Gravitational Twins

Engineering physics involves the transport of people and materials in space as well as practical applications here on Earth. So consider a pair of twins in free fall. Imagine that one twin is in circular orbit around a star and that her sister is shot out from this circular orbit location on a radial orbit that is, the traveling twin will fall back to meet up with the stay-at-home sister in circular orbit. For simplicity, let them meet after an integral number of revolutions around the circular orbit for the one left behind.

Any clock system in a gravitational potential, such as the clocks in the GPS system here on Earth, depends on two relativistic effects on the clock rate: (1) a clock ticks slower closer to a massive object than when far away, and (2) the faster-moving clock ticks slower than the slower-moving clock.

Initially, the clock rates of the twins are the same because they start out in the same circular orbit at the same radial distance from the star. The traveling twin moves away from the star along the radial line, all the while slowing down and eventually coming to a momentary stop and returning with ever-increasing speed until rejoining her sister in orbit. So on average, the traveling twin experiences a smaller amount of gravitational time dilation and a smaller amount of speed time dilation than her stay-at-home sister. Therefore the traveling twin returns home older than her sister, because her clock ticked faster on average. What do you think?

Answer

The traveling twin actually returns much younger than her stay-at-home sister. The argument given was correctly stated but incomplete. The local gravitational tidal effects are not the same for the twins that is, the rate of change of gravitational potential experienced was different. These tidal effects contribute to the clock rates and, when included in the calculations, contribute enough to change the result so that the stay-at-home twin ages faster and is older upon return of her sister. For a calculation, see the reference below.