B How can time dilation be the same for both observers?

[Prem1998]

Here, I'm not talking about the twin paradox in which one of the observer's frame gets accelerated. I'm talking about the case in which they just move relative to each other without accelerating. I'm having some confusion. Please tell me where I'm wrong:
1.There are two people, Dick and Jane on Earth. Dick goes on space travel on a spacecraft which moves with a velocity 0.8c relative to Earth.
2. He leaves when both their clocks read 07:00 AM, and Dick takes one clock along with him. Both of them have telescopes by which they can observe the time elapsed on another clock. When the time is 07:05 AM on Dick's clock, he views Jane's clock from his telescope and sees that it is reading 07:03 AM.
3. So, when Jane's clock reads 07:03 AM, Dick's clock reads 07:05 AM.
4. So, if Jane views Dick's clock from her telescope when her clock reads 07:03 AM, she should see 07:05 AM on Dick's telescope.
5. But Dick is also moving at 0.8c for Jane, so by applying time dilation formula, so when Jane's clock reads 07:03 AM, i.e. 3 minutes have passed on her clock, only 1.8 minutes should pass on Dick's clock, so she should see 07:1.8 AM on Dick's clock.
6. So, when Jane's clock reads 07:03 AM, Dick's clock reads 07:1.8 AM. But this contradicts my third point.
Please explain this to me. Thanks.

 

[Jonathan Scott]

The main missing factor here is that the definition of "at the same time" (that is, simultaneity) depends on the velocity of the observer, so events which are the same time according to one observer may be at different times according to another observer.
Also, any observation using a telescope or similar has to be adjusted for the light speed delay according to the reference frame of the observer.

A way of visualising this is to imagine two people walking in different directions. Each observes that the other one is going slower than themselves in their own direction, but there is no contradiction in this.

 

[Prem1998]

The main missing factor here is that the definition of "at the same time" (that is, simultaneity) depends on the velocity of the observer, so events which are the same time according to one observer may be at different times according to another observer.

I'm still not getting this. Can you please just explain the problem I've written here? I mean, how can Dick's clock read both 07:05 AM and 07:1.8 AM at the same time when Jane's clock reads 07:03 AM ? I'm putting this another way like this:
Let's just pause the universe when Jane's clock reads 07:03 AM. At this instant, Jane is seeing 07:1.8 AM on Dick's clock while Dick is seeing 07:05 AM on the same clock. And, let's just forget about the telescope part. Let's just assume they can see each others' clock instantaneously ( I know it's impossible, but let's just suppose).

 

[jbriggs444]

I'm still not getting this. Can you please just explain the problem I've written here? I mean, how can Dick's clock read both 07:05 AM and 07:1.8 AM at the same time when Jane's clock reads 07:03 AM ?

Because "at the same time" is not an invariant concept. It depends on one's choice of reference frame.

 

[Orodruin]

I suggest reading my Insight article https://www.physicsforums.com/insights/geometrical-view-time-dilation-twin-paradox/

 

[Prem1998]

Okay, so when Jane sees 07:03 AM on her clock is not 'the same time' when Dick sees 07:03 AM on the same clock. Then, what is the time on Jane's clock according to Jane when there is 07:03 AM on her clock according to Dick?

 

[Prem1998]

Okay, so when Jane sees 07:03 AM on her clock is not 'the same time' when Dick sees 07:03 AM on the same clock. Then, what is the time on Jane's clock according to Jane when there is 07:03 AM on her clock according to Dick?

 

[Jonathan Scott]

There is no problem with defining "at the same time" at the same location. The complication comes when trying to define whether two events at different locations are at the same time. The usual definition of "at the same time" for a given observer is effectively that if the observer sent a light-speed signal to a remote event and received a light-speed signal back again, the event where the signal was reflected is assumed to occur at a time half way between the time when the signal was sent and when it was received back. However, when this situation is viewed from a frame where the observer is moving, the two light trips will not be equal, so from that point of view the events are not at the same time.

