2 The Initial-Value Equations

Einstein’s equations, , represent ten independent equations. Since there are ten equations and ten independent components of the 4-metric , it seems that we have the same number of equations as unknowns. From the definition of the Einstein tensor (3), we see that these ten equations are linear in the second derivatives and quadratic in the first derivatives of the metric. We might expect that these ten second-order equations represent evolution equations for the ten components of the metric. However, a close inspection of the equations reveals that only six of the ten involve second time-derivatives of the metric. The remaining four equations are not evolution equations. Instead, they are constraint equations. The full system of equations is still well posed, however, because of the Bianchi identities

If we consider Einstein’s equations as a Cauchy problem¹, we find that the ten equations separate into a set of four constraint or initial-value equations, and six evolution or dynamical equations. If the four initial-value equations are satisfied on some spacelike hypersurface, which we can label with , then the Bianchi identities (4) guarantee that the evolution equations preserve the constraints on neighboring spacelike hypersurfaces.