I am trying to understand simple examples of space-time curvature.
Assume for the moment that $c$ is infinite (classical curvature due to Newton's laws). Also, I will only consider 1+1-dimensional space for simplicity: $(x,t)$.
Case I. I will consider uniform acceleration due to gravity g along the positive x-direction pervading the entire space. Then all particles will have $$x-x_0 = \frac{1}{2}g(t-t_0)^2.$$ For this case, we can use the transformation \begin{align} x' &= x - \frac{1}{2}g(t-t_0)^2 \\ t' &= t\end{align} to get a transformed space-time $(x',t')$ such that any path of the form $$x-x_0 = \frac{1}{2}g(t-t_0)^2$$ in the original co-ordinate system is equivalent to the form $$x' = vt' + x_0'$$ representing uniform motion in the new $(x',t')$ coordinate system. Thus, if space-time is warped as described by the transformation from $(x,t)$ to $(x',t')$ system, all objects just follow a straight line in the transformed system.
Case II. Now I will consider a slightly more complex scenario. Here, $$g(x) = -\omega^2 x.$$ Can we obtain transformation $x'=x'(x,t)$ and $t'=t'(x,t)$ such that uniform motion in $(x',t')$ is equivalent to Simple Harmonic motion in $(x,t)$?
Case III. Now assume that acceleration due to gravity is $g$ for $-1 \le x \le 1$ and is 0 everywhere else. What about this field?
I don't have any experience in differential geometry or any obscure mathematics. My current goal is to see how much we can understand relativity just with simple mathematics.