# Time Dilation Paradox (Not resolved)

bgq
Hi,

I am new to this forum, and I wish I get some help here.

I really can't find any solution to this paradox; I am going to deny all the special theory of relativity because of this. The paradox is very simple, here it is:

Consider two persons sitting on two adjacent trains at rest. Each person has a timer, and both of them are set to zero. The timers are designed in a way so that when the timer reaches 2 minutes, the other train will explode (and of course damaging the timer inside)! The two trains start moving in opposite directions with same constant speed; the timers start exactly as the trains starts to move.

Now here is the paradox: For each person frame of reference, he is at rest, and the other is in motion, then the other timer will run slower, so he will be very happy that his timer will reach 2 minutes first ... destroying the other train. The paradoxical question which of the trains will be destroyed?! Although the whole situation is completely symmetrical, yet each person will assure that his timer will reaches 2 minutes first destroying the other train.

Thanks in advance for any ideas.

## Answers and Replies

Mentor
The timers are designed in a way so that when the timer reaches 2 minutes, the other train will explode (and of course damaging the timer inside)!
How does train 1 decide "the other timer reached 2 minutes"? It depends on the position and velocity of the timer and train. Remember: There is no universal simultaneity in special relativity.
If you add specific position data, I can show you how that works. Both trains will explode at the same time, if you fix some method to determine "the time the other timer shows".

Relativity is self-consistent, if a paradox does not work out this just means you did something wrong.

bgq

The train may know that the timer reaches 2 minutes by electromagnetic signal for example. My point is whatever the mechanism of communication, the situation is completely symmetrical.

Anyway, each person can predict that his timer will reach 2 minutes before the other although the symmetry of the situation assures that both will be at the same instant (lets say with respect to a third observer which is in complete symmetry with respect to the two trains).

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Gold Member
The problem is your statement that "the other train will explode (and of course damaging the timer inside)!", as if this can happen in an instant of time. Assuming that you have explosives along the length of each train that can be set off by the other train at any point and then the explosion propagates along the rest of the train at some finite speed, then both trains will explode. It's perfectly symmetrical.

The train may know that the timer reaches 2 minutes by electromagnetic signal for example. My point is whatever the mechanism of communication, the situation is completely symmetrical.

Anyway, each person can predict that his timer will reach 2 minutes before the other although the symmetry of the situation assures that both will be at the same instant (lets say with respect to a third observer which is in complete symmetry with respect to the two trains).

Such a signal travels at a finite speed (<= c). In that case, both trains explode, each some time after their own timer passes 2 minutes. Each sees their own timer reaching two minutes, knowing the other train will explode. Each explodes before seeing the other explode.

Adding a little to this: each train sees their own clock reach 2 minutes first, knowing the other guy is doomed. However, some time later, each sees the other clock approaching 2 minutes, and knows they are doomed. If signals travel at light speed, each blows up the moment they see the other clock reach 2 minutes (and theirs reads later). If signals travel by current over long wires, each blows up some time after the other clock reaches 2 minutes, and theirs is well beyond two minutes.

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phyti
Hi,

Consider two persons sitting on two adjacent trains at rest. Each person has a timer, and both of them are set to zero. The timers are designed in a way so that when the timer reaches 2 minutes, the other train will explode (and of course damaging the timer inside)! The two trains start moving in opposite directions with same constant speed; the timers start exactly as the trains starts to move.

Now here is the paradox: For each person frame of reference, he is at rest, and the other is in motion, then the other timer will run slower, so he will be very happy that his timer will reach 2 minutes first ... destroying the other train. The paradoxical question which of the trains will be destroyed?! Although the whole situation is completely symmetrical, yet each person will assure that his timer will reaches 2 minutes first destroying the other train.

Thanks in advance for any ideas.

The situation is not symmetrical. It's another instance of the 'twin' scenario.
Each observes the other clock to run slower, but only a reunion will allow a comparison for accumulated time. There are three possible explosive outcomes, 1, 2, or both.

