Do the Principles of Special Relativity Require Synchronized Clocks in K and k?

[JM]

Consider the following properties of SR clocks. The coordinate systems are the usual K( X,Y,Z,T) and k( x,y,z,t) where k is the coordinate whose origin moves in the positive direction of K.
1.The clocks of K are synchronized with each other using the procedure given in the 1905 paper. Also the clocks of k are synchronized with each other similarly.
2.When the origins of K and k are located at the same position the clocks are set to zero.
3. A fundamental principle of physics requires all the terms of an equation to be expressed in the same units. Thus in the equation T=t/gamma, an example of time dilation, T and t must have the same units, and thus tick at the same rate.

Don't these properties lead to the conclusion that the clocks of K and k must always be synchronized?

 

[lugita15]

Consider the following properties of SR clocks. The coordinate systems are the usual K( X,Y,Z,T) and k( x,y,z,t) where k is the coordinate whose origin moves in the positive direction of K.
1.The clocks of K are synchronized with each other using the procedure given in the 1905 paper. Also the clocks of k are synchronized with each other similarly.
2.When the origins of K and k are located at the same position the clocks are set to zero.
3. A fundamental principle of physics requires all the terms of an equation to be expressed in the same units. Thus in the equation T=t/gamma, an example of time dilation, T and t must have the same units, and thus tick at the same rate.

Don't these properties lead to the conclusion that the clocks of K and k must always be synchronized?

There's two problems here. First, in point 2 you said that K and k are synchronized when they are in the same position at the same time. However, the meaning of "at the same time" depends on what coordinate system you're using. K may believe that the origins were at position X at the same time, but k will disagree. So you are presupposing that simultaneity is absolute, and you're using that assumption to try to disprove time dilation. But in relativity the question of whether two events are simultaneous depends on what inertial frame you're in.

Also, in response to point 3, just because two variables have the same units doesn't mean they're equal. Think about it: if T=5s and t=10s, then they both have the same unit, which is the second, but they have different magnitudes.

 

[Dale]

3. A fundamental principle of physics requires all the terms of an equation to be expressed in the same units. Thus in the equation T=t/gamma, an example of time dilation, T and t must have the same units, and thus tick at the same rate.

By this logic, since the perimeter, p, of a square is related to the length of a side, s, by the formula s=p/4, s and p must have the same units (true) and thus be the same length (false).

 

[JM]

By this logic, since the perimeter, p, of a square is related to the length of a side, s, by the formula s=p/4, s and p must have the same units (true) and thus be the same length (false).

I gather that you agree that T and t must be expressed in the same units. What that means is that the clocks used to measure T and t must tick at the same rate, not that T and t are equal. So, are the clocks ( not T and t ) synchronous?

 

[JesseM]

I gather that you agree that T and t must be expressed in the same units. What that means is that the clocks used to measure T and t must tick at the same rate, not that T and t are equal. So, are the clocks ( not T and t ) synchronous?

"Tick at the same rate" is meaningless unless you pick a coordinate system to measure the coordinate time between successive ticks of each clock so you can compare their rates. And there is no inertial frame where they tick at the same rate.

Consider two Cartesian coordinate systems on a 2D plane, with the x-y axes of one coordinate system rotated at an angle relative to the x-y axes of the other. Each coordinate system can measure distance in meters, but if you draw two dots on the plane and ask for the difference between the y-coordinates of the dots in each coordinate system, they can disagree even though they both use the same units to measure distance along the y-axis. Similarly, if you pick two events in spacetime and ask for the difference between the t-coordinates of the events in two different inertial frames, they can disagree, because their time axes are rotated relative to one another.

 

[Dale]

So, are the clocks ( not T and t ) synchronous?

No, they are not.

 

[GRDixon]

T and t must have the same units, and thus tick at the same rate.

Don't these properties lead to the conclusion that the clocks of K and k must always be synchronized?

