I tried to post the Delay factor of the Doppler effect relating to the acceleration under Gravitational - free fall, free draft, slingshot, but the audience did not have maturity, so if there is someone who understand my explanation, please let me know - this topic is noting new, just a clarification of the century old concent, before I can move on to the more advanced topics.

The key point is to understnad the acceleration / inertial frame of reference in precise definition.

A Local Inertial Frame of reference is defined by

This simply means that, when you are in a Gravitational acceleration that abides by (d^2)x/d(t^2) = g(local net value to the barycenter), then you are in an Inertial Frame of Reference regardless of your geometric acceleartion from another mass. Based on this principle, please read the below comments.

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Some claim that the Twin Paradox is explained by the fact that that the Symmetry is broken when taking the relative Doppler Shift into a consideration. Comparing the Earth and the rocket that made a round trip, the rocket travels along the signal from Earth and when it made a turn, it will experience an immediate shift from red to blue, while the Earth will experience a delay factor by t(del) = d/c because it takes time for the signal from the rocket to reach the Earth after the rocket has made a turn. Therefore, the rocket will have traveled fast forward in time with respect to Earth.

However, consider one scenario. Since the rocket cannot use booster all along for its acceleration, the rocket uses nearby celestial bodies to accelerate away from Earth but manages to stay relatively stationary to the center of the Galaxy, while the Earth, along the Sun, travels around the Galaxy. Both the Earth’s and rocket’s relative travel is due to a free fall towards the center of Galaxy and to the nearby celestial bodies, it is Gravitation. The Earth made a round trip and the rocket now maneuvers to land on the Earth.

In this case, which one made a round trip, and which went through an acceleration? Also, how does the Doppler Shift work for both parties?

My conclusion is that the Symmetry never breaks, or the Symmetry breaks in a symmetrical way to both the Earth and the rocket, and the Doppler Shift delay factor by t(del) = d/c to both parties because both stayed in their own “inertial” frame of reference. While the Earth will observe rocket’s time elapsed less, if you had been on the rocket, you will observe the Earth’s time elapsed less. Then, where is the agreement? See the Loaf Model of Space-Time, this explains well.

While Space-Time can have any vector mathematically, since we are locked in our own perceptual Space-Time, one can only meet the other for an infinitesimal period of moment and never again. The original Earth and the rocket never meet up again, each will experience the other newly constructed within each one’s frame of reference.

The Symmetry has never broken, and this is not a way to fast forward time travel.

Next topic will be on the Proper Acceleration, I call it “Inertial Acceleration”, and related phenomena we can observe, including the fast forward time travel.