Gravity at space like and null infinity - Section 2

I return to Friedrich's paper.

Section 2.1 continues the introduction but with some more details. The problem to be considered is an initial value problem given by a 3-dimensional smooth Riemannian manifold $S$ with a metric $\tilde{h}_{ab}$ and symmetric tensor $\tilde \chi_{ab}$. We assume that our space time is empty so the Einstein equations reduce to showing that the ricci tensor of the evoked metric vanishes. This implies two equations on the metric and the tensor (equations 2.3 and 2.4).

Further to this we assume that $\tilde S$ is asymptotically flat thatisasympototically eucliddan. This amounts to assuming that $\tilde S$ and the initial data satisfy some pealing properties and that they are conformally related to a smooth, compact riemannian manifold in a rather nice way.

Friedrich points out, further later use, that if we assume that the Yamabe number associated tot he conformally related metric $h$ is positive then wqe can assume that the riccir scalar of $h$ is negative (Friendrich uses a negative defiant signature for the metrics).

Lastlyhe proves a lemma showing that with mild assumptions on $\tilde h$ theree always exists a one parameter sequence of metrics that conformally converge to $\tilde h$ excluding some compact region.

Section 2.2 discusses the conformal constraint equations and how to solve them. Friedrich uses standard techniques along with the assumption that on $\tilde S$ we have $\chi=\tilde\chi=0$ and that $h$ is analytic in a neighborhood of "space like infinity", $i$. In principle he shows how to satisfy the constraint equations using these assumptions.

It is important to note the construction of the functions $U$ and $W$ which are used to construct the $\theta$ of equation 2.16 locally about $i$. The function $U$ gives the local geometry while $W$ gives the global geometry.

Section 2.3 goes over the initial data for the conformal field equation 2.1. I'm not exactly sure why these additional fields are required, but I suppose we'll find out. Four new piece of information are required. In each case it seems that they are used to restrict certain types of behavior in the unphysical metric and a new conformal connection. The new data are,

  1. A frame of vectors over $S$ that are orthonormal with respect to the unphysical metric
  2. A conformal connection that is unrelated to either metrics. Friedrich gives a reference to his paper, "Einstein's equations and conformal structure" as a justification for this addition.
  3. A condition on a tensor derived from the Ricci tensor of the conformal connection. I have no real idea of why this is introduced, I suppose we'll find out later.
  4. A condition on the rescaled Weyl tensor of the unphysical metric. Ostensibly I think this is so that we can get some local control over a power series expansion of thephysical Weyl tensor at $i$, but to be honest I'm once again unclear why this is needed.