# Is Einstein's characterization of "time" as "the position of the little hand of my watch" definitive and binding in the RT?

In Einstein's very first publication dealing with the Theory of Relativity, effectively as a preamble to all subsequent thought-experimental considerations and descriptions, Einsteín put the suggestion

"[... that instead] of time´´ we substitute the position of the little hand of my watch´´."
[Punctuation marks as in the German original: Ann. Phys. 17, 891 (1905)]

Is this the definitive and binding characterization of "time" within the RT?,

arguably together with (or based on) the understanding that any particular participant (such as "me", or "you", or "your watch" etc.) remains identifiable (as "the same" participant) even for distinct indications.

Is this characterization of "time" consistent or reconcilible with the notions "space-time" or "space-time continuum" which appeared (later) in certain other foundational literature on the RT ?

• What would "binding" even mean beyond uncritical hero worship? It is, in any case perfectly compatible with the notion of proper time as that is what is measured by the observer (on in this case by his watch). Apr 22, 2015 at 4:26
• @dmckee: "What would "binding" even mean beyond uncritical hero worship?" -- This is meant as Einstein's "time" definition being strictly bound to what's called "Theory of Relativity"; in possible distinction to its various interpreations (by Minkowski, Robb, Pauli, Weyl, Born ...). "perfectly compatible with the notion of proper time as that is what is measured by the observer [...]" -- Fine (consider expanding this into an answer). But keep in mind that any one "position of the little hand" is just one indication; not a duration of that observer/watch between a pair of indications. Apr 22, 2015 at 5:34
• p.s. ... v. Laue! Herglotz! ... Apr 22, 2015 at 5:40

Answering this requires a bit of a preamble, so bear with me ...

Any observer can construct a coordinate system to locate points in spacetime. By construct a coordinate system I mean they choose a ruler for measuring distance and a clock for measuring time, then they choose the directions of their three spatial dimensions.

Even before Einstein, coordinate systems were not universal because your coordinate system needn't necessarily match mine. For example you might be using different rulers (miles instead of kilometres) and your axes might point in different directions to mine. But matching up our spatial axes would be pretty trivial, and both of us would agree on the time dimension.

What was new with Einstein's theory of special relativity is that the split between spatial and time dimensions was observer dependant. Both you and I could construct coordinate systems with three spatial dimensions and one time dimension, but if we are in relative motion then my time dimension wouldn't match yours. More precisely, my time dimension would consist of a bit of your time dimension plus a bit of your spatial dimensions, and likewise your time dimension would contain a bit of my spatial dimensions. The effect of our relative motion has been to mix up space and time. The result is that there is no longer a universally agreed time - each observer will now disagree on what is meant by time.

But there is a way out of this. Special relativity introduced a concept called proper time, $\tau$, that is defined by the equation:

$$c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2 \tag{1}$$

Let me explain what this means.

Suppose I have constructed a coordinate system $(t, x, y, z)$ that I'm using for locating points in spacetime. Suppose I'm watching you and I see that in an infinitesimal time $dt$ you move a distance $dx$ in the $x$ direction, $dy$ in the $y$ direction and $dz$ in the $z$ direction. If I feed the $dt$, $dx$, $dy$ and $dz$ into equation (1) it will calculate the proper time associated with your movement.

The crucial thing about this is that the proper time $d\tau$ is an invariant i.e. all observers using any coordinate systems will agree on the value of $d\tau$. I can't emphasise enough how important this is - all the weird stuff in SR like time dilation and length contraction can be derived from the simple principle that $d\tau$ is invariant. This single equation tells you everything you need to know to understand special relativity.

OK that's the end of the preamble, and we are now in a position to answer your question.

Suppose I use equation (1) to calculate my own proper time. In my own coordinate frame I am not moving so $dx = dy = dz = 0$, and that means equation (1) simplifies to:

$$d\tau^2 = dt^2$$

In other words, in my own frame proper time is the same as coordinate time - the proper time is just the time shown by my watch.

And this is Einstein's point. Different observers will disagree about what the time is, but all observers in all frames will agree that my proper time is the time shown by the position of the little hand of my watch.

This is also true in general relativity. What happens in GR is that the equation (1) becomes more complicated. For example near a black hole equation (1) becomes:

$$c^2d\tau^2 = (1-\frac {2GM}{c^2r})c^2dt^2 - \frac1{1-\frac{2GM}{c^2r}} dr^2 - r^2 d\theta^2 - r^2\sin^2 \theta d\phi^2$$

But the same principle applies. The proper time $d\tau$ is still invariant.

• Waaaaaaaay better than my answer, +1 Apr 22, 2015 at 6:30
• Note that the proper time is basically Pythagoras's theorem (so it's the "distance" or "interval" between events) but with some of the terms negative. Commonly the squared space terms are negative and the squared time is positive as shown above, but this is just a convention and you can reverse them. I prefer the time term to be negative, as it's the odd one out. And then I can say that it's just ordinary Pythagoras except that time is imaginary! :) Apr 22, 2015 at 16:19
• John Rennie: "[Please] bear with me ..."-- Even a charitable reading seems to show your attempt, on a physics site, to address a question which is not explicitly about coordinates by conjuring up coordinates. "choose a ruler for measuring distance" -- "Distance" between what, or whom? Even more relevant: "a clock for measuring time" -- "Measuring time" (durations, or at least ratios) between what (of whom)?? (Can you conceive/admit that any "ruler" has identifiable "ends"?; that each "end", at any applicable "point in spacetime" has an identifiable [... cmp. OP title].) Apr 22, 2015 at 18:12
• @Earwicker: "I prefer the time term to be [...]" -- At least here you wrote "time term" (corresponding to "sq. space terms"), and not plainly "time". Note that the (temporal) correspondent of "distance" (or generally: of "spatial separation") is correctly/distinctly called "duration". "proper time is basically [the] "interval" between events)" -- By convention the "interval" $s^2[A,B]$ between two time-like related events is the maximum square of the duration of any participant at both events, from his/her/its indication at one event until his/her/its indication at the other. Apr 22, 2015 at 18:48
• @Lohoris: better still when someone mis-spells it and writes bare with me :-) Apr 23, 2015 at 5:08

Einstein says this, not because your watch is some ancient artifact with power over time, but because in the theory of relativity, time is relative. We can not longer say that the time is blah blah o clock everywhere. The time is different at different points. Therefore, your time is the time on your clock, and this is the "correct" time for your reference frame.

• Jimmy360: "time is relative [...] time is different at different points." -- Indeed; distinct "material points", or "principal identifiable points " (MTW, Box 13.1), or plainly "participants" all have distinct individual "positions of little hands" (generally: "indications"). "Therefore, your time is the time on your clock [...]" -- If you're referring to "coordinate time" (i.e. assigning real number values "$t$" to given distinct indications) then: which of my (possible) many different clocks?? Apr 22, 2015 at 5:51
• @user12262 Yes, I'm referring to coordinate time. The clock you have will obey your perception of time, for your reference frame. Apr 22, 2015 at 5:54
• @user12262 The one you brought with you, rather than the one you left at home. Apr 22, 2015 at 15:39
• Jimmy360: "Yes, I'm referring to coordinate time." -- Then you've missed the intended gist of my question. (But I'm still looking whether "Is there a difference between time and coordinate time, within RT? (If so: What is it?)" has been asked at PSE already.) "The clock you have will obey your perception of time, for your reference frame." -- It ? Will ?? ... (Not each real-valued parametrization of an ordered set (e.g. of your own indications, or of my own indications) is monotonous to that order; and any two such monotonous parametrizations are not necessarily affine to each other.) Apr 22, 2015 at 19:06