The equivalence principle states that constant acceleration inside a closed elevator is indistinguishable from gravitational force. But prior to its reaching constant acceleration, any elevator has positive jerk (the derivative of acceleration); and it also has negative jerk when it stops.
This leads to the following: suppose you are in a closed elevator in free fall. Say you are standing parallel to the $z$-axis. You experience a downward directed force $F(t)$ consistent with a normal elevator, $0\le t\le 1$. Say
- $F'(t)>0$ for $0<t<\epsilon$
- $F(t)>0; F'(t)=0$ for $\epsilon<t<1-\epsilon$
- $F'(t)<0$ for $1-\epsilon<t<1$
Can you infer that you are in an elevator that is being externally propelled, or could the accelerations you observe be due to the gravitational influence of (moving) external bodies?
You can assume the jerk is continuous, but it is also possibly interesting to consider the discontinuous case: we can think of $F$ as a step function (taking $\epsilon=0$). At first I thought discontinuous $F'$ could not induced from gravity, but maybe if one imagines a huge burst of energy traveling at the speed of light and moving past the elevator, the gravity of the energy would result in a discontinuous jerk? (I have no idea if this is correct).
But even for continuous jerks, are there limitations on what could be induced by gravity? Is there an "equivalence principle for jerk"?
Also, if you want you can just eliminate conditions (3) and (4) above to make the problem simpler.