Question

I understand that the Christoffel symbols associated with the metric will vanish locally once you perform the appropiate change of coordinates. These new coordinates correspond to an observer in free-fall.

Now, if this transformation is dependent on the velocity and acceleration of the free-falling observer, then any velocity should work, as long as the acceleration of this new observer works to cancel out the acceleration due to gravity.

But then, acceleration has only three parameters, so how are they enough to make all the Christoffel symbols go to zero? Aren't there like 40 of them? I have considered also rotation, but that only brings like three extra components for the angular velocity, which I'm not completely certain that gravity can produce.

Plus, if the metric is to also be converted into the Minkowski matrix, that's 6 extra components that must go to zero. Does not look good
A) The transformation dependent on the observer's velocity and acceleration leads to the complete elimination of all Christoffel symbols due to their interrelation with the observer's motion.

B) The finite parameters of acceleration are inadequate to nullify all the Christoffel symbols as there are over 40 of them; however, rotation's three extra components might assist in minimizing a subset.

C) Acceleration's three parameters are insufficient to nullify all Christoffel symbols; rotation contributes additional components via angular velocity, yet it remains unclear whether gravity alone generates these.

D) The reduction of the Christoffel symbols to zero through the transformation involving velocity and acceleration becomes challenging due to the multiple components involved, including the six necessary for the metric to align with the Minkowski matrix.

Answer 1

The transformation dependent on the observer's velocity and acceleration does not lead to the complete elimination of all Christoffel symbols. The three parameters of acceleration alone are insufficient to nullify all the Christoffel symbols.

Step-by-step explanation:

The transformation dependent on the observer's velocity and acceleration does not lead to the complete elimination of all Christoffel symbols, as there are over 40 of them. The three parameters of acceleration alone are insufficient to nullify all the Christoffel symbols. However, rotation can contribute additional components via angular velocity, although it is unclear if gravity alone can generate them. In addition, the reduction of the Christoffel symbols to zero through the transformation involving velocity and acceleration becomes challenging due to the multiple components involved, including the six necessary for the metric to align with the Minkowski matrix.