 

[Prem1998]

There is no problem with defining "at the same time" at the same location. The complication comes when trying to define whether two events at different locations are at the same time. The usual definition of "at the same time" for a given observer is effectively that if the observer sent a light-speed signal to a remote event and received a light-speed signal back again, the event where the signal was reflected is assumed to occur at a time half way between the time when the signal was sent and when it was received back. However, when this situation is viewed from a frame where the observer is moving, the two light trips will not be equal, so from that point of view the events are not at the same time.

If we suppose that the two persons are able to see each others' clocks instantaneously without any light pulse being sent, then will it be the same time when Dick sees 07:03 AM on Jane's clock and Jane sees 07:03 AM on Jane's clock? If not, then, what is the time on Jane's clock according to Jane when there is 07:03 AM on her clock according to Dick?

 

[Jonathan Scott]

If we suppose that the two persons are able to see each others' clocks instantaneously without any light pulse being sent, then will it be the same time when Dick sees 07:03 AM on Jane's clock and Jane sees 07:03 AM on Jane's clock? If not, then, what is the time on Jane's clock according to Jane when there is 07:03 AM on her clock according to Dick?

"Instantaneously" is not a valid concept when two events are at different locations. There is no global definition of "same time".

 

[A.T.]

If we suppose that the two persons are able to see each others' clocks instantaneously

Then relativity would be wrong, so it's pointless to ask what it would predict.

 

[Prem1998]

"Instantaneously" is not a valid concept when two events are at different locations. There is no global definition of "same time".

Just tell me what is the time on Jane's clock according to Jane when there is 07:03 AM on Jane's clock according to Dick.

 

[pervect]

I'm still not getting this. Can you please just explain the problem I've written here? I mean, how can Dick's clock read both 07:05 AM and 07:1.8 AM at the same time when Jane's clock reads 07:03 AM ? I'm putting this another way like this:
Let's just pause the universe when Jane's clock reads 07:03 AM. At this instant, Jane is seeing 07:1.8 AM on Dick's clock while Dick is seeing 07:05 AM on the same clock. And, let's just forget about the telescope part. Let's just assume they can see each others' clock instantaneously ( I know it's impossible, but let's just suppose).

"At the same time" is an ambiguous statement, as others have mentioned. You need to distinguish between "at the same time according to Dick" and "at the same time according to Jane". You also need to distinguish "what Dick sees" from "at the same time according to Dick". So there are four different concepts here which you are confusing and combining:

1) What Dick sees on Jane's clock through a telescope/
2) "At the same time" according to Dick, which is what Dick sees with the propagation delays compensated for.
3) What Jane sees on Dick's clock.
4) "At the same time" according to Jane.

It's hard to reword your example because I'm not sure what you meant due to the ambiguity of your language. So if you'd care to re-write it unambiguiously, we can give you a definite answer for all four observations.

 

[Prem1998]

Just forget all the above things. What I want to ask is: do both observers age at the same rate when they are moving at 0.8c relative to each other and Dick never comes back to Earth? If so, then how?

 

[PeroK]

Just forget all the above things. What I want to ask is: do both observers age at the same rate when they are moving at 0.8c relative to each other and Dick never comes back to Earth? If so, then how?

Ageing is the passing of elapsed time. So, ageing according to whom?

 

[jbriggs444]

Just forget all the above things. What I want to ask is: do both observers age at the same rate when they are moving at 0.8c relative to each other and Dick never comes back to Earth? If so, then how?

As has been suggested by @pervect previously, this is an ambiguous question. What would you mean by a rate at which someone ages?

You could firm up the question by considering Jane aging from 20 to 21 [by her personal biological clock] and Dick aging from 20 to 21 [by his personal biological clock] and how you could match these intervals up using frame-relative notions of "at the same time".

 

[Jonathan Scott]

"Age at the same rate" assumes global time.

