Planet W is 12ly from Earth. Anna and Bob are both 20 yr old. Anna travels to Planet W at 0.6c, quickly turns around, and returns to Earth at . How old will Anna and Bob be when Anna gets back?

Consider a Planet W is 12 light year from earth.

Consider the age of Anna and Bob are both 20 yr old.

Consider that Anna travels to Planet W at 0.6c.

Write the formula of Lorentz transformation equations of time dilation of Anna age.

${\mathit{t}}_{\mathbf{A}}\mathbf{}\mathbf{=}\mathit{y}\mathbf{\times }{\mathit{t}}_{\mathbf{B}}$ …… (1)

Here, ${\mathit{t}}_{\mathbf{B}}$is Bob’s time and $\mathit{y}$is relativistic transformation.

As you would in a classical issue, Bob determines the time for himself while seated on Earth in his inertial frame of reference.

$$\begin{gathered} \mathrm{t}=\frac{\mathrm{L}}{\mathrm{u}} \\ =\frac{12\mathrm{ly}}{0.6\mathrm{c}} \\ =20\mathrm{yr} \end{gathered}$$

Anna soon turns around, so we can ignore the time spent during this acceleration phase and assume that the travel was uniform throughout because this is only the first half of the voyage. As a result, we may estimate that the total voyage will take 40 yr, or twice as long as it will take to arrive.

Therefore, Bob’s age will be

40 yr +20 yr = 60 yr

Keep in mind that at the start of their journey, Bob and Anna are both 20 yr old. Once we start dealing with Anna's frame, the Lorentz transformation is put to use. We will evaluate the time dilation from Bob's frame because Anna isn't in an inertial frame (which is the whole point of the twin paradox), which implies that from Bob's perspective, Anna will mature more slowly on this journey.

Determine the Lorentz transformation equations of time dilation of Anna age.

Substitute $\sqrt{1-\frac{v^2}{c^2}}$for y and 40 for $t_B$into equation (1).

$$\begin{gathered} {\mathrm{t}}_{\mathrm{A}}=\sqrt{1-\frac{{\mathrm{v}}^2}{{\mathrm{c}}^2}}\times 40 \\ =\sqrt{1-\frac{{\left(0.6\mathrm{c}\right)}^2}{{\mathrm{c}}^2}}\times 40 \\ =32\mathrm{yr} \end{gathered}$$

Bob calculated that Anna will do her entire journey at this time, thus her age will be

$32\mathrm{yr}+20\mathrm{yr}=52\mathrm{yr}$

Therefore, the value of Anna’s age will be 52 yr and the value of Bob’s age will be 60 yr.