Is Proper Time Relative in Special Relativity?

[Shark 774]

The idea of proper time has been confusing me. Say for example we have a spaceship with John in it traveling past the Earth, towards a planet called NEC. Ann is on Earth and starts her timer as John travels past the Earth, towards NEC. Ann notes that it takes 2 years for John to arrive at NEC and that he is traveling at 0.9c, where gamma is 2.29. She therefore calculates that the time, by John's clock, will read t₀ = 2/2.29 = 0.87 years. She takes John's time to be "proper time" because he is in the moving frame of reference, relative to her. If we look at it from John's point of view the Earth is moving away from him at 0.9c and NEC is approaching him at 0.9c. He would therefore take Earth's (and NEC's) time to be proper time, because to him Earth is in the moving frame, and therefore he would calculate that the time taken on Earth for his journey is t₀ = 0.87/2.29 = 0.34years. Clearly this isn't right. What have I done wrong??!

 

[Mentz114]

Proper time is measured by a clock traveling along a 4D worldline between two events. To calculate it requires the Lorentzian length of the WL to be calculated between two points on the WL. It is explained quite well in this article

http://en.wikipedia.org/wiki/Proper_time

Proper time is not relative so all observers will agree on the calculation and also on the observation of the elapsed time on a clock.

 

[Shark 774]

To calculate it requires the Lorentzian length of the WL to be calculated between two points on the WL.

Thanks, but can you please explain this quote in more detail?

 

[morrobay]

Is it correct to say that proper time and the spacetime interval are numerically equal ?

 

[tiny-tim]

Hi Shark 774!

She takes John's time to be "proper time" because he is in the moving frame of reference, relative to her.
…
He would therefore take Earth's (and NEC's) time to be proper time, because to him Earth is in the moving frame

John's time is the "proper time" for John.

Earth's time is the "proper time" for Earth.

Both are the same no matter who calculates them.

Both are the time on a clock traveling with whoever's "proper time" it is … that's why it's called "proper time" ("proper" meaning "own" in this case, as in "property").

Is it correct to say that proper time and the spacetime interval are numerically equal ?

if the clock is moving uniformly, yes, otherwise, no

 

[Fredrik]

Is it correct to say that proper time and the spacetime interval are numerically equal ?

The "interval" is a number assigned to a pair of events, or equivalently, to the straight line segment connecting those two events. Proper time is a number assigned to an arbitrary timelike curve.

 

[PeterDonis]

The would calculate that the time taken on Earth for his journey is t₀ = 0.87/2.29 = 0.34years. Clearly this isn't right. What have I done wrong??!

You've left out the relativity of simultaneity (as well as being somewhat confused about the meaning of the term "proper time", which other posters have addressed). John's "elapsed time" of 0.34 years refers to the event on Earth that is simultaneous in his frame with his arrival at NEC, whereas Ann's "elapsed time" of 2 years refers to the event on Earth that is simultaneous in *her* frame (the Earth's frame) with John's arrival at NEC. Because of the relativity of simultaneity, these two events are *not* the same. I recommend drawing a spacetime diagram; it will help to clarify what's going on.

 

[GrayGhost]

Thanks, but can you please explain this quote in more detail?

Shark,

A worldline is the path of an observer (or his clock) thru space and time, ie thru spacetime. When we consider the proper time, we are considering the duration between 2 events, where the clock exists at both events.

You are standing stationary looking at your clock. You measure the time between 2 events that happen at your own location. Two such events might be this ... (1) your hour hand points to 2 with minute hand at 12, and (2) your hour points to 3 with minute hand at 12. Your proper time over said interval is exactly 1 hr. We might instead imagine that the events were marked by the chime of a bell you carry, one that occrs at 2pm and the next at 3pm. The 2 events could be the striking of lightning at your own location, the first at 2pm and the next at 3pm by your own clock. Etc.

Now consider a moving inertial observer (B) who carries a clock. Two event occur at his own location, one at 2pm by his clock and the next at 3pm by his own clock. The proper time between the 2 events is again 1hr, but by his clock, not by yours. His clock ticks slower per you, since you record him in motion. Therefore, you record the interval at >1hr by your own clock, maybe (say) 1.2 hr. This 1.2 hr does not represent proper time, because your clock does not exist at both events. It's the moving fellow's clock that existed at both events, and so the duration his clock ticks off between the 2 events mark the proper time of the interval ... ie 1 hr-B.

