2 The Initial-Value Equations

Einstein’s equations, G = 8πGT μν μν, represent ten independent equations. Since there are ten equations and ten independent components of the 4-metric gμν, it seems that we have the same number of equations as unknowns. From the definition of the Einstein tensor (3View Equation), 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

μν ∇ νG ≡ 0. (4 )
The four constraint equations appear as a result of the general covariance of Einstein’s theory, which gives us the freedom to apply general coordinate transformations to each of the four coordinates and leave the interval
ds2 = gμνdx μdxν (5 )
unchanged.

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 t = 0, then the Bianchi identities (4View Equation) guarantee that the evolution equations preserve the constraints on neighboring spacelike hypersurfaces.

 2.1 Initial data
 2.2 York–Lichnerowicz conformal decompositions
  2.2.1 Conformal transverse-traceless decomposition
  2.2.2 Physical transverse-traceless decomposition
 2.3 Conformal thin-sandwich decomposition
 2.4 Stationary solutions

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