# B Special relativity and sequence of events

1. Dec 18, 2016

### Glenn G

Sorry for my ignorance... still trying to get to grips...

If a lady in the middle of a moving train sends out beams to the front and back of the train. They reflect off mirrors back to her and arrive simultaneously because she can't do an experiment to give away that she is moving forward rather than being stationary relative to a platform.

A man on the platform (that discussed with the lady prior to the journey when to set off a bomb) will see the light hit the back mirror first then the front mirror as the light travelling at c has to catch up with the speeding train's front mirror. He will also see that the beams do not arrive back at the lady at the same time. So to her they arrive together and to him the don't. Now according to the lady she should and does set the bomb off, according to the man they don't arrive in synchs so she doesn't set the bomb off?

So this dichotomy confuses me. Would appreciate any help.
Glenn.

2. Dec 18, 2016

### Staff: Mentor

If you work out the math you will find that this is incorrect. He will see that the beams do arrive back at the lady at the same time.

3. Dec 18, 2016

### Staff: Mentor

The platform observer will find that the rays are emitted at the same time and that they get back to the lady at the same time, so he also expects the bomb to go off. he also find that both beams of light travel the same round-trip distance so it's not surprising that they get back at the same time.

Where he disagrees with the lady is that he does not find that the two beams reach the mirrors at the same time. The forward-moving beam has to chase its mirror on the outbound leg so travels farther and reaches its mirror after the rearwards-moving beam has reached its mirror. But then on their return leg the other way around: the reflection from the rear mirror chasing the lady so has farther to go. although it has farther to go, it starts sooner so ends up reaching the lady at the same time as the reflection from the front mirror, which started later but doesn't have as far to go.

Work out the math, like Dale says....

4. Dec 18, 2016

### robphy

In addition to the "math", you'll need some "physics" (if length contraction hasn't already been accounted for).

5. Dec 18, 2016

### Staff: Mentor

Actually, you don't need to allow for length contraction (or time dilation) as long as the only question you are answering is "Do the beams get back to the lady at the same time?". Length contraction and time dilation aren't needed until you ask "How long are the beams in flight?", a question that has a different answer in the two frames.

6. Dec 18, 2016

### robphy

7. Dec 18, 2016

### Glenn G

Thanks Dale, what got me thinking was a YouTube video that (yes I know!) was about a man on a platform seeing lightning striking the front and back of the train simultaneously but the lady inside the train receives the flash from the front before the flash from the back ... it was about the simultaneity of events not being the same for different observers in different inertial frames. Does this mean this was wrong and the lady would also note that the front and rear of the train were struck concurrently? Glenn
Great thanks, that makes sense now, what if the lady has agents at the front and back mirrors and they've decided to turn synchronous keys to explode the bomb when the beams arrive. From the lady on train both her agents turn the keys as both receive light at same expected times but from the point of view of the platform man the beams arrive at front and back agents at different times and so the synchro turning of keys to set off bomb doesn't happen?

8. Dec 18, 2016

### Staff: Mentor

Turning a key can explode a bomb right under the nose of the person doing the turning. But it can't trigger a bomb half a train away where the lady is sitting without sending some sort of signal from the key mechanism to the detonator. You have to allow for the travel time of that signal, and you won't be able to construct a mechanism in which the travel time for both signals is the same in both frames.

Depending on the exact mechanism that you're imagining, the analysis may be fairly complicated - you really will need some physics not just the math, as @robphy said - so we use light flashes in our thought experiments whenever possible. They make it easier to reason about the problem without getting bogged down in extraneous details.

9. Dec 18, 2016

### Glenn G

OK sorry let's say that the lady and her agents Fred and Bob over tea in the middle of the train using their metre stick and synched clocks calculate how long past noon (when light beams in opposite directions will be sent from the middle) the beams should arrive. Fred and Bob then go to the front and back of the train, Fred and Bob both receive the beams at the expected times on their clocks and turn their keys, bomb goes off somewhere in the train, say middle, triggered by the two key turns.
Platform man would see the beam arrive at Back man Bob first (key turned) then Front man Fred (sees key turned) then bomb goes off. Is this correct? So platform guy still sees explosion event but doesn't observe the simultaneous turning of the keys?
Thanks,
Glenn.

10. Dec 18, 2016

### Staff: Mentor

That's right. The events "Fred turns his key" and "Bob turns his key" happen at the same time according to anyone who is at rest relative to the train, but at different times according to the platform observer who is not at rest relative to the train.

In general, there is no universal and observer-independent way of defining "at the same time" for events that happen at different places. There is a universal and observer-independent way of establishing "A happened before B", but only if the times and distances are such that a light signal could have made it from A to B; this combined with the rule that nothing can move faster than light is sufficient to avoid any paradoxes of the "bomb exploded/did not explode" variety.

11. Dec 18, 2016

### Glenn G

Wow wow, that is so freakin weird it's sent a shiver down my spine!
Thanks Nugatory for explaining it.

12. Dec 18, 2016

### Staff: Mentor

You can set up the scenario in many different ways. If you set up the scenario so that the lady in the middle receives both flashes at once then she will receive both flashes at once in all frames. If you set up the scenario so that she receives them separately then she will receive them separately in all frames.

13. Dec 18, 2016

### FactChecker

Even without doing any math, one can see that the time difference that the platform man observes when the light beams are going out is exactly reversed when the light beams are coming back to the woman. So everyone agrees that the beams arrive back simultaneously. There is only a disagreement regarding the simultaneity of two events that are separated in the direction of relative motion.

