# Special Relativity - Simultaneity [closed]

Consider a railway carriage. It is moving in +ve x-axis with a velocity, v, relative to the platform. Let A',B' be end points and M' be the exact middle of the carriage.

Let the platform observer be O.

The observer situated at M' at rest relative to the carriage be O'.

When O and O' cross each other I set the clocks in both reference frames to 0.

I arrange a switch at M' such that if 2 photons hit M' "simultaneously" a bulb is switched on.

At time 0 (when O and O' cross) I shoot 1 photon each from A',B' towards M'.

Question - Will the bulb be 'on' or 'off'?

Based on some suggestions to include my thoughts.

As per O' which is at rest in the railway carriage - both photons travel same distance and hence hit M' simultaneously - hence the bulb should be 'on'

As per O photon from B' to M' travels less distance and hence reaches M' first - thus not simultaneous. So bulb should be 'off'

This is not a Q from any book - I was thinking about it as I was reading SRT and was not able to identify the flaw in my reasoning.

Thanks

• @PM2Ring - I did as you suggested, Thanks Commented Jun 6, 2019 at 7:14
• @PM2Ring - Thanks Commented Jun 6, 2019 at 10:59

Neat paradox. The flaw is in the innocent-looking line

"At time 0 (when O and O' cross) I shoot 1 photon each from A',B' towards M'."

The reason is that setting $$t=t'=0$$ when the origins coincide only applies at $$x=x'=0$$

For other values they are related by the Lorentz transformation $$t=\gamma (t'+v x'/c^2)$$, so if the carriage has the usual length $$L$$ then for the two photon-shooters, as seen from the platform, $$t=\pm v \gamma L/2c^2$$.

So O' says: The photons started at the same time, they travelled the same distance, so they arrived simultaneously and the bulb lit.

and O says: The photons travelled different distances, but you started the rear one earlier than the front one, so they arrived simultaneously and the bulb lit.

observer O is not necessary, unless you would have stated that he was in the exact center of a and b. if the photons are sent when O' is at the exact center of a and b (which was not specified) then the light will be off, by the time the rear photon hits the carriage will have moved ahead and already hit the front photon. if the photon emitters were moving with the carriage and O' were centered between them, then the light would be on because in o' reference frame both would move at same speed towards center, but to O reference frame rear photon would still hit later than front photon.