The geometry of an n dimensional complex composed of simplices is completely determined by the length of the 1D edges. If we compute the 4D mesh for GR by holding in the machine a 3D time slice which is itself a 3D complex and compute forward in time then we match the ordinary causality concepts of physics and computation. The first question is what information must accompany such a slice besides the 3D geometry. This leads directly to issues of strength of equations, a subject that I have found obscure.
Physics seems to specialize in second order differential equations. Including the first and second derivatives of the metric tensor
Einstein’s empty space equations
Rij + (1/2)gijR = 0 selects a subset of these possible spaces. That would seem to place n(n+1)/2 constraints at each point.
My intuition tells me that we need the full 4D stress-energy tensor Tab as well. Some conditions are imposed on Tab such as that the divergebce of the magnetic field be 0.
This rewarding presentation decomposes the 4D metric tensor into 1+3 dimensions thus:
ds2 = −(α − βaβa)dt2 + 2βadxadt + gabdxadxb
Scripts ‘a’ and ‘b’ range over three spatial dimensions.
and then the full Einstein equations for empty space:
∂tgab = −2αKab + ∇aβb + ∇bβa
∂tKab = −∇a∇bα + α(Rab + KabK − 2KacKcb) + βc∇cKab + Kca∇bβc + Kcb∇aβc
There are some things to say about these equations!: At each point in 3+1 space there are 6 values gab, three values βa and one value α which account for the ten values of the 4D metric tensor. I presume that βb = βagab, K = Kabgab and gabgbd = δac. ∇a is the covariant derivative in the 3D space of S where gab rules. It does seem that the given time evolution equations indeed predict the future (and past) for the 3D geometry but are silent on α and β. It seems clear to me that we are free to evolve α and β to minimize mesh distortion. Such additional equations amount to coordinate conditions. α paces and β steers. I think that in fact that α and β might be calculated afresh each time step.