To provide equations governing relativistic dynamics of matter, we use the Bianchi identities

On substituting Eq. (29) into Eq. (35), we get the dynamic formula for the Weyl conformal curvature[3,40,41]:

On decomposing Eq. (36) along and orthogonal to a 4-velocity vector, we obtain constraint ( ) and propagation ( ) equations of the Weyl fields in a form analogous to the Maxwell equations[10,42,43,44]:

The twice contracted Bianchi identities present the conservation of the total energy momentum tensor, namely

It is split into a timelike and a spacelike momentum constraints:

They provide the conservation law of energy-momentum, i. e., how matter determines the geometry, and describe the motion of matter.

2018-03-26