Strength of GR Equations

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_ijR = 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 c_ab. 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 c_jkdx^jdx^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 curl of the magnetic field be 0.