Question about gravity in 1 dimension Consider 2 pointmasses in 1D space. More specific ; object $A$ with mass $m_a$ and object $B$ with mass $m_b$. $A$ and $B$ are a distance $d$ away from eachother.
Let these pointmasses $A,B$ have no initial speed or rotation.
Let $d(t,m_a,m_b)$ be the distance between $A$ and $B$ after time $t$.
At collision $d(t,m_a,m_b)=0$ and at starting position ($t=0$): $d(t,m_a,m_b)=d$.
What is the closed form or integral equation or differential equation for $d(t,m_a,m_b)$ according to the different theories of gravity, such as, 


*

*Newtonian? 

*special relativity(*)? 

*general relativity?
(* I mean Newtonian + the velocity addition from special relativity ).
I also wonder about the analogues for the Laplacian for each gravity theory.
 A: Newtonian. First we should note, that in Newton's inverse square law for gravity $F\propto 1/r^2$ the power 2 is really $D-1$, where $D$ is the space dimension. So in 1D case the force $F$ between two point masses is independent of separation between them: $$m_1 \ddot{x}_1 = G m_1 m_2\, \mathop{\mathrm{sgn}}(x_2-x_1).$$ Therefore the closed form of equation for separation $d$ is
$$
\ddot{d} = - G (m_1+m_2).
$$
This simply gives the parabola known from the motion in constant acceleration field.
Newtonian gravity + special relativity. This is not an internally consistent model. Here I explain why this is so. 
General relativity. GR in one dimension (or, to be more precise, 1+1 dimension, with +1 denoting time) is really simple: Einstein equations for the empty space mean that the space-time must be locally flat. So there is no gravitational force between two point masses, and they would continue to move at constant speeds.
Dilaton gravity. Instead of 'ordinary' general relativity very interesting 'toy' model for gravitational forces in 1+1 dimensions is the dilaton gravity, which does provide many interesting features of higher dimensional general relativity including black holes and gravitational attraction between point sources. There are, in fact, several models of 1+1 dilaton gravity related to various choices for the terms like dilaton potential. If you are familiar with GR, I refer you to the review hep-th/0204253 for additional information. 
Now, the motion of point masses in these dilaton gravity is quite an interesting question. It turns out that the two body problem in such model allows explicit general relativistic treatment and produce result. The calculations could be found in the paper:

Mann, R. B., & Ohta, T. (1997). Exact solution for relativistic two-body motion in dilaton gravity. Classical and Quantum Gravity, 14(5), 1259. arXiv:gr-qc/9607016.

and a followup paper:

Mann, R. B., Robbins, D., & Ohta, T. (1999). Exact relativistic two-body motion in lineal gravity. Physical review letters, 82(19), 3738. arXiv:gr-qc/9811061

Though the solutions are 'exact', the equations are quite unwieldy and in terms of the Lambert W function, however the papers above do have figures such as this plot of separation vs. proper time for various masses (smaller masses means more relativistic solution):

