# Clocks in special relativity

One book on special relativity says:

Any observer at rest relative to his own timepiece will see that other clocks moving with respect to him run fast - the greater their speed, the faster they are.

Other book says:

Observers measure any clock to run slow if it moves relative to them.

Don't they contradict each other? If yes - who's right? If no - why are they both right? I assume it is a very newbie question, but relativity is one of the topics where you recheck every statement again and again, so I want to be sure.

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Maybe the statement about faster clocks included the Doppler effect, or something like that. It can be hard to understand special relativity because, in practice, observation is limited by the time taken for the light to travel, but the theory ignores this, assuming that observations are back-calculated. –  Alan Rominger Jun 12 '14 at 15:11
Which book claims that they run faster? –  George G Jun 12 '14 at 15:13
@user50381 could you give the title and page number? –  Danu Jun 12 '14 at 15:21
@Danu - "What Is Relativity? L. D. Landau, G. B. Rumer" (p. 47). See preview of the page: books.google.com/… –  user50381 Jun 12 '14 at 15:23
insti.physics.sunysb.edu/~siegel/sr.html "The analog in Euclidean geometry is that a board at an angle looks shorter than one standing up. So if you try to fit a board through a window, from the window's point of view the board looks like it will fit through better. But from the board's point of view it is the window that looks shorter, so the board will not fit. Here is a "paradox", and we didn't even need relativity! Needless to say, the solution is just as simple. " –  Count Iblis Jun 12 '14 at 18:15

In the first book you linked to in the comments, I think the author is trying to say that if an observer sets their watch by a clock in a train station, then gets in a train and travels to another train station, they will find that their watch is slow compared to the station's clock.

This is true because in order to travel from one station to another, the rider must accelerate. During the acceleration, the clock in the station will appear to tick much faster, and during the constant velocity part of the trip (if there is one) the clock in the station will tick more slowly.

These lecture notes do a good job of explaining how acceleration works in SR.

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Hmm, very interesting. It is unclear that they mean acceleration here. They could've said explicitly - "other clocks accelerating with respect to him" instead of "other clocks moving with respect to him". I would definitely have a look at those lectures. Thanks. –  user50381 Jun 12 '14 at 16:06
Yeah, the wording is not clear, but on the next page, there is a little discussion about acceleration, so I think this must be what the author meant. –  George G Jun 12 '14 at 16:08

Here's a basic, basic and rather heuristic explanation. When looking at velocity through spacetime with respect to the time of the person doing the traveling, (the person who left the train station's frame), we can use something called proper velocity. Proper velocity is the distance you travel as measured in the train stations frame, (the frame you will eventually return to), divided by the time in your moving frame, (called the proper time, $\tau$).

The magnitude of your proper velocity with respect to a given frame is fixed at c, the speed of light. If you're sitting at rest in the frame, you're hurtling through the time dimension at the speed of light. As you begin to increase your speed through space, your velocity through time with respect to the train station's fixed frame slows down. The entire time you're moving at this increased speed through space, the train station is moving through its dimension of time more quickly than you are. Here's an important point:

Acceleration matters in that you change your speed through time relative to the train station, but the interval that you remain at the new speed matters as much if not more. This was detailed in a fairly recent AJP article[2].

As for your original question... let's see. As usual in physics paradoxes, everyone is correct. Here's my Cinton-esque interpretation of the two statements above. It depends on what the word 'observer' means. In the first statement, the observer is the traveler. You can tell because he's at rest with respect to his own clock and concerned about the clock back at the train station. This comment fairly screams 'proper time' and 'proper velocity' once you've seen enough of these articles/books. Furthermore, L&L are bigtime into proper spacetime velocity which you'll see written as a two-vector a little further on in the text. I don't have my copy sitting right here, but they do tend to focus on proper velocity.

The observer in the second statement is the usual special relativity 'observer' used in most texts. He stays at the train station and measures everything with respect to his frame's distance and his frame's time. If he had a way to magically look at your clock, then yes, it would be moving slower than his. He can verify for certain that your clock ran slowly if you return.

I hope this helps as it rambled a bit and didn't include much of the underlying math at all. If you'd like to get into that aspect of it, or if you have any other questions, please let me know.

References

1. Geometrization of the Relativistic Velocity Addition Formula, Robert W. Brehme, Citation: Am. J. Phys. 37, 360 (1969); doi: 10.1119/1.1975576, View online: http://dx.doi.org/10.1119/1.1975576

2. Zero time dilation in an accelerating rocket, Ronald P. Gruber and Richard H. Price, Citation: Am. J. Phys. 65, 979 (1997); doi: 10.1119/1.18700 View online: http://dx.doi.org/10.1119/1.18700

