Abstract
We study the free data in the Fefferman–Graham expansion of asymptotically Einstein (n+1)-dimensional metrics with non-zero cosmological constant. We analyze the relation between the electric part of the rescaled Weyl tensor at {mathscr {I}}, D, and the free data at {mathscr {I}}, namely a certain traceless and transverse part of the n-th order coefficient of the expansion mathring{g}_{(n)}. In the case Lambda <0 and Lorentzian signature, it was known [23] that conformal flatness at {mathscr {I}} is sufficient for D and mathring{g}_{(n)} to agree up to a universal constant. We recover and extend this result to general signature and any sign of non-zero Lambda . We then explore whether conformal flatness of {mathscr {I}} is also neceesary and link this to the validity of long-standing open conjecture that no non-trivial purely magnetic Lambda -vacuum spacetimes exist. In the case of {mathscr {I}} non-conformally flat we determine a quantity constructed from an auxiliary metric which can be used to retrieve mathring{g}_{(n)} from the (now singular) electric part of the Weyl tensor. We then concentrate in the Lambda >0 case where the Cauchy problem at {mathscr {I}} of the Einstein vacuum field equations is known to be well-posed when the data at {mathscr {I}} are analytic or when the spacetime has even dimension. We establish a necessary and sufficient condition for analytic data at {mathscr {I}} to generate spacetimes with symmetries in all dimensions. These results are used to find a geometric characterization of the Kerr-de Sitter metrics in all dimensions in terms of its geometric data at null infinity.
Highlights
When written in terms of the conformal metric g, the Einstein equation of g is singular at I
It is noteworthy that associated to a conformal metric g solving the conformal Friedrich equations, there is a solution to the Einstein equations g which is “semiglobal”
We provide a geometric relation between g(n) and the tensor D, the electric part of the rescalled Weyl tensor at I, for spacetimes admitting a conformally flat I
Summary
When written in terms of the conformal metric g, the Einstein equation of g is singular at I. Theorem 2.4 is proven, which establishes that the nth order coefficient of the FG expansion coincides (up to a certain constant) with the electric part of the rescaled Weyl tensor in the case when I is conformally flat and n > 3 (for n = 3 this is true in full generality). This theorem finds immediate application in the Cauchy problem of Einstein equations at I with positive cosmological constant (cf Corollary 2.4.1). We will work in the general setup unless otherwise stated
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
