Twin Clock Paradox
"Everything likes to live where it will age the most slowly, and gravity pulls it there"  Kip S. Thorne ( 1940  )
"Relativity is a purely scientific matter and has nothing to do with religion"  Albert Einstein ( 1879  1955 )
The October 1971 Joe Hafele and Richard Keating Famous Experiment Using Four Cesium  Beam Atomic Clocks
testing general and special relativity physics time dilation on a British Airways Boeing aircraft
movie source: PBS's NOVA Science  "Time Dilation Experiment, October 1971"
source: J. C. Hafele and R.E. Keating, Science 177: 166  68 &168  70 ( 1972 )
Read: "Performance and Results of Portable Clocks in Aircraft", by J.C. Hafele
Also: "AroundtheWorld Atomic Clocks: Observed Relativistic Time Gains", by J.C. Hafele and Richard E. Keating
[ note: for those who cannot view this page whole, see quicktime or flash movie ]
Observed Facts: Flying eastward and then westward from the stationary U.S. Naval Observatory, General Relativity Physics predicted a time loss of 40 ± 23 nonoseconds on the eastward bound and a time gain of 275 ± 21 nanoseconds for the westward trip; the actual eastward loss was 59 ± 10 nanoseconds going eastward but gained 273 ± 7 nanoseconds for the westward trip!!
Rhetorical Question: Is the October 1971 Joe Hafele and Richard Keating clock experiment showing the effects of Special Relativity Physics or General Relativity Physics? Is the British Airways flight around the earth a straight  line, inertial flight subject to Special Relativity Physics analysis or is it rather a matter for General Relativity Physics analysis?
The Twin Clock Paradox is only inexplicable for the situation where the actual relativity physics mathematics are not fully understood.
The Problem: If a rocket leaves relatively stationary earth and travels a sufficiently great distance with a velocity at a significant fraction of the speed of light, and then later returns ostensibly "younger" than what was left behind, the question therefore arises if all motion is indeed relative, why then could not the earth be considered as having left the rocket and therefore upon the mutual return of earth to the rocket be shown equivalently also as "younger"?
The Solution: The solution comes in two parts:
The mathematics for accounting for differing gravitational potential fields is best understood in the general theory as opposed to the special theory of relativity physics
which reduces down to
in the conventional mathematics of special relativity physics.
And when the above Riemannian metric for General Relativity Physics is used for an accelerating, non  inertial spacetme system of bodies in relative motion, the Twin Clock Paradox is resolved and time dilation thus becomes mutually reciprocal as between the stationary observer versus the other moving person!
Notice: this generalized Riemannian equation uses only average mean square velocities and does not utilize either the strengths or durations of any bursts of acceleration !
"Everything likes to live where it will age the most slowly, and gravity pulls it there"  Kip S. Thorne ( 1940  )
Analyzing the Twin Clock Paradox using Doppler frequencies
However, we can still analyze the Twin Clock Paradox using Doppler frequencies that communicate between non  accelerated, relatively stationary earth, and our accelerated rocket as the following explanation provides with the added benefit of deriving the time dilation equation of special relativity physics using Minkowski's spacetime geometry!
One admission that must be stated is that earlier Doppler was itself derived from time dilation and, therefore, there is somewhat of a tautology ( circular reasoning; redundancy ) being expressed here. However, Doppler can and should be derived separately by means of the original Lorentz Transformation Equations which David Bohm's "The Special Theory of Relativity" accomplishes and to which in some later recapitulation of Relativity Physics and Science Calculator this author will submit a simplified version of this derivation. In the meantime, everything shown here is both mathematically and experimentally true and valid.
Example
Earth Observers vs. Astronauts v/c = 0.60 and v_{e}/c = 0.693 


Destination ( Galaxy or Star ) 
dl^{ ∗∗} ( earth distance ) 
dl_{e}^{[1]} ( astronaut distance ) 
dt ^{[2]} ( earth time ) 
dt_{e}^{[3]} ( astronaut time ) 

Milky Way Center  2.5 x 10^{4} ly ^{[a]}  2.311 x 10^{4} ly  4.166 x 10^{4} years  3.333 x 10^{4} years  
Proxima Centauri  4.223 ly  3.903 ly  7.0 years  5.6 years  
Sirius star  8.6059 ly  7.954 ly  14.34 years  11.47 years  
Procyon star  11.4043 ly  10.540 ly  19.0 years  15.2 years  
Andromeda ( M31 )  2.544 M ly ^{[b]}  2.351 M ly  4.24 x 10^{6} years  3.39 x 10^{6} years  

Using our upper limit for velocity of a photon rocket consisting almost entirely of photon energy in order to maximize deep interstellar Milky Way space travel towards, say, Proxima Centauri dwarf star [ see: derivation ], then let
^{∗∗∗}
note: this calculation assumes inertial ( non  gravitational, non  accelerating ) roundtrip motion which on its face is not possible since any "return motion" is gravitational; unless, of course, you wish to consider a sort of "slow motion turns" as an infinite series of inertial ( non  accelerating ) turns for the return trip.
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