Bob and Alice are twins. Bob accelerates rapidly away from Earth in a rocket that travels in a straight line until it reaches a velocity $v$ relative to the Earth. It then travels with constant $v$ for a long time before reversing its engines and decelerating rapidly until it is travelling at a velocity $-v$ relative to the Earth. After a further long period of time the rocket returns to Earth, decelerating rapidly until it is at rest. Alice remains on Earth throughout. Sketch the space-time diagram that describes Bob's world-line in Alice's rest frame, assuming that the periods of acceleration and deceleration are negligibly small compared to the total time, explain carefully why Bob ages less than Alice between his departure and his return and show that

$$\Delta t_{B}=\left(1-\frac{v^{2}}{c^{2}}\right)^{1 / 2} \Delta t_{A}$$

where $\Delta t_{B}$ is the time by which Bob has aged and $\Delta t_{A}$ is the time by which Alice has aged.

Indicate on your diagram how Bob sees Alice aging during his voyage.