We might imagine everyone who carries a clock, and ask ... who records the proper time and who doesn't? You first must "select the 2 events of interest". Once you do, you determine which of all those clocks resided at both events. In the scope of SR, the clock that resides at both events is owned by the observer who deems himself stationary and simply awaits the 2 events to occur at his own location. Again, the proper time between said 2 events is that measured by the clock that resided AT both events. So ...

Ask ... which clock existed AT both events? Only that clock marks the proper time of that interval, and that interval has a name ... "the spacetime interval". Since no observer disagrees that that 1 clock resided at both events, no observer disagress that the proper time was 1 hr. Only that clock can mark the proper time "of that particular spacetime interval". It's thus an invariant, since all observer's agree on its value.

One last point ...

You move relatively wrt another observer. You each carry a clock, and you each execute a flyby of each other whereby your clocks both happen to read 12am. You experience a 2nd flyby of another object subsequently. You experience your 2nd flyby at exactly 1hr later by your own clock. He experiences his 2nd flyby at exactly 2hr later by his own clock. If the spacetime interval is defined by your 2 flyby events, then the proper time is 1 hr, and all in the cosmos agree. If the spacetime interval is defined by his 2 flyby events, then the proper time is 2 hr, and all in the cosmos agree.

That's about it :)

GrayGhost

 

[morrobay]

if the clock is moving uniformly, yes, otherwise, no

Above answer to my question: Are proper time and the spacetime interval numerically equal ?

See the wikipedia page : http://en.wikipedia.org/wiki/Proper_time
In the twin paradox section there is an example with acceleration where the total proper time
is 5 years. The spacetime interval in this example is also 5 ?

s = sq rt ( 10² - 8.66²) = 5

 

[Fredrik]

Above answer to my question: Are proper time and the spacetime interval numerically equal ?

See the wikipedia page : http://en.wikipedia.org/wiki/Proper_time
In the twin paradox section there is an example with acceleration where the total proper time
is 5 years. The spacetime interval in this example is also 5 ?

s = sq rt ( 10² - 8.66²) = 5

No, because the (t,x) coordinates of the two events are (0,0) and (10,0), so the interval between them certainly can't be 5. (I would say that it's sqrt(10^2-0^2)=10, but as you can see below, I'm not 100% sure which quantity is called the "interval"). The proper time of the curve of constant x that connects the two events (the world line of the twin that stays on Earth) is 10. The proper time of the astronaut twin's world line is 5.

Can someone please tell me what exactly the term "interval" is supposed to refer to? If I use a -+++ metric and set c=1, is the interval between (0,0) and (t,x) defined by s^2=t^2-x^2 or s^2=-t^2+x^2, and is it s² or its positive square root s that's called the "interval". Does the answer (to the first part of the question) depend on whether (t,x) is timelike or spacelike?

 

[Mike_Fontenot]

[...]
Clearly this isn't right. What have I done wrong??!

It IS right.

https://www.physicsforums.com/showpost.php?p=2934906&postcount=7

Mike Fontenot

 

[morrobay]

See the wikipedia page : http://en.wikipedia.org/wiki/Proper_time
In the twin paradox section there is an example with acceleration where the total proper time
is 5 years. The spacetime interval in this example is also 5 ?
s = sq rt ( 10² - 8.66²) = 5
The above spacetime interval is from the Earth's frame and I just combined both legs of the trip.
To show that this is the invariant interval use the Lorentz Transformations to get
delta t' and delta x' for s =sqrt ( delta t'² -delta x'² )
gamma =2
x'=0
v =.866
x = 8.66
t'= gamma( t - vx/c) = 2(10-.866*8.66) = 5
So the spacetime interval from rocket frame , s = sqrt( 5² -0) =5

 

[Shark 774]

A worldline is the path of an observer...
GrayGhost

Thanks a lot GrayGhost, your answer seems most clear and it makes sense! Yet now I am still confused, as Tiny-Tim and Mike Fontenot seem to contradict what you said. You're saying that proper time is not relative, yet they are saying that it IS relative...

 

[tiny-tim]

… Tiny-Tim … seem to contradict what you said. You're saying that proper time is not relative, yet they are saying that it IS relative...

No, I most clearly said that they are not relative, they are the same no matter who calculates them …

John's time is the "proper time" for John.

