https://doi.org/10.1140/epjc/s10052-021-09704-6
Regular Article - Theoretical Physics
Free data at spacelike 
and characterization of Kerr-de Sitter in all dimensions
Instituto de Física Fundamental y Matemáticas, Universidad de Salamanca, Plaza de la Merced s/n, 37008, Salamanca, Spain
Received:
17
May
2021
Accepted:
1
October
2021
Published online:
18
October
2021
We study the free data in the Fefferman–Graham expansion of asymptotically Einstein 
-dimensional metrics with non-zero cosmological constant. We analyze the relation between the electric part of the rescaled Weyl tensor at , D, and the free data at , namely a certain traceless and transverse part of the n-th order coefficient of the expansion 
. In the case 
and Lorentzian signature, it was known [23] that conformal flatness at is sufficient for D and to agree up to a universal constant. We recover and extend this result to general signature and any sign of non-zero 
. We then explore whether conformal flatness of is also neceesary and link this to the validity of long-standing open conjecture that no non-trivial purely magnetic -vacuum spacetimes exist. In the case of non-conformally flat we determine a quantity constructed from an auxiliary metric which can be used to retrieve from the (now singular) electric part of the Weyl tensor. We then concentrate in the 
case where the Cauchy problem at of the Einstein vacuum field equations is known to be well-posed when the data at are analytic or when the spacetime has even dimension. We establish a necessary and sufficient condition for analytic data at 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.
© The Author(s) 2021
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