This is like the following question:
Dick and Jane start from the same spot.
If Dick walks 100m north and Jane walks 100m north-east, which one is further ahead?

 

[Prem1998]

As has been suggested by @pervect previously, this is an ambiguous question. What would you mean by a rate at which someone ages?

You could firm up the question by considering Jane aging from 20 to 21 [by her personal biological clock] and Dick aging from 20 to 21 [by his personal biological clock] and how you could match these intervals up using frame-relative notions of "at the same time".

Let me say it like this:
'Three years have passed on Earth since Jane's twin Dick went to space travel. Now, God pauses the universe and visits Earth to see Jane and then visits the spacecraft to see Dick. Now, which one will God find more older?'
I'm sorry if I'm asking stupid questions. But I'm not getting the concept.

 

[PeroK]

Let me say it like this:
'Three years have passed on Earth since Jane's twin Dick went to space travel. Now, God pauses the universe and visits Earth to see Jane and then visits the spacecraft to see Dick. Now, which one will God find more older?'
I'm sorry if I'm asking stupid questions. But I'm not getting the concept.

You're not listening. There is no universal "now". In your mind, you probably have the Earth's reference frame as God's reference frame in this case. But, you need a reference frame to define "now".

 

[Ibix]

Let me say it like this:
'Three years have passed on Earth since Jane's twin Dick went to space travel. Now, God pauses the universe and visits Earth to see Jane and then visits the spacecraft to see Dick. Now, which one will God find more older?'
I'm sorry if I'm asking stupid questions. But I'm not getting the concept.

Depends which frame God used when he defined the "now" at which he paused the universe. Assuming such a concept makes any sense. There's no way around the fact that "at the same time" is only unambiguously defined for two things at the same place, no matter how much you wiggle.

 

[jbriggs444]

Let me say it like this:
'Three years have passed on Earth since Jane's twin Dick went to space travel. Now, God pauses the universe and visits Earth to see Dick and then visits the spacecraft to see Dick. Now, which one will God find more older?'
I'm sorry if I'm asking stupid questions. But I'm not getting the concept.

God has a choice. He can pause the universe in a configuration where Jane is older than Dick. He can pause the universe in a configuration where Dick is older than Jane. And we cannot tell which one is "right". That is because we have no way to measure "at the same time" unambiguously to be sure that he paused all parts of the universe "at the same time".

 

[Prem1998]

Depends which frame God used when he defined the "now" at which he paused the universe. Assuming such a concept makes any sense. There's no way around the fact that "at the same time" is only unambiguously defined for two things at the same place, no matter how much you wiggle.

Let's just say, I, as a person on Earth, have power to stop everything, stop the time. I use my power and everything, wherever it was, whatever it was doing, whatever reference frame it was in, just stops. The spacecraft in which Dick is, stops, whatever process was going on, just stops. Except me. Now, I watch go to watch Dick and Jane separately. Who will I find older?

 

[jbriggs444]

Let's just say, I, as a person on Earth, have power to stop everything, stop the time. I use my power and everything, wherever it was, whatever it was doing, whatever reference frame it was in, just stops. The spacecraft in which Dick is, stops, whatever process was going on, just stops. Except me. Now, I watch go to watch Dick and Jane separately. Who will I find older?

You may need to take a step back. What do you think a reference frame is?

 

[Ibix]

Let's just say, I, as a person on Earth, have power to stop everything, stop the time. I use my power and everything, wherever it was, whatever it was doing, whatever reference frame it was in, just stops. The spacecraft in which Dick is, stops, whatever process was going on, just stops. Except me. Now, I watch go to watch Dick and Jane separately. Who will I find older?

Depends which frame's simultaneity convention you decided to use when you stopped everything.

 

[Prem1998]

I used Earth's simultaneity convention. Now, please, who will I find older?

