formulation discussed in the next chapter there arises no problem at all, in the second formulation this question has apparently not been analysed in detail but it also appears to be simpler.

If we want to analyse the freedom to prescribe boundary data for Einstein’s field equations on the boundary ${\cal T} \sim \mathbb{R}_0^+ \times \Sigma$, the first observation to be made in the model problem above is that the freedom to prescribe data on the boundary is only half as large as the freedom to prescribe initial data. Let us consider Einstein’s equations as equations of second order for the metric coefficients. In [*wave coordinates*]{}, characterized by the conditions $$\label{harm-coord}
0 = \Box_g\,x^{\mu} = - \Gamma^{\mu} = - g^{\lambda \rho}\,\Gamma_{\lambda}\,^{\mu}\,_{\rho},$$ the field equations take the form of a system of wave equations (referred to as the ‘reduced system’) for which only ten boundary conditions can be given. This leads to various complications. To get an idea what has to be achieved by the initial and the boundary conditions, we do some simple function counting.

On the space-like initial hypersurface $S = \{t \equiv x^0 = 0\}$ the Cauchy data $g_{\mu \nu}$ and $\partial_t g_{\mu \nu}$ have to be given. 