Consider Anna, Bob and Carl in the twin paradox.

(a) According to Anna, when Planet X passes her, clocks on Planet X and Earth tick simultaneously. What is the time interval between these two events in the Earth-Planet X frame?

(b) According to Carl, when Planet X passes, clocks on Planet X and Earth tick simultaneously. What is the time interval between these two events in the Earth-Planet X frame?

(c) What does the clock on Planet X read when Carl and Anna reach it? Show how your results from part (a) and (b) agree with Figure 2.20.

Speed of Anna and Carl with respect to Earth-Planet X frame is,

$v=0.8c$

The distance of planet X from Earth with respect to Anna and Carl is,

$l^{\text{'}}=24\text{\hspace{0.17em}ly}$

The time interval of two events measured by a stationary observer is related tothe time interval measured by a moving observer at velocity and the space interval of the two events measured by the moving observer as,

$t=\frac{t^{\text{'}}+\frac{v}{c^2}{x}^{\text{'}}}{\sqrt{1-\frac{v^2}{c^2}}}$ .....(I)

Here $c$ is the speed of light in vacuum.

The time interval of the two ticks with respect to Anna and Carl is $0$. The velocity of Anna with respect to Earth-Planet X frame is $0.8c$.Thus from equation (I), the time interval according to the stationary frame is,

$\begin{array}{c}t=\frac{0+\frac{0.8c}{c^2}24c\text{\hspace{0.17em}y}}{\sqrt{1-{0.8}^2}}\\ =32\text{\hspace{0.17em}y}\end{array}$

Thus, when the two ticks are simultaneous in Anna's frame, the time interval in the Earth-Planet X frame is $32y$.

(b)

The velocity of Carl with respect to Earth-Planet X frame is $-0.8c$. Thus from equation (I), the time interval according to the stationary frame is,

$\begin{array}{c}t=\frac{0+\frac{-0.8c}{c^2}24c\text{\hspace{0.17em}y}}{\sqrt{1-{0.8}^2}}\\ =-32\text{\hspace{0.17em}y}\end{array}$

Thus, when the two ticks are simultaneous in Carl's frame, the time interval in the Earth-Planet X frame is $-32y$.

(c)

The clock on Planet X is in a stationary frame and runs at the same rate as that of Bob's clock. According to that clock, it took $50\text{\hspace{0.17em}y}$ for Anna to reach Planet X.,So the clock reads $50\text{\hspace{0.17em}y}$. According to Anna, the Planet X clock is ahead of the Earth clock by $32\text{\hspace{0.17em}y}$. So, according to her, the Earth clock reads $18\text{\hspace{0.17em}y}$, which is her estimate of Bob's age. According to Bob, the Planet X clock is behind the Earth clock by $32\text{\hspace{0.17em}y}$. So, according to him, the Earth clock reads $82\text{\hspace{0.17em}y}$, which is his estimate of Bob's age.

Thus, when Carl and Anna reach Planet X, the clock reads $50y$ , which is in agreement with Figure 2.20.