Gravitational time dilation from the perspective of an accelerating rocket

Question

A common thought experiment illustrating gravitational time dilation involves Alice and Bob in an accelerated rocket. Alice is above Bob. Bob fires periodic signals upwards. Alice observes the signals at a lower rate due to kinematic Doppler and a link is made between acceleration and gravitational time dilation (Bob is at lower gravitational potential).

After some time has elapsed, Alice would presumably observe Bob a little younger than herself - they are twins. If she then moved (slowly) down the rocket to meet Bob, will Bob be younger? If so, how can this be understood from the perspective of someone observing the accelerating ship. If not, how does this fit in with the equivalence principle.

Answer 1

Yes, Alice will find she has aged more than Bob has.

The equivalence principle tells us that acceleration is locally indistinguishable from a gravitational field. We use the qualifier locally because gravitational fields always have tidal forces and these are not present in an accelerating frame. However if the setup is small enough that the tidal forces are undetectable then acceleration and gravity are equivalent.

Suppose the acceleration of the rocket is $a$, then the force on a unit mass is also $a$, and the work needed to lift that unit mass from Bob to Alice is just:

$$ W = ah $$

where $h$ is the vertical spacing between Bob and Alice. The gravitational potential energy per unit mass in a gravitational field with gravitational acceleration $g$ is:

$$ U = gh $$

And if $g = a$ then the two energies are the same. I mention this because for a weak gravitational field there is a simple equation that relates the relative time dilation of two observers to their difference in gravitational potential energy:

$$ \frac{dt_B}{dt_A} = \sqrt{1 - \frac{2(U_A - U_B)}{c^2}} $$

In this case $U_A - U_B = +ga$, because Alice is above Bob, so we get:

$$ \frac{dt_B}{dt_A} = \sqrt{1 - \frac{2ga}{c^2}} \lt 1$$

And that means that Bob's time measurements, $t_B$, are always less than Alice's time measurements $t_A$. In other words Bob ages more slowly.