### Some notes on General Relativity

I imagine space-time partitioned into simplexes.
In three dimensions a simplex is a tetrahedron with four (=3+1) vertices.
In four dimensional space-time it has five vertices.
I presume the space within a simplex is flat—its Riemann curvature is 0.
In n dimensions the curvature of a polyhedral complex is concentrated in its n-2 dimensional sub-complexes.
The curvature of a polyhedron in 3D is at its point vertices.
Regge has proposed the above ideas somewhere as a tool for studying Riemannian geometry and GR in particular.
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.

### Digression on Strength of Equations

From Riemannian geometry we know that any symmetric positive definite tensor field (signature (1,1,1,1)) defines a metric for some Riemannian manifold.
The above sloppy statement suggests the rigor that I plan here.
A signature of (1,1,1,−1) applies to our space-time.
That difference seems not to impact us for quite a while.
We can thus think of a general manifold as having n(n+1)/2 degrees of freedom at each point.
Physics seems to specialize in second order differential equations.
Including the first and second derivatives of the metric tensor

Einstein’s empty space equations

R_{ij} + (1/2)g_{ij}R = 0 selects a subset of these possible spaces.
That would seem to place n(n+1)/2 constraints at each point.

### The Initial Value Problem

Wald’s “General Relativity” says that a space-like 3-dimensional surface S cutting thru space-time has an intrinsic 3-dimensional curvature as well as an extrinsic curvature as it is embedded in 4-space.
The extrinsic curvature is a symmetric tensor K_{ab}.
Note that latin subscripts range over just three values.
Consider a point at x^{j} in S and a plane there tangent to S.
The distance (time interval) between another nearby point at x^{j} + dx^{j} in S and the plane is K_{jk}dx^{j}dx^{k} where j and k range over spatial dimensions.
My intuition tells me that we need the full 4D stress-energy tensor T_{ab} as well.
Some conditions are imposed on T_{ab} such as that the divergebce of the magnetic field be 0.

This rewarding presentation decomposes the 4D metric tensor into 1+3 dimensions thus:

ds^{2} = −(α − β^{a}β_{a})dt^{2}
+ 2β_{a}dx^{a}dt
+ g_{ab}dx^{a}dx^{b}

Scripts ‘a’ and ‘b’ range over three spatial dimensions.

and then the full Einstein equations for empty space:

∂_{t}g_{ab} = −2αK_{ab}
+ ∇_{a}β_{b} + ∇_{b}β_{a}

∂_{t}K_{ab} = −∇_{a}∇_{b}α
+ α(R_{ab} + K_{ab}K − 2K_{ac}K^{c}_{b})
+ β^{c}∇_{c}K_{ab}
+ K_{ca}∇_{b}β^{c}
+ K_{cb}∇_{a}β^{c}

There are some things to say about these equations!:
At each point in 3+1 space there are 6 values g_{ab},
three values β_{a} and one value α which account for the ten values of the 4D metric tensor.
I presume that β_{b} = β^{a}g_{ab},
K = K_{ab}g^{ab} and
g^{ab}g_{bd} = δ^{a}_{c}.
∇_{a} is the covariant derivative in the 3D space of S where g_{ab} 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.