Earth's time is the "proper time" for Earth.

Both are the same no matter who calculates them.

Both are the time on a clock traveling with whoever's "proper time" it is … that's why it's called "proper time" ("proper" meaning "own" in this case, as in "property").

 

[JesseM]

Thanks a lot GrayGhost, your answer seems most clear and it makes sense! Yet now I am still confused, as Tiny-Tim and Mike Fontenot seem to contradict what you said. You're saying that proper time is not relative, yet they are saying that it IS relative...

The proper time between any two specific events on a worldline is not-relative. However if John is trying to calculate the proper time on Ann's worldline, and he wants to calculate it between the time John departs from Ann and the time John arrives at planet NEC, then the second event isn't an event on Ann's worldline, so what you want is the event on Ann's worldline that's simultaneous with the event of John arriving at NEC, but this is relative to your choice of reference frame because simultaneity is relative. Different frames will pick different events on Ann's worldline when they are asked what event on her worldline is simultaneous with the event of John arriving at NEC.

Mike Fontetot has a definition of simultaneity that he prefers to use for observers who change velocities like John, and under his definition, as John arrives at NEC but hasn't changed velocities yet, this point on his worldline would be simultaneous with an event on Ann's worldline where she has only aged 0.34 years. Also, leaving aside definitions of simultaneity for observers who change velocities, if we just think about an inertial frame where the Earth is always moving at 0.9c, in this frame the definition of simultaneity is such that the event of John arriving at NEC is simultaneous with an event on Ann's worldline where she has only aged 0.34 years since John departed.

If on the other hand you have a twin paradox scenario where John departs from Ann, travels for a bit, then turns around and finally reunites with Ann, then the events (John and Ann depart) and (John and Ann reunite) are events on both worldlines, so you can calculate a single value for the proper time along each worldline between these events with no ambiguity.

 

[Shark 774]

Just to try and clarify: Say Tim and Bob are on two different spaceships and traveling at 0.9c relative to each other. Tim's clock reads that it takes 5 seconds for Bob to get from point A to point B. This means that Tim sees Bob's clock read t = t'/gamma = t'/2.29 = 2.18 seconds. However the same applies in Bob's frame of reference, i.e. Bob clock says that it takes Tim 5 seconds to get from A to B (actually B to A) but Bob sees that Tim's clock only reads 2.18. So when Tim uses the time dilation formula, he isn't calculating the actual time elapsed for Tim, in Tim's frame of reference, he is just calculating the time that he personally perceives for pass for Tim. Is this correct?! I am driving myself nuts trying to make sense of this.

 

[JesseM]

Just to try and clarify: Say Tim and Bob are on two different spaceships and traveling at 0.9c relative to each other. Tim's clock reads that it takes 5 seconds for Bob to get from point A to point B. This means that Tim sees Bob's clock read t = t'/gamma = t'/2.29 = 2.18 seconds. However the same applies in Bob's frame of reference, i.e. Bob clock says that it takes Tim 5 seconds to get from A to B (actually B to A) but Bob sees that Tim's clock only reads 2.18. So when Tim uses the time dilation formula, he isn't calculating the actual time elapsed for Tim, in Tim's frame of reference, he is just calculating the time that he personally perceives for pass for Tim. Is this correct?! I am driving myself nuts trying to make sense of this.

No, if you are calculating the time between some specific events on a person's worldline like the events of their passing some landmark A and their passing some landmark B, and you know the time between these events in your frame, then the time dilation formula can tell you the proper time experienced by that person between those events, i.e. (proper time) = (time in your frame)/gamma. And again, the proper time between events on a particular worldline is an objective fact that doesn't depend on your choice of reference frame.

The phrase "the time that he personally perceives to pass for Tim" doesn't really seem to make sense, you just said that "Bob clock says that it takes Tim 5 seconds to get from A to B", so the time Bob "perceives" between those events is 5 seconds, and the time he perceives Tim to age when passing from A to B is the proper time for Time which is objective and not a matter of perception.

 

[Shark 774]

No, if you are calculating the time between some specific events on a person's worldline like the events of their passing some landmark A and their passing some landmark B, and you know the time between these events in your frame, then the time dilation formula can tell you the proper time experienced by that person between those events, i.e. (proper time) = (time in your frame)/gamma. And again, the proper time between events on a particular worldline is an objective fact that doesn't depend on your choice of reference frame.