 

[PeroK]

Let's just say, I, as a person on Earth, have power to stop everything, stop the time. I use my power and everything, wherever it was, whatever it was doing, whatever reference frame it was in, just stops. The spacecraft in which Dick is, stops, whatever process was going on, just stops. Except me. Now, I watch go to watch Dick and Jane separately. Who will I find older?

That's not a bad definition of your reference frame. You pick a time and then wander round the universe looking at everything. That's your refernce frame.

Things are not in a particular reference frames. All things are in all reference frames at all times.

For example, if a particle at CERN could stop the universe and wander round, they would find you massively length contracted in the direction that you are moving at near light speed relative to it.

 

[PeroK]

I used Earth's simultaneity convention. Now, please, who will I find older?

In the Earth's reference frame Dick is younger. That's easy!

 

[Prem1998]

That's not a bad definition of your reference frame. You pick a time and then wander round the universe looking at everything. That's your refernce frame.

Things are not in a particular reference frames. All things are in all reference frames at all times.

For example, if a particle at CERN could stop the universe and wander round, they would find you massively length contracted in the direction that you are moving at near light speed relative to it.

I think I'm getting this now. So, when the particle at CERN resumes the universe on finding me, then it'll find me of my normal length suddenly, right? So, it depends on who pauses the universe and what was his motion relative to things he is observing now when he paused the time.
Another, very very stupid question: if two people in different reference frames (moving with a non zero velocity with respect to one another) pause the universe, and meet me together, will they see both see different lengths of me and will they see different events occurring in the same place of the universe?

 

[jbriggs444]

I think I'm getting this now. So, when the particle at CERN resumes the universe on finding me, then it'll find me of my normal length suddenly, right?

When the particle at CERN pauses the universe, the frozen "you" will be short. But then you always were short as viewed from the reference frame where that particle is at rest. Pausing the universe did not change that. Nor does resuming the universe change it. Your "length" as viewed from a particular reference frame is the length of a snapshot of you taken at an instant in time using the simultaneity convention for that reference frame.

 

[robphy]

(If you want to talk about length contraction, maybe that should go in a different thread...
Though there is a relation, it's convoluted to discuss time dilation and length contraction concurrently while trying to clarify one.)

A spacetime diagram may be helpful here.

https://www.desmos.com/calculator/ti58l2sair [my t axis is horizontal ]
Tune v1 and v2 and you can see that their senses of simultaneity disagree in general... and that the time dilation is symmetrical in the sense that each observer's dashed line of simultaneity cuts off the other observer's 4-velocity (unit vector) by the same fraction. [Slide E to -1 to see the Euclidean version, and 0 for the Galilean version.]

 

[Prem1998]

When the particle at CERN pauses the universe, the frozen "you" will be short. But then you always were short as viewed from the reference frame where that particle is at rest. Pausing the universe did not change that. Nor does resuming the universe change it. Your "length" as viewed from a particular reference frame is the length of a snapshot of you taken at an instant in time using the simultaneity convention for that reference frame.

What about this question of mine: if two people in different reference frames (moving with a non zero velocity with respect to one another) pause the universe, and meet me together, will they see both see different lengths of me and will they see different events occurring in the same place of the universe?

 

[jbriggs444]

What about this question of mine: if two people in different reference frames (moving with a non zero velocity with respect to one another) pause the universe, and meet me together, will they see both see different lengths of me and will they see different events occurring in the same place of the universe?

So they pause the universe, come over and look at their respective snapshots of you. One fellow looks at his snapshot, nods and sees you with one length. The other fellow looks at his snapshot, nods and sees you with another length. Each then looks at the other fellows snapshot, shakes his head and sees the simultaneity issue that led to the discrepancy in the other snapshot.

There is no single "event" here. An event has both a specific place and a specific time. The top of your head and the bottom of your feet are never in the same place at the same time.

