Spatial infinity in higher dimensional spacetimes

Motivated by recent studies on the uniqueness or nonuniqueness of higher dimensional black hole spacetime, we investigate the asymptotic structure of spatial infinity in n-dimensional spacetimes (n>~4). It turns out that the geometry of spatial infinity does not have maximal symmetry due to the nontrivial Weyl tensor ^(n-1)C_abcd in general. We also address static spacetime and its multipole moments P_{a1a2⋅⋅⋅as}. Contrasting with four dimensions, we stress that the local structure of spacetimes cannot be unique under fixed multipole moments in static vacuum spacetimes. For example, we consider the generalized Schwarzschild spacetimes which are deformed black hole spacetimes with the same multipole moments as spherical Schwarzschild black holes. To specify the local structure of the static vacuum solution we need some additional information, at least the Weyl tensor ^(n-2)C_abcd at spatial infinity.