Gold Member
Such a signal travels at a finite speed (<= c). In that case, both trains explode, each some time after their own timer passes 2 minutes. Each sees their own timer reaching two minutes, knowing the other train will explode. Each explodes before seeing the other explode.
I thought the whole idea here was that the timer (or the observer) on each train causes the other train to explode (or initiates the explosion) and that when the timer (and the observer) explodes, it prevents the other train from exploding. Wouldn't each observer see immediately the other train begin to explode at his 2-minute mark, not some time later? What do you mean by "Each explodes before seeing the other explode"? I guess you must mean each observer explodes before seeing the other observer explode, correct?

Gold Member
The situation is not symmetrical. It's another instance of the 'twin' scenario.
Each observes the other clock to run slower, but only a reunion will allow a comparison for accumulated time. There are three possible explosive outcomes, 1, 2, or both.
The situation is symmetrical. Why do you think it is not? Where is the asymmetry? This has nothing to do with the 'twin' scenario.

It also has nothing to do with each observer observing the other ones clock. Each observer only watches his own clock and when it reaches 2 minutes, it causes the other train to explode. Since the clocks are separated in space when they both cause the other train to explode, both trains will explode. There is only one possible outcome.

altergnostic
I don't think it is the same as with Twin Paradox. Here we have straight velocities in opposite directions starting at the same time, the situation is completely symmetrical. Insert an embankment at the origin (where the trains start moving). From this frame, you would observe both trains explode at the same.

There is no paradox: pick any train as your rest frame. As the trains depart (without outside references, you would only see the other as moving) you have to consider the time light from the other train takes to reach you. Say a light beam comes out of the train at a time T and position X, that light will take some time to reach you, and during that time, the train keeps moving forward. When the light reaches you, the time you will see on your clock will be:

T + the time it takes light to travel the distant between both trains

This light will bring you information from the position X, where the train was located when the light departed from it. But when this light reaches you, the train will not be at the position X anymore, it wil be at:

X + the distance the train traveled during the time light took to reach you

That's where length contraction and time dilation comes from. You will see the train at a shorter distance than it will see itself, at a longer time than the train would see itself to be when he sent the light beam. The relativity transforms allow you to transform your local measurements into proper numbers.

Thus, both trains would explode, when each clock ticks 2 minutes, the signals to detonate the bombs would be sent at exactly the same proper times, travel to the other during the same interval, and explode both trains at their proper times (which will be higher than 2).

I thought the whole idea here was that the timer (or the observer) on each train causes the other train to explode (or initiates the explosion) and that when the timer (and the observer) explodes, it prevents the other train from exploding. Wouldn't each observer see immediately the other train begin to explode at his 2-minute mark, not some time later? What do you mean by "Each explodes before seeing the other explode"? I guess you must mean each observer explodes before seeing the other observer explode, correct?

I was replying to the OP's clarification that instantaneous communication is not implied:

"The train may know that the timer reaches 2 minutes by electromagnetic signal for example. My point is whatever the mechanism of communication, the situation is completely symmetrical."

This means, that for each train, the sequence of events is:

1) I see my timer reach two minutes. I see the the other timer before two minutes. I now know the the other train will explode when it gets the signal from my timer reaching 2 minutes

2) If the detonation signal is light speed, then each train sees the other timer reach two minutes (while theirs is beyond two minutes) at the moment off their own detonation.

3) If they survive their own explosion, they see the other explosion some time after theirs.

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Gold Member
I guess when the OP talked about trains I assumed that they were very long and even after 2 minutes, they still hadn't passed each other and the signal to explode the other train didn't have very far to travel, just across to the next track. The way the rest of you seem to be interpreting this is that it would be the same as if we were talking about a single train car and a signal has to propagate over a long distance.

But the answer remains the same either way. In the frame that the OP defined, once each clock reaches 2 minutes, it initiates a signal to explode the other train and both signals will be in transit simultaneously and will reach the other train car after the 2-minute mark on each train. You can transform the scenario into any other frame but it won't change the result.

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bgq
2) If the detonation signal is light speed, then each train sees the other timer reach two minutes (while theirs is beyond two minutes) at the moment off their own detonation.

This is exactly my point: We have three possible outcomes, and each of them is obtained with respect to a certain inertial frame of reference. Each person will observe that his timer reaches 2 minutes before the other (thus the other explodes, but his own not), and for a third inertial frame (symmetrical to both), both timers reach 2 minutes at the same time, and so the both trains will explode.

Gold Member
2) If the detonation signal is light speed, then each train sees the other timer reach two minutes (while theirs is beyond two minutes) at the moment off their own detonation.

