# SR and synchronization of clocks!

1. Mar 4, 2014

### ShayanJ

Einstein's article "on the electrodynamics of moving bodies", starts with talking about synchronizing clocks at different places and it is through this that Einstein explains concepts like time dilation and length contraction but in the present books about SR you hardly find things about synchronizing clocks.Why is that?
There is another issue too.Not sure how to state it!
I wanna know what does it tell us about time?
I know that I shouldn't think about it like a problem in techniques of clock synchronization.My problem is I can't see such behavior that time is showing should lead me to what kind of a picture of it?you know?I mean...its right that we don't fully understand QM but at least it tells us what kind of a thing our nature is in microscopic world but Relativity doesn't lead me to a special kind of...character for the nature in the relevant situations!I hope I was clear enough!
Thanks

2. Mar 4, 2014

### WannabeNewton

What? Your post sounds like gibberish but I'll try to answer the question(s) I could gleam from it.

Clock synchronization conventions are simply ways of constructing global time coordinates relative to an observer, or a family of observers, using local times of clocks. The point is local time is specified entirely by the metric tensor (if we use standard clocks) but comparison of local times of distant clocks requires the adoption of a (more or less) arbitrary convention because a global time coordinate built from synchronized clocks is a synchronous time for space-like separated events, which lie outside of our local light cones and so cannot be cast as a physical observable.

3. Mar 4, 2014

### ShayanJ

Yeah...I know...I'm just confused.I don't know why I never succeeded in getting rid of this confusion about SR.All people seem to be thinking that the main puzzle is QM and SR isn't that strange but I'm better in QM than SR!
Anyway...that made things better.But Lorentz transformations don't seem to care about the fact that the two points we're comparing are time-like or space-like separated!
And...I'm still not sure about the connection between things you explained and SR.Because all of SR seems to be about observers moving uniformly relative to each other and I can't fit these things about clocks in different places into this picture.Of course I can see some little things but its hazy! Also about the Lorentz invariance.How does that relate to this bunch of clocks?
Sorry if I'm too confused to be clear,But I think confronting such students is a useful skill for someone who's going to be a professor!

4. Mar 4, 2014

### WannabeNewton

Well the Lorentz transformations assume we have a global time coordinate to start with relative to a given frame. You can't make sense of Lorentz transformations if all you have is local time. But how do we operationally build a global time coordinate relative to a given frame? We can use synchronized clocks. See below.

Well SR isn't just about inertial frames. SR is about accelerating and rotating frames as well but that's besides the point.

One of the Lorentz transformation equations is $t' = \gamma (t - vx)$ between two frames $O$ and $O'$. But $t$, which is a global time coordinate, is not given to you by nature; this isn't Galilean relativity anymore wherein there is a global time coordinate prescribed by nature. If you have a clock with you then all you can do is label the times of events in your direct vicinity-we call this local time. $t$ on the other hand requires you to know the times of events everywhere in space-time, not just of those in your vicinity. But how do you determine this? Your clock after all only tells you what time it is at an event happening right by you. You can't use only your clock.

To augment this, you can have a standard clock placed at each point in space at rest with respect to your clock. Now you synchronize each such clock with your own. What does this mean? Well if your clock hand reads $t$ then the clock hands of all the synchronized clocks in space will read $t$ as well. This means you can now use the reading of your clock, your local time, to label times of distant events of synchronized clocks. Doing this, you can now build a global synchronous time $t$ which lets you label times of all events in space-time. We can now make sense of Lorentz transformations, which require such a synchronous time for each inertial frame.

But there's an issue still lingering. How do we actually synchronize the clocks? Nature doesn't prescribe us with some synchronization convention. We must adopt an arbitrary one. For inertial frames there is a very natural one determined by the metric and is the same synchronization convention given by Poincare (and later Einstein). This is the convention that says to set clock hands by the time delay of light propagation between the clocks being synchronized. Doing this gives us the canonical form of the Lorentz transformations e.g. $t' = \gamma (t - vx)$.

But keep in mind this isn't the only synchronization convention we can adopt. It's just the most natural for inertial frames.

5. Mar 4, 2014

### A.T.

You only synchronize clocks that are at rest to each other. Then those clocks define the time coordinates of events in one frame. Lorentz transformation tells you how to transform these coordinates into another frame, with its own set of synchronized clocks.

