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
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.

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 Kab. Note that latin subscripts range over just three values. Consider a point at xj in S and a plane there tangent to S. The distance (time interval) between another nearby point at xj + dxj in S and the plane is Kjkdxjdxk where j and k range over spatial dimensions.

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 = −∇abα + α(Rab + KabK − 2KacKcb) + βccKab + Kcabβc + Kcbaβc

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.