Edit: But yes, different frames can see different events at the same place. If I am in a train tossing a baseball, I can throw it with my right hand and catch it with my right hand. The event of the throw and the event of the catch are in the "same place"

If a passer by on the railway embankment watches this, he may see the event of the throw and the event of the passenger ten rows back turning a page of his newspaper being in the "same place"

 

[Prem1998]

I'm sorry but I only have a plain english understanding of relativity. I current am not knowledgeable enough to know about 4 velocities, etc. But, is it true that both the people who paused the universe will see different events at the same place?
And, the thing that I've got so far in this thread that what's a object's length and when is an event happening, all depends on the observer's relative velocity and hence the observations depend on who paused the universe. Can it be true that everything ever remains the same as it was, it's just our senses fooling us?

 

[Orodruin]

I'm sorry but I only have a plain english understanding of relativity.

This is a big problem, bigger than you might realize. The English language simply is not accurate enough to give a good description of what is going on without being prone to misunderstandings and misconceptions. This is why we use maths to describe physics. If you want to obtain a deeper understanding, you will have to do some maths or at least look at the graphical representations of the math.

Also, please stop using the statement "two people in different reference frames". Observers are not in different frames. All frames contains all observers. However, any observer is at rest in one particular frame, its rest frame. This has been pointed out several times in this thread already.

 

[Prem1998]

This is a big problem, bigger than you might realize. The English language simply is not accurate enough to give a good description of what is going on without being prone to misunderstandings and misconceptions. This is why we use maths to describe physics. If you want to obtain a deeper understanding, you will have to do some maths or at least look at the graphical representations of the math. Also, please stop using the statement "two people in different reference frames". Observers are not in different frames. All frames contains all observers. However, any observer is at rest in one particular frame, its rest frame. This has been pointed out several times in this thread already.

I'm sorry. By observers in different reference frames, I meant any frame which is at rest relative to him/her.

 

[Jonathan Scott]

The universe is the same regardless of reference frame, but the way of identifying events in space and time depends on the choice of reference frame.

You are already familiar with an everyday part of the concept of a reference frame, which is that the coordinates of some event relative to you depend on where you are and which way you are facing, but it is easily understood that the events are the same even if the observer is in a different position or facing in a different direction. We know how to map between those frames using translations and rotations.

An extension of this concept also applies to the relationship between time and space in reference frames for observers traveling at different velocities. This mapping is called a "boost". It is mathematically closely related to rotations, but because of the difference between time and space the quantities involved are cosh and sinh rather than cos and sin. (Technically it is equivalent to a rotation through an imaginary angle).

 

[Prem1998]

The universe is the same regardless of reference frame, but the way of identifying events in space and time depends on the choice of reference frame.

It doesn't make any sense to say that the universe is same regardless of the reference frame. I mean, the universe is just the events which happen in it. And, if the events happening at any instant depend on the reference frame, then for different people, universe is different. Saying that the universe is same implies that all the relativistic effects are due to our incorrect observations of the universe due to our relative velocities but it's not the case.

 

[Jonathan Scott]

And, if the events happening at any instant depend on the reference frame, then for different people, universe is different.

The events which happen do NOT depend on the reference frame. The location and time used to identify an event depends on the reference frame, and aspects such as velocity, orientation and time dilation are relative to the reference frame, but the view from any reference frame can be systematically mapped to how it appears from any other reference frame by a suitable Lorentz transformation.

 

[Prem1998]

The events which happen do NOT depend on the reference frame. The location and time used to identify an event depends on the reference frame, and aspects such as velocity, orientation and time dilation are relative to the reference frame, but the view from any reference frame can be systematically mapped to how it appears from any other reference frame by a suitable Lorentz transformation.

I didn't say that for different observers different events happen. But, what one will observe in the universe when he pauses the time depends on which reference he was in when he paused the time. So, obviously different observers will see different universes, different lengths of the same thing, and different instants of the same happened event. So, the universe isn't the same and is different for different people.

 

[jbriggs444]

To most of us, the universe includes our past and our future [along with the "elsewhere" outside our light cone]. The fact that a snapshot of the universe taken at a particular instant does not include these pieces means that a snapshot of the universe not the same as the whole universe.

