# The Key to the Twins Paradox - Relativity of Simultaneity

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#### SpeedFreek

##### Guest
Bob is sitting in the middle of a train carriage, and above his head is a light bulb. It emits a burst of light in all directions. When the light from that bulb reaches each end of the carriage, it causes light bulbs at each end of the carriage emit a burst of light. Bob sees that the lights at each end of the carriage flash at the same time (he has eyes in the back of his head!). A simple scenario.

But the train is passing through a station when this all happens, and Tom, who is standing on the platform, happens to be right in line with that carriage. In fact, he is right in line with the centre bulb as it flashes. But the carriage is moving in relation to him, and because of this, his view of the scene is different.

From Toms point of view, the bulb at the trailing end of the carriage flashes before the bulb at the front of the carriage.

Now Tom knows what the speed of light is, so he does the calculations to subtract the time it took for the light from the bulbs at the ends of the carriage to reach him. After doing the math, he finds that the bulb at the trailing end still flashed before the bulb at the front!

So now he calculates the light travel time from the bulb at the centre of the train to the bulb at each end. He knows that light travels at c, and that the train was moving relative to him. Once the flash of light was emitted from the centre of the train, the light propagates at c relative to Tom, but not relative to the train, from Toms point of view! As the train was moving across his view, the light actually reached the bulb at the back of the train before it reached the front.

So to Bob, the bulbs at the ends of the carriage lit up simultaneously, but to Tom they did not. Both are reality. It is not about light-travel time, it is about the constancy of the speed of light. It is not that the light from simultaneous events took different amounts of time to reach different observers, it is that the speed of light is the same to all observers, whatever their relative motions, and therefore the distance between events is not absolute across frames of reference in relative motion. Simultaneity is relative - "now", is relative. Moreover, the relativity of simultaneity works symmetrically if each observer uses the constancy of the speed of light as their reference.

If we keep the relative motion but put the bulbs on the platform instead, then in Toms frame of reference the events were simultaneous, but in Bob's frame on the train the events were not simultaneous and the difference between the events would be entirely symmetrical to when the bulbs were on the train. This shows how there is no form of absolute motion, relative to c.

Due to the constancy of c between frames in relative inertial motion, light always propagates in a sphere centred on the origin and the origin always remains at rest in relation to the observer, regardless of the motion of the source.

So, from Bob's frame, the light from the central bulb above his head propagates at 300,000 km/s in all directions relative to him, and Tom (on the platform) is moving relative to that sphere.

But from Toms frame, the light from the central bulb propagates from the place in space it was emitted, at 300,000 km/s in all directions relative to him, and the train with Bob on it is moving through that propagating sphere of light.

This is ultimately where (through the Lorentz contraction) we get time-dilation from, and why time-dilation is symmetrical between observers in relative inertial motion. With relative motion and increasing distance, the notion of simultaneity between two observers shifts further and further apart in a symmetrical way - each observer calculates the others clock to be time dilated by the same amount. Each observer splits up time and space differently to the other by the same amount.

So, if the shift in simultaneity is symmetrical due to relative inertial motion, when would the shift in simultaneity be asymmetrical?

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#### Astro_Robert

##### Guest
I'll vote for non-inertial motion, ie an accelerating reference frame (train or rocket) per General Relativity. Don't ask me do to a pretty explanation like yours though.

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#### emperor_of_localgroup

##### Guest
SpeedFreek":3m1jcu9s said:
Bob is sitting in the middle of a train carriage, and above his head is a light bulb. It emits a burst of light in all directions. When the light from that bulb reaches each end of the carriage, it causes light bulbs at each end of the carriage emit a burst of light. Bob sees that the lights at each end of the carriage flash at the same time (he has eyes in the back of his head!). A simple scenario.
?

I was surprised to find that if Tom caculates the velocity of Bob, and Bob calculates his own velocity, they are equal. In other words, if Bob's speed is .9c with respect to Tom, and Bob calculates his velocity relative to a distant object ahead of him, Bob will find his speed is exactly .9c.

This will make special relativity people very happy, but I think it is kind of odd.

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#### SpeedFreek

##### Guest
I am not sure I understand you.

If Bob calculates his velocity relative to a distant object ahead of him, and that object is at rest in relation to Tom (which is what I assume you mean here), then all you are saying is that Bob and Tom have a relative velocity of .9c, isn't it? Each thinks the other is moving at .9c relative to themselves?

