Question

What is the radius of convergence of the Fefferman-Graham expansion?

Answer 1

What exactly do you mean by "radius of convergence"? Since the Fefferman Graham expansion is defined in a neighborhood of infinity, the proper distance from the boundary to any spacetime point with finite r must diverge. I assume you mean to ask something like "what is the greatest value of r for which the Fefferman Graham expansion converges?" This is a good question to which I don't know the answer. However, based on the fact what we have here is essentially a fancy Laurent series, my guess is that it is the greatest value of r at which a spacetime singularity exists.