I Which clock was slower in special relativity?

[entropy1]

So does my example have to be considered in flat spacetime, and why?

 

[Ibix]

Why do we have to consider this example in flat spacetime?

I think you edited this in - didn't see it before.

We consider flat spacetime because the maths is simpler (you don't need integrals right out of the gate) and the concepts are not so complicated. They remain complicated enough with just SR.

 

[PeterDonis]

does my example have to be considered in flat spacetime, and why?

If you're using SR, you're in flat spacetime. SR is only valid in flat spacetime.

 

[Dale]

So does my example have to be considered in flat spacetime, and why?

Your example was given in flat spacetime, but the answer can be generalized. In general, given an arbitrary spacetime equipped with metric $g$, expressed in coordinates $x$, the proper time $\tau$, along a given object’s worldline $P$, is given by
$$\tau_P =\frac{1}{c} \int_P \sqrt{g_{\mu \nu} \ dx^{\mu} \ dx^{\nu}}$$
This is true in any reference frame (inertial or not) in any spacetime (curved or not) for any massive object’s worldline.

So simply calculate this for both and the answer is obtained. It is invariant. This is the generalization of the formula posted by @PeroK in post 3.

 

[pervect]

I see this thread has already gotten very long, and I haven't had time to read it, alas. But there is a fundamentally simple point that I'd like to make, though others may have made it before.

That point is that the process of comparing clocks in an inertial frame is frame dependent.

It is convenient to compare clocks using the Einstein synchronization convention, as it is very standard, and most schemes are equivalent. Operationally, one way of describing this is that a light signal is emitted from the midpoint in the inertial frame, then the light signal is received "at the same time" , when it reaches its destination.

As is explained by the "Einstein's Train" thought experiment, this process is frame dependent. Google for it, or for "the relativity of simultaneity". This is one of the trickiest parts of SR to get across, by the way.

This is a necessary insight to understand how A can think B's clock is slow, and B can think A's clock is slow.

Since it was mentioned that the OP is not afraid of math, I'll go through an abstract argument about why symmetrical time dilation implies simultaneity must be relative. It's not much math - it's just the abstract notion of an invertible map, a 1:1 correspondence between sets, also known as a bijection. It may be overkill, but there's a wiki article on this at https://en.wikipedia.org/wiki/Bijection. For every element in one set (say A) there is one unique corresponding element in set B. And since the mapping is invertible, for every element in set B, there is one, and only one, corresponding element in set A.

The necessary assumption is that comparing times is done by such a one-one map, by a bijection.

Let's take a specific example. Let's say, for simplicity, that A thinks B's clock is running at half seed, and B thinks A's clock is running at half speed.

If there is only one mapping from A's time to B's time, this is logically impossible. With only one mapping, if B's clock is running at half A's rate, then A's clock is running twice as fast as B's clock.

A=1 corresponds to B=2. Logically, B=2 must correspond to A=1, as the map is unique and invertible - it's unique in both directions.

However, if the mapping from A's time to B's time is different from the mapping from B's time to A's time, this is perfectly possible.

In special relativity, this is the case. The mappings are done by clock syncrhonizattion convetions. A is in A's frame, and A uses the convention for this frame to make A's map. B is in B's frame, and B uses frame B's convention to make B's map. The important thing to realize is that A's map is NOT THE SAME as B's map.

If necessary, we can talk more about why we use Einstein's synchronization convention, but at this point it would distract from the main point, I think.

 

[Herman Trivilino]

But then we get to my point: SR clock differences should be the result of velocity, [...]

Differences are also due to position.

Suppose you are flying from Earth to the moon. I have set up beforehand a clock on the moon and I synchronized it with clocks on Earth. You will of course observe my clock running slow, but you will attribute my thinking that your clocks are running slow to an error I made when I did the synchronization.

(Of course, this situation is perfectly symmetrical. I will observe your clock running slow, but if you synchronized the clocks I would attribute your thinking that my clock is running slow to an error you made when you did the synchronization.)

You have to understand this effect, called the relativity of simultaneity, and how it explains the symmetry of time dilation (that is, how each thinks the other's clocks are running slow).

Once you do that you can start understand the answer to the original question you asked. Of course, you have to understand that the two people involved share the same location twice, and can therefore compare the proper time that elapsed between. I recommend that you learn how to draw spacetime diagrams (it's easy) and see that the clock that took the shorter path through spacetime is the clock that shows the smaller elapsed time.

 

[PeterDonis]

It is convenient to compare clocks using the Einstein synchronization convention

This only works if the clocks are at rest relative to each other. But the OP's question is about clocks that aren't at rest relative to each other. The Einstein synchronization convention does not work for that case.

 

[pervect]

This only works if the clocks are at rest relative to each other. But the OP's question is about clocks that aren't at rest relative to each other. The Einstein synchronization convention does not work for that case.

In order to compare the two clocks, one needs a synchronization system. This is a notion of "now". In order to compare the clocks at all, one must pick a frame with an associated clock synchronization mechanism, a notion of "now". It is usual to pick one of the frames in which one the clocks are at rest, but it's also possible to pick a third frame in which neither clock is at rest.

The basic issue that confuses people is that in SR, clock synchronization, the notion of "now", is a frame dependent process in special relativity.

 

[Ibix]

The basic issue that confuses people is that in SR, clock synchronization

Also, "how are you synchronising your clocks" is one of those things that sounds like hopeless pedantry when first raised (of course, it's actually critical to understanding relativity). I suspect that's a big part of why it bounces off people's mental filters the first couple of times you mention it.

 

[PeterDonis]

In order to compare the two clocks, one needs a synchronization system.

