cosmos”, both published in 2021. The paperback edition of his new book, titled “Interstellar”, was published in August 2024.