Yes, we all agree that the contents of two hyper-surfaces of simultaneity sliced out of the universe at different angles will not be identical.

 

[Jonathan Scott]

... So, the universe isn't the same and is different for different people.

By that logic, you'd have to claim that the universe is different if you're facing in a different direction, which isn't a useful definition of "different".

If I'm looking at something from one direction I might claim that its "width" from my point of view is different from what someone sees when looking at it from a different direction, but that doesn't mean we are observing different events.

Similarly, if we are looking at things from different velocities, we may disagree about things such as lengths and time intervals, but those depend on our point of view and there's a simple rule for mapping between those points of view. So the view of the universe is different, but we all observe the same events and the same physics, and can easily map any view to any other.

 

[pixel]

How can time dilation be the same for both observers?

Your confusion is arising due to an incorrect interpretation of time dilation. I only read through about half of the responses so sorry if I'm repeating here.

Following your scenario, when Dick reads his clock at 07:05 AM, 5 minutes have elapsed in his frame of reference. This is what’s known as a "proper time interval" because the clock is in the same location in his frame at the beginning and end of the interval. The time dilation formula tells us what time interval would be measured in another frame of reference moving with respect to Dick’s, in which the clock is not in the same location at the start and end of the interval. For v = 0.8c, it's 1.67 x 5 = 8.35 minutes in Jane’s frame (hence “moving clocks slow down.”).

Now the fallacy is to say, well if it 8.35 minutes in Jane’s frame, then it must be 1.67 x 8.35 minutes in Dick’s frame etc. etc. ad infinitum, but the 8.35 minutes in Jane’s frame is not a proper time interval so you can't apply the time dilation formula to it.

 

[PeroK]

It doesn't make any sense to say that the universe is same regardless of the reference frame. I mean, the universe is just the events which happen in it. And, if the events happening at any instant depend on the reference frame, then for different people, universe is different. Saying that the universe is same implies that all the relativistic effects are due to our incorrect observations of the universe due to our relative velocities but it's not the case.

This is where a knowledge of classical physics is an advantage rather than learning relativity first. A simple example is that the sun rises at different times around the world. If I'm in London and phone someone in New York, it may be dawn in London, but it won't be in New York. So, does that make the universe "different" for us?

The frequency of a siren on an ambulance depends on how fast you are moving with respect to the ambulance.

Or, I travel to New York on one flight and record the distance as 5,600km. Someone else travels on a different flight on a different path and measures the distance as 5,900km. So, that's two different distances from London to New York. You're familiar with the concept that we can both start from A and travel to B and measure different distances we have travelled. Once you understand relativity, you find we may also have measured different elapsed times since we last met. What seems obvious for distances may seem world-shattering for time. But, once you understand the structure of spacetime, it's not so world-shattering.

The point is that you really have to think through what you need to be "different" to get an inconsistency. None of these examples shows an inconsistent universe. Measurements are different, but not in an inconsistent way. Measurements, even in classical physics, depend on your reference frame.

 

[stevendaryl]

Okay, so when Jane sees 07:03 AM on her clock is not 'the same time' when Dick sees 07:03 AM on the same clock. Then, what is the time on Jane's clock according to Jane when there is 07:03 AM on her clock according to Dick?

Let me make this a little more concrete by imagining that Jane and Dick are each in a long spaceship, and the spaceships are passing each other. There is a clock in the front of each spaceship, and a clock in the rear of each spaceship. Let's suppose that the spaceships are so long that it takes 5 minutes for the front of Dick's spaceship to travel from the rear of Jane's spaceship to the front of Jane's spaceship, and let's suppose that Dick's spaceship is traveling at 0.8 c, relative to Jane, and that Dick's front clock and Jane's rear clock both say 7:00 when they pass each other. Then the time dilation rules tell us the following:

1. When Dick's front clock passes Jane's rear clock, they both say 7:00.
2. When Dick's rear clock passes Jane's rear clock, Dick's rear clock says 7:05, while Jane's rear clock says 7:03.
3. When Dick's front clock passes Jane's front clock, Dick's front clock says 7:03, while Jane's front clock says 7:05.

