Strange Paths

Infrared-light composite filtered image of M104 Galaxy taken by Spitzer Space Telescope in june 2004. Galaxy M104 (”Sombrero galaxy”) is located in the Virgo cluster, at a distance of about 30 million light-years. Its giant ring of dust spans over 50000 light years. It is believed that a supermassive black hole of a billion solar masses is located at its center.

Click image to zoom¹

1. © NASA/JPL-Caltech/R. Kennicutt (University of Arizona), and the SINGS Team [↩]

We can understand the paradoxes by Zeno of Elea (ca. 470 BC)¹ in two ways.

The first interpretation is that Zeno is not denying movement, but rather questioning its continuity, which is what actually leads to the paradoxes. In this sense, we can consider that Zeno is experiencing a kind of technical difficulty, and that the problem can be easily solved with calculus or as a converging sum of an infinite series. This interpretation is however short-sighted in its way of arbitrarily postulating the existence of movement, and just concentrating on the technical argument of the consistency of continuity, which is truly a mathematical problem and not a physical nor philosophical one. It shall be noted that one cannot prove that Zeno intended to contradict that the sum of an infinite series can be finite, as the mention of ‘finite time’ appearing in the report of the paradoxes² could be merely an interpretation by Aristotle.

The second interpretation is that Zeno basically denies movement, in the extraordinarily modern meaning of Parmenides, who considered change as illusory and the world as static and eternal. Zeno is not denying the appearance of movement, but rather its reality. The paradoxes thus appear at a deeper level, from the comparison between the phenomenon of movement and its disappearance implied by a thorough analysis of its model - either it being continuous (dichotomy paradox) or discontinuous (arrow paradox) -. The question becomes a purely physical question, which must be answered within a physical theory : why the experience of movement if movement appears logically impossible?

In the classical continuous model, the arrow must assume an infinite number of states in order to move from a point to another point. If such an infinite separation between two events, modelled by the absence of the successor of a real number, is equivalent or not to their physical dissociation, is a physical question, on the same level of reasoning as the ultraviolet catastrophe ideas which brought to quantum mechanics.³ If infinite divisibility is mathematically consistent, it is not necessarily physically meaningful (see also the Banach-Tarski paradox).⁴ This picture further changes with quantum mechanics as, per Heisenberg principle, a particle in a determined motion does not have a determined position. Interestingly, Zeno also gives his name to a quantum effect described by the Misra-Sudarshan theorem :⁵ if it is observed continuously whether a ‘quantum arrow’ has left the space it occupies, then it indeed never leaves this space.

In a discrete model (arrow paradox), Zeno’s argument is even stronger, and we can find a similar formulation of the argument in loop quantum gravity on the basis that time, being a pure gauge variable, is fundamentally nonexistent.⁶

1. • DICHOTOMY: Motion is impossible, because before arriving to the end, that which is moved must first arrive at the middle, and so on ad infinitum.
   • ACHILLES: The slower tortoise cannot be overtaken by the quicker Achilles, as he must first reach the point where the tortoise started, from which it has already left, and so on ad infinitum.
   • THE ARROW: An arrow shot from a bow occupies an equal space when at rest, and when in motion it always occupies such a space at any moment, the flying arrow is therefore motionless. [↩]
2. Aristotle, “Physics”, VI:9 [↩]
3. A. Einstein, “Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt” (”On a Heuristic Viewpoint Concerning the Production and Transformation of Light“), Annalen Der Physik, 1905. [↩]
4. S. Banach, A. Tarski, “Sur la décomposition des ensembles de points en parties respectivement congruentes”, Fundamenta Mathematicae, 6, 244-277 (1924) [↩]
5. B. Misra, E. C. G. Sudarshan, “The Zeno’s paradox in quantum theory“, Journal of Mathematical Physics, 18, 4, 756-763 (1977) [↩]
6. J. Barbour, “The end of time“, Oxford University Press (2001) [↩]

Three-dimensional map of the large-scale distribution of dark matter in the observable universe, from Hubble Space Telescope data (NASA, Jan 7, 2007). The map, determined by analysis of gravitational distortions of light coming from distant galaxies, reveals a network of filaments intersecting at the locations of normal matter in galaxy clusters. Clumping of dark matter appears more pronounced from right (distant regions in space and time) to left (nearest and recent regions).

Click image to zoom¹

Dark matter is a theoretical form of matter currently observed only by its gravitational effects and representing in the standard cosmological model about 20% of the energy density of the universe. It was first postulated to explain some motions of galaxies and other cosmological data, and confirmed by gravitational lensing observations of the Bullet Cluster of galaxies in August 2006.