6. Mar 4, 2014

### Staff: Mentor

Lorentz transformations are about coordinates, which are the labels that we assign to events.

Time-like and space-like are properties of the relationship between events, and these relationships hold no matter what labels we attach to the events - the events don't care whether we slap labels on them, nor how we transform these labels from one representation to another.

Looking to the Lorentz transforms for an insight about space-like or time-like separation is like looking to the transforms between polar and cartesian coordinates for an insight into why the ratio of the diagonal of a square to the side is $\sqrt{2}$.

7. Mar 4, 2014

### vanhees71

Well, you can be sure, quantum theory is much more mind boggling than classical (classical=non-quantum here) relativistic physics.

Nowadays we are pretty used to high-precision time keeping, i.e., nearly everywhere we have some standard time available. This is provided by the national bureaus of standard nearly everywhere on the world and practically awailable in many forms, e.g., by ntp servers for your computer or the GPS etc.

This is, however by far not a trivial thing! In Einstein's time, around 1900, every city had its local time. Maybe that's why Einstein came to his clock-synchronization idea to address the problem of finding a spacetime compatible with (a) the principle of inertia (i.e., the (apparent) existence of a special class of reference frames, where Newton's principle of inertia holds) and at the same time (b) Maxwell's equations, describing electromagnetic phenomena.

First of all one cannot overemphasize the importance of the paradigm change Einstein brought into the thinking of physics: Before, everybody took the Newton-Galilei space-time model as given and unavoidable and thought, one has to postulate the existence of an omnipresent "aether or ether" to accomodate Maxwell's equations to the principle of inertia, according to which the physics is invariant under changes from one inertial reference frame to another in terms of the Galileo transformation of Newton-Galilean mechanics. Their way out was to postulate that there is this substance named ether defining an absolute inertial reference frame as its rest frame, which would imply that you could measure the absolute velocity of a body relative to this absolute inertial restframe. Others had thought, the Maxwell equations would have to be modified such as to fulfill the principle of inertia in the Galilei-Newton sense, but it was quickly clear that this destroys the success of Maxwell's equations in describing all electromagnetic phenomena known at the time with great accuracy. Also the properties of the hypothetical ether became more and more obscure the longer the physicists tried to make sense of it.

So it came as a great surprise, how simply a quite unknown clerk at the Bern patent office could solve this outstanding problem in the foundations of physics! The idea was to take Maxwell's equations as correct as given and demand the validity of the principle of inertia at the same time. Maxwell's equations together with the principle of inertia, however immediately then imply that the phase velocity of electromagnetic waves are the same (in a vacuum), independent of the speed of their source. Now, this implies the existence of a universal speed in the vacuum, the speed of light. The important thing is that this speed of light is finite, i.e., it takes time to transmit a signal from one place to another (at least as long as you use light).

Now to define a (global) reference frame you must establish an origin for time and space and some periodic thing that provides the standard unit for time intervals. Nowadays this is provided by a certain hyperfine transition radiation of Cesium to define the second as the unit of time. Now you have defined how to measure time at one point in space we take as the origin of our reference frame. Now you also have to define a distance measurement, and that's pretty easy now: Since there is the universal speed of light, you just send out a spherical wave packet, whose wave front defines the distance $c t$ from the origin.

Then, if you want to tell a distant observer, how much ticks your standard clock has made measured from the time-zero point to make him synchronize his clock with yours, you have to take into account that you can do this only by sending out a light signal and thus you have to correct for the finite time $\Delta t=L/c$ this signal needs to reach him. In this way you can place synchronized clocks at any point in space, even very far from the origin.

Thinking than about how an observer from another frame of reference that is moving to the original one with a constant speed relative to this original frame, you come to the conclusion that the Galileo-transformation rules must be abandoned and the Lorentz-transformation rules must be applied.