14. Dec 19, 2016

### Jeronimus

The following picture contains two x-t diagrams, representing how Bob would measure/calculate the events to be happening within his inertial frame of reference in the case of the left diagram and how Alice would measure/calculate the spacetime coordinates within her IFR.

The two diagrams represent a somewhat more complex scenario.

Think of Bob inside a wagon floating in empty space, with nothing else being nearby. According to special relativity, there is no way for Bob to build a device which could tell him if he is moving or not.

Alice which is also floating in empty space inside another wagon, has no means to tell if she is moving or not as well.

At some point the wagons get closer to each other. Now both Alice and Bob can build devices which could tell them at which speed they are moving relative to each other. In the case of the two diagrams, the speed is half the speed of light. 0.5c

When Alice and Bob meet at some point and are local to each other, they release two yellow laser beams simultaneously to each side of their wagon. Those would be the yellow diagonals in both diagrams, travelling at c ~ 300000km/s.
Since one space unit in both diagrams is 1 lightsecond = 300000km and each time unit in the diagram is 1 second, one can easily see that lightbeams have to be diagonals at a 45° angle within both diagrams.

The blue line in the left diagram is Bob's worldline and the numbers on this line are the clock counts on a watch he is wearing (local to him). Bob considers himself at rest and therefore his worldline is parallel to the t axis and also overlaps with the t-axis since draws the x-t diagram such that he is in the middle of it.

The green lines in the left diagram are the endpoints of his wagon. The numbers on the lines are clock counts of clocks two terrorists are wearing which agreed to trigger two bombs whenever the light signal reaches them simultaneously (at the same t-position within the left x-t diagram). Hence when both their synced clocks display a clock count of 5.

The red stars around the clock count of 5 represent the event of the light beam reaching the two terrorists and them triggering the bomb blast.

In the right diagram, the worldline of Bob and the worldlines of his wagon's endpoints cannot be parallel to the t' axis, because from the perspective of Alice, Bob and the wagon is moving at 0.5c. Hence they are diagonals with their angle depending on the velocity Alice measures for Bob/his wagon.

In the right diagram, the purple line is Alice's worldline and the purple numbers are the clock counts of a clock which is local to Alice. The white lines are the endpoints of Alice's wagon and the white numbers are clock counts of two terrorists at the endpoints within Alice's wagon. They will also trigger a bomb when the laser light signals reach them simultaneously. The orange stars representing the event of the blast at t'=5s.

I could take this further now, but since this is a "B" question i just wanted to give you a "picture" of how the events would occur in detail without using too much math.

If you ever decide to study SR, you will be able to use the Lorentz transformation formulas and calculate at which x'(space) and t'(time) position an event happens for Alice, which for Bob happens at a given x(space) and t(time) position.

For example. Let's take the event represented by the right red star(bomb exploding at the right side of Bob's wagon) in the left diagram. That event happens at x=5ls (lightseconds) and t=5s.

The Lorentz transformation formulas to get x' and t' are

$x' = γ(x-vt)$
$t' = γ(t-(vx/c^2))$

$γ = √(1-v^2/c^2)$

v - the velocity Alice measures for Bob/his wagon
x - the space position Bob measures for an event within his frame
t - the time position Bob measures for an event within his frame
c - the speed of light ~300000km/s

so we get

$x' = 1.1547...(5ls - 0.5c*5s) = 2.88675...ls$
$t' = 1.1547...(5ls - (0.5c * 5ls) / c^2) = 2.88675...s$

So now we know the exact space and time location of where and when Alice will measure one of the bombs inside Bob's wagon triggering, which is x' = 2.88675...ls and t' = 2.88675...s. Something we cannot easily see from just watching the diagram.

Last edited: Dec 19, 2016
15. Dec 19, 2016

### Glenn G

Jeronimus, thanks for that. I think I need to print it out and work through it.

16. Dec 19, 2016

### Jeronimus

Np, just one correction i was too late to edit.

v - the velocity Alice measures for Bob/his wagon should be v - the velocity Bob measures for Alice/her wagon

Since from the diagrams we can see, Bob observes Alice to be moving towards the positive x-axis. Her velocity is therefore 0.5c. (only with 0.5c the calculations would be correct for x' and t')

Alice observes Bob to be moving towards the negative x-axis. She observes him to be moving at v=-0.5c.

The reason for why they don't observe each other to be moving at the same velocities is because of how they chose to draw the x-t diagrams, as in which direction they decided to place the positive and which direction the negative x axis.

We could draw the diagrams in such a way, that both observe each other to be moving at v=-0.5c, or both observe each other to be moving at v=0.5c but i have been told that there is a deeper reason of why it is done this way, you might be able to figure out later.

17. Dec 19, 2016

### FactChecker

Although the diagrams and math are important for further understanding of SR, they are not necessary for correcting the conceptual error in the OP. And I think they make an intuitive understanding of the OP more difficult.

A fundamental fact is that both reference frames will agree on the simultaneity of events that are not separated in the direction of relative motion. (Even though they will disagree if two events are separated in that direction)

In the case of the OP, this is easy to see:
The woman on the train sees the light beams arriving back at the same time. Her results are not changed by her own inertial motion.
Consider the time measurements that the man on the platform makes. Let F1 and F2 denote the times he measures for the beam going forward to the mirror and returning, respectively. Let B1 and B2 denote the times he measures for the beam going backward to the rear mirror and returning, respectively. The distances and velocities associated with F1 and B2 are identical, so F1 = B2. Similarly F2 = B1. So the man on the platform sees the forward beam return in F1 + F2 = B2 + B1, which is the time he sees the backward beam return. Therefore he sees their return to be simultaneous.