Gravitational multisoliton solutions on flat space

It is well known that, for even $n$, the $n$-soliton solution on the Minkowski seed, constructed using the inverse-scattering method (ISM) of Belinski and Zakharov (BZ), is the multi-Kerr-NUT solution. We show that, for odd $n$, the natural seed to use is the Euclidean space with two manifest translational symmetries, and the $n$-soliton solution is the accelerating multi-Kerr-NUT solution. We thus define the $n$-soliton solution on flat space for any positive integer $n$. It admits both Lorentzian and Euclidean sections. In the latter section, we find that a number, say $m$, of solitons can be eliminated in a nontrivial way by appropriately fixing their corresponding so-called BZ parameters. The resulting solutions, which may split into separate classes, are collectively denoted as $[n-m]$-soliton solutions on flat space. We then carry out a systematic study of the $n$- and $[n-m]$-soliton solutions on flat space. This includes, in particular, an explicit presentation of their ISM construction, an analysis of their local geometries, and a classification of all separate classes of solutions they form. We also show how even-soliton solutions on the seeds of the collinearly centered Gibbons-Hawking and Taub-NUT arise from these solutions.

Figure 1

Rod sources of the static $n$-soliton solution on flat space for even $n$, all with mass one-half per unit length.

Figure 2

Rod sources of the static $n$-soliton solution on flat space for odd $n$, all with mass one-half per unit length.

Figure 3

The rod structure of the $n$-soliton solution: the $n$ turning points $z_{1,\dots ,n}$, corresponding to the locations of the $n$ solitons, divide the $z$ axis into $n+1$ rods, each assigned with a direction placed above it. The directions $(a_k,b_k)$ for $1\le k\le n+1$ are explicitly given by (4.17) and (4.20).

Figure 4

The locations of the turning points and rods in the prolate spheroidal coordinates $(p,q)$ and the Weyl-Papapetrou coordinates $(\rho ,z)$ for solutions with two free solitons ${\mu }_{\mathrm{I},\mathrm{II}}$.

Figure 5

The locations of the turning points and rods in the C-metric-like coordinates $(u,v)$ and the Weyl-Papapetrou coordinates $(\rho ,z)$ for solutions with three free solitons ${\mu }_{\mathrm{A},\mathrm{B},\mathrm{C}}$.