From the perspective of K, more time elapses between ticks of the k clocks than between ticks of the K clocks. From the perspective of k, more time elapses between ticks of the K clocks than between ticks of the k clocks. That is the general meaning of time dilation, and time dilation is completely symmetric among inertial frames. Similar remarks apply to the synchronization of distributed clocks. In the opinion of K, all of the clocks at rest in K are synchronized, and the clocks at rest in k are out of synch. It's all implicit in the Lorentz transformations or, if you prefer, the Lorentz transformations can be derived on the assumption of length contraction, time dilation, and the relativity of simultaneity.

 

[jason12345]

Consider the following properties of SR clocks. The coordinate systems are the usual K( X,Y,Z,T) and k( x,y,z,t) where k is the coordinate whose origin moves in the positive direction of K.
1.The clocks of K are synchronized with each other using the procedure given in the 1905 paper. Also the clocks of k are synchronized with each other similarly.
2.When the origins of K and k are located at the same position the clocks are set to zero.
3. A fundamental principle of physics requires all the terms of an equation to be expressed in the same units. Thus in the equation T=t/gamma, an example of time dilation, T and t must have the same units, and thus tick at the same rate.

Don't these properties lead to the conclusion that the clocks of K and k must always be synchronized?

All clocks tick at the same rate in their proper frame. That doesn't mean they all measure the same time interval between the same two events.

According to your logic, if the sun casts a shadow of a stake in the ground, then all rulers should measure the same width of the shadow at different locations because their units of measurement are identical.

 

[JM]

All clocks tick at the same rate in their proper frame.

Jason, can we examine this statement a bit? Do you mean that the rate of ticking of the clocks of K with respect to the K frame is the same as the rate of ticking of the clocks of k with respect to the k frame? If so, that is the point I am trying to make.

This property doesn't make T and t equal, perhaps I should have said that the relation T = t/gamma implies that the clocks measuring T and t are ticking at the same rate.
JM

 

[JM]

No, they are not.

Would you mind explaining in what respect you disagree with the properties of clocks?

 

[JM]

"Tick at the same rate" is meaningless unless you pick a coordinate system to measure the coordinate time between successive ticks of each clock so you can compare their rates. And there is no inertial frame where they tick at the same rate.

Similarly, if you pick two events in spacetime and ask for the difference between the t-coordinates of the events in two different inertial frames, they can disagree, because their time axes are rotated relative to one another.

Jesse, Think about it this way: If we want to compare the duration of an event as seen by two observers isn't it necessary that both observer use clocks that tick at the same rate? If the clocks tick at different rates how can we tell whether the disagreement is due to axis rotation or tick rate?

 

[Dale]

Would you mind explaining in what respect you disagree with the properties of clocks?

You asked if the clocks in the different reference frames were synchronized. The answer is "no". This is called the relativity of simultaneity. A system of clocks which is synchronized in one frame will not be synchronized in any other frame.

 

[jeblack3]

There's two problems here. First, in point 2 you said that K and k are synchronized when they are in the same position at the same time. However, the meaning of "at the same time" depends on what coordinate system you're using. K may believe that the origins were at position X at the same time, but k will disagree.

That's incorrect. "At the same position at the same time" is a frame-invariant concept. Simultaneity is relative, but when two events are simultaneous in frame K, the time difference between the events in another frame k is proportional to the distance between the events. If two events occur at the same time and place in one frame, they do so in all frames; the theory would hardly make sense otherwise.

 

[lugita15]

That's incorrect. "At the same position at the same time" is a frame-invariant concept. Simultaneity is relative, but when two events are simultaneous in frame K, the time difference between the events in another frame k is proportional to the distance between the events. If two events occur at the same time and place in one frame, they do so in all frames; the theory would hardly make sense otherwise.

I'm not so sure about that. Suppose you have a train of length L traveling speed v to the right. If two beams of light are emitted from either end of the train, at the same time according to the train's reference frame, then they will arrive at the center of the train at the same time according to the train's reference frame. But if I'm not mistaken, according to an observer on the ground they will have arrived at the center of the train at different times. Correct me if I'm wrong, though, since I haven't really studied special relativity in great detail. What I know comes mainly from popular books, which can be misleading.