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#### emperor_of_localgroup

##### Guest
SpeedFreek":28usaehs said:
I am not sure I understand you.

If Bob calculates his velocity relative to a distant object ahead of him, and that object is at rest in relation to Tom (which is what I assume you mean here), then all you are saying is that Bob and Tom have a relative velocity of .9c, isn't it? Each thinks the other is moving at .9c relative to themselves?
Let me ask you this question. I could draw a better diagram if I knew how to insert a jpg image.

TOM+++++++++++10LY++++++++BOB+++++++++++10LY++++++++++STAR

Say Tom is on earth or on some rest frame. Distance of the star with respect to Tom is 20LY. If Bob's spaceship is at rest, he'll measure distance of the star is 10LY, also distance of Tom is 10LY. But if Bob is moving towards the star with speed .9c with respect to Tom, and Bob measures distance of the star from exactly the same point, he'll find the distance is 4.36LY (rounded off).
But what would he find if Bob measures distance of Tom from exactly the same point?
Is it 4.36 LY or 10 LY or 15.64LY? Note, Bob is moving away from Tom.

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#### SpeedFreek

##### Guest
An interesting question.. but first I need to know how Bob is making his "measurements". What I can say in the meantime is that Carl is moving at .9c in the other direction with respect to Tom, he is heading towards Tom, and as he passes Bob he will calculate that the distance to Earth is 4.36 ly.

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#### emperor_of_localgroup

##### Guest
SpeedFreek":3jazvl01 said:
An interesting question.. but first I need to know how Bob is making his "measurements". What I can say in the meantime is that Carl is moving at .9c in the other direction with respect to Tom, he is heading towards Tom, and as he passes Bob he will calculate that the distance to Earth is 4.36 ly.
We are assuming an idealistic case where all measurements are accurate from experimenter's reference frame.

The reason I am asking this question, I want to guess what causes time dilation and space contraction. I was comparing this space contraction with Doppler phenomena. In case of Doppler it is the 'sound wavefronts' that get squeezed in the direction of motion and expanded in opposite direction.

In case of extremely high speed, does the 'space wavefront' (for the lack of a better term) get squeezed or get wrinkled to make the distance shorter? Does the space in opposite direction get expanded, like Doppler?

I know these are weird thinking, but who knows..

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#### SpeedFreek

##### Guest
Apparent effects like Doppler shift and the aberration of light are completely separate from time-dilation.

Time-dilation is symmetrical between observers in relative inertial motion - each calculates the other's clock to be running slower than themselves, and it doesn't matter whether the other clock is moving towards you or moving away from you.

But Doppler shift would show an apparent slowing as the other clock moves away, and an apparent speeding up as the clock moved towards you.

You can think of it in terms of the siren on a train. It is only apparently changing pitch as the train passes you.

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#### nettsight

##### Guest
the difference between the events would be entirely symmetrical to when the bulbs were on the train. This shows how there is no form of absolute motion, relative to c.
only relative to motion! :roll: its a question of duel directionality - that is the back of the carriage moving toward the light and the front moving away from it and is also true for the moving observer watching the lights on the static platform.

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#### SpeedFreek

##### Guest
nettsight":22hp0bac said:
the difference between the events would be entirely symmetrical to when the bulbs were on the train. This shows how there is no form of absolute motion, relative to c.
only relative to motion! :roll: its a question of duel directionality - that is the back of the carriage moving toward the light and the front moving away from it and is also true for the moving observer watching the lights on the static platform.
Relative to c. And motion relative to other motion is no different from motion relative to someone "at rest", relative to c, as all observers in inertial frames of reference can consider themselves to be at rest, in relation to c, and that it is everything else that is moving, relative to them. The shift in simultaneity is always symmetrical between them.

If light always propagates at 300,000 km/s relative to you, whatever relative speed you have to anything else, then the relativity of simultaneity shows us that events that are simultaneous in one frame may not be simultaneous in the other and, because that shift in simultaneity is not due to the difference in the distance to events and it is not due to moving towards the light from events, it has to be that the events happened at different times from one frame than they did in another.

It doesn't matter if you are in an inertial frame and you are moving towards the light as it moves towards you, as the speed of light is always 300,000 km/s relative to you. If it was emitted 300,000 km away it will reach you 1 second later, regardless of how fast you are "moving towards it". Whilst in inertial motion you can feel no movement and consider yourself at rest and that the light is doing all the moving.

This is how, if you are on the train in the middle of the carriage, and the bulb above your head lights up, its light illuminates both ends of the carriage at the same time, but to an observer in the platform the light hits the back of the carriage before it hit the front.

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