= Solution
The distinction between the evolution equation and the constraint is essential. For strings alone, the exact equations reduce to
$$
\ddot a=0,\qquad\dot a^2=\frac{8\pi G}{3}C_s-K.
$$
Linearizing the acceleration equation gives $\delta a=A+Bt$. At the exactly tuned static $C_s$, however, the full <Friedmann equation> requires $\dot a^2=0$, so $B=0$ for an admissible exact perturbation.
A small nonzero velocity $v$ can be admissible if its initial density also changes:
$$
C_s=C_E+\frac{3v^2}{8\pi G},\qquad
\delta\rho(t_E)=\frac{3v^2}{8\pi Ga_E^2},\qquad
\boxed{a(t)=a_E+v(t-t_E).}
$$
Arbitrarily small $v$ and the corresponding second-order density change therefore produce a secular departure. There is no <Lyapunov stability> against all nearby constrained initial data for the <static closed cosmic-string universe>. Neutral radius shifts at fixed normalization and drift when the normalization changes are different perturbation classes.
Back to article page