This is exactly my point: We have three possible outcomes, and each of them is obtained with respect to a certain inertial frame of reference. Each person will observe that his timer reaches 2 minutes before the other (thus the other explodes, but his own not), and for a third inertial frame (symmetrical to both), both timers reach 2 minutes at the same time, and so the both trains will explode.
Different reference frames do not allow observers to see different things or for different things to actually happen. They only allow us (or the observers) to calculate different coordinates for events. What you are overlooking is the fact that both signals to detonate the other train are in transit at the same time and once the signals are sent, there is no way to prevent the explosion of the other train and so they both explode. Once you set up the problem in one frame, that is all you need to determine the outcome. Just for the fun of it, you can transform the same scenario into any other frame to see how that frame also predicts the same outcome but you have to do this correctly using the Lorentz Transformation process. You can't just redefine the scenario in a different reference frame and assume a different outcome.

altergnostic
2) If the detonation signal is light speed, then each train sees the other timer reach two minutes (while theirs is beyond two minutes) at the moment off their own detonation.

This is exactly my point: We have three possible outcomes, and each of them is obtained with respect to a certain inertial frame of reference. Each person will observe that his timer reaches 2 minutes before the other (thus the other explodes, but his own not), and for a third inertial frame (symmetrical to both), both timers reach 2 minutes at the same time, and so the both trains will explode.

I have to disagree. You will explode before you see the other train exploding. When the signal of detonation reaches you, it will be reaching the other at the same time (but you won't be able to measure or observe this anyway). Both will explode, but by the time light from the other explosion reaches you, you're already dead (so you can't see the other explosion, unless you believe that your spirit lives on, but that's outside the scope of relativity).

phyti
The situation is symmetrical. Why do you think it is not? Where is the asymmetry? This has nothing to do with the 'twin' scenario.

It also has nothing to do with each observer observing the other ones clock. Each observer only watches his own clock and when it reaches 2 minutes, it causes the other train to explode. Since the clocks are separated in space when they both cause the other train to explode, both trains will explode. There is only one possible outcome.

You don't know what their speed is before parting. Say it's u.
A leaves B at speed u+v, B leaves A at u -v.
The speed difference is (u+v)-(u-v) = 2v. This is true for any value of u
that keeps the result <c. Time dilation before parting is a function of u/c.
After parting what each observer measures is the difference in their absolute
speeds, thus they cannot determine the time dilation of each, unless there is a
clock comparison. The clock rate is symmetrical because it's doppler effects.
Each sees the other clock running fast when converging, and slow when diverging.
The speeds (u+v) and (u-v) cannot be equal, therefore we cannot expect equal
time dilation for each. One will accumulate 2 minutes before the other.

You don't know what their speed is before parting. Say it's u.
A leaves B at speed u+v, B leaves A at u -v.
The speed difference is (u+v)-(u-v) = 2v. This is true for any value of u
that keeps the result <c. Time dilation before parting is a function of u/c.
After parting what each observer measures is the difference in their absolute
speeds, thus they cannot determine the time dilation of each, unless there is a
clock comparison. The clock rate is symmetrical because it's doppler effects.
Each sees the other clock running fast when converging, and slow when diverging.
The speeds (u+v) and (u-v) cannot be equal, therefore we cannot expect equal
time dilation for each. One will accumulate 2 minutes before the other.

Nonsense. The OP clearly stated:

"
Consider two persons sitting on two adjacent trains at rest.... The two trains start moving in opposite directions with same constant speed; the timers start exactly as the trains starts to move."

Then, in clarification, the OP asserted complete symmetry. The OP's reasoning that symmetry must mean either neither explodes or both explode is correct reasoning. The only flaw was believing the SR said anything different, and also that, for stated scenario, SR clearly says both trains explode.

[EDIT: I missed that phyti has a much bigger problem than reading the OP. Velocity addition in SR is not u+v. Instead it is (u+v)/(1-uv). Further, if you analyze from a frame moving at u relative to the initial trains, it will have no effect on the result at all - that is a fundamental feature of SR.]

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Shall we do some maths?