1. © NASA, ESA, R. Massey (Caltech) [↩]

Image of the Sun taken through the Earth, in “neutrino light”, at the Super-Kamiokande detector (Japan). The image has been obtained with a 503 days exposure, by registering neutrinos emitted from the solar core and detected in a 50 000-ton water pool located 1 km underground. At night, neutrinos were transparently traversing the whole earth before being registered in this image.

Click image to zoom¹

A neutrino is an elementary matter particle of almost zero mass, only interacting through weak nuclear forces and gravity, leading to its unimpeded traveling through ordinary solid matter at almost the speed of light. During a rare interaction between a neutrino and an electron in the water, the electron is accelerated at a speed greater than the speed of light in water, producing a pulse of light -called Cherenkov radiation- similar to a supersonic boom. These pulses are detected by thousands of light amplifiers disposed everywhere on the pool surface.

Click image to zoom²

1. © R. Svoboda, K. Gordan [↩]
2. © Kamioka Observatory, ICRR (Institute for Cosmic Ray Research), The University of Tokyo [↩]

Most theories predict that at some time in the future, matter, structures, and/or the universe will have an end. Protons could decay within about 10⁵⁰ years according to Grand Unification theories.¹ The whole universe should approach an absolute zero temperature within 10¹⁰⁰ years, even black holes having evaporated.² And time itself could have an end, in a final imploding big-crunch singularity³ or in a diverging expansion cutting the universe in smaller and smaller chunks up to each particle.⁴ If a form of intelligent life would still exist by such times (certainly not based on current biology), how could it adapt to handle such fundamental limits?

Living systems could optimize toward less redundancy and less energy consumption. However, such optimizations being finite, they would not solve the problem, unless time happens to be infinite.⁵⁶

In end-of-time singularity scenarios, life could try to slow down perceived time by thinking faster. For example, the same amount of thinking and experience of billion years of human life could happen in a second. However, the energy required would accordingly increase, and physical time would subjectively slow down but not stop. In order to make eternity fit into a finite time interval, so as to escape an end of time singularity, infinitesimally small durations should exist, but time could be discrete at Planck scales, and even if it was not, infinite amount of energy would need to be pumped in the “eternity” process, so as it cannot happen “before” the singularity. Some argue that it could happen “during” the singularity itself.⁷

So how to escape the system? Logically this should be impossible. However, the system could prove to be more deep and resourceful than we perceive it today.

My favorite dream is about an infinite “sub-time”, perfectly still into each instant of time, where life could ultimately translate itself. There is also the idea of a cyclical universe,⁸ where the same finite combinatories of experience would come up again and again, which is similar, as if the whole universe was cyclical, time included, then it would not cycle within time but within another variable, so there would be no way to make a difference within time between it being cycling or not cycling. The same would apply if we live in a multiverse, like the one modeled by the chaotic inflation theory, where any singularity is local and the multiverse is made of a non-countable infinity of inflationary domains.⁹

So this fundamental question appears to be an introductory question about the nature and meaning of time.

1. H. Georgi, S. L. Glashow, “Unity of All Elementary-Particle Forces“, Phys. Rev. Lett. 32, 438-441 (1974) [↩]
2. S. W. Hawking, “Particle creation by black holes“, Comm. Math. Phys., 43, 3, 199-220 (1975) [↩]
3. Although there is experimental evidence for an universe presently undergoing an accelerated expansion, the dark energy could still be an oscillating scalar field leading to future recollapse [↩]
4. R. R. Caldwell, M. Kamionkowski, N. N. Weinberg, “Phantom Energy and Cosmic Doomsday“, Phys. Rev. Lett. 91 (2003). [↩]
5. F. J. Dyson, “Time without end: Physics and biology in an open universe“, Rev. Mod. Phys. 51, 447 - 460 (1979) [↩]
6. K. Freese, W. H. Kinney, “The Ultimate Fate of Life in an Accelerating Universe“, Phys. Lett. B 558, 1-8 (2003) [↩]
7. F. Tipler, “The Physics of Immortality“, Anchor (1997) [↩]
8. P. J. Steinhardt, N. Turok, “A Cyclic Model of the Universe“, Science, 296, 5572, 1436 - 1439 (2002) [↩]
9. A. Linde, “Eternally Existing Self-Reproducing Chaotic Inflationary Universe“, Phys. Lett. B175, 395 (1986) [↩]