One needs a choice of simultaneity convention. Einstein clock synchronization corresponds to a particular choice of simultaneity convention that works for a pair of clocks that are at rest relative to each other.

You can still choose a simultaneity convention for clocks that are not at rest relative to each other, but that convention won't correspond to Einstein clock synchronization.

 

[pervect]

The Einstein conventiion applies to frames of reference, in particular inertial frames of reference, which can be modeled as an infinite array of clocks at rest relative to each other. They fill space-time, so there is a clock present at every possible event. Then these clockss are all synchronized in that frame via the Einstein convention. All the clocks in an inertial frame run at the same rate.

One typically is invited to imagine these "frame clocks" held at a constant distance apart by a rigid structure or framework of some kind.

Using the frame concept, a moving clock can be compared to a co-located "frame" clock. Of course the moving clock will generally be found to run at a different rate than the frame clocks, this is what's called time dilation.

This is all standard SR. GR needs to use more general notions. I tend to use the idea that all that is necessary is to assign unique labels to events. Physically, it is necessary and sufficient to know the unique Lorentz intervals between all sufficiently nearby events to define the space-time geometry.

 

[Dale]

Using the frame concept, a moving clock can be compared to a co-located "frame" clock.

This was, in fact, explicitly done in Einstein's seminal "On The Electrodynamics Of Moving Bodies" paper. It is one of the more confusing aspects of the paper, but it is perfectly legitimate.

 

[Sagittarius A-Star]

It is one of the more confusing aspects of the paper, but it is perfectly legitimate.

In §2, Einstein creates also real confusion:

Let us furthermore suppose that the two clocks synchronous with the clocks in the system at rest are brought to the ends A and B of a rod, i.e., the indications of the clocks correspond to the "time of the stationary system" at the places where they happen to arrive; these clocks are therefore "synchronous in the stationary system".
...
Therefore the observers moving with the moving rod, thus would not find the clocks synchronous, though the observers in the stationary system would declare the clocks to be synchronous.

Source:
https://en.wikisource.org/wiki/Translation:On_the_Electrodynamics_of_Moving_Bodies#%C2%A7_2._On_the_relativity_of_lengths_and_times%2E

In the commented German version of the paper, the comment says:

The clocks described in the 2nd clause are not only needless, but the experimental set-up is impossible. As Einstein shows consecutively, clocks moving relative to each other tick differently fast. Therefore, the clocks attached to both ends of the moving rod cannot be synchronous with the "rest frame".

 

[Dale]

Well, I disagree with the "impossible" comment. It is entirely possible to construct such clocks. In fact, we do something similar with GPS where we simply add a "counter time dilation" factor to the clock frequency. In its rest frame such a clock will not keep correct time, but the goal is specifically to have it keep time in a frame where it is not at rest. It can in fact be done.

I do agree with the "needless" comment. I would even add "unhelpful" or "distracting" or "confusing".

 

[Janus]

Well, I disagree with the "impossible" comment. It is entirely possible to construct such clocks. In fact, we do something similar with GPS where we simply add a "counter time dilation" factor to the clock frequency. In its rest frame such a clock will not keep correct time, but the goal is specifically to have it keep time in a frame where it is not at rest. It can in fact be done.

I do agree with the "needless" comment. I would even add "unhelpful" or "distracting" or "confusing".

For example, it is perfectly possible to to have an arrangement like this, according to an observer at rest with respect to the lower row of clocks:

But according to an observer at rest with respect to the upper row of clocks, this is what is happening:

 

[Dale]

@Janus nice graphics!

 

[Herman Trivilino]

Well, I disagree with the "impossible" comment. It is entirely possible to construct such clocks. In fact, we do something similar with GPS where we simply add a "counter time dilation" factor to the clock frequency.

I'm not sure this is more than a quibble, but if we say a clock is something that measures time, these "clocks" don't qualify in the sense that they don't measure their own proper time. That is perhaps the context in which the "impossible" claim was made.

 

[Nugatory]

I'm not sure this is more than a quibble, but if we say a clock is something that measures time, these "clocks" don't qualify in the sense that they don't measure their own proper time.

I don't think that's a quibble - there's a subtlety here about how we measure and use time in our daily lives.

Consider the clocks we use most often: A wall clock synchronized by the 50Hz or 60Hz line voltage signal from the power utility, anything synchronized to a standard time signal broadcast over the air or the internet, the time displayed on airport monitors, anything that shows UTC with or without a timezone offset... None of these are measuring proper time. They are displaying a coordinate time that is never too far off from local proper time and that's what makes them useful to civilization.

Everyone calls these "clocks" but you're right that in the context of a discussion of special relativity, they aren't clocks. That's why finding them called that in Einstein's writing is so confusing. At the turn of the 20th century the distinction between civilization's coordinate time and proper time was not recognized; Einstein of course understood it, but would have no reason not to use the word "clock" as it was generally used then.

 

[Dale]

I'm not sure this is more than a quibble, but if we say a clock is something that measures time, these "clocks" don't qualify in the sense that they don't measure their own proper time. That is perhaps the context in which the "impossible" claim was made.

Yes, I can see that viewpoint. Certainly that could be.

In any case, you can construct a device that behaves as Einstein described and use it as he described using it. But I can see the viewpoint that would refuse to call that device a clock for the reason you mention.

 

[vanhees71]

I also think it helps a lot to define a clock as a device that measures its proper time in any frame of reference. That doesn't mean that we can't use other measures of time. The GPS is an example that the spacetime structure of GR is the best description we have today. It wouldn't work, if the precise synchronization of the clocks of the satellites and on ground were not considering all relativistic effects.