Let's consider the rest frame of train A. In this frame, train A is stationary and train B is doing velocity v in the x direction. At time t=2, train A emits a signal at speed V (not necessarily the speed of light, but V>v or it'll never catch the other train), which catches up with train B and causes it to explode at time tB. That time is easily calculated by equating the distance travelled by the train (vtB) with that travelled by the signal in two minutes' less (V(tB-2)):
$$t_B=\frac{2V}{V-v}$$

What time (according to train A) does train B emit its signal? Train B emits its signal when its clock reads two minutes. Per the time dilation formula, then, train A says that train B emits its signal at time T:
$$T=\frac{2}{\sqrt{1-v^2/c^2}}$$

Do the trains explode?

Train B does - since T>2 for any non-zero v, train A emits its signal first and train B is doomed (unless it can outrun the detonation signal).

Train A only explodes if train B receives the signal from train A after it emitted its own signal - in other words, train A explodes if T<tB. That is:
$$\begin{eqnarray} \frac{2}{\sqrt{1-v^2/c^2}}&<&\frac{2V}{V-v}\\ 1-\frac{v}{V}&<&\sqrt{1-v^2/c^2} \end{eqnarray}$$
Square both sides, rearrange, cancel the ones, and pull out a common factor of v to get:
$$v\left(v\frac{c^2+V^2}{c^2V^2}-\frac{2}{V}\right)<0$$
The left hand side has two roots and is negative between, so train B gets to emit its signal as long as the velocity of train B lies between those roots:
$$0<v<\frac{2V}{1+V^2/c^2}$$
Note that the upper bound is never less than V unless V>c, which isn't possible. That means that if the train is going fast enough to exceed the upper bound, it is also going fast enough to outrun the detonation signal. So train A also explodes unless it is fast enough to outrun the detonation signal.

Conclusion: Both trains explode unless they are going fast enough to outrun the detonation signals. Observers may differ about which one emits a signal first, and which one explodes first, but the delay between the detonate signal being sent and being received always allows the later train time to send its detonation signal.

altergnostic
But isn't the beams sent when each train's proper time equals 2? Picking either train frames as your rest frame will result in different local and relative times, that's correct, but locally both will send their beams at t=2 proper. If you pick a third frame at the origin, and assuming all signals travel at the same speed, and both trains at the same speed relative to the origin, every event on each train will be observed as simultaneous. That is sort of a preferred frame, in the sense that it is the only frame of the three that can observe simultaneity, and is at the midpoint of every other event, isn't it? Of course it is not a preferred frame in any other sense, since you can choose any and all calculations from it will be valid. I think einstein proposed this very scenario, with two mirrors at 45 degrees so you could see all events at the same time. In this sense, both trains are observed as late by the same amount, and both would explode at the same proper time from the origin. Is this correct?

Mentor
In this sense, both trains are observed as late by the same amount, and both would explode at the same proper time from the origin. Is this correct?

If I'm understanding you properly, the answer is "yes".

You will also notice that Ibix carefully worded his conclusion as "Observers may differ"; he did that because it is possible to select a frame in which the two events are simultaneous.

Of course that frame is not preferred or special in any way - I can find such a frame for any two space-like separated events, and this is no more interesting than the observation that I can draw a straight line between any two points in a Euclidean plane.

As altergnostic says, yes, there exists a frame in which the signals are sent simultaneously and received simultaneously. But as Nugatory says - not redundantly at all - that is only a special frame in that it happens to reflect the symmetry of the problem. It's often a good frame to work in because things are often simpler in this frame (altergnostic apparently correctly analysed the situation in this frame without any maths at all, for example), but that doesn't make it the 'right' frame. Merely the convenient frame.

I don't think Nugatory's line analogy is quite right, however. A better one would be that I can always rotate my coordinate frame so that a straight line is parallel to the x axis. It's a useful frame, but it's not the One True Frame in any sense.

Gold Member
This means, that for each train, the sequence of events is:

1) I see my timer reach two minutes. I see the the other timer before two minutes. I now know the the other train will explode when it gets the signal from my timer reaching 2 minutes

2) If the detonation signal is light speed, then each train sees the other timer reach two minutes (while theirs is beyond two minutes) at the moment off their own detonation.

3) If they survive their own explosion, they see the other explosion some time after theirs.
I really like this list of events and I would like to show how we can determine the timings of them using just Doppler analysis which is independent of any particular frame. All we need to know is the relative speed (as a fraction of c) between the two trains (assuming instant acceleration to that speed).

