Twin Paradox: Still a Paradox? Alright, so David Griffiths in his "Introduction to Electrodynamics" states that the Twin Paradox is not a paradox at all since the traveling twin returns to Earth. By returning to Earth, the twin had to reverse direction, thus undergoes acceleration, and therefore cannot claim to be a stationary observer. 
However, what if the traveling twin simply Skypes the twin that is on Earth. The twin on earth will still appear older, which would make no sense since in that case the rocket can be seen as the stationary frame of reference while the Earth "travels" at a speed close to the speed of light. No acceleration is undergone, yet the paradox remains. 
Is Griffiths just completely glossing over important nuance again?
 A: 
The twin on earth will still appear older

No, that is not correct.  If the twin on the rocket never reverses course and remains inertial, the twins never meet to compare ages at the some location.
Since the twins remain spatially separated, their ages must be compared by spatially separated clocks.
For example, when the twins are separated by 1 light-year, the observation of the age of the twin on the rocket, as observed by the twin on Earth must be made with a clock, synchronized with the clock on Earth but co-located with the rocket, i.e., located 1 light-year from Earth.
Similarly, the observation of the age of the twin on Earth, as observed by the twin on the rocket, must be made with a clock, synchronized with the clock on the rocket but co-located with Earth, i.e., located 1 light-year from the rocket.
But, as is well known, clocks synchronized in the Earth's frame of reference are not synchronized in the rocket's frame of reference and vice versa.
Thus, due to this relativity of simultaneity (synchronization), each twin observes the other to have aged less without contradiction.
See this answer for a useful diagram.
A: Another nuance that is sometimes skipped over is the doppler shift: that is, the number of wave-crests of light emitted by one twin and seen by the other is different. Imagine that each twin has an atomic clock that is counting the number of wave-crests emitted by an atomic clock held by the other twin.
When the travelling twin starts the journey, both twins see eachother's atomic clocks radiation red-shifted and so they both see the other moving more slowly though time.
When the travelling twin turns around some vast distance later, the frequency of the Earth bound atomic clock is blue-shifted immediately and Earth time is seen to speed up. The Earth twin has to wait some time for the travelling space clock becomes blue-shifted and so for this short period of time, the travelling twin sees Earth time running faster than normal and the Earth twin sees spaceship time running slower than normal.
For the remaining part of the journey, both twins see blue shifted light from the other and so see the other moving faster through time, but really they are just catching up on light already emitted. 
When the travelling twin finally arrives back, the middle sequence is never made up: the Earth bound atomic clock has emitted more wavecrests than it has counted leaving the spaceship clock (and visa versa). This is why the Earth twin looks older and has gained more life experience.
Satellites are already in our future: gravity red-shifts our light as it reaches the satellites and blue-shifts satellite light when it falls back to Earth. Blue shifted light means more CPU clock cycles and so satellites are travelling through time at a faster rate than we on the Earth's surface are.
A: The twin paradox and time dilation more generally are widely misunderstood, even by some physicists. The key points to bear in mind are as follows:
If two people start together, move away and then return, their ages will have changed by different amounts upon re-meeting if their respective paths through spacetime have not been entirely symmetrical. In the classic version of the twin paradox, one twin (on Earth) remains in an internal frame throughout while other twin does not, so the arrangement is clearly asymmetric.
At any point when both twins are moving inertially, the effects of SR are entirely symmetrical. For example, on the outbound leg of the travelling twin's journey, the time on the travelling twin's watch falls increasingly behind the local time in the Earth's frame, and the time on the stay-at-home twin's watch falls increasingly behind the local time in the travelling twin's frame. To put it another way, each twin appears time dilated to the other.
The difference in the age of the twins when they re-meet is accounted for by the turn-around event, when the travelling twin shifts from the outbound inertial reference frame to the in-bound one, and it is a consequence of the relativity of simultaneity.
At the turnaround point, the orientation of the plane of simultaneity of the outbound twin rotates with the change of reference frame. The consequence of that rotation is a step-change increase in the time back on Earth which is simultaneous with 'now' on the travelling twin's watch. There are countless diagrams on the Internet that show this if you care to look.
When you ask 'what does each twin see happening to the other?' you must take into account the Doppler effect. That effect will generally cause each twin to see the other as slowed down  when they are moving apart and as speeded up when they are moving towards each other. However, there is another important asymmetry here to do with the timing of changes to the value of Doppler effect, which I will now explain.
