In the 5th century B.C., Zeno of Elea argued that motion is impossible. Although few philosophers have accepted Zeno’s conclusion (but Bertrand Russell described his logic as “immeasurably subtle and profound”) it must be conceded, I believe, that Zeno did raise some profound questions. Zeno’s paradoxes have puzzled, challenged, influenced, inspired, infuriated, and amused philosophers, mathematicians, and physicists for over two millennia. The most famous are the arguments against motion described by Aristotle in his Physics, Book VI, the well-known ones as the “Achilles and the tortoise”, and less-known ones such as the dichotomy (or the infinite divisibility of a finite quantity: if we assume that movement exists, we contradict ourselves because it is impossible to cover infinite sections of length in a finite time). Anyway, if we humbly admit that motion does indeed occur, and that it constitutes a pervasive and fundamental feature of the Universe, then we will immediately recognize that the concept of velocity, the very expression of motion, plays a key role in our description and understanding of nature.
For centuries, philosophers and physicists have been fascinated by the phenomenon of light, and have speculated on the speed with which it is propagated. During the ancient and medieval periods, light was generally believed to travel at infinite speed. So strongly did Descartes endorse this view that in 1634 he wrote to a friend: “For me this was so certain that if it could be proved false, I should be ready to confess that I know absolutely nothing about philosophy”. Galileo was somewhat more cautious in the matter; he proposed an experiment to determine whether light requires any time to traverse a distance. When the experiment was performed – not by Galileo himself, but by the Florentine Accademia del Cimento – no time lapse was detected. With our modem knowledge of the speed of light, we are not surprised at this null result. Any supposed delay on account of transmission over terrestrial paths would be totally indiscernible by any methods of time measurement then available. Terrestrial measurements of the speed of light were not successfully accomplished until the middle of the nineteenth century, but the methods used were in principle quite analogous to those of Galileo’s experiment.
Such a measurement was precisely the objective of the famous Michelson-Morley experiment, first undertaken in 1881. Although the apparatus they employed was capable of detecting a velocity which was a small fraction of any velocity it was reasonable to postulate for the Earth with respect to the ether, the experiment was a crashing failure, as it failed to detect any motion of the Earth with respect to the ether. It was repeated at different seasons of the year (in case the Earth might happen to be virtually at rest in the ether just at some point, in its annual orbit around the Sun), and with more refined equipment. Still the outcome was null, and the poor Michelson was baffled at the result (Michelson famously was a strong supporter of the theory of ether, and his dislike of the relativity theory was such that he referred to it, even in Einstein’s presence, as a “monster”). The apparatus for this experiment consists basically of two paths situated at right angles to one another. By comparing the times required for light rays to make round-trips (and not just one-way trips) back and forth between mirrors along each of these paths, it would be possible, according to classical physics, to ascertain our state of motion with respect to the ether, because of a directional dependency which would exist for the time of the round trip if the apparatus is in motion through the ether.
In classical physics the concept of rest (that is, being at the same place at different times) and the concept of simultaneity (occurring at the same time at different places) both had a clear and non ambiguous significance. Einstein instead saw that both concepts had to be questioned, which led him to reject both the notion of absolute rest and absolute simultaneity. By calling into question these basic tenets of space and time, Einstein was presenting us with a new kinematics – a new theory of motion.
If an object moves from A to B, and if we wish to determine its average velocity for the trip, then we must know how far B is from A and we must have clocks at A and B to establish the time of departure from A and the time of arrival at B respectively. The average velocity is, by definition, the ratio of the distance covered to the elapsed time. If, however, two distinct clocks at A and B are used to establish the times at which the trip begins and ends, these clocks can meaningfully be said to furnish the duration of the trip only if they are synchronized – that is, only if the two clocks, located at the two different places register the same time. If we are to be able to treat motion in even the most rudimentary way, we must be able to synchronize clocks located at a distance from each other. To say that two clocks are synchronized means that they show the same reading at the same time: this is the problem of distant simultaneity. Note that this does not involve the reading of clocks in different frames of reference, which may be in motion relative to one another. We are simply attempting to synchronize clocks in a single frame of reference.
