Priya, however, made an assumption. It’s a perfectly ordinary assumption, one that seems sufficiently obvious that you probably didn’t notice, but an assumption nonetheless. Priya assumed that the explosions were simultaneous.

Didn’t Ryan find them to be simultaneous? Indeed, he did. Suppose we call Ryan’s reference frame S, the explosion on the right event R, and the explosion on the left event L. Ryan found that $t_{\text{R}}=t_{\text{L}}$. But Priya has to use a different set of clocks, the clocks in her reference frame $\text{S}\prime$, to measure the times $t_{\text{R}}^{\prime }$ and $t_{\text{L}}^{\prime }$ at which the explosions occurred. The fact that $t_{\text{R}}=t_{\text{L}}$ in frame S does not allow us to conclude that $t_{\text{R}}^{\prime }=t_{\text{L}}^{\prime }$ in frame $\text{S}\prime$.

In fact, in frame $\text{S}\prime$ the right firecracker must explode before the left firecracker. Figure 36.16b, with its assumption about simultaneity, was incorrect. FIGURE 36.17 shows the situation in Priya’s reference frame, with the right firecracker exploding first. Now the wave from the right reaches Priya and the box first, as Ryan had concluded, and the light on top turns green.

FIGURE 36.17 The real sequence of events in Priya’s reference frame.

One of the most disconcerting conclusions of relativity is that two events occurring simultaneously in reference frame S are not simultaneous in any reference frame $\mathbf{\text{S}}\prime$ moving relative to S. This is called the relativity of simultaneity.

The two firecrackers really explode at the same instant of time in Ryan’s reference frame. And the right firecracker really explodes first in Priya’s reference frame. It’s not a matter of when they see the flashes. Our conclusion refers to the times at which the explosions actually occur.

The paradox of Priya and Ryan contains the essence of relativity, and it’s worth careful thought. First, review the logic until you’re certain that there is a paradox, a logical impossibility. Then convince yourself that the only way to resolve the paradox is to abandon the assumption that the explosions are simultaneous in Priya’s reference frame. If you understand the paradox and its resolution, you’ve made a big step toward understanding what relativity is all about.