High School Examining Time Dilation and the Effects of Velocity on Perception of Time

[PeterDonis]

they are the same two events and they are measuring the same space-time interval.

Two "measurements" that yield different results cannot possibly be measuring the same spacetime interval, since spacetime intervals are invariant. Trying to describe it that way just obfuscates what is going on.

There is an alternative way of describing how $t_A$ is defined (assuming we all agree that $t_B$ is just the proper time along B's worldline, i.e., it is the spacetime interval between the events B1 and B2). We could say that $t_A = t_{B2} - t_{B1}$, i.e., it is just the coordinate time difference. But that is not the spacetime interval between the events.

Also, you have to be much more elaborate in specifying how the coordinate time difference is measured, since now we are not talking about two events along A's worldline, for which the proper time on A's clock is a simple measurement, and we are not talking about the proper time along B's worldline, since that is $t_B$ and not $t_A$. So we now have to set up a whole "reference frame" of measuring rods and clocks all at rest relative to A, such that $t_{B1}$ is the time on the reference frame clock at event B1 and $t_{B2}$ is the time on the reference frame clock at event B2, and these will be two different clocks. But those two times by themselves, taken at those two events, don't define any spacetime interval at all, and it's just asking for confusion to use that term in reference to the "measurement" made in this way.

Nor does this measurement measure the "elapsed time for A", which was the term that was used by Elnur Hajiyev--that is the proper time along A's worldline. The fact that A's proper time between events A1 and A2 happens to be numerically equal to $t_{B2} - t_{B1}$ does not mean the latter is a measurement of the former.

 

[Andrew Mason]

Two "measurements" that yield different results cannot possibly be measuring the same spacetime interval, since spacetime intervals are invariant. Trying to describe it that way just obfuscates what is going on.

They yield different results for the time component but the spacetime intervals are the same: S² = x² + y² + z² - c²t² = x'² + y'² + z'² - c²t'²

There is an alternative way of describing how $t_A$ is defined (assuming we all agree that $t_B$ is just the proper time along B's worldline, i.e., it is the spacetime interval between the events B1 and B2). We could say that $t_A = t_{B2} - t_{B1}$, i.e., it is just the coordinate time difference. But that is not the spacetime interval between the events.

The single clock in B measures proper time in B. A measures successive readings on B's clock at different locations so A needs two synchronized clocks to make this measurement. In B, E1 co-ordinates, in the form (t,x,y,z), are (0,0,0,0) and E2's are: (t,0,0,0). In A, these same two events have co-ordinates in the form (t',x',y',z') of E1=(0,0,0,0) and E2= (γt,γ(-vt),0,0). The space time interval in A is $S^2 = -c^2t^2$ and in B it is $S^2 = \gamma^2(v^2t^2 - c^2t^2) = -\gamma^2t^2( c^2-v^2) = -c^2t^2$.

AM

 

[Justintruth]

Sorry for being pedantic, but it's not correct to say that "60 seconds pass between successive ticks of Bs clock, 100 seconds pass for between successive ticks of A's clock". There are two different ways of measuring elapsed time between two events:

1. If you have a single clock that moves inertially from event to the other, then you can use that clock to measure the proper time between the events.
2. If you have a pair of synchronized clocks at rest relative to each other, then you can use one clock to record the time of one event, and the other clock to record the time of the other event. By subtracting the two times, you can compute the time between the events.

Time dilation is a matter of comparing the one way of computing time between events to the other way. The way that uses a single clock always shows less elapsed time than the way that uses two clocks. But it's not a matter of comparing ticks on one clock to ticks on another clock.

I don't think you are right. One way of measuring two ticks on a single clock in a moving frame is to use two clocks in the stationary frame. But those two stationary clocks must be synchronous and so the result is a measure of how many ticks occurred on any single synchronous clock in the stationary frame between ticks in the moving frame. We therefore usually leave out the details of synchronization and reporting and say, for example, "as measured by the clock of the synchronous frame of reference." You cannot measure the elapsed time between ticks with two clocks unless you synchronize them first. Once synchronized any clock in the frame can be used to tell the time of an event as all clocks always show the same value at any time relative to that frame.