The phrase "the time that he personally perceives to pass for Tim" doesn't really seem to make sense, you just said that "Bob clock says that it takes Tim 5 seconds to get from A to B", so the time Bob "perceives" between those events is 5 seconds, and the time he perceives Tim to age when passing from A to B is the proper time for Time which is objective and not a matter of perception.

Ok but here's where I am getting stuck: 5 seconds pass for Bob, he calculates that only 2.18 seconds pass for Tim. Tim observes Bob to be moving past him and therefore calculates that it takes Bob 2.18/2.29 = 0.95 seconds. Obviously I am doing something totally wrong, as this can't be right.

 

[JesseM]

Ok but here's where I am getting stuck: 5 seconds pass for Bob, he calculates that only 2.18 seconds pass for Tim. Tim observes Bob to be moving past him and therefore calculates that it takes Bob 2.18/2.29 = 0.95 seconds. Obviously I am doing something totally wrong, as this can't be right.

The problem is you're being too vague, 5 seconds between what event and what? "It takes Bob" 0.95 seconds to do what? If you pick two specific events on Tim's worldline, and you know the time between these events in Bob's frame, you can calculate the proper time Tim experiences between this specific pair of events. Likewise if you pick two specific events on Bob's worldline, and you know the time between these events in Tim's frame, you can calculate the proper time Bob experiences between this other specific pair. But your statement above gives no clue as to what events you're talking about on either worldline.

 

[Shark 774]

The problem is you're being too vague, 5 seconds between what event and what? "It takes Bob" 0.95 seconds to do what? If you pick two specific events on Tim's worldline, and you know the time between these events in Bob's frame, you can calculate the proper time Tim experiences between this specific pair of events. Likewise if you pick two specific events on Bob's worldline, and you know the time between these events in Tim's frame, you can calculate the proper time Bob experiences between this other specific pair. But your statement above gives no clue as to what events you're talking about on either worldline.

Ok great, that helps. Thank you all!

 

[JesseM]

Ok but here's where I am getting stuck: 5 seconds pass for Bob, he calculates that only 2.18 seconds pass for Tim. Tim observes Bob to be moving past him and therefore calculates that it takes Bob 2.18/2.29 = 0.95 seconds. Obviously I am doing something totally wrong, as this can't be right.

To add to what I said above, let me pick some specific pairs of events here. Suppose we have the events of Tim passing one object A and then the event of Tim passing another object B, and in Bob's frame this takes 5 seconds. So, he calculates that Tim will experience 2.18 seconds between the event of passing A and the event of passing B, which is the objective proper time for Tim between these events. Now Tim picks the event E1 on Bob's worldline that is simultaneous in Tim's frame with the event of Tim passing A, and the event E2 on Bob's worldline that is simultaneous in Tim's frame with the event of Tim passing B. Naturally in Tim's frame the time between E1 and E2 is 2.18 seconds, so he calculates that that Bob will experience 0.95 seconds of time between E1 and E2, and this is indeed the objective proper time for Bob between those events. However, in Bob's frame E1 is not simultaneous with the event of Tim passing A, and E2 is not simultaneous with the event of Tim passing B, for Bob these are just two meaningless events on his worldline where nothing of significance happened.

 

[Mike_Fontenot]

Ok but here's where I am getting stuck: 5 seconds pass for Bob, he calculates that only 2.18 seconds pass for Tim. Tim observes Bob to be moving past him and therefore calculates that it takes Bob 2.18/2.29 = 0.95 seconds. Obviously I am doing something totally wrong, as this can't be right.

It IS right!

You are getting all confused by bringing the term "proper time" into this problem. You can talk about EACH of the twin's conclusions about the correspondence between their ages without talking about "proper time" at all. My previous response had NOTHING to do or say about "proper time" at all ... I ignored it because it was just causing you confusion.

At any instant in each twin's life, he can determine the current time in his twin's life, and he can plot his twin's age as a function of his own age ... call that his "simultaneity diagram".

If you compare each of those two simultaneity diagrams (constructed by each twin), you will find that the two diagrams DON'T agree. That's what is important here.

Mike Fontenot

 

[JesseM]

It IS right!

You are getting all confused by bringing the term "proper time" into this problem.

The question (and the thread title!) is specifically about proper time, so you're not being helpful by trying to promote a particular coordinate-dependent notion of simultaneity here, when in relativity no notion of simultaneity is more "right" than any other.