If the stated problem defines the speed that each train reaches from their initial rest state, then we can calculate the Doppler factor as (1+β)/(1-β).

Or we can calculate the relative speed based on the defined speed in this example as 2β/(1+β2) and then calculate the Doppler factor as √((1+β)/(1-β)).

When the two observers (and their clocks) are moving away from each other, the Doppler factor is the ratio of the time on each observers clock to that of the observed time on the other observer's clock, assuming that they both start at zero.

Event 1) This gives us an easy way to calculate the time each observer sees on the other ones clock at the time they each transmit the detonate signal. We simply divide 2 minutes by the Doppler factor.

Event 2) We simply multiply the Doppler factor by 2 minutes to see what time on their own clock each observer sees the other ones clock reach 2 minutes and the time they each experience the explosion.

Event 3) Since the time of Event 2 is also the time the other ones clock is at when their explosion occurs, we multiply the time for Event 2 by the Doppler factor to get the time on their own clock that each observer (assuming they survive) sees the explosion of the other one.

For example, let's assume that you and I are the observers and the defined speed that we each move away from our common starting point is 0.5c. We calculate the Doppler factor as (1+0.5)/(1-0.5) = 1.5/0.5 = 3.

We could have also first calculated our relative speed as 2*0.5/(1+0.52) = 1/(1+0.25) = 1/1.25 = 0.8 and then calculated the Doppler factor as √((1+0.8)/(1-0.8)) = √(1.8/0.2) = √9 = 3 and we get the same Doppler factor as before.

Event 1) At the time I see 2 minutes on my clock, I see 2/3 minutes which equals 40 seconds on your clock. This is when I send my detonate signal to you.

Event 2) At the time I see 2 minutes on your clock, I see 6 minutes on my clock which is when my train explodes (but I survive, traveling at the same speed as before).

Event 3) Since your train will also explode when your clock reaches 6 minutes, the time on my clock will be 18 minutes.

Note that this is all done without resorting to calculating the coordinates of any events which can be different in different frames but will have no bearing on what we each actually observe of our own clock and the other ones clock.

However, just for the fun of it, I'm going to do all those calculations first in our initial mutual rest frame and then in my rest frame after I reach the target speed. I'm going to assume that everyone knows how to calculate gamma and do the Lorentz Transformation. I'm going to use units where c=1, time is in minutes and length is in light minutes.

OK, at the speed of 0.5c, gamma is 1.154701 so when the Proper Time on each of our clocks is 2 minutes, the coordinate time is 2*1.154701 or 2.309402 minutes. Since we are each traveling at 0.5c, we will be located 0.5*2.309402 or 1.154701 light-minutes away from the origin. My x-coordinate will be -1.154701 and yours will be +1.154701.

Now we need to calculate how long it will take for my detonate signal to reach you, keeping in mind that you continue to travel away from me. If we denote this time as Δt, then the position of my signal will be Δt-1.154701 (because the signal is traveling at c which equals 1 and it starts at -1.154701). Your position will be 0.5*Δt+1.154701 (because you are traveling at 0.5c and you start at 1.154701). Equating these two functions and solving for Δt, we get:

Δt-1.154701 = 0.5*Δt+1.154701
Δt-0.5*Δt = 1.154701+1.154701
0.5*Δt = 2.309402
Δt = 4.618804

Adding this to the previous coordinate time we get 2.309402+4.618804 or 6.928206 minutes. Your coordinate location is 0.5*6.928206 or 3.464103. Dividing the coordinate time by gamma gives the Proper Time on your clock when your train explodes: 6.928206/1.154701 = 6 minutes. All that work just to get the same answer we got using Doppler analysis.

There are other events that we could calculate but they take even more work and I just wanted to point out that this is the time of Event 2.

Now we will transform these two events into my rest frame. Note that I am traveling at -0.5c. Event 1 in the first frame had coordinates of t=2.309402 and x=-1.154701. This transforms into t=2 and x=0. No surprise there.