 

[JesseM]

Jesse, Think about it this way: If we want to compare the duration of an event as seen by two observers isn't it necessary that both observer use clocks that tick at the same rate? If the clocks tick at different rates how can we tell whether the disagreement is due to axis rotation or tick rate?

I don't understand what you mean by "due to axis rotation or tick rate"--how are those two distinct possibilities? The fact that the axes of one frame appear rotated in a particular way from the perspective of another frame, combined with the fact that a physical clock at rest in a given frame always keeps pace with that frame's time coordinate, implies that a clock which is moving in some frame must tick at a different rate relative to the frame's time coordinate than a clock at rest in that frame. There's no way the first part could be true without the second part also being true! It's not meaningful to ask whether the difference in time intervals between two events is "due to axis rotation or tick rate" unless you can explain what it would mean to have a situation where it was due to axis rotation but not tick rate, or a situation where it was due to tick rate but not axis rotation.

 

[JM]

I think you all are missing the point. Your comments don't seem to be related to the properties listed in the first post. Why jump to Relativity of Simultaneity? Does the idea that k sees Ks clocks as out of synch change the property that K sees his own clocks as in synch? Do you think that T and t can be expressed in different units in the equation t = T/m?

 

[JesseM]

I think you all are missing the point. Your comments don't seem to be related to the properties listed in the first post.

My analogy of spatial coordinate systems in 2D directly addressed your first post, can you tell me if you think it's not analogous to the situation you were discussing involving clocks?

Why jump to Relativity of Simultaneity? Does the idea that k sees Ks clocks as out of synch change the property that K sees his own clocks as in synch?

It's not something they "see", it's how they each define what it means for two clocks to be "synchronized" (i.e. how they assign time-coordinates to events, since two clocks are synchronized in a given frame if they both show the same reading at the same time coordinate). In much the same way, if you have two x-y coordinate systems on a 2D plane, with one system's y-axis rotated relative to the other system's y-axis, then they have a different definition of what it means for two points to have the "same y-coordinate".

Do you think that T and t can be expressed in different units in the equation t = T/m?

If that's supposed to be the time dilation equation where t is the time between events in the frame where they happen at the same location, T is the time between the same events in another frame moving relative to the first, and m is the relativistic gamma-factor, then in that case t and T must be in the same units. In terms of the spatial analogy, this is just like how if you have two points on a 2D plane, different coordinate systems can disagree on the difference between the y-coordinates of those points, with dy = A*dy' (where A is some constant), in spite of the fact that both dy and dy' are expressed in the same units. If you think there's some problem with the idea that two frames can use the same units yet disagree on the time between a pair of events, you need to explain whether you also have a problem with the idea that two 2D coordinate systems can use the same units yet disagree on the difference in y-coordinates between a pair of dots, and if you don't have a problem with the second you need to explain what the relevant difference is.

 

[JM]

My analogy of spatial coordinate systems in 2D directly addressed your first post, can you tell me if you think it's not analogous to the situation you were discussing involving clocks?

I'm not sure which part of my first post you are referring to. If its point 3, I think my statement is not clear. My point there is that the units of t and T must be the same, not that T and t are the same. If that's the point of your analogy then we agree.

It's not something they "see", it's how they each define what it means for two clocks to be "synchronized" (i.e. how they assign time-coordinates to events, since two clocks are synchronized in a given frame if they both show the same reading at the same time coordinate). In much the same way, if you have two x-y coordinate systems on a 2D plane, with one system's y-axis rotated relative to the other system's y-axis, then they have a different definition of what it means for two points to have the "same y-coordinate".

Again, my question is 'does ks definition of the synch of Ks clocks change the fact that Ks clocks are in synch according to K?'

