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 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_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 divergebce of the magnetic field be 0.

This rewarding presentation decomposes the 4D metric tensor into 1+3 dimensions thus:
ds² = −(α − β^aβ_a)dt² + 2β_adx^adt + g_abdx^adx^b
Scripts ‘a’ and ‘b’ range over three spatial dimensions.

and then the full Einstein equations for empty space:
∂_tg_ab = −2αK_ab + ∇_aβ_b + ∇_bβ_a
∂_tK_ab = −∇_a∇_bα + α(R_ab + K_abK − 2K_acK^c_b) + β^c∇_cK_ab + K_ca∇_bβ^c + K_cb∇_aβ^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^ag_ab, K = K_abg^ab and g^abg_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.