Abstract
We present a 2+1 decomposition of the vacuum initial conditions in general relativity. For a constant mean curvature one of the momentum constraints decouples in quasi isotropic coordinates and it can be solved by quadrature. The remaining momentum constraints are written in the form of the tangential Cauchy-Riemann equation. Under additional assumptions its solutions can be written in terms of integrals of known functions. We show how to obtain initial data with a marginally outer trapped surface. A generalization of the Kerr data is presented.
Highlights
Vacuum initial data in general relativity consist of a Riemannian metric g = gi j d xi d x j and a symmetric tensor K = Ki j d xi d x j given on a 3-dimensional manifold S
Where ∇ ̃ i are covariant derivatives corresponding to g, Ris the Ricci scalar of gand
Solving (62) with respect to ω is equivalent to finding integral lines of the vector field v = γ,2∂1 + (γ,1 + f )∂2 in R2. The latter problem can be reduced to an ordinary differential equation with an arbitrary initial condition
Summary
Vacuum initial data in general relativity consist of a Riemannian metric g = gi j d xi d x j and a symmetric tensor K = Ki j d xi d x j given on a 3-dimensional manifold S These data have to satisfy the constraint equations. Equation (5) were solved analytically or reduced to a simpler system only for conformally flat initial metrics [4,5,6,7] or symmetric data [8,9,10,11,12]. Operator ∂ defines the Cauchy–Riemann (CR) structure (see [22] and references therein) on the initial manifold, not unique since locally there are many systems of coordinates in which metric takes the form (7). Existence of solutions of this system is much more difficult to prove [3,19]
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