If you are receiving a light signal, the extent to which it is Doppler shifted can change for one of two independent reasons- either you can accelerate or the source can accelerate. While both will eventually bring about a change in the Doppler shift affecting the received signal, only in the first case is the change immediate. If the source accelerates you will not see any effect until light from the accelerated source has reached you; whereas if you accelerate you will see the frequency change straight away.
If you now apply that to the twin paradox, when the travelling twin accelerates at the turnaround point, they immediately see light from Earth changing from being red shifted to being blue shifted, or, to put it in other words, if they were viewing a clock on Earth through a telescope they would see the clock immediately switch from being slowed down to being speeded up. However, on Earth it takes time for light from the newly accelerated travelling twin to reach Earth, so a person on Earth watching the travelling twin's clock would not see it change from being slowed down to being speeded up until much later.
In terms of what each twin sees, each twin sees the other slowed down by the same amount on the outbound leg, and each twin eventually sees the other speeded up by the same amount when the are moving together. However, the outbound twin sees the switch happen immediately at the half-way turning point, whereas the Earth twin does not see the shift until later. The Earthbound twin therefore sees that the traveller spends more of the journey slowed down than speeded up, while the travelling twin sees the Earth twin slowed down for half the journey and speeded up for the remainder. As a consequence, the travelling twin sees the Earth twin age more.
A: Actually, the paradox in the "Twin Paradox" was never the paradox you so often see today.
Einstein presented the first example of two clocks in his 1905 paper on special relativity. One clock remains stationary and another clock makes a round trip. Upon return, the traveling clock will be behind the stationary clock. He called the result peculiar in that it was paradoxical to our sense of time. Some call this the clock paradox.
https://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf
In 1911, Langevin wrote a paper in which he gave a similar example, except instead of clocks, he considered a person making the round trip. The result was that the traveller would age less than those who remained at home. This was even more paradoxical to our sense of time as it involved living organisms. This was the origin of the twin paradox, although Langevin did not use twins.
https://en.wikisource.org/wiki/Translation:The_Evolution_of_Space_and_Time
Neither Einstein or Langevin questioned the validity of the result. Nor did they propose anything as absurd as switching the roles of who travelled and who did not (keep in mind they understood the problem and theory very well). The paradox was simply that relativity had real consequences on clocks and people that conflicted with our sense of time.
And that is what paradoxes are. You solve the problem correctly, and the result conflicts with your senses or intuition.
Somehow, the notion of "why does my ignorance of relativity contradict itself" paradox took the place of the actual paradox. And that is why many say "this isn't a paradox" because it isn't. An incorrect analysis of the problem isn't a paradox, it's just an incorrect analysis.
Thus the original paradox wasn't about incorrect analyses, it was about the result of a correct analysis seeming "peculiar". And later incorrect analyses aren't paradoxes, they are just incorrect analyses.
David Griffiths isn't glossing over anything. He feels that the problem is solved correctly and the fact that someone is unable to follow the solution (which is obviously common) isn't a paradox, it is simply the case of someone not understanding.
Additionally, your specific (skype) example is not a paradox, it is just an incorrect analysis. The skype messages are essentially round trips. Indeed, when you examine the twin problem in a more continuous fashion (such as the twins are in constant radio contact) you get the same solution. In a real life scenario, on a mission to some star, distances and relativity will be taken into account. Imagine that the ship has a panel with Earth Time and Ship Time on it, and Earth Time will be running fast compared to Ship Time continuously. Imagine a similar panel on earth, and on that panel Earth Time will also be running fast compared to Ship Time. And these times are updated with radio messages sent back and forth and relativistic effects and distances have been accounted for.
We do this now with GPS satellites.
https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6074806
A: I believe Griffiths explanation at the end of example 12.2 is correct when he says that the paradox is broken when the traveling twin cannot be considered a stationary observer because of his acceleration. More can be said about this if we dip our toe into general relativity where acceleration or gravity can enter the analysis.
I will tell a story of what I think happens if a twin named Cain leaves his brother Able at a constant speed and then abruptly reverses direction and returns to Able at the same speed. There can be other observers in this picture (Eve and Sarah) who match Cain's velocity on the outward and return trips. I will omit their important stories.
During the trip, Able and Cain skype each other and can see the clocks on the walls behind them. Both of them hear delays in their conversation responses. They both realize that the time shown on the other's clock occurred half way through the conversation response time (deduced real time). The deduced time intervals on the other's clock become longer. However, when signals arrive, the observed time intervals, colors of their clocks, and pitch of their voices depends on their relative speeds (doppler shift).