Event 2 in the first frame had coordinates of t=6.928206 and x=3.464103 which transforms into t=10 and x=8 in my rest frame. However, in my rest frame you are traveling at 0.8c (remember we did that calculation a long time ago?) which gives a gamma of 1.666667. Dividing 10 by 1.666667 gives 6 minutes. So we see that in both frames we get the same answer for the time on your clock when the detonate signal reaches you, which is also the time on my clock when your detonate signal reaches me.

The bottom line is that both frames arrive at the same conclusion which is that both observers emit their detonate signals when their own clock reaches 2 minutes but they don't receive the detonate signal from the other observer until a long time later because it takes time for the signal to propagate over the long distance.

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bgq
The main problem is:

1) Person in train A can truly deduce that he will survive as follows: I am at rest, B is moving, so his clock is slower than that of me, then my timer reaches 2 min before him; therefore, he will be dead, and I will survive.

2) Similarly, Person in train B can truly deduce that he will survive and A will be dead.

3) But for the frame of reference at rest at the midpoint of the two persons, both will be dead.

The point is that it seems from this example that life and death seems to depend on the frame of reference which is impossible, it is out of logic that someone could be alive with respect to a frame of reference, yet he is dead with respect to another frame!!!

Gold Member
The main problem is:

1) Person in train A can truly deduce that he will survive as follows: I am at rest, B is moving, so his clock is slower than that of me, then my timer reaches 2 min before him; therefore, he will be dead, and I will survive.

2) Similarly, Person in train B can truly deduce that he will survive and A will be dead.

3) But for the frame of reference at rest at the midpoint of the two persons, both will be dead.

The point is that it seems from this example that life and death seems to depend on the frame of reference which is impossible, it is out of logic that someone could be alive with respect to a frame of reference, yet he is dead with respect to another frame!!!
As I have repeatedly said, different frames assign different coordinate times to the same events but they will all agree on the outcome of the scenario. I have also repeatedly said that you are overlooking the time it takes for a signal to traverse the distance from you to the other person.

Your deduction in your first point assumes that there is an absolute reality to the concept of simultaneity between two clocks separated by distance as evidenced by your statement "my timer reaches 2 min before him". Even if this were true, the signal that you send out at 2 minutes does not reach his train instantly, it takes time to get there.

Did you understand any of my points with regard to the Doppler analysis or the Lorentz Transform analysis?

The main problem is:

1) Person in train A can truly deduce that he will survive as follows: I am at rest, B is moving, so his clock is slower than that of me, then my timer reaches 2 min before him; therefore, he will be dead, and I will survive.
No. You can correctly conclude he will die. You can correctly conclude that as you see it, your clock reached two minutes first. However, you must reason more. At two minutes you send a signal to detonate his train. When does it get there according his clock? Using the numbers ghwellsjr did, even though when your clock reads two minutes and you see 40 seconds on his clock, you can calculate that your detonation signal will arrive when his clock reads 6 minutes. Since his clock has long past 2 minutes before he explodes, you will also explode - when your clock reads 6 minutes.
2) Similarly, Person in train B can truly deduce that he will survive and A will be dead.
Nope.
3) But for the frame of reference at rest at the midpoint of the two persons, both will be dead.
Both will die as correctly calculated in every reference frame. Further, in every reference frame, all conclusions as to what clock readings will be for each observer at particular events will be the same.
The point is that it seems from this example that life and death seems to depend on the frame of reference which is impossible, it is out of logic that someone could be alive with respect to a frame of reference, yet he is dead with respect to another frame!!!
Nope, what this example shows is that you need to study at least a little relativity before proclaiming paradoxes.

bgq
As I have repeatedly said, different frames assign different coordinate times to the same events but they will all agree on the outcome of the scenario. I have also repeatedly said that you are overlooking the time it takes for a signal to traverse the distance from you to the other person.

Your deduction in your first point assumes that there is an absolute reality to the concept of simultaneity between two clocks separated by distance as evidenced by your statement "my timer reaches 2 min before him". Even if this were true, the signal that you send out at 2 minutes does not reach his train instantly, it takes time to get there.

Did you understand any of my points with regard to the Doppler analysis or the Lorentz Transform analysis?

Hi George, first of all thank you for your time to write this.
I don't claim that I fully understand relativity (else why I need to post question). I just try to understand relativity, not just memorize and apply equations. I am until now astonished if this theory is true, how things such as time dilation could happen without any mechanism, is it just faith?!