Jane and Dick have two different explanations for these phenomena. The following picture illustrates Jane's story. According to Jane, there are three odd facts about Dick's ship:

• Dick's clocks advance slower than Jane's--they only advance 3 minutes for each 5 minutes that Jane's clocks advance.
• Dick's clocks are not correctly synchronized: His rear clock is set 3 minutes and 12 seconds ahead of his front clock.
• Dick's ship is shorter than Jane's, by the factor 0.6.

She explains the three facts above in the following way:

1. Initially, both of Jane's clocks are set to 7:00. Dick's front clock is set to 7:00, but his rear clock is set to 7:03:12
2. By the time Dick's rear clock has reached Jane's rear clock, her clocks have advanced 3 minutes, while Dick's clocks have only advanced 1 minute and 48 seconds. So Dick's rear clock advanced from 7:03:12 to 7:05, while Jane's rear clock advanced from 7:00 to 7:03.
3. By the time Dick's front clock has reached Jane's front clock, her clocks have advanced 5 minutes, while Dick's clocks have only advanced 3 minutes. So Dick's front clock shows time 7:03, while Jane's front clock shows time 7:05.

Dick has a completely different explanation. From his point of view:

• Jane's clocks run slower, advancing only 3 minutes to his 5.
• Jane's clocks are not synchronized; her front clock is ahead of her rear clock by 3 minutes and 12 seconds.
• Jane's ship is shorter than Bob's by a factor of 0.6.

His story explaining the same three events is illustrated by the following picture:

His explanation of the three facts is:

1. Initially, Dick's two clocks are set to 7:00. Jane's rear clock is set to 7:00, but her front clock is set to 7:03:12
2. When Dick's front clock reaches Jane's front clock, his front advances 3 minutes, to 7:03, while Jane's front clock advances only 1 minute 48 seconds, to 7:05
3. When Dick's rear clock reaches Jane's rear clock, his rear clock will have advanced 5 minutes, from 7:00 to 7:05, while Jane's rear clock will have advanced 3 minutes, from 7:00 to 7:03.

So each can consistently believe that it is the other ship that has time-dilated, out-of-synch clocks.

 

[pervect]

For a good reference on the following exercise, try Bondi's old (but very good) book, "Relativity and Common Sense". The numbers may be slightly different than yours, but they're easy to work with.

Dick and Jane are together at 7:00. At 7:01, Dick sends out a bright red flash, which illuminates Jan'es clock. This signal returns to Dick at 7:04, at which point Dick "sees" Jane's clock, and jots down the reading.

Dick concludes that at 7:02.30, Jane was 90 light seconds away (1 minute and 30 seconds, one half of the round-trip time of 3 light minutes). This is because, in Dick's frame, light travels at a constant velocity, so the time it took the light to get to Jane must be the same time it took for it to return.

This is worth working out for yourself, it's one of the main points of this thought exercise. Using mathematical notation, If Dick sends a signal at time t1, and receives it at time t2, dick concludes that at time (t1+t2)/2, the signal was reflected off an object that was at a distance of c*(t2 - t1)/2 , c being the speed of light.

Going back to the example - we can say that Dick "sees" Janes clock illuminated at 7:04, but the time, according to Dick, at which it was illuminated was 7:02.30. The difference is due to the travel time of light.

GIven this information, you can work out Jane's speed relative to Dick - you should get 3/5 the speed of light.

Now, here's the kicker. According to relativity, when Jane's clock was illuminated, it reads 7:02, not 7:02:30.

The argument works just as well if Jane sends out a blue flash of light towards Dick at 7:01 according to Jane's clock.

 

[Prem1998]

Thanks a lot for all the answers. It was really helpful.