Lyapunov stability of Schwarzschild circular orbits in the fixed-angular-momentum radial system

By [F1], the reduced vector field is $F_{\mathrm{\ell }}(r,w)=(w,-W_{\mathrm{\ell }}^{\mathrm{\prime }}(r)\mathrm{/}2)$ on the open set $(s,\mathrm{\infty })\times \mathbb{R}$. It is smooth there and [F4] gives a unique local solution through every state; uniqueness lets one glue all solutions through the same initial state on the union of their intervals to obtain its maximal solution. The change of variables $(q,v,T)=(r\mathrm{/}s,w\mathrm{/}c,c\tau \mathrm{/}s)$ gives $dq\mathrm{/}dT=v$ and $dv\mathrm{/}dT=-s{W}_{\mathrm{\ell }}^{\mathrm{\prime }}(sq)\mathrm{/}(2c^2)$ on $(1,\mathrm{\infty })\times \mathbb{R}$. This invertible scaling preserves maximality, compact containment and existence for all nonnegative time. Differentiating $W_{\mathrm{\ell }}=f(c^2+{\mathrm{\ell }}^2\mathrm{/}{r}^2)$ gives $W_{\mathrm{\ell }}^{\mathrm{\prime }}=c^{2}s\mathrm{/}{r}^2-2{\mathrm{\ell }}^2\mathrm{/}{r}^3+3s{\mathrm{\ell }}^2\mathrm{/}{r}^4$; substituting the circular value of ${\mathrm{\ell }}^2$ from [F2] yields $W_{\mathrm{\ell }}^{\mathrm{\prime }}(r_c)=0$, so $F_{\mathrm{\ell }}(r_c,0)=0$. Along every solution, $H_{\mathrm{\ell }}(r,w):=\frac{1}{2}(w^2+W_{\mathrm{\ell }}(r))$ satisfies $d{H}_{\mathrm{\ell }}\mathrm{/}d\tau =w(-W_{\mathrm{\ell }}^{\mathrm{\prime }}\mathrm{/}2)+(W_{\mathrm{\ell }}^{\mathrm{\prime }}\mathrm{/}2)w=0$. Thus $H_{\mathrm{\ell }}$ is constant on each solution interval.

Suppose $r_c>3s$. By [F2], $W_{\mathrm{\ell }}^{\mathrm{\prime }\mathrm{\prime }}(r_c)>0$, so continuity gives an interval about $r_c$ on which $W_{\mathrm{\ell }}^{\mathrm{\prime }\mathrm{\prime }}>0$. Since $W_{\mathrm{\ell }}^{\mathrm{\prime }}(r_c)=0$, the derivative is negative to the left and positive to the right there; hence $W_{\mathrm{\ell }}$ has a strict local minimum. Fix any dimensionless phase-space radius $\epsilon >0$ whose closed ball about $(r_c\mathrm{/}s,0)$ lies in $(1,\mathrm{\infty })\times \mathbb{R}$. Choose a length $0<\eta <s\epsilon \mathrm{/}2$ so $[r_c-\eta ,r_c+\eta ]\subset (s,\mathrm{\infty })$ and both boundary values of $W_{\mathrm{\ell }}$ exceed $W_{\mathrm{\ell }}(r_c)$. By continuity of $H_{\mathrm{\ell }}$, choose $0<\delta \le \epsilon$ so every state at distance $d<\delta$ from $z_c$ has $\mathrm{\mid }r-r_c\mathrm{\mid }<\eta \mathrm{/}2$, energy below both boundary potential halves, and $H_{\mathrm{\ell }}-W_{\mathrm{\ell }}(r_c)\mathrm{/}2<c^2{\epsilon }^2\mathrm{/}8$. Conservation in 1.1 prevents such a solution from reaching either radial boundary, since there $H_{\mathrm{\ell }}\ge W_{\mathrm{\ell }}(r_c\pm \eta )\mathrm{/}2$. Inside the resulting radial interval $W_{\mathrm{\ell }}(r)\ge W_{\mathrm{\ell }}(r_c)$, so $w^2\mathrm{/}2=H_{\mathrm{\ell }}-W_{\mathrm{\ell }}(r)\mathrm{/}2<c^2{\epsilon }^2\mathrm{/}8$ and $\mathrm{\mid }w\mathrm{\mid }\mathrm{/}c<\epsilon \mathrm{/}2$. The state stays in a compact subset of the open domain; [F5] rules out a finite forward maximal endpoint. With $\mathrm{\mid }r-r_c\mathrm{\mid }\mathrm{/}s<\eta \mathrm{/}s<\epsilon \mathrm{/}2$ and $\mathrm{\mid }w\mathrm{\mid }\mathrm{/}c<\epsilon \mathrm{/}2$, it remains inside the prescribed phase-space ball for every $\tau \ge 0$. This is Lyapunov stability by [F3].