In the Special Theory of Relativity, all observers are assumed to measure the speed of light c to be the same. However and again, this notion refers to the round-trip speed of light, where a clock at the origin times both the outward and return trips of light reflecting off a distant mirror. Hans Reichenbach framed this issue more formally: if the light signal is sent from A at time t1 and it arrives back at A (after being reflected at B) at time t3, then its time of arrival at B is t2, which is given by the simple formula t2 = t1 + e(t3 – t1 ). The parameter e can take a value between 0 and 1. If we set e=1/2 we have Einstein’s standard signal synchrony. It is worth noting that Einstein himself acknowledged that the physical predictions of the theory would be unchanged if the speed of light was anisotropic, as long as the average round-trip speed is equal to c: the claim that simultaneity is conventional means that we may choose any values for e different from 1/2, without making a factual error. Notably, a value other than 1/2 implies that the speeds of light on the two opposite one-way legs of the round trip are not equal to one another. In the light (pun intended) of Michelson’s attempts to measure the motion of the Earth relative to ether in different directions, such a possibility is more than just an intellectual curiosity. Then the obvious question is: can we actually measure the speed of light one-way only?
As it turns out, such a measurement seems impossible in our universe, despite several attempts have been made to devise (more or less) ingenious schemes to circumvent the theoretical and practical difficulties. For example, in the October 2009 issue of the American Journal of Physics, Greaves, Rodriguez and Ruiz-Camacho reported a measurement of what they identified as the one-way speed of light (https://doi.org/10.1119/1.3160665). A light beam was sent from a laser to a photosensor, and then the signal from the photosensor was transmitted through a coaxial cable back to the vicinity of the laser. The length of the cable was 23.73 m, and the authors measured the fixed time delay (79 nanosecond) of the transmission through the cable. The authors pointed out that all timing is performed in a single place (the vicinity of the laser) so no convention for the synchronization of distant clocks seemed to be necessary. However, an old fox like Jerome Finkelstein mercilessly shot down the experiment (https://doi.org/10.1119/1.3364868), showing that the assertion of a known time delay through the cable is only meaningful if one imagines having synchronized clocks at the two ends of the cable: the assumption that the second leg is accomplished with a known speed is what allows the speed of the first leg to be determined, therefore the measurement is again a two-way determination of the speed of light.
Measuring the one-way speed of light is inevitably fraught with the issues of clock synchronisation, and, as long as the average speed of light remains equal to c, the speeds on the outward and return legs could be different. One objection to such an anisotropic speed of light, is that views of the distant Universe would be different in different directions, especially with regards to the ages of observed astronomical objects, and to the smoothness of the Cosmic Microwave Background. In a paper recently appeared in the Journal of the Australian Astronomical Society (https://doi.org/10.1017/pasa.2021.2) the well-known astrophysicist Geraint Lewis and his long-time coworker Luke Barnes studied the possibility of anisotropic light speed in a Milne-type universe. For the less cosmology-oriented among you, I must precise that Milne’s universe is a zero-order approximation to the Fridman-Robertson-Walker (broadly accepted) model of the homogeneous universe: in this approximation, the universe is empty, no matter, energy or radiation exist. It is a toy-model of the universe, whose usefulness lies in the property that it describes a curved space while its space-time remains flat, like in special relativity, thus making for a simple solution of the gravitational field equations. What Lewis and Barnes found in their calculations, is that the anisotropic speed of light results in anisotropic time dilation effects that compensate for the differing light travel times. In this universe, any observer would see an isotropic universe around them, even if the speed of light was not. However, the possibly anisotropic speed of light could be ascertained in the effect that galaxies at a given distance away should have a faster recession speed in one direction than in the other, and the universe could be expanding faster “to the right” or “to the left”. In such a peculiar universe, the question of the one-way speed of light could be directly addressed. Of course, for more general cosmological models closer to the actual appearance of our Universe, where the presence of mass and energy results in curved space-time, the picture is more complicated as there is no simple mapping of the modified Lorentz transformations into the general relativistic picture. Lewis and Barnes are now studying this extension, and hopefully they will provide us with some deeper answer soon.