MODEL Let the explosion on Ryan’s right, the direction in which Priya is moving, be event R. The explosion on his left is event L.

VISUALIZE Priya and Ryan are in inertial reference frames. As FIGURE 36.30 shows, Priya’s frame $\text{S}\prime$ is moving with $v=0.8c$ relative to Ryan’s frame S. We’ve defined the reference frames such that Priya and Ryan are at the origins. The instant they pass, by definition, is $t=t\prime =0\text{s}$. The two events are shown in Ryan’s reference frame.

FIGURE 36.30 A pictorial representation of the reference frames and events.

SOLVE

1. The two burn marks tell Ryan that the distance between the explosions was $L=600\text{m}$. Light travels at $c=300\text{m}/\mu \text{s}$, and the burn marks are 300 m on either side of him, so Ryan can determine that each explosion took place $1.0\mu \text{s}$ before he saw the flash. But this was the instant of time that Priya passed him, so Ryan concludes that the explosions were simultaneous with each other and with Priya’s passing him. The spacetime coordinates of the two events in frame S are $(x_{\text{R}},t_{\text{R}})=(300\text{m,}0\mu \text{s})$ and $(x_{\text{L}},t_{\text{L}})=(-300\text{m,}0\mu \text{s})$.

2. We already know, from our qualitative analysis in Section 36.5, that the explosions are not simultaneous in Priya’s reference frame. Event R happens before event L in $\text{S}\prime$, but we don’t know how they compare to the time at which Ryan passes Priya. We can now use the Lorentz transformations to relate the spacetime coordinates of these events as measured by Ryan to the spacetime coordinates as measured by Priya. Using $v=0.8c$, we find that $\gamma$ is

   $$\gamma ={\displaystyle \frac{1}{\sqrt{1-v^2/c^2}}}={\displaystyle \frac{1}{\sqrt{1-0.8^2}}}=1.667$$

   For event L, the Lorentz transformations are

   $$\begin{array}{l}{x}_{\text{L}}^{\prime }=1.667\left(\left(-300\text{m}\right)-\left(0.8c\right)\left(0\mu \text{s}\right)\right)=-500\text{m}\\ t_{\text{L}}^{\prime }=1.667\left(\left(0\mu \text{s}\right)-\left(0.8c\right)\left(-300\text{m}\right)\text{/}{c}^2\right)=1.33\mu \text{s}\end{array}$$

   And for event R,

   $$\begin{array}{l}{x}_{\text{R}}^{\prime }=1.667\text{}\text{}\text{}(300\text{ m})-\left(0.8c\right)(0\mu \text{s})\text{}\text{}\text{}=500\text{ m }\\ t_{\text{R}}^{\prime }=1.667\text{}\text{}\text{}(0\mu \text{s})-\left(0.8c\right)(300\text{ m)/}{c}^{2\text{}}=-1.33\mu \text{s}\end{array}$$

   According to Priya, the two explosions occur 1000 m apart. Furthermore, the first explosion, on the right, occurs $1.33\mu \text{s}$ before Ryan passes her at $t\prime =0\text{s}$. The second, on the left, occurs $1.33\mu \text{s}$ after Ryan goes by.

REVIEW Events that are simultaneous in frame S are not simultaneous in frame $\text{S}\prime$. The results of the Lorentz transformations agree with our earlier qualitative analysis.