Solitonic metrics and harmonic maps

We investigate the relationship between solitonic metrics $g_u= - (sin^2 u/2 ) c^2 dt^2 + (cos^2 u/2 ) sum_{i=1}^n (d x^i)^2$ with $u in C^infty ( mathbb{R}^{n+1} )$ and stationary points of the functional $E_Omega ( Phi ) = (1/2) int_Omega || d Phi ||^2 d^{n+1} mathbf{x}$ with $Phi in C^infty (mathbb{R}^{n+1} , S^2)$. Building on work by F.L. Williams (cf. Williams, in: Milton (ed), Quantum field theory under the influence of external conditions, Rinton Press, Princeton, pp. 370–372, 2004; in: 4th International Winter Conference on Mathematical Methods in Physics, Rio de Janeiro, http://pos.sissa.it/archive/conferences/013/003/wc2004_003.pdf (http://pos.sissa.it/archive/conferences/013/003/wc2004_003.pdf), 2004; in: Chen (ed.), Trends in Soliton Research, Nova Science Publications, Hauppauge, pp. 1–14, 2006; in: Maraver, Kevrekidis, Williams (eds.), The sine-Gordon modeland its applications, Springer, Berlin, pp. 177–205, 2014) we show that a map $Phi = (cos beta sin u/2 , sin beta cos u/2 )$ with $beta (mathbf{x} , t) = m (1 + | mathbf{v} |^2 )^{-1/2} (t + mathbf{v} cdot mathbf{x}$ is harmonic if and only if $u_{tt} + Delta u = m^2 sin u$ (the sine-Gordon equation) and $u_t + mathbf{v} cdot nabla u = 0$ (the convection equation). In the spirit of work by B. Solomon (cf. Solomon in J. Differ. Geom. 21:151–162, 1985) we build a 1-parameter variation of $Phi$ which singles out [from the full Euler–Lagrange system of the variational principle $delta E_Omega (Phi)=0$] the convection equation. Williams’ harmonic maps $Phi^{pm} = ( 1 + v^2 )^{-1/2} ( tau cos beta , tau sin beta , pm (1+ v^2 )^{1/2} tanh rho )$ are shown to be harmonic morphisms of dilation $lambda = m ( 1 + v^2 )^{-1/2} tau $, further explaining the relationship between Jackiw–Teitelboim 2-dimensional dilation-gravity theory (cf. Jackiw, in: Christensen (ed), Quantum theory of gravity, MIT, Cambridge, pp. 403–420, 1982; Teitelboim, in: Christensen (ed), Quantum theory of gravity, Adam Hilger Ltd, Bristol, pp. 403–420, 1984) and harmonic map theory (cf. Baird and Wood, Harmonic morphisms between Riemannian manifolds, London Mathematical Society monographs, new series, 29, The Clarendon Press, Oxford University Press, Oxford, ISBN 0-19-850362-8, 2003). We show that geodesic motion in a weak solitonic gravitational field $g_{u_epsilon}$ [ $u_epsilon = u_0 + 2 epsilon rho$ with $rho in C^infty (mathbb{R}^{n+1} )$ bounded and $epsilon <<1$ ] in the Newtonian velocity limit ( $ ||mathbf{u} ||/c << 1$) obeys to the law of motion $d^2 mathbf{r}/ dt^2 = - nabla phi$ in a central force field of potential $phi equiv epsilon c^2 tan (u_0/2) rho$. To justify the use of geodesic equations as equations of motion we address a relativistic mechanics problem i.e. the rotation of a disc in a solitonic gravitational field and exhibit a class of metrics $g_u$ one of whose geodesic equations yields a relativistic generalization $d^2 r/d s^2 = (r w^2)/c^2$ of centrifugal accelerations in classical mechanics.