Equations of Motion of the Connection

Consider all the curves passing through two points p and q of a metric space. It is a theorem that at least one curve exists whose length is equal to the distance between p and q, and no curve is shorter. That curve is called the geodesic through p and q. Conversely, given any vector V at p, there is a geodesic whose tangent at p is that vector. Around p there is a neighborhood such that for all q therein the geodesic between p and q exists and is unique, though for more distant points it may not be unique, as in the case of gravitational lensing.

If V[i] is the tangent vector to a geodesic, then

	0 = V[i][/j]V[j] = V[i][!j]V[j] + V[m] g'[i][n] C[m][j][n] V[j]

A major goal is to identify physical objects in this mathematical model and then to work out the rules of their motion and interaction. For this, we need to set up and then solve the equations of motion of the connection itself. In his General Theory of Relativity, Einstein postulated that pointlike particles and light rays follow geodesics of the connection. We're expecting that particles will turn out to be persistent structures in the connection, and hence the analogous statement here is that the connection is parallel propagated on (some of) its own geodesics. This should turn into the equations of motion we want.

Any legitimate connection can be split up by Kodaira's theorem into a maximally nontensor part and a part that can then be differentiated to give a conserved source field. Conversely, it is likely that any conserved source field that satisfies additional inequalities (like not being too big) has a connection which yields it when differentiated. This is not helpful for establishing equations of motion.

At the opposite extreme, suppose that the connection is parallel propagated along every geodesic. Then all components of the covariant derivative of the connection have to be zero. So, given arbitrary initial conditions at one point, the ordinary derivative of each connection component in each direction can then be computed, and the resulting system can be solved to give a solution throughout a neighborhood of the initial point, which in some cases may extend over the whole space. This rule is rather restrictive.

Therefore, let us assume that at each point there is a special direction, which is interpreted as the local zero velocity. Remember that the space having these local velocities is 8-dimensional, so that the same 4-space point can have various zero velocities according to the location in momentum space. The velocity field is restricted so its integral curves are geodesics, and the connection is restricted so it is parallel propagated just along those geodesics. Then, given initial values for both fields on a 7-D slice (which is nowhere parallel to the velocities), the ordinary derivatives in the 8th dimension (time) can be computed and the equations can be solved for the connection and velocity field on a neighborhood of the initial slice. This is exactly what equations of motion are.

Here are the equations of motion in coordinate form. V[i] is the zero velocity field.

    0 = C[i][j][k][/L] V[L] 
        = C[i][j][k][!L] V[L] 
        - C[m][j][k] g'[m][n] C[i][L][n] V[L]
        - C[i][m][k] g'[m][n] C[j][L][n] V[L]
        - C[i][j][m] g'[m][n] C[k][L][n] V[L]
    0 = V[i][/L] V[L]
        = V[i][!L] V[L] + V[i] g'[m][n] C[i][L][n] V[L]

These are easily solved for the time derivatives. It is necessary that V[t] be nonzero everywhere.