I am currently working through The International Winter School on Gravity and Light lecture series and have had some trouble understanding the following claim made in lecture $9$ at the 22:30 minute mark.
Dr. Schuller says that asking
"$(1)$ Can gravity be encoded in the curvature of space, such that its effects show if particles under the influence of (no other) force are postulated to move along straight lines in this curved space?"
is the same as asking:
"$(2)$ Is $$\ddot{x}^{\alpha}(t)-f^{\alpha}(x(t)) = 0$$ of the form of a autoparallel equation?"
Note $\ddot{x}^{\alpha}$ is the ($\alpha^{th}$-component of the) acceleration of the test particle and $f^{\alpha}$ is the ($\alpha^{th}$-component of the) gravitational acceleration.
My question is:
why is asking question $(1)$ the same as asking question $(2)$? How is an autoparallel equation (an equation of the form $\ddot{x}^{\alpha}(t) + \dot{x}^{\beta}(t)\dot{x}^{\gamma}(t)\Gamma^{\alpha}_{\beta\gamma}(x(t))$) related to question $(1)$?
Ironically, I understand the answer to this question from a mathematics point of view which is a resounding "no", you need to ask this question about spacetime not just space, but I'm just trying to connect the dots. Here is a similar question on this lecture and time stamp posted on Physics.SE.