Hello, normally the twin paradox is formulated and solved as follows:
Alex goes on a space trip around earth at relativistic speed while John, his brother twin, stays at home. Due to relativistic time dilation, Alex ages at slower rate than John, therefore when he comes back, he is still young while john is very old.
The paradox arises as in the reference frame of Alex, Alex is standing still and the Earth is moving at relativistic speed, therefore when Alex comes back home, he should be older and John should be younger. The paradox is usually solved by observing that in order to make a closed round trip, Alex must accelerate so that its frame reference is not inertial, therefore special relativity does not apply in the frame reference of Alex.
Now, let's make the scenario a bit more interesting. Let's assume that the trip of Alex is made at exactly one g of acceleration. Then, both twins feel the same gravitational field, and both twins see the other one moving at relativistic speed. Who has aged more at the end of the trip in this case?
General relativity predicts that Alex returns even younger relative to John, since he can reach a very high speed (as described by John) with 1g acceleration for a few months. The gravitational effect essentially depends on difference in gravitational potential, which is quite small between the earth's surface and interstellar space.
Now, Alex need a little more than 1g to leave the earth, and he must turn around at some point. But these are small corrections if the trip is long enough.
No paradox. The resolution is the same, as in special relativity: between two points in a fixed spacetime the geodesic equation has a unique solution; so, if one twin follows a geodesic, another doesn't. The unique solution of the geodesic equation extremizes the proper time, in that spacetime, so any other path doesn't.
Once more, one shouldn't mix up space and spacetime, that only leads to more awkward descriptions in the best case and confusion in the worst.
(Example: in a curved spacetime relative velocity isn't uniquely defined. The resolution doesn't have anything to do with the magnitude of relative velocity. It has to do with spacetime curvature, that the twin, that doesn't follow a geodesic, experiences.)
It all depends on the values of the velocity (SR) and of the gravitational potential (GR).
The general relativistic term has a sign opposite to the special relativistic one (the Langevin twin paradox). This has been seen quite directly a long time ago. In 1975, Alley {C. Alley, "Proper Time Experiments in Gravitational Fields with Atomic Clocks, Aircraft, and Laser Light Pulses," in Quantum Optics, Experimental Gravity, and Measurement Theory, eds. Pierre Meystre and Marlan O. Scully, Proceedings Conf. Bad Windsheim 1981, 1983 Plenum Press New York, ISBN 0-306-41354-X, pg 363–427.} sent 4 atomic cesium clocks for a 15-hour flight on an airplane and compared them with 11 clocks that stayed on Earth. Those clocks should have been late according to the traditional twin paradox. Not at all; they they had gained 47.10^{-9} s! The special relativistic effect is ∆t/t = -{v2}/{2c^2} => -6 10^{-9} s in the conditions of the flight, the general relativistic effect is ∆t/t = ∆V_G/c^2 => 53 10^{-9} s in those conditions.
A much more sophisticated experiment was carried out, a bit later, by R.F.C. Vessot and collaborators " Test of Relativistic Gravitation with a Space-Borne Hydrogen Maser'', Phys. Rev. Lett. {\bf 45}, 2081, (1980).
Indeed, the gravitational potential isn't so direct basis for time dilation. In free fall the relative basis for time dilation is the distance twins fall in the same metric coordinate system between they meets and syncronize their clocks.
You can comove with the other twin and count the mean velocity in respect of that inertial reference frame. Which of them have greater mean velocity magnitude is younger - in comoving coordinates it can be positive, negative or zero...
Hence the gravitational potential effect is relative in principle, only circular orbits can be measure as absolute dilation. I think the angle of incidence of clock syncronizations has something to do with relative aspects... Of course you can constitute a function of gravitational potential but linearly it's observable of mean velocity i.e. a distance gap between fallers.
The proper acceleration instead affects always absolute time dilation, which seems to be the source of inertia and gravity too...
Disclaimer: my text is consists of intuitions and free associations in addition of learnings and experiences. ;)
Giuseppe Papari "Then, both twins feel the same gravitational field, and both twins see the other one moving at relativistic speed."
This is exactly a wrong assumption. The stationary one is measurening same relativistic speed difference, but no acceleration.
I descovered new 6 relativistic paradoxes and solved them as also I solved the old twins paradox. This is solved when we assume, that gravitons are to be treated like photons by a Doppler effect, a Gravitational Doppler effect - which is my discovery too.
Thesis Graviton Doppler and Level Relativity solved a discovered mo...
This is first really understanding deeper of relativistic effects and therefore it is a new Relativity Theory, displacing the SRT. The GRT is also effected and replacedby a GravitonsRelativity theory, which unites dynamic and static relativity in one universal principle of gravitation level relativity.
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