I read the equations, but we can choose the speed of the trains so that the signal - of the speed of light - can travel very fast before the other timer reaches 2 min. So in this case, what makes A conclusion wrong?

Thank you again.

bgq
No. You can correctly conclude he will die. You can correctly conclude that as you see it, your clock reached two minutes first. However, you must reason more. At two minutes you send a signal to detonate his train. When does it get there according his clock? Using the numbers ghwellsjr did, even though when your clock reads two minutes and you see 40 seconds on his clock, you can calculate that your detonation signal will arrive when his clock reads 6 minutes. Since his clock has long past 2 minutes before he explodes, you will also explode - when your clock reads 6 minutes.

Nope.

Both will die as correctly calculated in every reference frame. Further, in every reference frame, all conclusions as to what clock readings will be for each observer at particular events will be the same.

Nope, what this example shows is that you need to study at least a little relativity before proclaiming paradoxes.

Hi, I studied relativity, and I think I am not bad in applying the related equations. However I don't post this to claim a paradox, I just post this to understand what seems to be a paradox to me, and I am very thankful to all replies.

H

I read the equations, but we can choose the speed of the trains so that the signal - of the speed of light - can travel very fast before the other timer reaches 2 min. So in this case, what makes A conclusion wrong?

Thank you again.

Nope. The slower the trains, the less time difference observed by each. For any number you pick, the result is the same: both trains blow up (as long as light signals are used; for slower signals, as Ibix pointed out, you have to allow for the possibility of the trains outrunning the signals, so neither explodes).

I really thank ghwellsjr for his beautiful writeup, because now I can refer to it. Note in his writeup there is one doppler factor used throughout. Change the train speed, change this factor, but no matter what one value greater than 1 you put in the computations, you always find the same result: both trains blow up at some time after 2 minutes on their own clocks.

Gold Member
Hi George, first of all thank you for your time to write this.
I don't claim that I fully understand relativity (else why I need to post question). I just try to understand relativity, not just memorize and apply equations. I am until now astonished if this theory is true, how things such as time dilation could happen without any mechanism, is it just faith?!

I read the equations, but we can choose the speed of the trains so that the signal - of the speed of light - can travel very fast before the other timer reaches 2 min. So in this case, what makes A conclusion wrong?

Thank you again.
If you think you can choose a speed of the trains so that a signal traveling at the speed of light can get to the other clock before it reaches 2 minutes--then I'd like to see what speed that is. I gave you the equations that you need, first to calculate the Doppler factor, (1+β)/(1-β) and then from that to calculate the Proper Time on each clock when the signal reaches it, it should be obvious that for any value of β, the Doppler factor will be greater than 1 which means that when we multiply 2 minutes by the Doppler factor we will always get a value greater than 2 minutes. This means that each clock must always get the detonate signal from the other clock after it sends its own signal (at 2 minutes).

Keep in mind that the Doppler analysis is frame independent. That means that it doesn't matter what frame you choose, you will always get the same result. You don't even have to choose a frame with the Doppler analysis as I tried to show earlier.

Mentor
I am until now astonished if this theory is true, how things such as time dilation could happen without any mechanism, is it just faith?

Not just faith.
There's a rigorous and quite logically convincing argument that starts with Einstein's two postulates, derives the Lorentz transformations from them, and then uses the transformations to demonstrate relativity of simultaneity, time dilation and length contraction.

What trips up a lot of beginners, and may be tripping you up, is that many of the popular "relativity made simple" sorts of books skip this argument and jump straight to the conclusions about length contraction, time dilation, and the like. Then it really does feel a lot like you're being asked to take things on faith with no supporting arguments or logic about WHY the world should work this way.

It's worth trying to dig up some of the real explanations; there are some good ones in threads here and there on PF, some excellent ones on the web, and you could do a lot worse than digging up a copy of Einstein's book "Relativity: The Special and General Theory" and working through it.

bgq
Not just faith.
There's a rigorous and quite logically convincing argument that starts with Einstein's two postulates, derives the Lorentz transformations from them, and then uses the transformations to demonstrate relativity of simultaneity, time dilation and length contraction.

What trips up a lot of beginners, and may be tripping you up, is that many of the popular "relativity made simple" sorts of books skip this argument and jump straight to the conclusions about length contraction, time dilation, and the like. Then it really does feel a lot like you're being asked to take things on faith with no supporting arguments or logic about WHY the world should work this way.

