Photoelectric Effect and Relativity

Question

Einstein postulated that light consists of discrete packets of energy called photons. Using this, he successfully explained the photoelectric effect.

Another work of Einstein's is that of relativity. According to the special theory of relativity, simultaneity is not absolute.

Let's consider a set-up for the photoelectric effect. The lamp is switched on at $t=0$. The galvanometer registers a deflection at $t=\tau$. We know that $\tau$ is very small, as the deflection of the galvanometer and the switching on of the lamp are often termed as simultaneous (when explained in contrast to classical theory predictions). The $\tau$ may be ignored as per convenience, as far it doesn't affect the answer considerably.

Consider an observer travelling at relativistic speeds, looking at the set-up. Depending on his direction relative to it, he will either see:

• The lamp is switched on. The galvanometer shows a deflection a certain time after this.

• The galvanometer shows a deflection. The lamp is switched on a certain time after this.

How is this discrepancy due to the observed non-simultaneity explained by the observer?

Answer 1

This is because the events $A$ lamp switching on and $B$ galvanometer deflexion (idealizing for this exercise that both happen in no time, which is far from true) are of timelike separation. That is, the time $\Delta t$ between the two events fulfills $\Delta t\geq d/c$, where $d$ is the distance between the lamp and galvanometer.

It is well known that the sign of the time separation between timelike separated events cannot be changed by a less than lightspeed boost. So, even though the time between events will change for different observers, the order cannot. I prove this fact in another answer, but I suspect you won't yet have the background for it. If you do look into this more, see my answer here; a variation on the same proof is given here

Indeed the reason we postulate that there can be no faster than light signalling is precisely because of the property described above and the means it gives us to salvage a notion of causality in relativity: as long as events are timelike separated, then we won't run into problems with causes coming before effects for some observers. We therefore postulate that causal links can only exist between timelike separated events.