Regarding the apparent paradox, perhaps this explanation will help. Here
The last paragraph is interesting as it takes a Doppler approach. This is a little like the business analogy where “add-on” and “mark-up” differ. A 50% add-on to $1 is $1.50, but a 50% mark-up is $2, allowing the seller to make 50% profit. [I think this applies.]
iPhone with rib sauce.
Helio, That is a really good pictorial explanation of the time difference effect for a round trip, as described by Special Relativity. This result is counter-intuitive, because it shows that a person would age less due to traveling fast, even though he ends up back at the starting point at zero velocity.
But, that in itself is not the "twin paradox" that people keep talking about. That twin paradox tries to use the idea or "relativity" to imply that a calculation done the same way for the twin who stays at the starting point should be considered to be moving in the other direction while the actual traveling twin is assumed to be at-rest. The problem with that is, as you have pointed out, the twin that is accelerated can be determined to be different from the twin that remains at rest because the forces involved are detectable, and while they are being applied,
violate the condition of Special Relativity that it applies to observations between two inertial frames of reference.
I agree with that, but the problem is that the calculation does not include the effects of the accelerations on the mathematical result.
So, we are left in the situation where we agree that the Theory of Special Relativity cannot be applied to this whole experiment without modification, but we do not have the formula that we think does apply to the whole experiment. That is what leaves in-doubt that veracity of the result.
However, I seem to remember that some experiments were done by putting clocks in orbit around the Earth, and comparing them to clocks left on earth once the orbiting clocks were returned to Earth, and that those experiments verified the time difference calculated by "relativity". Unfortunately, that would involve both Special and General Relativity, and the media reports that I remember were not really adequate to describe the details of the experiment.
But, we currently use those theories to make verifiable predictions with our satellite GPS systems. See https://www.astronomy.ohio-state.edu/pogge.1/Ast162/Unit5/gps.html
. That says Special Relativity makes the clocks in the GPS satellite orbits seem
to run 7 microseconds per day slower than our clocks here on Earth's surface, but General Relativity says that clocks here on Earth's surface should run 45 microseconds per day slower than the clocks in orbit because of the difference in the gravitational field force at the two different altitudes. The net effect is that the clocks on the GPS satellites need a 38 microsecond per day correction to avoid the calculated positions on Earth shifting by 10 kilometers per day! That does seem like good experimental validation of the theories, as applied together, for observations of things in 2 different locations.
But, that is really not represented by the out-and-back time lines in your example, because the distance between the sender and receiver is not being monotonically increased and decreased by the velocity in orbit. So, I am having a hard time thinking about it as vectors. The Special Relativity part is really a scalar calculation, so it seems
to work no matter what direction(s) the travel is going in, compared to the observer.
The problem is that an orbit is not an inertial frame of reference
. It is a constant acceleration frame of reference with the acceleration vector changing the direction of travel along the constantly shifting radial direction (relative to an actual inertial frame of reference).
So the same criticism (motions of subjects not always being in inertial frames of reference) that applies to the "twin paradox" seems to also apply to this experimental verification for Special Relativity. It seems inconsistent to use the same concept in 2 diametrically opposite ways, logically.
So, I am still looking for a pure
empirical demonstration that a clock that is sent into orbit and returned to Earth has actually shown less time passage than a clock left on Earth when the difference is corrected for the altitude effects of General Relativity.
Because that seems like an experiment that can be done, I expect that it has been done. I just have not found it, yet. This link https://en.wikipedia.org/wiki/Time_dilation
talks about experimental verifications using things like sub atomic particle half-life measurements. It also claims a solution to the twin paradox with Minkowski diagrams, but I am not following that clearly, (yet?).