Mt + c1 M Mx + c2 M My + c3 M Mz = 0 (not certain on this without finding my earlier reasoning, will say, meant to essentially function as a diffusion wave equation with Turing patterns and I can't find anything resembling this in a bit of hasty research I've done now)
Where M=M(x,y,z,t).
(x,y,z)=(0,0,0) is essentially the centre of the Earth, a point under the central pole, and each c is a constant to be determined by experimentation.
x,y are between -R and R, where R is the radius of the Earth. z between -h and h where h is... complicated to explain quickly, but basically the ellipsoid region formed by this (x,y,z) marks the point up to a significant discontinuity (which are after all pretty common when it comes to flows). t>0 signifies time, naturally.
M is essentially the amount of spacetime at a coordinate. The tricky part of this is that we're kind of inventing a coordinate system underlying spacetime, as we can't measure it with itself, but it is strictly mathematical. Mx My Mz are the rate of change in that direction, basically the flow. Mt is the rate of change in one location over time.
With boundary conditions:
M(x,y,z,t)=M(-x,-y,-z,t)
Mx(x,y,z,t)=Mx(-x,-y,-z,t)
My(x,y,z,t)=My(-x,-y,-z,t)
Mz(x,y,z,t)=Mz(-x,-y,-z,t)
Mx(x,y,0,t)= ek1(x+y)cos(x*n*pi)
My(x,y,0,t)= ek2(x+y)sin(y*m*pi)
Mz(0,0,z,t)=-g(h-z)-2
M(x,y,z,t)=M(x,y,z,t+P)
Each k, m, n are also constants to be found by experimentation.
(Beyond this region, M will be constant. Equally, M integrated over the region (x,y,z) to determine the net 'amount' within will also be constant and not depend on t).
So, basically there's a kind of reflective symmetry over the disk, rotational motion that forms shrinking inwards circular, spiralling motion at z=0, and a straight downwards force following the inverse square law over the centre, with z=9.8 approx. Also M is periodic over time. Though they tie more to my FE model than to GR.
The ideal situation would be some kind of solution, and the c values would determine things like whether M got larger or smaller towards the edge of the boundary, and that they could be approximately determines to give a gauge for the behaviour of spacetime. To be honest I don't expect that to happen, but wanted to give some indication as to the potential long term goals.
M itself admittedly is of less use due to its mathematical construction, but its derivatives give us a more constant rate of change that would have observable results relative to the Earth.
I think that's right at least, some of that's just copying things I noted down years back and I haven't yet tracked down where I got them from so there could be mistakes.