*M*_{t} + *c*_{1} *M* *M*_{x} + *c*_{2} *M* *M*_{y} + *c*_{3} *M* *M*_{z} = 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. M_{x} M_{y} M_{z} are the rate of change in that direction, basically the flow. M_{t} is the rate of change in one location over time.

With boundary conditions:

M(x,y,z,t)=M(-x,-y,-z,t)

M_{x}(x,y,z,t)=M_{x}(-x,-y,-z,t)

M_{y}(x,y,z,t)=M_{y}(-x,-y,-z,t)

M_{z}(x,y,z,t)=M_{z}(-x,-y,-z,t)

M_{x}(x,y,0,t)= e^{k1(x+y)}cos(x*n*pi)

M_{y}(x,y,0,t)= e^{k2(x+y)}sin(y*m*pi)

M_{z}(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.