 

[ghwellsjr]

The idea of proper time has been confusing me. Say for example we have a spaceship with John in it traveling past the Earth, towards a planet called NEC. Ann is on Earth and starts her timer as John travels past the Earth, towards NEC. Ann notes that it takes 2 years for John to arrive at NEC and that he is traveling at 0.9c, where gamma is 2.29. She therefore calculates that the time, by John's clock, will read t₀ = 2/2.29 = 0.87 years. She takes John's time to be "proper time" because he is in the moving frame of reference, relative to her. If we look at it from John's point of view the Earth is moving away from him at 0.9c and NEC is approaching him at 0.9c. He would therefore take Earth's (and NEC's) time to be proper time, because to him Earth is in the moving frame, and therefore he would calculate that the time taken on Earth for his journey is t₀ = 0.87/2.29 = 0.34years. Clearly this isn't right. What have I done wrong??!

Clearly what you are doing has to be wrong because if you keep doing your repetitive calculation, you will eventually come to the conclusion that the time on each other's clocks is approaching zero, corrrect? Is that what is concerning you?

Your confusion is coming about by mixing what observers see or measure and what they calculate or interpret by what they see. When two observers have a relative speed between them, they can measure that relative speed and they can calculate the relative time dilation factor but they cannot see or measure that time dilation factor directly. What they can see and measure is the relativistic doppler factor and from that, they can calculate or interpret the relative time dilation factor. The relativistic doppler factor is the perceived measurement of the time dilation rather than the actual time dilation factor.

Let's lay out some facts about your scenario that you didn't specifically mention but are nevertheless true. First, we'll describe and analyze everything in the commom rest frame of Earth and NEC. You say that it takes 2 years for John to travel from Earth to NEC traveling at 0.9c. This means that the distance between Earth and NEC is 2 years multiplied by 0.9c or 1.8 light years, correct? Let's also assume that there is a clock on NEC that has been previously synchronized using Einstein's convention to a clock on Earth. Let's suppose that the time of the clock on Earth reads 0 years at the moment that John passes by Earth. Since it takes 1.8 years for light to go from NEC to Earth, Ann on Earth will see the clock on NEC as reading -1.8 years at the time John passes her. John will also see the clock on NEC as reading -1.8 years at that same moment.

Then as John travels over the next two years from Earth to NEC, Ann will observe him only a little more than half way there during that interval of time. It will take another 1.8 years for her to actually see him arrive at NEC at which time she will see the clock on NEC read 2 years and her own clock will read 3.8 years.

Now, what will she see of John's clock? We have agreed that his clock starts at zero when he passes her. She will see his clock running slower than her clock but not by the factor of 1/gamma. Instead we need to use the relativistic doppler formula for clocks moving away from each other which is:

√[(1-β)/(1+β)]

In our case, β = 0.9 so the relatistic doppler factor is:

√[(1-0.9)/(1+0.9)]
√[(0.1)/(1.9)]
√[(0.1)/(1.9)]
√[0.05263]
0.2294

This means that Ann will see John's clock running slow by a factor of 0.2294 so that instead of his clock advancing by 2 years for his trip during the 3.8 years that she is observing his trip, she will see it advance by 3.8 times 0.2294 or 0.8718 years in agreement with the time dilation factor that you calculated.

Everything OK so far?

Now let's do the same thing all over again from the frame of reference in which John is stationary:

From this frame of reference, Earth and NEC are traveling toward John at 0.9c but they are not 1.8 light years apart, Instead, we can use the length contraction factor 1/gamma to calculate their distance apart and it will be 0.7846 light years, since gamma is 2.294. We can also calculate how long it will take for John to make the trip from Earth to NEC (actually for NEC to get to John) and it will be 0.7846 light years multiplied by 0.9c or 0.8718 years. This is his proper time for the trip and is in agreement with what Ann observed of his proper time.

Now remember, we said that at the moment of John passing Earth (actually Earth passing John), he will observe the clock on Earth reading zero as well as his own clock, and the clock on NEC as reading -1.8 years. During the 0.8718 years of the trip, John will see the clock on NEC running faster than his own clock using the relativistic doppler formula for clocks moving towards each other (which is the reciprocal of the value for clocks moving away from each other):

√[(1+β)/(1-β)]

We could do the calculation as we did before but we can also just take the reciprocal of the previous calculation, 0.2294, and we end up with 4.359.

