## June 30, 1905: ‘Time’ for Einstein…

As a tribute to Einstein’s original contribution to our understanding of time 103 years after he made the contribution, here is an effort to make the concept of time, as he conceived it, more accessible and intelligible to the reader. For the reader’s information, this writer now has a more evolved and considered opinion of “time”, as generally understood, and “time”, as in the strict sense of the word – that expounded in special relativity and, hence, physics.

This write-up will focus on the concept of time as Einstein conceived it in his June 30, 1905, paper “On the Electrodynamics of Moving Bodies”, and as I have absorbed the concept. While every attempt has been made to make things clear, I don’t deny that there might be inadvertent loopholes in this write-up. Though I have made a fairly thorough “first study” of the Special Theory of Relativity (STR), I have not as yet been able to form a clear “picture” (if that is the right word) of time. There is a lot of haziness enshrouding “time”. I write in an effort to clear that haze as well. Genuine criticism or questions are most welcome.

Time is a physical quantity unlike any other – mass, charge, energy et al. A physical quantity is measurable. And time, therefore, is measurable. But when we say time is measurable, we must understand what exactly it is that we are measuring. To describe the motion of a body, we specify its position at different instants of time – that is to say, at different ticks of the clock that we are using. So really, “the different instants of time” are “the different ticks of the clock” and nothing more. Nothing other than the clock ticks are observed while marking different ticks of the clock as being simultaneous, with the corresponding positions of the body. Essentially, therefore, our measurements of time hinge on the simultaneity of events – the clock ticks and so do the corresponding positions of the body. To quote from Einstein’s paper:

“If, for instance, I say, “That train arrives here at 7o’clock,” I mean something like this: “The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events.” It might appear possible to overcome all the difficulties attending the definition of “time” by substituting “the position of the small hand of my watch” for “time.” And in fact such a definition is satisfactory when we are concerned with defining a time exclusively for the place where the watch is located; but it is no longer satisfactory when we have to connect in time series of events occurring at different places, or—what comes to the same thing—to evaluate the times of events occurring at places remote from the watch.”

Here it is important to be clear on what we mean when we use the word “event”. Event, as used in special relativity and physics, means a specific point in space-time, that is, a specific point in a given space (region, to put it more colloquially) at a given instant of time. Corresponding to each point in space there is a “local time” indicated by the clock at that position. There is, therefore, nothing like a “global time” common to all points. Each point has a clock of its own, running at a fixed rate. How then do we relate these clocks and evolve a standard of time measurement? For this we have to consider inertial frames of reference.

In physics, a frame of reference is something like (to put it colloquially) a “way” of looking at events from a well-defined vantage point. For instance, if your vantage point is a particular window of a moving train, you observe (if you look out through the window) the world outside moving in the direction opposite to the direction in which your train moves. While, if your vantage point is the railway station then, of course, you see the train moving; depending on your vantage point you either see the “world” moving or the “train” moving. What the principle of relativity (one of the two fundamental postulates of special relativity, the other one being the constancy of the speed of light) says is that if the train “moves” at a constant speed along a straight path, then it is impossible to detect which of the two (the train or the railway station) is “really” moving.

To simplify our discussion, let us consider two observers, A and B, each with a clock of his own, moving away from each other at relative velocity v along a straight line. That is, for A, B is moving away, and for B, A is moving away. Neither of them can ascertain through any experiment, whatsoever, whether he is “really” moving. Nor can any observer make any comment as to which of the two is “really” moving. All we can say is, they are moving apart at a constant relative velocity v. And they are, therefore, inertial frames. As a parenthetical remark, non-inertial frames are those that accelerate in some way or the other.

Einstein suggested a way of synchronizing two clocks in a given inertial reference frame. This allows you to define a “common time” for the clocks, as distinct from a different “local time” for each clock. That is, the “local time” of all clocks in a given inertial frames becomes equal to the “common time” (as it should be, if it is to be called a “common time”). Consider a clock P that is at rest relative to A’s clock. To synchronize the two clocks (A and P), so that both read the same time corresponding to the occurrence of an event, A sends out a light signal towards P; this light signal is reflected at P and comes back to A. The instant at which A sends out the light signal, A sets his clock to zero. For clock P to be synchronous with clock A, P must also read zero at that very instant and agree with A for all subsequent instants. However, the fact that the light from A takes a finite time to reach P (call it the “A-P” time) means that P knows of the zero instant of A only after “A-P” time has elapsed. This is because P can only see A’s reading once the light signal from A reaches him, which happens “A-P time” after the emission of light signal from A. So, P upon observing A’s clock must set his clock, not to zero (which was the past reading of A’s clock), but to “A-P time” (which is the present reading of A’s clock). This “A-P time” is assigned a value L/c (that is, L divided by c), where L is the distance between A and P while c is the speed of light which is a universal constant for all inertial frames.

The universal nature of c is the second fundamental postulate of special relativity. Similarly we can synchronize another clock Q, that is at rest relative to clock B, with the clock B. Having thus synchronized clocks A and P, and clocks B and Q, we now turn to the behaviour of clock B (and, therefore, Q) with respect to clock A (and, therefore, P).

Since clock B moves at a constant velocity v relative to clock A, something peculiar is observed. Without going into the real details of what are known as, the Lorentz Transformations, and how they lead to these peculiar results, I will explain what the results are and what they mean. Say, at some instant, to begin with, clock B’s position coincides with that of clock A. At this instant, A sets his clock to zero, and so does B. So, initially, clocks A and B agree on their readings. The real drama begins now.

To put the results (I will present hereafter) into perspective, I must make one thing explicit: that the time and length scales for B contract. What this means is, as observed by A, B’s clocks runs slower compared to A’s clock (even though, by construction, they are identical), and if A and B carry identical measuring rods, B’s measuring rod contracts compared to A’s measuring rod. This is when A observes B. Similarly, when B observes A, A’s time runs slower compared to B’s time and A’s measuring rods contracts in comparison to B’s measuring rod. The key point is: for A, since his length and time scales contract as observed by B, his measuring rod is of the same length as it was when compared to B’s measuring rod while both were, say, at relative rest. And his time also runs at the same rate. A is not able to sense any change in length or time scales because all the lengths contract and all processes go slower, so that everything is normal to him. It’s only for B that A’s length and time scales contract. A can’t sense any such contraction. Instead A, reciprocally, observes contraction of B’s length and time scales.

These are results that follow from two deceptively simple postulates of special relativity which I will quote from Einstein’s paper:

“1. The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion.

2. Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body.”

If this write-up has been able to elicit from the reader any deeper level of understanding of the nature of time, I strongly urge the reader to read Einstein’s paper (at least the first few non-mathematical pages) for a still deeper understanding.

Ravi Kunjwal

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