# Einstein's relativity of simultaneity - train

Einstein's thought experiment has two lightning flashes at either end of a moving train as seen by an observer on the train, and a platform observer. They disagree on the simultaneity of the flashes.. But if we alter the experiment as follows they will agree on simultaneity. Have the flashes located in front of the train on the rails, front left and front right, with the on train observer equidistant from the flashes. He will see them as simultaneous whether the train is moving or stationary.

The platform observer is positioned on a footbridge in front of the train also equidistant from the flashes, so he also sees the flashes as simultaneous.

Both observers agree on simultaneity. How is this possible ? DAC

• Note: The Lorentz boost in the $x$-direction (where $w=ct,$ $\beta = v/c,$ and $\gamma=1/\sqrt{1-\beta^2}$) says that $w' = \gamma~(w - \beta x)$ and $x' = \gamma~(x - \beta w)$ while $y' = y$ and $z'=z.$ You have given the two events the same $x$-coordinate and $w$-coordinate (albeit different $y$ coordinates) and are surprised that they have the same $w'$-coordinate, but in fact that is the only possibility allowed by the Lorentz boost. – CR Drost Jan 2 '17 at 15:55

## 3 Answers

Your interpretation of your modification is correct, and it's also entirely consistent with Special Relativity. What appears to be incorrect here is your assumption that two observers in different frames must disagree on simultaneity of events. This is not actually true (as you have demonstrated), nor does Special Relativity say it is true.

What Special Relativity actually claims, on the other hand, is that observers in different frames can disagree on simultaneity (i.e. it is possible, but not guaranteed). The original thought experiment was designed to illustrate a case where observers in different frames disagreed (which is usually the non-intuitive case). Your modification is a case where they agree.

You have moved the line supporting the flashes from parallel to the train direction to perpendicular to that direction. The situation has become symmetric for both observers (a symmetry around the vertical plane including the train direction leaves the flashes and the observers unchanged), hence their agreement on the simultaneity.

There is often confusion about this thought experiment because textbooks fail to explain how the two observers "see" simultaneity or lack of it. One can imagine that either observer possesses so many (synchronous) clocks, spread everywhere in his system, that two of them happened to be exactly where ligtning strikes occurred, and registered the times. Then, if the speed of light is independent of the speed of the source (Einstein's 1905 light postulate is correct), the two observers disagree - the two clocks of one of them show simultaneity, the two clocks of the other don't. If the speed of light does depend on the speed of the source, the two observers agree on simultaneity.