Analyzing this situation brings you pretty easily to the conclusion that the transformation must involve both spatial coordinates and time, that the transformation must be linear and such that it leaves the "Minkowski form" invariant, i.e., if $(t,\vec{x})$ are the time and space coordinates in the original inertial frame and $(t',\vec{x}')$ are those in the new one, you must have
$$c^2 t'^2 -\vec{x}'^2=c^2 t^2 - \vec{x}^2,$$
where I have assumed that the origin of space and time of both reference frames are identical (for simplicity). That the transformation must be linear becomes clear from thinking about what the principle of inertia tells you: If a free mass point runs a straight line in one inertial frame this should be the case in any other inertial frame, no matter how fast they move against each other and in which direction. So you must map linear motion of a mass point with constant velocity in one inertial frame to a linear motion of this same mass point with (another) constant velocity in the other inertial frame. Thus the transformation must be linear.

Now it is pretty easy to derive the Lorentz transformation for the motion of the new frame with velocity $v$ in $x$ direction in the old frame:
$$c t'=\frac{c t-\beta x}{\sqrt{1-\beta^2}}, \quad x'=\frac{-\beta c t + x}{\sqrt{1-\beta^2}}, \quad y'=y, \quad z'=z.$$
Here, I've used the usual abbreviation $\beta=v/c$.

One remarkable conclusion is that this only makes sense if $|\beta|<1$! This means the speed of the new frame as measured in the old frame must be always smaller than the speed of light. As long as you obey this "absolute speed limit", there is no big obstacle in the conclusions to be drawn from this updated transformation law (i.e., using the Lorentz transformation instead of the Galileo transformation to change from one inertial frame to another).

Take, e.g., the famous phenomenon of "length contraction". Suppose you have a rod of length $L$ along the $x$ axis of the original inertial frame of reference and at rest relative to this frame. Then the one end of the rod is at the origin $x_1=0$ and the other point then necessarily at $x_2=L$. Now, how do you measure the length of the rod in the new inertial frame? You look at the same time $t'=0$ on the coordinates of the two end points in the new frame!

Now the first point is at $t=t'=0$ at $x=x'=0$. However, now the new length is read off by looking at the $x'$ coordinate of the second end point at $t'=0$. In order for that to be compatible with the Lorentz transformation, the read-off of the $x'$ coordinate, which is $L'$, i.e., the length of the rod in the new frame must have been at a time $t_2 \neq 0$ as seen from the old coordinate frame since we must have
$$c t_2'=0=\frac{c t_2-\beta L}{\sqrt{1-\beta^2}} \; \Rightarrow \; c t_2=\beta L.$$
Then the Lorentz transformation tells us that the $x'$ coordinate of this end of the rod in the new frame must be
$$L=\frac{-\beta c t_2+L}{\sqrt{1-\beta^2}}=\frac{L(1-\beta^2)}{\sqrt{1-\beta^2}}=L \sqrt{1-\beta^2}.$$
This shows that the rod, in the reference frame, where it is moving appears to be shorter due to the necessity to read off the coordinates of the ends of the rod at the same time in the new frame.

As soon as you have the Lorentz transformation derived, there's no much trouble anymore to get all these kinematical effects (length contraction, time dilation, relativity of simultaneity, twin paradox...) which seem so unusual to us, we are dealing with speeds very small relative to the speed of light, since then the finite signal time doesn't play a role, but it's only unusual due to this everyday experience, it's not unusual or strange given the fact that there is a universal speed limit at place and that we have to synchronize our clocks taking this speed limit into account.

8. Mar 4, 2014

### WannabeNewton

Sorry I missed this question the first time around. A rigid coordinate system is just a lattice of synchronized clocks and rigid rods so synchronization is just one part of the construction of a rigid coordinate system relative to an observer. Coordinate systems are by themselves not fundamental therefore neither is synchronization. Once we motivate how to construct a coordinate system using synchronized clocks and rods, there's no need for them anymore. Coordinate systems are by themselves just convenient computational tools, nothing more. This is the route usually taken in SR books.

Just to reiterate, simultaneity and synchrony are not fundamental concepts and neither are coordinate systems. You and I can send light signals to one another and make sure the hands of our clocks are always in the same relative position but this has no impact whatsoever on the physical observables that you or I actually make measurements of at the end of the day.

But if you want more detail on clock synchrony and the likes there are a number of philosophy books you can read; these issues are usually treated with much more care in philosophy books. My favorite is the following: https://www.amazon.com/Foundations-Space-Time-Theories-Relativistic-Philosophy/dp/0691020396

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