# Relation between coordinate time and proper time

1. May 1, 2013

### ash64449

Hello friends,

If we consider ${T}$ as coordinate time and ${\tau}$ as proper time, the relationship between them is:

$\frac{T}{\tau}= \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

so,

${T}= \frac{\tau}{\sqrt{1-\frac{v^2}{c^2}}}$

So we can consider this expression like this: If In IRF,An Observer "A" sees another Observer "B" moving,then ${T}$ of Observer B is dilated by the factor of $\frac{\tau}{\sqrt{1-\frac{v^2}{c^2}}}$ where ${\tau}$ is the proper time of Observer A

So we can consider the time dilation as the ratio of coordinate time of one observer to the proper time of another observer....

Am I correct?

Last edited: May 1, 2013
2. May 1, 2013

### ash64449

I found the earlier Relationship from this forum.. I just connected this equation and with the explanation of time dilation given in this article .

And then i got the conclusion that i posted in this thread.

Explanation of time dilation in that article is in Chapter 12

3. May 2, 2013

### ghwellsjr

No. The time on any clock is Proper Time. You should not think of Coordinate Time as the time on a clock but rather it is the time for the coordinate system. The Proper Time on any clock applies only to that one clock at whatever location it happens to be. The Coordinate Time applies simultaneously to every location in the coordinate system.

Of course, any clock that is stationary in the coordinate system and set to the Coordinate Time will also display the coordinate time and that is what Einstein does in his derivation of the ratio of Coordinate Time of one system (K) to the Proper Time on a clock fixed at the origin of another system (K') moving with respect to the first system.

And that is what I demonstrated to you in your other thread asking about the same thing. I thought we had made a lot of progress on that thread, including that one observer cannot see the Time Dilation of another observer's clock. We talked about Relativistic Doppler which describes what each observer sees of the other observer's clock.

The implication is that Observer "A" is stationary because you say that Observer "B" is moving. Therefore you should be talking about the Coordinate Time of System "A" not Observer "A". Then you should not be talking about the Coordinate Time, ${T}$, of Observer "B" but rather the Coordinate Time, ${T}$, of System "A" produces a larger time than the Proper Time, ${\tau}$, of Observer "B" by the ratio $\frac1{\sqrt{1-\frac{v^2}{c^2}}}$

4. May 2, 2013

### ash64449

Yes.we have analysed what is time dilation but i didn't understand what is coordinate time and proper time.so i thought i have lots of things to learn..

5. May 2, 2013

### ash64449

Thank you for providing the definition of coordinate time and proper time. I really didn't understand what these terms meant but simply made assumptions based on that book of Einstein's.

Thank you for providing more understandings from Einstein's Chapter.

I know that we made lots of progress in the other thread and then i got into these two new concepts and i got confused a bit.. I really agree that we cannot see Time dilation and instead see clocks ticking faster or slower because of relativistic Doppler Effect.

Yes.This is Exactly what i meant but couldn't express as i don't know many important terms that is required in order to discuss Relativity..

6. May 2, 2013

### ghwellsjr

EDIT: I see you delete the following post while I was responding so maybe you were able to figure it all out but I'm going to leave my response as is. Who knows? It might help someone else.

Sometimes people talk about the rest frame of Observer "A" (or just the frame of Observer "A") and they mean a frame in which Observer "A" is at rest at the spatial origin of a frame which we could also call Frame "A" or System "A" or Coordinate System "A". But it's not Observer "A"'s frame just because he is at rest in it and being at rest in it does not provide him with any of the Coordinate Time information going on remotely to him. So, to make things clear, especially when discussing the ratio of Coordinate Time to Proper Time, I prefer to have just one frame and one clock. There doesn't have to be any observers involved at all, except us, of course.

I go back to the diagrams I provided earlier. They exactly correspond to Einstein's analysis of how the Proper Time on a clock is dilated when it is moving in an IRF. Here's the beginning of Einstein's analysis:

And here is my diagram that corresponds to a clock at rest at the spatial origin of a frame that we will call frame K' to be consistent with Einstein's nomenclature:

Note that x'=0 is the Coordinate Distance of the clock which stays at 0. The first two blue dots at the bottom correspond to the Coordinate Times of t'=0 and t'=1. Any questions about this so far?

