Spatial infinity in higher dimensional spacetimes

TL;DR

This paper explores the asymptotic structure of spatial infinity in higher-dimensional spacetimes, revealing that it generally lacks maximal symmetry due to the Weyl tensor, and discusses how static vacuum solutions are not uniquely determined by multipole moments alone.

Contribution

It demonstrates that the geometry at spatial infinity in higher dimensions is non-maximally symmetric and shows the necessity of additional data, like the Weyl tensor, to uniquely specify static vacuum solutions.

Findings

Spatial infinity geometry is generally non-maximally symmetric in higher dimensions.

Static vacuum solutions are not uniquely determined by multipole moments alone.

Additional information, such as the Weyl tensor at infinity, is required for unique specification.

Abstract

Motivated by recent studies on the uniqueness or non-uniqueness of higher dimensional black hole spacetime, we investigate the asymptotic structure of spatial infinity in n-dimensional spacetimes($n \geq 4$). It turns out that the geometry of spatial infinity does not have maximal symmetry due to the non-trivial Weyl tensor {}^{(n-1)}C_{abcd} in general. We also address static spacetime and its multipole moments P_{a_1 a_2 ... a_s}. Contrasting with four dimensions, we stress that the local structure of spacetimes cannot be unique under fixed a multipole moments in static vacuum spacetimes. For example, we will 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 static vacuum solution we need some additional information, at least, the Weyl tensor {}^{(n-2)}C_{abcd} at spatial infinity.