Also it is possible using signals to just use one clock in the stationary frame. You need to record the initial tick and then the time of arrival of a singnal anouncing the second tick then back out when the signal originated. Again you get the number of ticks in the stationary frame between ticks in the moving frame.

 

[stevendaryl]

I don't think you are right. One way of measuring two ticks on a single clock in a moving frame is to use two clocks in the stationary frame. But those two stationary clocks must be synchronous and so the result is a measure of how many ticks occurred on any single synchronous clock in the stationary frame between ticks in the moving frame. We therefore usually leave out the details of synchronization and reporting and say, for example, "as measured by the clock of the synchronous frame of reference." You cannot measure the elapsed time between ticks with two clocks unless you synchronize them first. Once synchronized any clock in the frame can be used to tell the time of an event as all clocks always show the same value at any time relative to that frame.

That was exactly my point; to measure the time between two events that take place at different locations requires clock synchronization, so you are not really measuring relative rates of clocks. Time dilation is a comparison of the elapsed time on a single clock with a computed time between events using clock synchronization. Yes, you can use light signals to compute times between distant events, as well, but that implicitly requires a synchronization convention, also.

It's because you're comparing different types of time measurements, a direct measurement using a single clock, and an indirect one using a synchronization convention, that mutual time dilation is possible.

 

[PeterDonis]

They yield different results for the time component but the spacetime intervals are the same

Here is what you said before:

He was comparing A's measurement of the time between two events that occur at the same location in B's frame

"A's measurement of the time between two events that occur in the same location in B's frame" is not a measurement of a spacetime interval. It's a measurement of a coordinate time difference. To measure the spacetime interval between these two events in A's frame, you would have to measure the distance between them as well as the time. That's not what you described. (And of course if you did do that, the answer would end up being $t_B$, not $t_A$; but "A's measurement of the time" is $t_A$, not $t_B$.)

I know you know the actual physics; that's not what I'm complaining about. But this whole thread has been about all the pitfalls of trying to express the physics in ordinary language instead of math. That's why I'm being so insistent about not adding another pitfall.

 

[Herman Trivilino]

"Elapsed time for A" is a spacetime interval between a different pair of events than "elapsed time for B". The time dilation formula compares these two intervals.

I don't think that is what Elnur was saying. He was comparing A's measurement of the time between two events that occur at the same location in B's frame to the proper time interval ie. shown by a single clock in B at that location. In this case, the two events are 1. B enters toilet and 2. B exits toilet. The toilet, hopefully, was at rest in B's frame of reference. So they are the same two events and they are measuring the same space-time interval.

@Andrew Mason I know you understand that those two events occur in the same place for B, but they don't occur in the same place for A. Thus the proper time $\Delta \tau$ that elapses between them for B is equal to the interval $\sqrt{(\Delta t)^2-(\Delta x)^2}$ measured by A. Which therefore means, of course, that $\Delta t$ cannot equal $\Delta \tau$, but rather the two are related by the time dilation formula $\Delta t=\gamma \Delta \tau$.

Note that $\Delta t$, a difference between two clock-readings, is not an elapsed time between the two events discussed above. Not for A, and not for anyone. It could be an elapsed time (that is a proper time) between a pair of events for A, but those two events would be different events..

 

[Justintruth]

That was exactly my point; to measure the time between two events that take place at different locations requires clock synchronization, so you are not really measuring relative rates of clocks.

I think this is a non-sequitur. Synchronization allows you to measure the relative rates of clocks in the two frames it does not prevent it. Why do you think that synchronization prevents you from measuring the relative rates of clocks?

But you did not say "measure the relative rates of the clock" did you? You said "really measure the relative rates of the clock". Are you making a distinction between a real measurement and one the theory allows? Because clearly the theory allows a measurement and they have in fact been made.

