Does the twin paradox require both twins to be far away from any gravity field?

Question

If one twin is on earth at 1 g and the other twin accelerates away from earth following a great big elliptical counterclockwise trajectory. He travels at .9 g for 20 years(according to earth time) as well as some small amount of left acceleration (.44 g since $\sqrt{0.9^2+0.44^2}=1$) to make the first semicircle . He then turns around and decelerates at .9 g in the opposite direction for 20 years(according to earth time) and now experiences some small amount of rightward acceleration (.44 g) to complete the semicircle. He then arrives at earth. Will they both be the same age?

Although the twin paradox has been discussed on this site on 42 pages of questions, there are only 4 pages that address the question of gravitational acceleration compared to motion acceleration. My intuition suggests that gravitational acceleration should have the same effect as motion acceleration so they should be the same age. In reading some of these questions I find contradictory answers. For example, this question

Gravitational Time Dilation vs Acceleration Time Dilation

suggests

a higher acceleration would yield the same results as more gravity

But answers to this question

Why does only one twin travel in the twin paradox?

says

"that the earth twin experiences the same RELATIVE acceleration as the space twin (in the opposite direction) this is incorrect."

So which is the correct interpretation?

My twist to this question is the elliptical orbit. The direction of the principle component of acceleration (from the rear of the space ship) remains unchanged yet the lateral acceleration does change. That the lateral acceleration changes direction should not affect time dilation because that is a scalar quantity. Am I overlooking something when I make that statement?

Aside from the focus of the question in the title, there is a slight difference in the geometry. Gravitational acceleration gives a tidal effect whereas motion acceleration does not.This distinction though does not seem to enter into the calculation of time dilation though.

Answer 1

Will they both be the same age?

No, they will not. The traveling twin will be substantially younger. To calculate the age of each twin simply integrate the metric over their worldline: $$\tau = \int d\tau = \int \sqrt{g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}}d\lambda$$ (in units where c=1). This procedure is general, it works for any twin motion and any spacetime, with or without gravity.

For example, this question Gravitational Time Dilation vs Acceleration Time Dilation suggests a higher acceleration would yield the same results as more gravity

Unfortunately, this one is worded a little poorly in a way that appears to be contributing to your confusion. Gravitational acceleration does not cause time dilation. Gravitational time dilation is caused by the gravitational potential. Also, the equivalence principle only applies over small enough regions of spacetime that spacetime curvature can be neglected.

So what would be true is that a clock on the ground on earth would tick slower than a clock raised 1 m off the ground on earth, and a clock on the back of the rocket would tick slower than a clock raised 1 m off the back of the rocket, and the difference in tick rate would be the same for both cases. That is how the equivalence principle would apply in this scenario.

That the lateral acceleration changes direction should not affect time dilation because that is a scalar quantity. Am I overlooking something when I make that statement?

Since the change in lateral acceleration is experimentally detectable it is in itself sufficient to break the symmetry. However, in this case it is rather irrelevant since acceleration does not cause time dilation anyway. But the two twins are in no way symmetric in this version of the problem.

Gravitational acceleration gives a tidal effect whereas motion acceleration does not.

This is correct. In fact, the tidal effect you mention is spacetime curvature. So the equivalence principle only is valid over regions of spacetime small enough that tidal effects are negligible.

Answer 2

Gravitational time dilation on the surface of the Earth vs. life at infinity is minuscule and has no impact on the Twin Paradox.

Time dilation due to acceleration does not play a role in the age difference for the traveling twin: all that matters is that he changes direction after traveling at high speed. Including linear acceleration only confuses the matter, and adding a big elliptical loop just adds another dimension.

It's best to understand the idealized twin paradox 1st: that is the one with instant acceleration. If Earth twin says each leg lasts $T$, then he ages $2T$ for whole trip.

Meanwhile he sees space-twin age $T/\gamma$ on each leg of the trip.

The paradox arises because space twin also sees himself age $T/\gamma$ on each leg of the trip, but he sees Earth twin age $T/\gamma^2$ on each leg of the trip.

Note that:

$$ 2T/\gamma^2 \ne 2T $$

so that space twin has a discrepancy of $\Delta T = 2T(1-1/\gamma^2)$.

Once can look too gravitational time dilation during the acceleration, but the problem is that space twin took $0$ seconds to turn around in his reference frame. He also to $0$ seconds to turn around Earth's reference frame.

Note that:

$$ 0 - 0 \ne \Delta T $$

However, at the turn around event, the Earth's clock is both at $T/\gamma$ (for the outgoing twin) and at $T/\gamma + \Delta T$ (for the ingoing twin) at the same time.

When the space twin switches reference frames, the Earth clock advances by $\Delta T$. You can work that a gravitational time dilation back on Earth would correspond to $\Delta T$, but that has the unfortunate property of being reversible, if the twin turns around again...and nobody wants to accept time going backwards, so considering it as gravitational time dilation is tricky.

I think it's better to remain in flat space time and be cognizant of the fact that the time on Earth at the turn event is not well defined, and depends on the velocity of the space twin. If he turns around, Earth time jumps forward; however if he decides to accelerate away from Earth even faster, then time on Earth can jump backwards.