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Notice that the first citation says "other clocks moving with respect to him run fast". Obviously, you can switch the moving frames in SR, however, if you want it all to make any sense, you also have to switch the "direction" of time dilatation. Otherwise it all looses all appearance of science and becomes just magic tricks. So, if you are right the authors of this book are just ... magicians. –  bright magus Jun 12 '14 at 19:19
If I had a nickel for every time someone called one of my special rel explanations magic, I'd have 10 cents so far in the last year :) L&L are in fact at the caliber of what some might call magic, in the Arthur Clarke sense of the word, in their conciseness and elegance. The 'direction' of time dilation is a twin paradox misdirection. The attribute that determines whose clock runs slow is what frame is 'started from' and 'returned to'. L&L refer to proper velocity, i.e. moving frame. The second ref. almost certainly refers to laboratory frame. –  dolphus333 Jun 12 '14 at 20:15
Whoops! @bright magus, my bad! I hadn't realized the book referred to wasn't L&L, but L&R? Looks like an interesting book, and thanks to the post's author for the cool reference! I'm still holding to the proper velocity frame though. –  dolphus333 Jun 12 '14 at 20:19
What you are saying boils down to SR being bogus. Looks like Einstein's magic is not working anymore. Fine with me. Pity though other gods are taking over apparently. Not much room left for old good physics. What can one do? Magic is easier than reality, though not necessarily more fascinating. –  bright magus Jun 12 '14 at 20:33
Actually, what I'm saying isn't as clear as I'd hoped it might be. I need to take more time to say it more precisely, which I hope to be able to do in the near future. Thanks for pointing out one of the blocks to clarity. –  dolphus333 Jun 12 '14 at 21:00

Well, if both statements are interpreted sympathetically (and both are so short and improperly phrased that they can use a lot of sympathetic interpretation) then they are arguably consistent with each other and are referring to the same fairly simple experimental situation, described from opposite perspectives:

We have two participants, say $A$ and $B$, who are and remain at rest to each other, and another participant, $J$, who moved from $A$ to $B$; uniformly, with speed $\beta~c$. (This short description is sufficient to describe the setup unambiguously.)

Corresponding to these three participants in this setup there are three durations of particular relevance:

• the duration of $A$ from $A$'s (own) indication of having been left by $J$ until $A$'s (own) indication simultaneous to $B$'s indication of having been met by $J$; symbolically: $\tau A[ \circ_J, \circledS B_J ]$,

• the duration of $B$ from $B$'s (own) indication simultaneous to $A$'s indication of having been left by $J$ until $B$'s (own) indication of having been met by $J$; symbolically: $\tau B[ \circledS A_J, \circ_J ]$, and

• the duration of $J$ from $J$'s (own) indication of having been left by $A$ until $J$'s (own) indication of having been met by $B$; symbolically: $\tau J[ \circ_A, \circ_B ]$.

Obviously (due to $A$ and $B$ being at rest to each other)

$$\tau A[ \circ_J, \circledS B_J ] = \tau B[ \circledS A_J, \circ_J ];$$

and it is not difficult to derive (by appealing to the notions of "mutual rest" and "duration" and "speed", as defined within the theory of relativity) that

$$\tau J[ \circ_A, \circ_B ] = \sqrt{1 - \beta^2} \times \tau A[ \circ_J, \circledS B_J ];$$

and therefore (due to $0 \lt \beta^2 \lt 1$)

$$\tau J[ \circ_A, \circ_B ] \lt \tau A[ \circ_J, \circledS B_J ].$$

The suggested interpretation of the first statement is then to identify $J$ as "any observer (incl. his timepiece)" and $A$ and $B$ as the "other clocks";
while the suggested interpretation of the second statement is to identify $A$ and $B$ as the "observers" and $J$ as "any clock".

There's one more "subtlety" to note:
Earlier in the section "Clocks and Rulers Play Tricks" of Landau/Rumer's brochure (namely in the second paragraph of that section) it is pointed out:

But the watchmaker assured the traveller that his clock is perfectly alright.
[My translation from a German edition of Landau/Rumer's brochure, which I happen to have available at the moment.]

Therefore:

1. All clocks considered in Landau/Rumer's examples are (arguably) "running at equal rates"; there aren't some "running slow(er)" and/or others "running fast(er)",
but instead, corresponding to the inequality shown above, it could be said more correctly that $P$'s duration (or "run") was shorter than the corresponding durations (or "runs") of $A$ and $B$. And

2. It can be noted that the equations and the inequality shown above (incl. their derivation) are only concerned with comparing durations, not "rates" or "readings". These relations are independent of the "rates" of the various clocks being equal and "alright (in comparison to each other)", or not. Instead, these relations are useful for determining in the first place whether the "rates" of different clocks remained equal (as any watchmaker may have readily promised), or not, especially if the clocks to be compared were moving with respect to each other.

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The Special Relativity Theory says that the moving clock is slower. It results from the the transformation equation for time that shows time dilatation:

$$\Delta t' = \Delta t \gamma$$

where $\Delta t'$ is time measured in a reference frame considered stationary, and $\Delta t$ is measured in a reference frame considered moving (with respect to the stationary one) and $\gamma>0$*.

As you can see, whatever time period you choose as $\Delta t$, $\Delta t'$ will be always greater, because $\Delta t$ will be multiplied by $\gamma$. This means that the stationary clock will always measure larger number of seconds for a given number of seconds measured by the moving clock, and therefore the moving clock will always be the slower one according to SR.

*$\gamma = \frac{1}{\sqrt{1 - \tfrac{v^2}{c^2}}}$

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The statement is

"Any observer at rest relative to his own timepiece will see that other clocks moving with respect to him run fast - the greater their speed, the faster they are".

According to one source, Don Koks (a physicist text book author)this statement is true..... on the condition A orbits B.

http://math.ucr.edu/home/baez/physics/Relativity/SR/movingClocks.html

Let A very closely orbit B. The orbit is so close that A is almost touching B; therefore time delays in signal exchanges can be neglected. The faster A orbits B, the FASTER B's clock runs in A's view. Conversely, the SLOWER A's clock runs in B's view.

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