At the turnaround point, Cain blinks and deduces that Able's clock has jumped way ahead. Cain may believe he hibernated for an extended period. As he returns to Able, Able's slower clock remains ahead of his own.
I have presented more details and math here: http://www.astro.uvic.ca/~jchapin
Click on "Asymmetry of the Twin Paradox"
A: Tyrion Lannister,
All answers to the problem claiming that there is no such thing as the twins paradox I saw were based on acceleration. Acceleration is said to be actually responsible for the difference in time-flow between the two twins.
However, before heralding all over the world to have found the solution to the twins paradox, one should simply take a look at the equation that lead to the conclusion that the paradox must exist:
$$ \Delta t' = \frac {\Delta t} {\sqrt{1 - \frac{v^2}{c^2}}} $$
Immediately a question arises: Where is the acceleration in this equation? If acceleration is the culprit, it should appear somewhere in the equation that shows time dilatation. And yet, all we can see is just constant $v$. So how come acceleration is responsible for time dilatation, if there is no acceleration at all ...?
We can also approach the problem from another angle. Claiming that acceleration solves the problem of twin paradox is equivalent to claiming that all time dilatation in Special Relativity is due to acceleration, isn't it? If acceleration solves the paradox, than there must be no other source of time dilatation in this theory than acceleration, right? 
But this would mean that Einstein simply didn't notice the fact that all the unusual phenomena in his own theory are all due to acceleration. This would mean that Einstein must have been plain wrong saying that the Special Relativity pertains to inertial frames only. Claiming that SR (time dilatation) simply boils down to acceleration must mean that SR is actually equivalent to General Relativity - that, in fact, there is no SR at all. 
So, the dismissal of the twins paradox based on acceleration means simply the dismissal of the very Special Relativity Theory. But for this to be a legitimate claim, someone needs to prove it. Is anyone able to provide such a proof? I haven't seen any so far ...
On the other hand, one obviously cannot take off Earth and travel in space without accelerating. No doubt about it. But then, it is not acceleration that lead somebody to ponder what happens to twins after one of them went on a space travel. Therefore, any acceleration contributing to time dilatation in the twins paradox is only a part of the problem. It must be so, unless we see the proof that SR = GR.
EDIT: Any comment as to what in my answer is incorrect? I just love to learn more and more about SR.
A: No absolute relativity as long as a constant reference exists this reference is speed of light (this reference will be responsible for defining  the moving body
Earth twin will see that other twin's clock is slow because it is really slow
But space twin will not see that other twin's clock is slow because his time 
Perception is also slow
Space twin will see earth moves far from him but by shorter distance than the distance seen by earth twin, so if we supposed that space twin is moving far from earth with the exact speed of light he will never see that earth is moving because he is moving with the light that is moving from earth so he will not receive image updates
A: I believe the twin paradox is truly a paradox in that the same problem approached different ways gives different answers. So if the travelling twin accelerates from near earth at 1g, he will be close to the speed of light (relative to earth) in one year. If he then turns 180 degrees and continues acceleration at 1g, he will return to earth in one year. Both twins will have experienced the same 1g acceleration. They are symmetrical and will see time slowing down for the other twin as they recede from each other and time speeding up as they approach each other. They are both the same age when they reunite.  So there is no way we can use time dilation to explore the galaxy or ever get to other than the nearest stars. The indirect proof? We have not been visited by aliens. There are hundreds of billions of stars and many more planets in our galaxy. There must be millions of earth like worlds that have existed for as much time or longer than our earth. Surely we are not the most advanced civilization among these. If time dilation allowed travel throughout our galaxy, then why haven't we been visited? We have been transmitting radio and TV signals for more than 80 years now, and should be detectable to alien civilizations within 80+ light years. Where are they????
A: Not for me.
Provided we assume that the travelling twin is encapsulated in an environment that is in the same frame of reference as him/herself (e.g. the living compartment of the spaceship), the twins will age at the same rate.
The aging process must surely be dependent on speed of travel w.r.t. living  environment.
I have heard about experiments involving radioactive particles travelling at great speed through the earth's atmosphere, in which those particles were seen to have the expected greater radioactive half life. That makes sense to me as I compare it to a swimmer at sea who gets hit by waves at a slower rate the faster she swims (up to a point) giving the effect of slowing the waves.
