A thought experiment

1. Dec 12, 2012

cosmic dust

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:

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?

2. Dec 12, 2012

Staff: Mentor

Yes. In fact, it's a good way of describing in fairly simple terms how to detect spacetime curvature. However, there is a key proviso to that; see below.

For the connection that's used in GR, the Levi-Civita connection, yes; this is the definition of that connection. There are other connections that could be used that don't correspond to parallel transport, but they aren't used in GR.

This brings up a key point. You say this earlier in your post:

In order to evaluate spacetime curvature by parallel transporting a vector around a small closed curve, as you describe, each segment of the curve has to be a geodesic. Along a geodesic, the observer's proper acceleration is zero, so there is no "non-inertial movement". The observer does have an easy way to distinguish inertial from non-inertial movement: use an accelerometer. He can ensure that he stays on a geodesic by ensuring that his accelerometer reads zero.

(Btw, the equivalence principle says that the observer can't distinguish non-inertial movement, such as accelerating under rocket thrust, from *being held at rest* in a gravity field. An object held at rest in a gravity field is not freely falling, so an accelerometer moving with such an object will not read zero. Freely falling in a gravity field can easily be distinguished from non-inertial movement, by using an accelerometer; what it can't be distinguished from, locally, is *inertial* movement; in fact, according to GR it *is* inertial movement. The "locally" qualifier is there because there will be some distance over which curvature can be detected if it's present.)

So to run your thought experiment properly, the first requirement is that the observer's accelerometer always reads zero. Then *any* change in a vector when it's parallel transported around a closed curve can *only* be due to spacetime curvature. Thus, if the "accelerometer = zero" requirement is met, then yes, your assumption is valid: all "kinematic" adjustments must cancel, leaving only the change in the vector due to curvature (if any).