During John's "trip", he will observe the clock on NEC as advancing 4.359 times his own clock, 0.8718 years, which turns out to be 3.8 years, so he will observe NEC's clock go from -1.8 years to 2.0 years during the trip in complete agreement with what was calculated from the Earth-NEC rest frame.

 

[JesseM]

Just so Shark doesn't get more confused, note that ghwellsjr's calculations concern apparent visual rates of clock ticking which are influenced by the Doppler effect, rather than pure time dilation in a given frame which isn't. The rate that you see a clock ticking in the visual sense is different from the rate that it is ticking relative to coordinate time in your rest frame.

 

[ghwellsjr]

Just so Shark doesn't get more confused, note that ghwellsjr's calculations concern apparent visual rates of clock ticking which are influenced by the Doppler effect, rather than pure time dilation in a given frame which isn't. The rate that you see a clock ticking in the visual sense is different from the rate that it is ticking relative to coordinate time in your rest frame.

I thought I already made that very clear:

Your confusion is coming about by mixing what observers see or measure and what they calculate or interpret by what they see. When two observers have a relative speed between them, they can measure that relative speed and they can calculate the relative time dilation factor but they cannot see or measure that time dilation factor directly. What they can see and measure is the relativistic doppler factor and from that, they can calculate or interpret the relative time dilation factor. The relativistic doppler factor is the perceived measurement of the time dilation rather than the actual time dilation factor.

 

[JesseM]

OK, I guess I must have skimmed that part but in any case I don't really agree with the idea that there is some basic distinction between "measurements" and "calculations", after all even the Doppler shift involves a minimal calculation (finding the ratio of number of ticks seen vs. number of ticks on your clock), and you can also "measure" time dilation with very minimal calculations using synchronized clocks at different locations which are at rest in your frame, then you see the times T1 and T2 a moving clock reads when it passes a pair of stationary clocks and compare with the times t1 and t2 on each stationary clock at the moment the moving clock passed it, the gamma-factor of the time dilation equation is then (t2 - t1)/(T2 - T1).

 

[ghwellsjr]

OK, I guess I must have skimmed that part but in any case I don't really agree with the idea that there is some basic distinction between "measurements" and "calculations", after all even the Doppler shift involves a minimal calculation (finding the ratio of number of ticks seen vs. number of ticks on your clock), and you can also "measure" time dilation with very minimal calculations using synchronized clocks at different locations which are at rest in your frame, then you see the times T1 and T2 a moving clock reads when it passes a pair of stationary clocks and compare with the times t1 and t2 on each stationary clock at the moment the moving clock passed it, the gamma-factor of the time dilation equation is then (t2 - t1)/(T2 - T1).

You must have skimmed the rest of my post because that's exactly what I illustrated. I put a clock on NEC synchronized to the Earth clock and showed how t1 and T1 were both zero and how t2 = 2 years and how T2 = .8718 years and I said that this agrees with the time dilation factor that Shark calculated in his original post. I just didn't bother to repeat the same calculation that Shark did because he clearly already understands it:

This means that Ann will see John's clock running slow by a factor of 0.2294 so that instead of his clock advancing by 2 years for his trip during the 3.8 years that she is observing his trip, she will see it advance by 3.8 times 0.2294 or 0.8718 years in agreement with the time dilation factor that you calculated.

Shark is not having a problem with understanding time dilation between one observer and another. His problem is that when he applies the factor repeatedly from observer A to observer B and then from observer B back to observer A, he gets an illogical result. That is what his question was about:

...
She therefore calculates that the time, by John's clock, will read t₀ = 2/2.29 = 0.87 years.
...
He would therefore take Earth's (and NEC's) time to be proper time, because to him Earth is in the moving frame, and therefore he would calculate that the time taken on Earth for his journey is t₀ = 0.87/2.29 = 0.34years. Clearly this isn't right. What have I done wrong??!

and he repeated it in a second post with different numbers:

Ok but here's where I am getting stuck: 5 seconds pass for Bob, he calculates that only 2.18 seconds pass for Tim. Tim observes Bob to be moving past him and therefore calculates that it takes Bob 2.18/2.29 = 0.95 seconds. Obviously I am doing something totally wrong, as this can't be right.