Next, Einstein uses the Lorentz Transformation process to see what the new Coordinates, x and t, are in a new frame, K, moving at speed v with respect to the first one. I used a specific value of v = -0.6c to make the second diagram for frame K:

Just in case you're not familiar with the Lorentz Transformation process, I will go through the details:

First we calculate gamma, γ, from the speed beta, β, the ratio of v/c, using the equation;

γ = 1/√(1-β2) = 1/√(1-(-0.6)2) = 1/√(1-0.36) = 1/√0.64 = 1/.8 = 1.25

Now we use the form of the LT where c=1 to calculate the new values of the coordinates:

x = γ(x'-βt')
t = γ(t'-βx')

Don't be confused by Einstein's interchanging of the prime and unprimed terms. We accomplish the same thing by changing the sign of the velocity.

So when x'=0 and t'=0 we get:

x = γ(x'-βt') = 1.25(0-(-0.6)*0) = 0
t = γ(t'-βx') = 1.25(0-(-0.6)*0) = 0

Just as Einstein got except he only did it for t. We need both x and t to be able to plot the events on the diagram. You can see that the first dot goes at the Coordinates of x=0 and t=0.

And when x'=0 and t'=1 we get:

x = γ(x'-βt') = 1.25(0-(-0.6)*1) = 1.25(0.6) = 0.75
t = γ(t'-βx') = 1.25(1-(-0.6)*0) = 1.25

Again, just as Einstein got if we plug the velocity into his formula. You can see that the next dot up is at the Coordinates of x=0.75 and t=1.25.

The Proper Time of the clock is dilated because it takes longer for it to tick out 1 second when it is moving. As Einstein said it:

I think the issue that you are dealing with is that we say the ratio of Coordinate Time to Proper Time is gamma which is greater than 1 and indicates Time Dilation but yet we say the moving clock is ticking slower than a stationary clock so it seems like we should be saying the Coordinate Time is the one that is dilated. But we do it this way to be consistent with the concept of Length Contraction. When we depict objects and clocks on a spacetime diagram, we see that the length of a moving object takes up less coordinate space and the ticking of a moving clock takes up more coordinate time.

Last edited: May 2, 2013
7. May 2, 2013

### Popper

The proper time between two events is the time as measured by a clock whose which moves through both events whereas coordinate time is the time measured by synchronized clocks and which the time between the two events is the differene between the readings on two different clocks.

8. May 2, 2013

### pervect

Staff Emeritus
Except for the fact that proper-time is independent of any observer, that's exactly right.

Time dilation is the ratio of coordinate time to the observer-independent proper time. Hence time dilation is always coordinate dependent.

I see that another poster told you that you were wrong, I don't understand why he thinks it's wrong. I hope I can convince him civily not to post misinformation like that :-(.

9. May 2, 2013

### ash64449

Exactly.this is the one that confused me..That Coordinate time should be the one that is Time Dilated.Now i understand Time Dilation. It is the Proper Time that changes when It is moving with respect to coordinate system A... So this Proper Time needs 1.25 seconds in coordinate time of A to tick 1 second..

Can you explain How Arranging Time Dilation like this helps us to Explain the concept of length contraction easier?

10. May 2, 2013

### ash64449

Yes.Sorry that i deleted the post. I got the answer from your earlier comment itself.I thought a little bit hard. And Your presentation of answering the post is really great.

11. May 2, 2013

### ash64449

Ya.Proper Time is the one that change. But it appears invariant because we see co-ordinate time of our system relative to that observer change... That is why Time dilation is coordinate dependent..

12. May 2, 2013

### ghwellsjr

I don't know why you would express Proper Time in this way. It makes it sound like there is a single Proper Time between two events but as you correctly point out, it is measured by a clock which moves through both events, but what you didn't point out is that it is dependent on the path of that clock between those two events so two different clocks taking two different paths can end up with different accumulated times on them.

It is sufficient to say that Proper Time is what any clock measures.

