# Simultaneity and special relativity

Suppose, in inertial reference frame $$F_1$$, observers A and B are at rest, each having torch, and are separated by some distance and we have put machine M at middle of A and B.

Machine M has light bulbs on both sides ,right and left, so that if it catches light from A which is at left ,then machine M glows left light bulb, similar with right bulb.Also, if it senses both reaching at same instant of time then it start to make noise.

Now consider another inertial reference frame $$F_2$$ which is moving at constant speed $$v$$ with respect to $$F_1$$ to the right.

Now ,in frame $$F_1$$ , both A and B turn on their torches at same instant of time, say $$t=0$$ and both rays reach at M at $$t=t_1$$, and machine M makes noise indicating that those events were "simultaneous" in $$F_1$$.

Now we know these events are not simultaneous in $$F_2$$, in other words ,person seating in $$F_2$$ will say ,"I should not hear sound from machine M." But somehow machine makes noise.(or it doesn't make noise?)

So does this mean according to $$F_2$$ ,machine is malfunctioning?

• Hi Pratik, welcome to Physics StackExchange! Please split the textbook recommendation into a separate question on the site - this is considered best practice for some reasons detailed here. May 23 '20 at 10:25
• This is the second question along these lines that this site has received in the last day. Is there a particular source that has prompted these questions, out of curiosity? In general, what brought you to ask this question? May 23 '20 at 22:31

The machine M responds to events right there at the machine---the events of light arriving from left, light arriving from right. So the machine is reporting that the light arrival events are simultaneous at M. This is fine; all reference frames will agree that two things happening at the same place and time do indeed happen at the same place and time.

But when we interpret M to be reporting that the emission events are simultaneous, now we have a frame-dependent interpretation. What M is really saying is "well the two light beams reached me simultaneously, so what I can claim is that if the emitters are at distances $$d_1$$ and $$d_2$$, then the emission times were $$d_1/c$$ and $$d_2/c$$ before now. So if $$d_1=d_2$$ then the emission events were simultaneous. And if $$d_1 \ne d_2$$ then the emission events were not simultaneous."

What happens in your scenario is that in frame $$F_1$$ the two distances are equal, whereas in frame $$F_2$$ they are not equal.

By the way I would always recommend learning to use spacetime diagrams when learning special relativity.

In $$F_1$$ the light from A and the light from B arrive simultaneously at M and it makes a sound.

In $$F_2$$ the light from A and the light from B arrive simultaneously at M and it makes a sound. No paradox. Simultaneity is OK if (and only if) it refers to 2 things happening at the same place.

The difference is that in $$F_1$$ the light is sent simultaneously from A and from B, and takes the same time to travel from A to M as it does from B to M. In $$F_2$$ the light from B (if A is on the left and B is on the right) starts a bit earlier than the light from A but also takes longer to travel the longer distance, so the signals arrive at the same time.

In more detail, if needed: an observer at M in $$F_2$$ as the signals arrive reckons that A and B are equidistant. But A is travelling away from them and B is travelling towards them. So whenever either signal started, B was further away and as they arrive at the same time, B's signal must have started first.

The machine is in reference frame F1, not F2, so It depends who you ask if it's malfunctioning.

Assuming that you know that the box works in this way - then the fact that the machine made a noise only tells you that in F1 - the beams of light reached the machine at the same time.

I can say that the machine is malfunctioning only if I have no knowledge of the relativistic effect. So yes - it is malfunctioning according to most people on the planet. But for those who do know something about relativity effects, it is not.

-- It get's a bit more complicated than that if you are trying to imagine what would you see. Try not to go that path yet, most Youtube videos and even PBS got that wrong. That's why the assignment uses noise, not "what would you see".

A machine $$M$$ is not equidistant between $$A$$ and $$B$$ in $$F_2$$, so receiving their signals at the same time means they didn't send them at the same time, and they didn't. Totally self consistent.