But nobody on this thread agreed with him that his result wasn't correct and some even said his result was correct.

But Shark is clearly confused on the issue of what each observer sees:

Just to try and clarify: Say Tim and Bob are on two different spaceships and traveling at 0.9c relative to each other. Tim's clock reads that it takes 5 seconds for Bob to get from point A to point B. This means that Tim sees Bob's clock read t = t'/gamma = t'/2.29 = 2.18 seconds. However the same applies in Bob's frame of reference, i.e. Bob clock says that it takes Tim 5 seconds to get from A to B (actually B to A) but Bob sees that Tim's clock only reads 2.18. So when Tim uses the time dilation formula, he isn't calculating the actual time elapsed for Tim, in Tim's frame of reference, he is just calculating the time that he personally perceives for pass for Tim. Is this correct?! I am driving myself nuts trying to make sense of this.

So I'm addressing Shark's question and hopefully reducing his confusion rather than increasing it.

 

[JesseM]

You must have skimmed the rest of my post because that's exactly what I illustrated. I put a clock on NEC synchronized to the Earth clock and showed how t1 and T1 were both zero and how t2 = 2 years and how T2 = .8718 years and I said that this agrees with the time dilation factor that Shark calculated in his original post.

OK, I didn't see that you explicitly calculated the time dilation factor for the ship. But this doesn't get at the reciprocity question with time dilation, since you didn't have multiple clocks at rest relative to the ship that could show why the Earth or the NEC clock would also be ticking slower in the ship's frame. And if you intended these readings to show that time dilation could be measured directly, why say that stuff about "Your confusion is coming about by mixing what observers see or measure and what they calculate or interpret by what they see"? Clearly if you have multiple clocks, time dilation is something you "measure" just as much as Doppler shift, no?

Shark is not having a problem with understanding time dilation between one observer and another. His problem is that when he applies the factor repeatedly from observer A to observer B and then from observer B back to observer A, he gets an illogical result.

But it's not an illogical result when you take into account the relativity of simultaneity and the fact that they are calculating the time between different pairs of events on the other's worldline. If we consider the events of John leaving Earth and arriving at NEC, his time between the events is 0.87 years, and if we consider the event E1 on Ann's worldline that's simultaneous in Ann's frame with John arriving at NEC, the time for Ann between John leaving and E1 is 2 years, whereas if we consider the event E2 on Ann's worldline that's simultaneous in John's frame with John arriving at NEC, the time for Ann between John leaving and E2 is 0.34 years. That's why I emphasized the relativity of simultaneity and being careful to keep track of what specific events you want to calculate proper time between, that way you can see how both "Ann experiences 2 years" and "Ann experiences 0.34 years" are correct in different senses.

That is what his question was about:

...
She therefore calculates that the time, by John's clock, will read t0 = 2/2.29 = 0.87 years.
...
He would therefore take Earth's (and NEC's) time to be proper time, because to him Earth is in the moving frame, and therefore he would calculate that the time taken on Earth for his journey is t0 = 0.87/2.29 = 0.34years. Clearly this isn't right. What have I done wrong??!

and he repeated it in a second post with different numbers:

Ok but here's where I am getting stuck: 5 seconds pass for Bob, he calculates that only 2.18 seconds pass for Tim. Tim observes Bob to be moving past him and therefore calculates that it takes Bob 2.18/2.29 = 0.95 seconds. Obviously I am doing something totally wrong, as this can't be right.

But nobody on this thread agreed with him that his result wasn't correct and some even said his result was correct.

They are correct in the sense above. And I don't think either of these questions was meant to be about visual appearances, I think they were about the physical meaning of this sort of double application of the time dilation equation.

But Shark is clearly confused on the issue of what each observer sees:

Just to try and clarify: Say Tim and Bob are on two different spaceships and traveling at 0.9c relative to each other. Tim's clock reads that it takes 5 seconds for Bob to get from point A to point B. This means that Tim sees Bob's clock read t = t'/gamma = t'/2.29 = 2.18 seconds. However the same applies in Bob's frame of reference, i.e. Bob clock says that it takes Tim 5 seconds to get from A to B (actually B to A) but Bob sees that Tim's clock only reads 2.18. So when Tim uses the time dilation formula, he isn't calculating the actual time elapsed for Tim, in Tim's frame of reference, he is just calculating the time that he personally perceives for pass for Tim. Is this correct?! I am driving myself nuts trying to make sense of this.

