# Train experiment problem

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1. Jul 30, 2015

### Stephanus

Dear PF forum,
I have a problem with Train experiment.
Perhaps someone can take a look at it and tell me where did I go wrong?
V = 0.6c;
Gamma = 1.25
Train length = 1500
Platform length = 1200, so the light can reach the observer at the same time.
I draw the space diagram for that problem:

Where the train (Green) is at rest first. The cars in train synchronize their clock. And preprogrammed at S1 (for the train Stern) and B1 (for the train Bow), it moves to the station.
But in platform frame, S and B can't reach the platform at the same time wrt platform.
See, the zoomed section.
And if I draw this as Pic 2

Then, S1 and B1 don't start at the same time wrt train rest. This means that S1 (stern) starts first then the bow follows. Shouldn't it derail or destroy the train if the back car moves first?
Where did I go wrong?
Can anyone help?

Thanks.

2. Jul 30, 2015

### Staff: Mentor

The train will break if both ends try to accelerate at the same time (as synchronized before). This is Bell's spaceship paradox.

3. Jul 30, 2015

### Janus

Staff Emeritus
The problem is that you are trying to assume instantaneous acceleration and that can be confusing.

As the train gains speed it contracts as seen in the frame of the tracks, so in effect, the rear of the train does accelerate faster than the front, but since the train is undergoing length contraction at the same time, there is no "squeezing" of the train in the train's frame.

The world lines of the two train ends will look something like this in the embankment frame with the rear of the train's world line line curving slightly differently than the front of the train. Notice how they get closer together. ( the yellow "light" lines, I'll touch on later.)

At any point, the distance between the two ends will equal to the length contracted length of the train while moving at that speed.

In the train's frame, the length of the train stays fixed. If we want to look the space time diagram for the train we are stuck with picking a particular "moment" during the acceleration. If we want to see the whole acceleration period we would need to look at a series of images. In each image, the distance between the ends of the train at that instant will always be the same.
Here's the moment close to when the lights emitted when the train started accelerating reach the opposite ends.

I picked this moment as it demonstrates what happens to clocks in the train frame when the train is under acceleration. The two sets of light pulses were sent with the same time separation. ( shown in the first image), but on arrival, the pulses hitting the rear of the train are closer together,and the pulses hitting the front are spaced further apart. In effect, the rear of the train will see light coming from the front of the train as blue-shifted and the front of the train will see light coming from the rear as red-shifted. Further someone in the rear will see a clock in the front running fast, and someone in the front sees a clock in the rear run slow.

Now typically when we deal with a Doppler shift, even Relativistically, there is a component of that shift that is due to the changing distance between the sender and receiver. It is after you factor this component out that you are left with the time dilation component. But in this situation, this isn't the case, as far as occupants of the train are concerned, the distance from front to back remains constant at all times, so there is no changing distance component to account for, and all that is left is the time dilation component. Ergo, as far as anyone in the train is concerned, the clock in the front of the train really runs faster than the one at the rear. If the clocks started in sync before acceleration, they will be out of sync afterwards. (this means that if you want the clocks to be in sync for the up coming experiment, you will have to re-sync them after the acceleration. )

Last edited: Jul 30, 2015
4. Jul 30, 2015

### Stephanus

Thanks a lot! That makes things clear for me. No I know what I don't know.
I think it's about time to study accelerating world line?

5. Jul 30, 2015

### Stephanus

Yes, there's no such thing as instanteous acceleration in real world. We "talk" instant acceleration to understand the concept of SR. Instant acceleration even as small as 0.0%c per second -> 30km/sec2 will kill any passenger in the rocket. 3000g?