If that's supposed to be the time dilation equation where t is the time between events in the frame where they happen at the same location, T is the time between the same events in another frame moving relative to the first, and m is the relativistic gamma-factor, then in that case t and T must be in the same units. In terms of the spatial analogy, this is just like how if you have two points on a 2D plane, different coordinate systems can disagree on the difference between the y-coordinates of those points, with dy = A*dy' (where A is some constant), in spite of the fact that both dy and dy' are expressed in the same units. If you think there's some problem with the idea that two frames can use the same units yet disagree on the time between a pair of events, you need to explain whether you also have a problem with the idea that two 2D coordinate systems can use the same units yet disagree on the difference in y-coordinates between a pair of dots, and if you don't have a problem with the second you need to explain what the relevant difference is.

Doesn't this refer to point 3, discussed above?

 

[JesseM]

Doesn't this refer to point 3, discussed above?

Points 1 and 2 can also have analogues in the geometric analogy. In place of a bunch of clocks at rest in K, you can have a bunch of rulers which are aligned parallel to the y-axis of one of the 2D coordinate systems (we can call this coordinate system K as well). In this case, saying that the clocks at rest in K are synchronized is equivalent to saying that these parallel rulers are all arranged so that any given marking--say, the 3-meter mark on each ruler--is at the same level relative to the y-axis of K, so that identical markings on each ruler have the same y-coordinate in K. In this case the parallel rulers are analogous to the worldlines of the clocks, and the marks on each ruler are analogous to the tick-events on the worldline of each clock. And of course, you can have a different set of rulers which are arranged to be parallel to the second 2D coordinate system k.

 

[JM]

Points 1 and 2 can also have analogues in the geometric analogy. In place of a bunch of clocks at rest in K, you can have a bunch of rulers which are aligned parallel to the y-axis of one of the 2D coordinate systems (we can call this coordinate system K as well). In this case, saying that the clocks at rest in K are synchronized is equivalent to saying that these parallel rulers are all arranged so that any given marking--say, the 3-meter mark on each ruler--is at the same level relative to the y-axis of K, so that identical markings on each ruler have the same y-coordinate in K. In this case the parallel rulers are analogous to the worldlines of the clocks, and the marks on each ruler are analogous to the tick-events on the worldline of each clock. And of course, you can have a different set of rulers which are arranged to be parallel to the second 2D coordinate system k.

So, where do we stand? Given my clarification of point 3, do you agree with the properties listed in post 1, and do you think that they lead to the conclusion that the clocks of both frames K and k are in synch?

 

[JM]

There's two problems here. First, in point 2 you said that K and k are synchronized when they are in the same position at the same time. However, the meaning of "at the same time" depends on what coordinate system you're using. K may believe that the origins were at position X at the same time, but k will disagree. So you are presupposing that simultaneity is absolute, and you're using that assumption to try to disprove time dilation. But in relativity the question of whether two events are simultaneous depends on what inertial frame you're in.

Also, in response to point 3, just because two variables have the same units doesn't mean they're equal. Think about it: if T=5s and t=10s, then they both have the same unit, which is the second, but they have different magnitudes.

In response to your first comment, point 2 can be made using the Lorentz transforms. The position of the origin of k is identified by x = 0, enter this in the transform x = m( X -vT) to get X = vT, so when T = 0, X = 0 also. So when the origins coincide ( x = X = 0 ) , T = 0. Enter X = T = 0 in the time transform ct = m( cT -vX/c) to get t = 0. Thus when the origins coincide the clocks of K and k are = 0.
OK?

 

[JM]

You asked if the clocks in the different reference frames were synchronized. The answer is "no". This is called the relativity of simultaneity. A system of clocks which is synchronized in one frame will not be synchronized in any other frame.

What I'm doing is separating the properties of clocks from the properties of light, as expressed by the light postulate. Relativity of Sumultaneity ( Ros) is a property of light. It can be stated as 'Clocks at rest with k and synchronized by exchange of light signals, are not synchronized with respect to K because with respect to K the clocks of k are moving and the light is still moving at c.' This requires that the clocks of k be referred to the frame K. In my properties the clocks of k are not referred to K, but only to k itself. Ros does not deny that ks clocks are in synch with respect to k, does it?

