# How to theoretically define a concrete operation to perform in order to measure the length of an object?

EDIT: Some answerers pointed out that the title of this question is in contrast with what is actually been asked in the body of the question, so I changed the title accordingly. The original title was "How to theoretically define a measure of length?". This formulation has a mathematical answer, namely that the length of an object is the maximum distance between two of its points, where the notion of distance derives from the particular metric the space is equipped with. The more physical-flavoured question follows.

Is there a way of defining the "length" of some object? We agree that the length of an object is the distance between two points $A$ and $B$. The naive approach would be using a meter stick in the usual way: this method does not take into account a possible problem, that of the simultaneity of the measurement; that is, information from point $A$ could be obtained in a different time with respect to information from point $B$.

So, the next logical thing to do is setting up some detector in a point $C$ such that information travelling from point $A$ and $B$ will be received at $C$ in the same moment.

Is there a way of doing this? In relativity such measurements are obtained via flashes of light. How do we know that this is the best method? I am really confused, up to the point that I am not sure of what exact meaning to attach to the term "length".

• You should more more precisely define, what the problem is about. There is no issue with simultaneity if the object being measured is located in the same inertial frame of reference as the measurer. – bright magus Nov 21 '14 at 11:06
• The question is: what is a definition of "length" in current physics? – marco trevi Nov 21 '14 at 11:52
• @Alba, those definitions depend on the notion of distance between two points, which again depends on the geometry of the space you are working in. – marco trevi Nov 21 '14 at 15:15
• @marcotrevi if you want to edit the title to better reflect what you're actually asking, go right ahead and do it yourself. No need to flag for moderator attention. :-) – David Z Nov 22 '14 at 6:43

The naive approach would be using a meter stick in the usual way: this method does not take into account a possible problem, that of the simultaneity of the measurement;

Well, the usual way seems to keep the two "meter stick" ends (or markings) in touch with the two ends (or elements) of the object under consideration sufficiently long (i.e. typically for much longer than the ping duration between these ends), so that this particular simultaneity problem does not arise.

There is however a problem with this naive approach, namely:
how to determine whether the opposite ends (i.e. either the two ends of the object under consideration, or likewise the two ends of the "meter stick") had actually remained at rest to each other while the "meter stick had been alongside the object", such that some particular distance value (e.g. "one meter") could rightfully be attributed to either pair at all.

Of course, that particular problem is rarely considered in "everyday technical" applications of (what's perhaps calibrated and sold as) "meter sticks"; but it's nevertheless a profound conceptual problem to experimental physicist.

But let's now turn to the simultaneity problem, i.e. specificly:
considering two separate participants who are (known to be) at rest to each other, how to determine in mutual agreement which indication of one had been simultaneous to a given indication of the other.

the next logical thing to do is setting up some detector in a point $C$ such that information travelling from point $A$ and $B$ will be received at $C$ in the same moment.

Right. (First to note: observations by any one participant, such as $C$, "at the same moment" are also called "coincident observations" of that participant.)
That's the setup described by Einstein in his coordinate-free definition of "(how to determine) simultaneity" from 1917 ( "Relativity. The Special and the General Theory."; chap. 8: "On the Idea of Time in Physics". (There exist earlier and independent descriptions of such a setup, for instance by Poincaré; but its definitive importance had apparently first been expressed by Einstein.)

Is there a way of doing this?

The question is (and it is an excellent question, as far as I understand):
How exactly to identify a suitable participant $C$ (or in Einstein's description rather: participant $M$) uniquely?,
because, in general, there can be plenty of different separate participants whose observation of some indication of $A$ had been coincident with some indication of $B$.

The specification given by Einstein is of course

• that $M$ should be identified as "middle between $A$ and $B$",

• that the distance ratios determined between these three participants should satisfy: $\frac{AM}{AB} + \frac{MB}{AB} = 1$
(which relates to your question title) and

• implicitly: that $A$, $B$, and $M$ should (pairwise) have been at rest to each other (which relates to the discussion at the beginning of this answer).

In relativity such measurements are obtained via flashes of light.

Right. (That is, for addressing the entire list of measurement tasks above.)
Besides giving the simultaneity defintion as specific example case ("If the observer perceives the two flashes of lightning at the same time, then [...]") Einstein also noted that

How do we know that this is the best method?

Because this method is the most perfectly unambiguous and comprehensible we know.
If anyone reports "I have seen a particular flash indication of $A$, and a particular flash indication of $A$; and my very first observations of these two flash indications had been at the same moment." then we can comprehend outright that report; and we may grant that anyone reporting that has at least the comprehension "to know what he's talking about", just as you and me.