Also, when you are talking about coordinate time, you should not be connecting it with actual clocks. Of course, you could always put synchronized clocks at the two events in question but then when you do a Lorentz Transformation on the situation, those two clocks will not be synchronized and you will have to create two more synchronized clocks to put at those two events. And how will you know what time to put on them? You look at the Coordinate Times of the two events and you set the clocks accordingly. So what have you accomplished by this sort of explanation?

The whole point of Time Dilation is that it maintains the Proper Time of all events on real clocks even though the Coordinate Time of those events can be different.

13. May 2, 2013

### ghwellsjr

Didn't you just point out that his statement wasn't completely correct? That's what I did, except I provided a great many more details.

If you think something in any of my posts is misinformation, you should quote it and point out what you think is wrong. You won't have any problem convincing me to not post misinformation but you have to point out specifically what it is. And please don't take anything out of context, read my entire posts.

14. May 2, 2013

### ghwellsjr

OK, here's a spacetime diagram for an IRF in which the observer and mirror are at rest showing one tick of a light clock:

At time 4 nanoseconds, the observer in blue at location 0 sends a flash of green light to a red mirror that is six feet away from him. He gets the reflection back at time 16 nanoseconds so each tick is 12 nanoseconds long.

Now let's see what happens if we view the same thing in an IRF moving at -0.6c with respect to the original IRF:

Now the observer and his mirror are moving at 0.6c. Notice that the distance to the mirror is Length Contracted. Instead of six feet it is only 4.8 feet. Be sure to measure this along a horizontal line where the Coordinate Time is a constant. Also notice that the observer's time is dilated, that is, it takes longer in the diagram to mark of the same Proper Time from the first IRF. Finally note that the flash of light propagates at c along a 45-degree angle in both diagrams and so the observer continues to experience exactly the same thing in this diagram as he did in the first one. He sends the light signal out at 4 nanoseconds of his Proper Time and receives the reflection at 16 nanoseconds of his Proper Time.

15. May 2, 2013

### ash64449

Yes.This Answers my question. I have another question. Is Length Contraction Observed?

Well,You said Time Dilation is not observed.Instead Relativistic Doppler Effect..

16. May 2, 2013

### ghwellsjr

No, Length Contraction is also not observed. How can it be? It can be different in different IRF's. Just look at the two examples in the above diagrams. Can the blue observer detect anything different as he looks for the reflection from his mirror as determined from either IRF?

17. May 2, 2013

### ash64449

no.i said the other observer who observes the blue observer... Yes,like time dilation,length contraction changes with different frames of reference.
If length contraction is not observed,then what change is observed instead of it?
Just like this:time dilation is not observed,instead relativistic doppler effect....

18. May 2, 2013

### ash64449

and i do agree that blue cannot identify length contraction.but i asked whether other observer in a coordinate system observe length contraction of the blue observer?

19. May 2, 2013

### ghwellsjr

Usually when we are talking about Doppler and especially when it is shown on a spacetime diagram, we are only considering relative motion between observers that are directly in line with each other because the formula is very simple and because we can only show one dimension of space on a normal spacetime diagram.

If we do the same thing with Length Contraction, that is, only consider in line motion, it becomes very difficult to visually determine the length of an object along that dimension. So usually, when this subject comes up, we consider the appearance of an object that is traveling at right angles to our line of sight but some distance away. And it turns out that the analysis is extremely difficult to ascertain because we cannot just take the Length Contraction along the direction of motion and say that an object will appear the way it would be drawn on a diagram because the observer has to wait for the light signals coming from the different portions of the object to arrive at his eyes simultaneously and since the object is in motion at a speed comparable to that of light, it is a complicated subject.

However, the subject has been dealt with, most notably by Terrell, who has determined that the shape of a sphere traveling at high speed will still appear as a sphere. That is rather surprising, don't you think? Anyway, for more information you can read the wikipedia article or see this thread.

20. May 3, 2013

### ash64449

George,why can't we consider "real" situations?? Just like Einstein's Thought Experiment?

Whenever i am providing with examples,it means that i am considering "real" situations..