There is only ever one “time period” to communicate in. This one. The “now”.
In the classic so called “twins paradox”, the two are always in the same time line.
Visualize the traveling twin doing this travel always in sight of Earth. Orbiting the Sun at near the spped of light. He is always visible from Earth via telescope and radio. Although there will be some doppler shifting in his image and radio signals, communication is nevertheless still possible. Just difficult. He is traveling for what seems to be his Earth bound twin’s span of 50 years. During his voyage, he ALSO counts the Earth rotating around the Sun 50 times! Yup. That’s the way it works. Yet when he finally lands back on Earth, he has aged only 5 years! Relativity has slowed his BIOLOGICAL clock, relative to his twin. Not time itself. Just like it slows the atomic clock on the moving jet plane, relative to the one left on the ground. Gravity has little or nothing to do with it. It is the difference in RELATIVE velocity that produces what gets measured as time dilation. The same is tru for the twins. Actually, neither twin left the other’s time line. Both were always in the same one. Both counted 50 revolutions of the Earth around the Sun, but the travelers biological CLOCK ran slower. That’s all.
There is no paradox. And communiction across “time periods” is only ever communication across the same time period.
Where I am now, it’s 11:30 AM. But in Japan, it’s 4:30 TOMMORROW MORNING. But me and the Japanese are still in the same time line, time period (in the greater sense), and can communicate with each other just fine.
The past has gone by. The future hasn’t happened yet. And all the near light speed travel or power sources that alter matter can’t change that. It’s a rather fundamental law of the universe.
Even Einstein only ever said time running backwards during faster than light travel, MIGHT be possible. He NEVER said for sure that it was true or for certain WOULD occur. And even withdrew further and further from this original position as he got older.
In the pure scientific sense, we need to always be vigilant as to the difference in what constitutes hypothesis, and what constitutes theory.
Good science always tries to DISPROVE hypothesis. Not prove it. Failure to DIS-prove it is what then results in what becomes evolving theory. And still only theory at that.