Perhaps you are setting up a distinction between "really measuring" and "not really measuring"? Do you think the measurement made of the relative rates of the clocks is "not real" because it is relative? The notion of relativity being real can cause confusion. We are used to assuming there is one "reality" that can "be imagined at any time". This reality is absolute and we separate observational facts from it. For example if you look at a distant clock in Newton's physics the time it shows can be described as not being the "real" time because of the light transit time. But Einstein's theory set's up a different "reality" that can "be imagined at any time." for each frame. Then there are those same additional observational affects that are distinguished from and are on top of the relativity of time. Confusing the relativity of time with and observational effect is a mistake. There is a difference in the ontology of a relative world and a Newtonian world. You will not see a time on a distant stationary clock that matches the time on a local clock in either theory but Newton's theory, after you remove the effects of the transit time, results in everyone agreeing on a description of what is happening or happened at some time. But in Einstein's theory this does not happen. The relativity of time establishes that there is no single present 3d way that the universe is. Rather there is a separate 3d way that it is for each frame of reference. Both are slices of the same space time but they are not the same slice! Now on whether these "ways" are real the theory is silent but the theory does agree with experiment and we are therefore used to calling its result real. Minkowski saw this clearly and announced the fact that four dimensional space-time intervals replaced the old "reality" in the sense that they were in fact the same for every frame. But he did not do away with the fact that the resulting understanding showed that there was no description of "the way the universe is right now" (an image of a 3d space) that exists and that is the same for every frame of reference and from which one can derive the predicted observations. Instead there are many.

It is possible to set such a theory up but the laws of electricity and magnetism are then very complex and there is no way to distinguish which frame is the "real" one so such a possibility is considered unscientific.

So I am sure we can measure the relative rates of the clocks and am less sure that we can "really" measure the relative rates of the clock but my vote is we can!

There just is no single three d image that we can image the universe to be at some time and then derive all observations in every frame from it. Instead there is Minkowski's space-time and we can derive create multiple three-D images from it by slicing space-time in the correct way for each frame and then derive all of the observations for each frame from the frame's own "present".

So I guess I still think we can really measure the relative rate of the two clocks. The theory tells us how and I even believe it has been done.

 

[Andrew Mason]

\
Note that $\Delta t$, a difference between two clock-readings, is not an elapsed time between the two events discussed above. Not for A, and not for anyone. It could be an elapsed time (that is a proper time) between a pair of events for A, but those two events would be different events..

If you are saying that:

1. A difference between two clock readings (the events) on the same inertial clock is the definition of a proper time interval.
2. But to a moving inertial observer, those two events are measured at different locations in the moving frame and the time interval is the difference in co-ordinate times of the two events.
3. The co-ordinate time interval is determined by applying the Lorentz transformation and taking the difference in the time co-ordinates.

then I agree with you.

AM

 

[stevendaryl]

I think this is a non-sequitur. Synchronization allows you to measure the relative rates of clocks in the two frames it does not prevent it. Why do you think that synchronization prevents you from measuring the relative rates of clocks?.

Because different synchronization conventions give different answers to the question of which clock is running faster.

Suppose you have four clocks, A and B at rest in one frame and C and D at rest in another. Each pair is synchronized in their own rest frame. With a speed of .866 c, we can arrange the following situation:

1. Clocks A and D pass each other when they both read 12:00 am

2. Clocks A and C pass each other when A reads 12:30 and C reads 1:00

3. Clocks D and B pass each other when D reads 12:30 and B reads 1:00

Facts 1&2 would lead you to think that clock A is running at half the rate of clocks C and D (A advanced 30 minutes between events 1 & 2, while C&D advanced 1 hour) Facts 1&3 would lead you to think that clock D is running at half the rate of clocks A and B. But all 3 facts are true. It seems like a paradox, until you realize that synchronization is frame-dependent; In one frame, A and B are synchronized but C is 45 minutes ahead of D. In another frame, C and D are synchronized but B is 45 minutes ahead of A.

I'm only bringing up the role of clock synchronization in comparisons of clock rates because otherwise, mutual time dilation seems paradoxical.

 

[Ibix]

To add to @stevendaryl's comment, you can adopt different simultaneity conventions such as Rindler coordinates. Such systems don't agree with Einstein's scheme about who is at rest with respect to whom, so they don't agree about who agrees on coordinate time differences between events, let alone agreeing about the actual differences.

Unless you are talking about the proper time along a specified path you must always specify a simultaneity convention. In SR the assumption is Einstein's convention unless otherwise stated. But it is an assumption.