It's worth trying to dig up some of the real explanations; there are some good ones in threads here and there on PF, some excellent ones on the web, and you could do a lot worse than digging up a copy of Einstein's book "Relativity: The Special and General Theory" and working through it.

Mentor
Hi, I studied relativity, and I think I am not bad in applying the related equations. However I don't post this to claim a paradox, I just post this to understand what seems to be a paradox to me,

Which equations have you been applying?

Try writing down the (x,t) coordinates in A's frame of three events: trains depart in opposite directions; A's clock reaches 2 minutes and light signal is sent; and A's light signal reaches B. Then apply the Lorentz transformations to find the coordinates of these events in B's frame.

The paradox will disappear.

someGorilla
I read the equations, but we can choose the speed of the trains so that the signal - of the speed of light - can travel very fast before the other timer reaches 2 min.

No, you can't. Try to do it - in numbers. Or better, set your equations to find the speed the trains must have in order for it to happen that way. You'll see it's impossible.
Did you try to draw a space-time diagram for each of the 3 observers, with the trains' trajectories, the signals' paths and the clock readings?

harrylin
[..] I really can't find any solution to this paradox; I am going to deny all the special theory of relativity because of this. The paradox is very simple,[..]

[..] I don't claim that I fully understand relativity (else why I need to post question). I just try to understand relativity, not just memorize and apply equations. I am until now astonished if this theory is true, how things such as time dilation could happen without any mechanism, is it just faith?! [..]

Hi bqq, I know exactly what you're going through as a number of years ago I experienced the same, and even worse: concerning one paradox I only received erroneous explanations and as a result I struggled to find the right answer myself. In contrast, you immediately received correct answers on the one that you presented here.

SR is not based on a physical model, and thus it doesn't provide a physical mechanism. For example the principle of energy conservation doesn't provide a mechanism either. That is not just faith but experience from experiments (except if for you experience is "just faith").

Your questions have some similarities with discussions that we have in parallel:

- a bomb explosion paradox:

- understanding physical process of time dilation:

Cheers,
Harald

PS some links to good introductions:
http://www.fourmilab.ch/etexts/einstein/specrel/www/
http://www.bartleby.com/173/

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altergnostic
Bgq, let's try to leave math out of it and work with a logical approach:
Don't think of the other clock as literally slowing down, think of it as going at exactly tue same rate as yours. Imagine that every tick of the clock sends a light signal to you, a blink. When both of you are at rest relative to each other, the blinks occur simultanesously with your clock's seconds. They are in sync, ticking at the same rate of 1 second per second.

Now, when the other train is moving away, the disance increases and now light has considerable distance to travel. In contrast with when both of you were at the same place (the origin), now light takes time to get to you, it has to cover the distance between you. You can't see the blink at the same moment that it blinks as seen from that train, you can't see light at a distance. Light must be right in your eyes for you to see something.

That means that you will see the first blink at 1 second + the time it took for the blink to get to you. If light took half a second to reach your eyes, you will think it blinked late compared to your clock, you will think the other clock slowed down. But it only appears to be slow, from your perspective. Special Relativity is basically a set of equations that take the speed of light in consideration in distant measurements.

bgq
Bgq, let's try to leave math out of it and work with a logical approach:
Don't think of the other clock as literally slowing down, think of it as going at exactly tue same rate as yours. Imagine that every tick of the clock sends a light signal to you, a blink. When both of you are at rest relative to each other, the blinks occur simultanesously with your clock's seconds. They are in sync, ticking at the same rate of 1 second per second.

Now, when the other train is moving away, the disance increases and now light has considerable distance to travel. In contrast with when both of you were at the same place (the origin), now light takes time to get to you, it has to cover the distance between you. You can't see the blink at the same moment that it blinks as seen from that train, you can't see light at a distance. Light must be right in your eyes for you to see something.

That means that you will see the first blink at 1 second + the time it took for the blink to get to you. If light took half a second to reach your eyes, you will think it blinked late compared to your clock, you will think the other clock slowed down. But it only appears to be slow, from your perspective. Special Relativity is basically a set of equations that take the speed of light in consideration in distant measurements.