I think you're probably reading too much significance into the word "see", often in relativity words like "see" and "observe" are used as shorthand for what happens in a particular observer's frame rather than what is actually seen visually. But no big deal either way, both approaches are useful to understand, whatever Shark's original intentions were.

 

[Mike_Fontenot]

The question (and the thread title!) is specifically about proper time, so you're not being helpful by trying to promote a particular coordinate-dependent notion of simultaneity here, when in relativity no notion of simultaneity is more "right" than any other.

I'm not arguing that the concept of "proper time" is NEVER useful, or that it WASN'T the original subject of this thread. But when Shark_774 said:

"Ok but here's where I am getting stuck: 5 seconds pass for Bob, he calculates that only 2.18 seconds pass for Tim. Tim observes Bob to be moving past him and therefore calculates that it takes Bob 2.18/2.29 = 0.95 seconds. Obviously I am doing something totally wrong, as this can't be right.",

he was describing a calculation that he was doing, which is actually about the relativity of simultaneity, and has nothing to do with "proper time". My goal was to get him on track to an understanding of the relativity of simultaneity, and I felt his possible misunderstanding of the concept of proper time was interfering with his ability to understand the relativity of simultaneity.

 

[JesseM]

I'm not arguing that the concept of "proper time" is NEVER useful, or that it WASN'T the original subject of this thread. But when Shark_774 said:

"Ok but here's where I am getting stuck: 5 seconds pass for Bob, he calculates that only 2.18 seconds pass for Tim. Tim observes Bob to be moving past him and therefore calculates that it takes Bob 2.18/2.29 = 0.95 seconds. Obviously I am doing something totally wrong, as this can't be right.",

he was describing a calculation that he was doing, which is actually about the relativity of simultaneity, and has nothing to do with "proper time". My goal was to get him on track to an understanding of the relativity of simultaneity, and I felt his possible misunderstanding of the concept of proper time was interfering with his ability to understand the relativity of simultaneity.

Well, I agree the calculation is correct from the perspective of each observer's inertial frame, I shouldn't have said you were "trying to promote a particular coordinate-dependent notion of simultaneity here" since your comment didn't really require the use of your CADO, just the common notion that we take it for granted that each inertial observer calculates things from the perspective of their inertial rest frame unless explicitly stated otherwise. But I don't think ignoring proper time altogether is a good way of addressing the question since Shark emphasized confusion about the meaning of "proper time" in the question, I think it's better to discuss how simultaneity and proper time relate to one another as in my last comment to ghwellsjr:

'If we consider the events of John leaving Earth and arriving at NEC, his time between the events is 0.87 years, and if we consider the event E1 on Ann's worldline that's simultaneous in Ann's frame with John arriving at NEC, the time for Ann between John leaving and E1 is 2 years, whereas if we consider the event E2 on Ann's worldline that's simultaneous in John's frame with John arriving at NEC, the time for Ann between John leaving and E2 is 0.34 years. That's why I emphasized the relativity of simultaneity and being careful to keep track of what specific events you want to calculate proper time between, that way you can see how both "Ann experiences 2 years" and "Ann experiences 0.34 years" are correct in different senses.'

 

[Shark 774]

To add to what I said above, let me pick some specific pairs of events here. Suppose we have the events of Tim passing one object A and then the event of Tim passing another object B, and in Bob's frame this takes 5 seconds. So, he calculates that Tim will experience 2.18 seconds between the event of passing A and the event of passing B, which is the objective proper time for Tim between these events. Now Tim picks the event E1 on Bob's worldline that is simultaneous in Tim's frame with the event of Tim passing A, and the event E2 on Bob's worldline that is simultaneous in Tim's frame with the event of Tim passing B. Naturally in Tim's frame the time between E1 and E2 is 2.18 seconds, so he calculates that that Bob will experience 0.95 seconds of time between E1 and E2, and this is indeed the objective proper time for Bob between those events. However, in Bob's frame E1 is not simultaneous with the event of Tim passing A, and E2 is not simultaneous with the event of Tim passing B, for Bob these are just two meaningless events on his worldline where nothing of significance happened.

This is exactly what I needed to know. I'm no longer confused. Thanks a lot to everybody, this has really helped me a lot in trying to learn Special Relativity in high school without a teacher! Thanks again everybody.