 

[Justintruth]

Because different synchronization conventions give different answers to the question of which clock is running faster.

Suppose you have four clocks, A and B at rest in one frame and C and D at rest in another. Each pair is synchronized in their own rest frame. With a speed of .866 c, we can arrange the following situation:

1. Clocks A and D pass each other when they both read 12:00 am

2. Clocks A and C pass each other when A reads 12:30 and C reads 1:00

3. Clocks D and B pass each other when D reads 12:30 and B reads 1:00

Facts 1&2 would lead you to think that clock A is running at half the rate of clocks C and D (A advanced 30 minutes between events 1 & 2, while C&D advanced 1 hour) Facts 1&3 would lead you to think that clock D is running at half the rate of clocks A and B. But all 3 facts are true. It seems like a paradox, until you realize that synchronization is frame-dependent; In one frame, A and B are synchronized but C is 45 minutes ahead of D. In another frame, C and D are synchronized but B is 45 minutes ahead of A.

I'm only bringing up the role of clock synchronization in comparisons of clock rates because otherwise, mutual time dilation seems paradoxical.

We make a distinction sometimes in engineering between a counter and a clock. A clock is something that ticks at a regular interval. Its sometimes called an oscillator. A counter is something that counts the ticks. The time values you are using are really counters. You can use counters but you must know that two counters can be counting the ticks of a single clock and reading different values. This is because the counter can be reset at different times. We do it all the time. New York is ahead of California but both New York and California's clocks tick at the same rate. The "paradoxes" you mentioned work because the counters in the moving frame are not set to the same initial values at the same time relative to the rest frame. This has nothing to do with the rate of the clock in the moving frame. That is why I recommended that the number of counts in the reference frame between successive ticks in the moving frame be counted. If the number of counts in the rest frame between two ticks in the moving frame is one there is no dilation or contraction. If it is more than one (even by a fraction), there is time dilation of the moving frames clock as measured by the rest frame clock. If it is less than one there is time contraction. This can be measured and when it is one finds dilation is the factual situation always and from both frames.

I get that.

If you took a moving frame of reference in a Newtonian world, two trains let's say, and miss - set the counters to the same values that you would see had they been synchronized in the relativistic world, you would not produce time dilation. The clocks would read different times but not tick at different rates. The difference is not just that the clocks read different numbers. Its that they actually slow down.

Let me tell you something that actually happened to me. I had a prominent professor of Marxism who in fact was a dialectical materialist and actually a naïve materialist argue that all of the "effects" of relativity were merely observational. Now I know you don't believe that the effects are purely observational. But when you suggest that the relativity of time comes down to how time is measured it can imply - or rather obscure - the fact that the effect is real. It can seem that relativity is just a different way of measuring the same situation as we thought existed at the time of Newton.The person trying to understand the theory then goes down the path of trying to construct an image of how things "really are". That is a big mistake when you are trying to understand this theory. This is literally a picture in his head of how reality is at some moment and unfortunately it constitutes an absolute frame of reference even though the person trying to imagine does not realize it. So he is trying to "imagine" an absolute frame of reference and then see how by measurement he can get the relativity of time. He tries to derive observations from his imagined reality that are consistent with relativity theory. They have a lot of trouble - in fact the theory cannot be understood that way. I know this happens because it happened to me. The relative universe is not imaginable - or better not imaginable as existing some 3d way right now. There is more to it than the measurement process. If by "ontology" I mean what the philosopher Quine thought it meant, namely, "what is". Then there is an ontological difference between the frames of reference. One frame may have one set of things that are, while the other frame's list of what is has some that are not yet, and others that already were. And either list is as right as the other. This is why relativity is so counter intuitive - because we all are programed biologically to believe it must actually be in some imaginable way and so if I can just imagine it and see how it is viewed I will understand the theory. The first step is to give up trying to imagine how things are. Unless you use Minkowski. You see my point? Once you try to imagine how it is and then derive how it will look you are dead. It isn't one way or the other. Nor is it both. It is one for one frame and the other for the other.

