I presume you have some notions about general relativity, especially the Einstein equations
As these equations are basically hyperbolic for a suitable metric, it is reasonable to study the Cauchy problems for them. Under the Gauss and Codazzi conditions, we have two constraints called Hamiltonian and Momentum constrains. Cauchy problem is to determine the solvable of these constrains of variables -the extrinsic curvature and -the spatial metric. Interestingly, the conformal method says that we can start with an arbitrary metric then we recast the constrain equations into a form which is more amenable to analysis by splitting the Cauchy data. In this method, we try to solve within the conformal class represented by the initial metric. So, in general, the conformal factor is chosen so that we eventually have a simplest model.
This idea is given via the following theorem.
Theorem. Let be a conformal initial data set for the Einstein-scalar field constraint equations on . If
for a smooth positive function , then we define the corresponding conformally transformed initial data set by
Let be the solution to the conformal form of the momentum constrain equation w.r.t. the conformal initial data set and let be the solution to the conformal form of the momentum constrain equation w.r.t. the conformal initial data set (we just assume both exist). Then is a solution to the Einstein scalar field Lichnerowicz equation for the conformal data with
if and only if is a solution to the Einstein scalar field Lichnerowicz equation for the conformal data with
We refer the reader to a paper due to Yvonne Choquet-Bruhat et al. [here] published in Class. Quantum Grav. in 2007 for details. We adopt this theorem from that paper, however, there is no proof there.
Outline of the proof. Let
Obviously we have (go to this entry for further information)
However, it is clear to see that
which also yields
Keep in mind that
Next we have
We refer the reader to this entry showing that how to calculate the norm of tensors. It is also clear to see that
since and remain unchanged under the conformal change. Concerning , from its definition
we see that
Thus adding all gives
The proof now follows.