 

[JM]

I think you all are missing the point. Your comments don't seem to be related to the properties listed in the first post. Why jump to Relativity of Simultaneity? Does the idea that k sees Ks clocks as out of synch change the property that K sees his own clocks as in synch? Do you think that T and t can be expressed in different units in the equation t = T/m?

Note to All
Several of you have commented on my point 3 and on Relativity of Simultaneity. If I haven't answered you specifically, I have tried to answer someone on the topic. So please check all the posts for your answer, and feel free to respond to questions I have posed to others. I value all your contributions.

 

[JesseM]

So, where do we stand? Given my clarification of point 3, do you agree with the properties listed in post 1, and do you think that they lead to the conclusion that the clocks of both frames K and k are in synch?

You still aren't addressing my analogy. Again, all your points 1-3 seem to be directly analogous to the situation with two 2D coordinate systems with axes rotated relative to one another, and each coordinate system having a set of rulers parallel to that system's y-axis with a given set of rulers being "level" (analogous to clocks being synchronized) in the sense that a given marking on one ruler will have the same y-coordinate as all the other parallel rulers in that set (so for example the 3-meter marks on each ruler in the set which are parallel to the y-axis of K would all have the same y-coordinate in K, and likewise the 3-meter marks on each ruler in the set which are parallel to the y-axis of k would all have the same y-coordinate in k). But this would not imply the conclusion that the rulers in the set parallel to the y-axis of K are "level" with the rulers in the set parallel to the y-axis of k, so the logic of your argument doesn't work when applied to this analogous situation, implying it doesn't work in the situation with clocks either. If you don't understand the analogy, or you think there is some important way that it's not analogous to the situation involving clocks, please explain.

 

[Dale]

What I'm doing is separating the properties of clocks from the properties of light, as expressed by the light postulate. Relativity of Sumultaneity ( Ros) is a property of light. It can be stated as 'Clocks at rest with k and synchronized by exchange of light signals, are not synchronized with respect to K because with respect to K the clocks of k are moving and the light is still moving at c.' This requires that the clocks of k be referred to the frame K. In my properties the clocks of k are not referred to K, but only to k itself. Ros does not deny that ks clocks are in synch with respect to k, does it?

I'm sorry, but I don't really understand what you are saying here. It might be helpful if you re-do your description of the setup.

K and k are different inertial coordinate systems, correct? K and k are related to each other via the Lorentz transform, but each can be defined independently of the other via a system of clocks and rods at rest wrt each other and synchronized using Einstein's convention. If two clocks are synchronized wrt K then they are not synchronized wrt k and vice-versa. Is that clear enough?

 

[JM]

I'm sorry, but I don't really understand what you are saying here. It might be helpful if you re-do your description of the setup.

K and k are different inertial coordinate systems, correct? K and k are related to each other via the Lorentz transform, but each can be defined independently of the other via a system of clocks and rods at rest wrt each other and synchronized using Einstein's convention. If two clocks are synchronized wrt K then they are not synchronized wrt k and vice-versa. Is that clear enough?

Yes, that's clear but how is it relavent?
I wonder if you would be agreeable to examining the properties given in post 1. Can we take them one at a time . Please read property 1. Do you agree with it?

 

[Dale]

Yes, I said that each system of clocks was synchronized using Einstein's convention.

 

[JM]

Yes, I said that each system of clocks was synchronized using Einstein's convention.

Good. Can we move to property 2. I have elaborated on it in a recent post to lagita. Do you agree with this property?

 

[Dale]

Yes, as I said K and k are related to each other by the Lorentz transform so for (X,Y,Z,T) = (0,0,0,0) we have by the Lorentz transform that (x,y,z,t)=(0,0,0,0) also. Lugita's point is also correct that for (X,Y,Z,T) = (A,0,0,0) we have by the Lorentz transform that t does not equal zero for A not equal to zero due to the relativity of simultaneity.

