Hello! In order to deepen my understanding of GR foundations, I tried to create something like thought experiment. I would like to post it so you criticize it and tell me if this a correct thinking or just a delusion I have created in order to fill my mind gap. Here it is:(adsbygoogle = window.adsbygoogle || []).push({});

Suppose there is an observer in free of gravity spacetime, which is equipped with accelerometers and gyroscopes, in order to measure accelerations and rotations of his coordinates system. Now, suppose that this observer is tasked to carry a 4-vector and parallel-transport it as he moves through spacetime in a closed orbit. From his readings of the accelerometers and gyroscopes he has, he adjusts the direction (w.r.t. his coordinate system) of this vector in order to keep it parallel. For example, if his gyroscope tells him that his coordinate system has been rotated by some angle about some axis, then he rotates the vector about that axis by an equal angle but with opposite direction. So this observer has the ability to parallel-transport vectors through spacetime. When he returns to the point he started and compares the parallel-transported vector with the initial vector, he will find that the two vectors are parallel (I assumed that the adjustments he made during his travel on the closed orbit, have cancelled each other, because these adjustments are of kinematical nature).

Now, suppose that the same observer, which is equipped with the same instruments, moves in spacetime with the presence of gravity field. He is tasked to do the same thing: to parallel-transport some vector. This time, the instruments are not influenced only by the accelerations and rotations of his coordinate system, but also from the gravity field. According to equivalence principle, the observer cannot distinguish if the readings of his instruments are due to his non-inertial movement in empty space or due to gravity field, so he is obliged to correct the vector’s direction according to the readings, without processing them. When he returns to the point he started, the adjustment he has made have not cancelled each other, because they are not only of kinematic nature. So he finds that the parallel-transported vector is not the same as the initial. This failure of successful parallel-transportation could be explained by spacetime curvature. That is why gravity and curvature are the same thing.

And my questions are:

-Is the thought experiment I described compatible with the foundations of GR?

-Is yes, then the connections (that define parallel transport) is nothing more than the mathematical description of the process “adjust the vector according to accelerometer and gyroscope readings”?

-I assumed that adjustments of kinematic nature cancel each other when one returns to his point of departure. Is this assumption valid?

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# A thought experiment

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