(Contrast this with other, perhaps "more practical" or "more traditional" methods where we and anyone is supposed to understand outright which clocks were and remained "good" and/or "undisturbed", which clocks had been "transported slowly", which artefacts had "preserved their shape", who "felt free", or somesuch.)

Now at least to begin to address the above list of measurement tasks by means of the relativistic method based on signalling and judging coincidence (or else: sequence) of one's observations, we can require that

• for any indication which $M$ had stated $M$ observed $A$'s corresponding reflection indication coincident to observing $B$'s corresponding reflection indication,

• for any indication which $A$ had stated $A$ observed $B$'s corresponding reflection indication coincident to observing that $M$ had observed that $A$ had observed $M$'s corresponding reflection indication,

• for any indication which $B$ had stated $B$ observed $A$'s corresponding reflection indication coincident to observing that $M$ had observed that $B$ had observed $M$'s corresponding reflection indication.

(These conditions are still not sufficient for establishing that $M$ had been at rest with respect to $A$ and $B$, and thus for identifying $M$ uniquely as the "middle between $A$ and $B$". But at least this illustrates all conditions which can properly evaluated by $A$, $B$, and $M$ themselves. Further conditions necessarily involve additional participants.)

Finally, the length attributed to the pair $A, B$, as characterizing their geometric relation to each other (besides having been at rest to each other) is expressed in terms of the ping durations between them (i.e. $A$'s duration for any indication until observing $B$'s corresponding reflection indication, or likewise $B$'s duration for any indication until observing $A$'s corresponding reflection indication): $$AB := \frac{c}{2} \tau A_{\circ}^{\circledR B} = \frac{c}{2} \tau B_{\circ}^{\circledR A},$$ which is therefore indeed a distance, rather than a quasi distance;
and where "$\frac{c}{2}$" is foremost a symbolic, non-zero factor for distinguishing ping durations from other incidental durations which may be determined. Due to the definition of "(how to measure) speed", the symbol $c$ is only subsequently identified as universally representing "signal front speed".

Everything you said seems correct. The only thing that looks like it needs to be resolved is this:

How do we know that this is the best method?

Einstein synchronization gives us a way of measuring times and lengths, but it's equivalent to various other methods. For example, Einstein-synchronizing two clocks at a distance is equivalent to synchronizing the two clocks next to each other, then transporting one of them slowly to the other location. (Measuring time and length are equivalent things, since they can always be related by $c$, which is invariant.)

This is a common situation where one applies the operationalist approach: http://plato.stanford.edu/entries/operationalism/ Operationalism says that we define a thing by specifying how to measure it. We usually have more than one way of measuring the same thing, and we need to make sure that these different methods are consistent in the regimes where they are both applicable.

So really you have a family of methods of measuring length, all of them consistent with each other, and the question would be why all these methods are good, as opposed to some other method that is bad. Basically we choose these methods because they have nice transformation properties. Length transforms according to the usual relation $L=L_0/\gamma$, which is simple.

• Ben Crowell: "Length transforms according to [...]" -- This seems to promote an improper notion of "length". (And I doubt that's what the OP had in mind.) "We usually have more than one way of measuring the same thing" -- Yes: different measurement operators may be applied to one-and-the-same given set of observational data. But No: genuinely different measurement operators are genuinely different. "and we need to make sure that these different methods are consistent in the regimes where they are both applicable." -- "Making sure" is a slippery slope in experimental physics. – user12262 Nov 19 '14 at 13:47
• The question was "how do you theoretically define length" does your post answer the OP question? – bobie Nov 19 '14 at 14:10
• @user12262 and bobie: You might benefit from reading the linked article on operationalism. – user4552 Nov 21 '14 at 14:45
• Ben Crowell: "You might benefit from reading the linked article on operationalism." -- I agree that reading plato.stanford.edu/entries/operationalism is beneficial; and I've done so several times in the past couple of years. (Even versions from before 2009, if my memory serves.) And I'm always puzzled by the "Critiques" and by Bridgman's ultimate defeatism. Which part of "[All our well-substantiated space-time propositions amount to the determination of space-time coincidences]()" did he not understand, not know about, or not willing to suppose as universally shared ability ? ... – user12262 Nov 21 '14 at 16:33
• "... A with C is only simultaneous with the coincidence of B with D in one particular frame..." , "..See the paragraph beginning with "Einstein synchronization..." , ".bobie: You might benefit from reading the linked article.." – Ben Crowell , you might benefit if, before citing a philosophy article, if you studied philosophy so that you might fully realize its conceptual implications, and also discern when it is the case to quote Einstein and simultaneity. :) – bobie Dec 3 '14 at 11:41