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For the Newtonian (gravitational) $n$-body problem in the Euclidean$d$-dimensional space, $d\ge 2$, the simplest possible periodic solutions areprovided by circular relative equilibria, (RE) for short, namely solutions inwhich each body rigidly rotates about the center of mass and the configurationof the whole system is constant in time and central (or, more generally,balanced) configuration. For $d\le 3$, the only possible (RE) are planar, butin dimension four it is possible to get truly four dimensional (RE).  A classical problem in celestial mechanics aims at relating the(in-)stability properties of a (RE) to the index properties of the central (or,more generally, balanced) configuration generating it. In this paper, weprovide sufficient conditions that imply the spectral instability of planar andnon-planar (RE) in $\mathbb R^4$ generated by a central configuration, thusanswering some of the questions raised in \cite[Page 63]{Moe14}. As acorollary, we retrieve a classical result of Hu and Sun \cite{HS09} on thelinear instability of planar (RE) whose generating central configuration isnon-degenerate and has odd Morse index, and fix a gap in the statement of\cite[Theorem 1]{BJP14} about the spectral instability of planar (RE) whose(possibly degenerate) generating central configuration has odd Morse index.  The key ingredients are a new formula of independent interest that allows tocompute the spectral flow of a path of symmetric matrices having degeneratestarting point, and a symplectic decomposition of the phase space of thelinearized Hamiltonian system along a given (RE) which is inspired by Meyer andSchmidt's planar decomposition \cite{MS05} and which allows us to rule out theuninteresting part of the dynamics corresponding to the translational and(partially) to the rotational symmetry of the problem.

Spectral stability, spectral flow and circular relative equilibria for the Newtonian $n$-body problem