Twin Paradox: Why does't the traveling twin measure faster-than-light photons on Earth? Sorry if this question has been asked, but it was difficult to search through given the volume of Twin Paradox questions.
Anyway. 
My question isn't with the twin paradox per se; it's with what the traveling twin observes when they're changing their direction of travel.
So, the twin heads away from Earth at, say, $0.8c$ for $t$ seconds. All this time, things on Earth are moving slowly--less time appears to have elapsed on Earth than has on their spaceship. However, after $t$ seconds the time comes to change direction and head back to Earth at $0.8c$ for another $t$ seconds. When this happens, their plane of simultaneity changes:

However, let's say that the traveling twin isn't able to somehow accelerate instantaneously, so their plane of simultaneity doesn't abruptly change its angle but rather sweeps through the Minkowski diagram. As it's sweeping it stands to reason that the traveling twin would see Earth on something akin to fast-forward, until they finished accelerating and once again observe ("future") Earth as moving slower than they are (at least I think--I'm a layman, so it's completely possible that I'm totally wrong at any point).
While they are under accleration, and Earth appears to be on fast-forward (IF it appears to be on fast-forward), why don't they observe faster than light photons racing across the Atlantic in undersea cables? Does the size of Earth get perfectly squished down due to length contraction? That only happens in the direction of travel, correct?
 A: 
While they are under accleration, and Earth appears to be on fast-forward (IF it appears to be on fast-forward), why don't they observe faster than light photons racing across the Atlantic in undersea cables?

The observer on the ship would see that photons in the cable are:


*

*Moving at exactly the speed of light relative to the cable and other observers on the earth

*Moving much faster than the speed of light in a reference frame where the ship is at rest


This shouldn't be surprising.  The frame shift caused by the acceleration of the ship doesn't just affect the photons, it affects the cables, the residents, and all the other objects on earth.  So the photons don't appear to be violating the local maximum speed of $c$
A: I'll throw my penny in here as well. When considering the "fast forwarding" it is important to keep in mind that the plane of simultaneity is a theoretical construct, that basically tell you that you should consider being simultaneous with you in order for light to arrive a the right time.
As such changing planes of simultaneity does not have any effect on any other points in the space-time except you own.
Rather what you need to keep an eye on is the speed at which photons from earth reaches the travelling twin.
In the picture bellow I have drawn both the planes of simultaneity as well as the light rays that emanate from the earth  and eventually reach the travelling twin.


Note that what the travelling twin is seeing as he is turning is light that comes from the earth past. (The red arrow) It is only when he turns around and goes back to earth that he meets all the light from earth, and thus sees the clocks on earth run at fast forward.
Note that this is a simple geometric effect. Nothing on earth is moving at  super luminal speed, only the image of what is happening on earth is moving fast.
Conversely, the light that is emitted from the twin at the reversal will  reach earth at the red arrow (right hand picture). Note that the light is sent put with the same interval both before and after the reversal of direction. AS a consequence one earth is will be perceived that the cloc is running much slower when the twin is going away, and the fast forwarding when the twin is going back.
As a side effect, the precise details of how the change of speed happens, is not very important. As most of the light will reach the twin while he is travelling back to earth.
For completeness consider the "non relativistic" version where the planes of simultaneity does not change (but light still travels with speed $c

You'll note that also on this case, will the twin experience how the clocks on earth slows down as he his moving away, and then speeds up as he has reversed and goes back.
The main difference between the two is that in the relativistic case the eigen-time that has passed for the travelling twin is shorter (this is the resolution of the twin paradox)
A: Stand in the middle of the United States and look north.   Canada is 1000 miles ahead of you.  Now turn and face west.  It takes you a tiny fraction of an instant to do this, and after that fraction of an instant, Canada is 1000 miles to your right.  
Do you want to conclude that Canada just moved a great distance in a fraction of an instant, and hence at an enormous speed?
A: Accelerating twin sees things like this (he's thinking like this):
Now at this moment on the Earth it's year 2017 and photons are crossing the Atlantic very fast, while I am here looking at light that left the Earth 10 years ago, when it was year 2000 on the Earth. Photons were not crossing the Atlantic very fast at that time, that is why I am not seeing photons crossing the Atlantic very fast. Photons are crossing the Atlantic very fast now, but I am not seeing it.
After 10 years I will be seeing the Earth that was fast-forwarding. I will see it in slow-motion, because the events were recorded in fast moving stream of photons, and those photons will be moving at normal speed when they reach me, because probably I will not be accelerating any more, and because fast-forwarding applies to distant objects only. Seeing the fast-forwarding Earth in slow-motion will result in everything appearing to happen at normal speed.
A: In 1918 Einstein informed the world that, during the turning-around acceleration, a HOMOGENEOUS gravitational field appears:
http://sciliterature.50webs.com/Dialog.htm 
 Albert Einstein 1918: "A homogeneous gravitational field appears, that is directed towards the positive x-axis. Clock U1 is accelerated in the direction of the positive x-axis until it has reached the velocity v, then the gravitational field disappears again. An external force, acting upon U2 in the negative direction of the x-axis prevents U2 from being set in motion by the gravitational field. [...] According to the general theory of relativity, a clock will go faster the higher the gravitational potential of the location where it is located, and during partial process 3 U2 happens to be located at a higher gravitational potential than U1. The calculation shows that this speeding ahead constitutes exactly twice as much as the lagging behind during the partial processes 2 and 4."
This HOMOGENEOUS gravitational field is crucial - without it, the twin paradox becomes an absurdity. The problem is that the HOMOGENEOUS gravitational field itself is an absurdity - Einsteinians know that and never discuss it. Sometimes the HOMOGENEOUS gravitational field is mentioned euphemistically - e.g. here it is referred to as "enough strangeness":
http://www.people.fas.harvard.edu/~djmorin/chap11.pdf 
 David Morin, Introduction to Classical Mechanics With Problems and Solutions, Chapter 11, p. 14: "Twin A stays on the earth, while twin B flies quickly to a distant star and back. [...] For the entire outward and return parts of the trip, B does observe A's clock running slow, but enough strangeness occurs during the turning-around period to make A end up older."