 

[JM]

Yes, as I said K and k are related to each other by the Lorentz transform so for (X,Y,Z,T) = (0,0,0,0) we have by the Lorentz transform that (x,y,z,t)=(0,0,0,0) also. Lugita's point is also correct that for (X,Y,Z,T) = (A,0,0,0) we have by the Lorentz transform that t does not equal zero for A not equal to zero due to the relativity of simultaneity.

I accept Lugitas point.
Re point 3. Do you agree with this point as amended? The tick rates are the same but T and t are not necessarily the same.

 

[Dale]

I don't know what you mean by that. Also, what is your amended version of your third point.

 

[JM]

I don't know what you mean by that. Also, what is your amended version of your third point.

The early replies to my first post seemed to interpret point 3 to say that T and t must be the same. So I amended ( explained ) that I meant only that the units of measure of t and T must be the same. If the units are the same then the clocks measuring T,t must advance at the same rate. Does this help?

 

[Dale]

The early replies to my first post seemed to interpret point 3 to say that T and t must be the same. So I amended ( explained ) that I meant only that the units of measure of t and T must be the same.

Yes, if t is measuring in seconds and T is measuring in hours then you cannot use equation 3 directly, you must include a conversion factor. Is this what you mean?

If the units are the same then the clocks measuring T,t must advance at the same rate.

In their respective rest frames, yes.

Think of it this way. When we say that we have two identical rulers we mean that if we measure the same distance with both rulers we get the same result. A clock is just a ruler which measures timelike intervals. So when we say that we have two identical clocks we mean that if we measure the same interval with both clocks we get the same result.

If we use one ruler to measure the base of a right isosceles triangle and the other identical ruler to measure the hypotenuse then we will find that the measurements differ by a factor of $\sqrt{2}$. We attribute this difference to a difference in the thing being measured instead of to the rulers, since they are identical. Similarly, if we have one clock measure a vertical line through spacetime (at rest) and another clock measure a diagonal line through spacetime (in motion) then we will find that the measurements differ by a factor of $\gamma$. We again attribute this difference to a difference in the thing being measured instead of to the clocks, since they are identical.

 

[JM]

Yes, if t is measuring in seconds and T is measuring in hours then you cannot use equation 3 directly, you must include a conversion factor. Is this what you mean?

In their respective rest frames, yes.

Think of it this way. When we say that we have two identical rulers we mean that if we measure the same distance with both rulers we get the same result. A clock is just a ruler which measures timelike intervals. So when we say that we have two identical clocks we mean that if we measure the same interval with both clocks we get the same result.

If we use one ruler to measure the base of a right isosceles triangle and the other identical ruler to measure the hypotenuse then we will find that the measurements differ by a factor of $\sqrt{2}$. We attribute this difference to a difference in the thing being measured instead of to the rulers, since they are identical. Similarly, if we have one clock measure a vertical line through spacetime (at rest) and another clock measure a diagonal line through spacetime (in motion) then we will find that the measurements differ by a factor of $\gamma$. We again attribute this difference to a difference in the thing being measured instead of to the clocks, since they are identical.

Cheers!, DaleSpam, I think you have expressed my idea very well. I would express the result as 'all the clocks of both K and k are in synch'. I see no contradiction with relativity of simultaneity because ros envisions each observe judging the others clocks, instead of jusst looking at his own.
So the time dilaton eqn t = t/gamma does not mean 'moving clocks run slow' but means the times t and T represeent the behavior of the light rays under consideration.
OK?

 

[Dale]

I would express the result as 'all the clocks of both K and k are in synch'.

No. The clocks of K are in synch with each other. The clocks of k are in synch with each other. But the clocks of K are not in synch with the clocks of k. Therefore it is not true that "all of the clocks of both K and k are in synch".