There are two types of images. The ones observers actually see - that is the same for any two collocated observers even if they are in different frames but they are very different even within one frame for observers at different places because of the transit times. Then there are ones they calculate. These are the same for all observers in the same frame but different between frames. This second set is the notion of "what is" in one frame vs the other. Then there is Minkowski which is strictly speaking not an image. Perhaps you believe that two images produced by calculations and different between observers should not be called real. But these images are real in the sense that they replace the single real image we used in Newtonian physics as they can then be used to predict what observations will be for different observers in each frame.

There are two takeaways for me: First: the rate of the clocks in a moving frame of reference can be measured by the clock(s) in a stationary frame of reference. When that is done it will be found that the clocks in the moving frame are ticking slower than in the rest frame. And second: That the relativity of time is a natural phenomenon that cannot be transformed away. There is something about nature and not just our way of measurement involved.

To me the best way to motivate mutual time dilation - for you are right that it is a problem - is to take two yard sticks and rotate one relative to the other. Then show how dropping a perpendicular from one yardstick to the other and projecting perpendicular to the ruler being measuring will result in a dilation of length and that that same dilation will occur if the second yardstick measures the first by the same method. Both rulers measure each other as larger using the same procedure. By changing the definition of perpendicular to being perpendicular to the measured ruler instead of the measuring you can see how mutual contraction can also occur and be relative which is another source of confusion. Then you use a Minkowski space to show how a space time interval that is entirely temporal will be rotated so that it is partly in space and partly in time in the other frame. This has the advantage of motivating both time dilation and space contraction in one stroke and does not imply that the relativity of time is just an effect of how time is measured. Good luck explaining however, the negative sign in the metric. The interval between two points on a light cone is zero?! What?! Then what separates them?

Still you can show a light cone in two d and then show the world line of the origin of a moving frame and then draw a diagonal line between the rays of the cone oriented so that the world line of the moving frames origin bisects the diagonal. Then by sliding the line along the world line of the moving frames origin you can show that the intersections of the diagonal line with the light cone are always the same distance along this diagonal from the origin of the moving frame while one side is getting farther away when measured by a horizontal line! Rotating the line that defines the present is the key.

Anyway, if you actually read this my complements. I think I understand your point. I am just trying to keep it straight in my head.

I still think we can measure the time dilation and that it occurs and is real.

 

[stevendaryl]

To me the best way to motivate mutual time dilation - for you are right that it is a problem - is to take two yard sticks and rotate one relative to the other. Then show how dropping a perpendicular from one yardstick to the other and projecting perpendicular to the ruler being measuring will result in a dilation of length and that that same dilation will occur if the second yardstick measures the first by the same method. Both rulers measure each other as larger using the same procedure. By changing the definition of perpendicular to being perpendicular to the measured ruler instead of the measuring you can see how mutual contraction can also occur and be relative which is another source of confusion.

That's a very good analogy. Disagreeing about what "perpendicular" means in Euclidean geometry is exactly analogous to disagreeing about what "simultaneous" means in SR.

As to time dilation being "real", there are two aspects to time dilation:

1. Measuring the "rate" of a clock at rest in one frame using the coordinate system of another frame.
2. Measuring the total elapsed time for a clock to go between two events.

#1depends on a synchronization convention, but #2 does not. So in a sense, #2 is more "real" than #1, even though you can use effect #1 to compute effect #2.

 

[GeorgeDishman]

A clock is something that ticks at a regular interval. Its sometimes called an oscillator. A counter is something that counts the ticks. The time values you are using are really counters.

A clock is a counter which counts the ticks of an oscillator. Synchronisation consists of both setting the count and the phase of the oscillator to defined values. To measure the elapsed time, you should include not only the number of ticks but also the amount of phase change from the set value.

 

[fieldofforce]

No matter how the time dilation pans out, you still have to have an electric and a magnetic wave undulating in the applicable space. If the time dilation prevents the electric and magnetic waves from undulating then something is wrong.

 

[jbriggs444]

No matter how the time dilation pans out, you still have to have an electric and a magnetic wave undulating in the applicable space. If the time dilation prevents the electric and magnetic waves from undulating then something is wrong.

If you think of yourself as riding upon a photon and that undulations in the corresponding fields must be visible from such a viewpoint, you are mixing models and stacking misconceptions on misconceptions.