There seems to be a lot of confusion about time dilation and length contraction due to a lack of distinction between perception and reality. In reality time and length are constant v speed, but when we try to observe very fast moving objects, we can only do so by communication of some kind (optical or otherwise) and it is the communication that alters our perception of time and length.
I am assuming a universal measurement of time based on the distance travelled by light.
A: If they skype each other, then because the skype signals travel at the speed of light, as long as they are moving apart they will each symmetrically see the other one as slowed down by the amount predicted by the relativistic Doppler effect equation. For example, suppose they are moving apart inertially, with a relative velocity of 0.6c. Then according to the equation, visually each will see the frequency of the other's clock slowed down by a factor of:
$\sqrt{(1 - 0.6)/(1 + 0.6)} = 0.5$
(Note that this is different from the time dilation factor, which is designed to factor out the effects of signal delays)
You can see how this works if you actually imagine following the path of successive signals sent by each one. Let's analyze things in the frame of the Earth twin, after the traveling twin has started moving away from the Earth. Let's say each starts their stopwatch at the moment of departure, so at that moment the Earth twin's watch reads $T_e = 0$ seconds and the traveling twin's watch reads $T_t = 0$ s. Then when each one's clock reads 10 seconds, they send a light signal showing their clock reading 10 second to the other, and when each one's clock reads 20 seconds, they send a light signal showing their clock reading 20 seconds to the other. 
First we can figure out when the signals from the Earth twin will reach the traveling twin. This signal is sent at coordinate time t = 10 s in the inertial rest frame of the Earth, and if Earth is defined to be at position x = 0 light-seconds in this frame and the traveling twin is moving in the +x direction, then at this moment the traveling twin is at x = 6 ls in this frame. Then at 15 seconds later at t = 25 s in this frame, the light ray will be at x = 15 ls, and the traveling twin moving at 0.6c will be at x = 6 + (0.6 * 15) = 15 ls as well, so t = 25 s will be when the first signal catches up to the traveling twin in this frame. But in this frame the traveling twin's clock is running slow by a factor of $\sqrt{1 - 0.6^2} = 0.8$ due to time dilation, so the traveling twin's clock reads $T_t = 25 * 0.8 = 20$ seconds when the first signal reaches him. 
The Earth twin sends the second signal at t = 20 s in this frame, when the traveling twin is at position x = 12 ls. 30 seconds later at t = 50 s, the second signal from the Earth twin will have reached x = 30 ls, and the traveling twin will have reached x = 12 * (0.6 * 30) = 30 ls as well, so this is when the second signal catches up to the traveling twin in this frame. Since the traveling twin's clock is running slow by a factor of 0.8 in this frame, the traveling twin's clock reads $T_t = 50 * 0.8 = 40$ seconds when the second signal reaches him. So the traveling twin sees signals sent 10 seconds apart by the Earth twin's clock reach him 20 seconds apart according to his own clock, meaning he that visually he sees the Earth twin's clock running slow by a factor of 0.5, just as predicted by the Doppler formula.
Now we can also analyze when the signals sent by the traveling twin will reach the Earth twin, again using the Earth rest frame. The traveling twin sends signals when his clock reads $T_t = 10$ and $T_t = 20$ seconds, but because his clock is running slow by a factor of 0.8 in this frame, the coordinate times of his sending the signals are t = 10/0.8 = 12.5 s and t = 20/0.8 = 25 s. Since he is traveling at 0.6c, at t = 12.5 s he is at position x = 0.6 * 12.5 = 7.5 ls when he sends the first signal, so the signal traveling at the speed of light will take 7.5 s to reach the Earth twin, reaching him at t = 12.5 + 7.5 = 20 s in this frame, when the Earth twin's clock reads $T_e = 20$ seconds. And at t = 25 s when the traveling twin sends the second signal, he is at position x = 0.6 * 25 = 15 ls when sending the second signal, so the signal takes another 15 s to reach the Earth twin, arriving at t = 25 + 15 = 40 s, when the Earth twin's clock reads $T_e = 40$ seconds. So you can see that everything is totally symmetrical--each one sends signals at 10 seconds and 20 signals on their clocks, and receives the other one's signals at 20 seconds and 40 seconds on their clock, meaning each sees the other one slowed down visually by a factor of 0.5. Only if one of them accelerates is this symmetry broken, just as Griffiths suggests.
