# The Problem of Measuring Time for the Physicist: From Poincaré to Einstein In his 1905 book The Value of Science, French mathematician Henri Poincarè considers the difficulties of measuring time and discovers a number of obstacles in its fitness for rigorous, scientific use. The central problem is how we quantify that thing we refer to as ‘time’ in a physically meaningful way—that is, in such a way that the scientist can measure some event’s temporal location—when we only experience time phenomenologically. It is important to understand what is meant for a quantitative measure of something to be ‘physically meaningful’. Einstein, in his 1911 lecture “The Theory of Relativity”, develops this concept through the consideration of space. He notes that in a specified, three-dimensional coordinate system, a point’s three orthogonal coordinates have a verifiable validity: by using a measuring rod that counts as one unit of length in the coordinate system, one can physically count out the lengths along lines perpendicular to the coordinate planes, and the results constitute the point’s three spatial coordinates.

In this way of measuring space, then, the scientist has a completely physically determinable meaning of spatial position. What it means for an event to be at some point in the spatial coordinate system, or for it not to be at that point, is easily identifiable. The problem with trying to do the same for time is, essentially, that we have no measuring rod. The study of physics requires the notions of duration and equality of duration, but, Poincarè notes, we have no “direct intuition” of equality of duration. The conscious subject does not perceive, in a wide sense of the term, any quantified timeline of phenomena. Rather, she perceives phenomena that are simultaneous or antecedent to each other in a non-exact, qualitative way, judging the events non-scientifically. The question, then, is how does physics develop a unit of time that has been rigorously defined, a quantified measuring rod for time?

Poincarè first looks at the use of “chronometers”—indicators of time that work off of a natural periodicity, like pendulums. He comes to the conclusion that the use of a chronometer’s beats as our unit of time always relies, implicitly or not, on the supposition that the same causes take the same time to produce the same effects. But, he reasons further, physicists can only approximately determine causation (as opposed to rigorously, because that would require bringing every body in the universe into your calculation), and so the above supposition can only assert approximate causal relationships. Therefore any definition of time built on this supposition will itself be merely approximate.

What about the common practice of using astronomical observations as measurements of duration, via Newtonian mechanics? Poincarè points out the unscientific circularity of such calculations: if the observed quantity of duration does not match the quantity predicted by Newton’s laws of mechanics, the clock is adjusted so that it does match. In other words, if a rotation was measured to be faster than Newton’s laws predicted, then we conclude that we must adjust our means of measurement to fit the prediction. But this leads Poincarè to the troubling conclusion that the experimental physicist merely defines duration in such a way that in using it the laws of nature are verified. As in the case of the chronometer, however, this means that the scientific unit of time is a mere approximation, evolving with ever-changing Newtonian laws of motion, given that these laws are only experimentally (i.e., inductively) determined, and so, are not in theory absolutely necessary.

One final problem with this Newtonian way of measuring time is that there is any number of different units of measurement that satisfy the definition of the experimental scientist if only one just slightly alters the laws of nature such that they are expressed differently, and yet still validly. The scientist, therefore, does not, in fact, choose how to measure time only according to its ability to verify the laws of nature, but also how conveniently it does so. One way of measuring time is not truer than any other as long they both verify the natural laws equally well.

From this line of reasoning, Poincarè realizes that our notion of time comes down to rules. There is no strict definition of the unit of time in use, only rules that tell one, for instance, how to correct chronometers according to some observational data, and even worse, these rules have not been adopted due to their truth, but for their simplicity and convenience, such that if we measured time in any other way, the application and/or testing of the physical laws would be a bit more difficult.

This is where Einstein and his theory of special relativity come in. What Einstein ultimately does in special relativity is develop a rule by which the physicist can come to a standard measurement of time. Let there be some specified coordinate system k. According to Einstein, for the physicist to be able to measure time in a physically meaningful way, it would mean being able to determine the temporal location of an event at any position in the coordinate system k. This is what his rule allows for. Einstein notes, as Poincarè did, that we have no “direct intuition” of duration (i.e., the interval between two events), and that the difficulty with our common definition of time as some clock ticking absolutely independent of events is that it makes time impossible to measure. One of Einstein’s goals, then, is to make time measurable—that is, to define time in such a way that an event can be measured in a physically meaningful way. After all, as Poincarè wrote, the study of physics requires a rigorous definition of duration. Interestingly, Einstein discovers that the problem with measuring duration lays not in any of the Newtonian laws per se, but in certain assumptions underlying the entire scientific interpretation of events in space and time. In a coordinate system k, we cannot scientifically measure the time between two distant events without having already measured the velocity of light. But this measurement is only possible given two other measurements: (1) the distance between the points between which the light propagates, and (2) the time of the emission of the light at the first point and the time of the arrival of the light at the second point. This is, though, a circular string of measurements. Einstein concludes, then, that it is in principle impossible to rigorously measure the velocity of light, and that it becomes possible only under arbitrary suppositions. For Einstein, this supposition is that the velocity of light from some spatial point A to another spatial point B is the same as it is from B to A, and it is only by virtue of this supposition that Einstein’s rule for measuring time in special relativity actually works.