Suppose $3s\mathrm{/}2<r_c<3s$. By [F2], $W_{\mathrm{\ell }}^{\mathrm{\prime }\mathrm{\prime }}(r_c)<0$; shrink $\eta >0$ so $W_{\mathrm{\ell }}^{\mathrm{\prime }}(r)<0$ on $(r_c,r_c+\eta ]$. For any proposed initial-state radius $\delta >0$, choose $r_0\in (r_c,r_c+\min \{\eta \mathrm{/}2,s\delta \mathrm{/}2\})$ and initial data $(r_0,0)$. [F4] supplies a local solution. Its acceleration is $-W_{\mathrm{\ell }}^{\mathrm{\prime }}(r_0)\mathrm{/}2>0$, and conservation of $H_{\mathrm{\ell }}$ gives, on the outgoing branch, $w=\sqrt{W_{\mathrm{\ell }}(r_0)-W_{\mathrm{\ell }}(r)}$. The travel-time integral to $r_c+\eta$ is ${\int }_{r_0}^{r_c+\eta }[W_{\mathrm{\ell }}(r_0)-W_{\mathrm{\ell }}(r){]}^{-1\mathrm{/}2}dr$, whose only endpoint singularity is bounded by a constant times $(r-r_0{)}^{-1\mathrm{/}2}$ because $W_{\mathrm{\ell }}^{\mathrm{\prime }}(r_0)<0$. Before reaching that radius, the energy identity bounds $w$ on the compact radial interval $[r_0,r_c+\eta ]$, so the state stays in a compact subset of $(s,\mathrm{\infty })\times \mathbb{R}$ near any finite maximal endpoint. The compact-continuation result [F5] therefore prevents finite-time termination before the finite travel time; the solution reaches $r_c+\eta$ and leaves the fixed $d$-ball of dimensionless radius $\eta \mathrm{/}s$. Since $d((r_0,0),z_c)=\mathrm{\mid }{r}_0-r_c\mathrm{\mid }\mathrm{/}s$ can be made smaller than every proposed $\delta$, this contradicts [F3].

At $r_c=3s$, the circular relation gives ${\mathrm{\ell }}^2=3c^2{s}^2$. Differentiating $W_{\mathrm{\ell }}(r)=c^2(1-s\mathrm{/}r)+{\mathrm{\ell }}^2(r^{-2}-s{r}^{-3})$ at this fixed $\mathrm{\ell }$ gives $W_{\mathrm{\ell }}^{\mathrm{\prime }}(r)=c^{2}s(r-3s{)}^2\mathrm{/}{r}^4$ and $W_{\mathrm{\ell }}^{\mathrm{\prime }\mathrm{\prime }}(r)=-2c^{2}s(r-3s)(r-6s)\mathrm{/}{r}^5$. Thus $W_{\mathrm{\ell }}^{\mathrm{\prime }}$ is positive except at $3s$ and $W_{\mathrm{\ell }}^{\mathrm{\prime }\mathrm{\prime }}$ changes sign there, so the equilibrium is a stationary inflection. Fix $0<\eta <s$ and, for any $\delta >0$, choose $r_0\in (3s-\min \{\eta \mathrm{/}2,s\delta \mathrm{/}2\},3s)$ with initial data $(r_0,0)$. [F4] supplies a local solution. The acceleration is negative. On the inward branch energy conservation yields $w=-\sqrt{W_{\mathrm{\ell }}(r_0)-W_{\mathrm{\ell }}(r)}$ for $3s-\eta \le r<r_0$, since $W_{\mathrm{\ell }}$ is strictly increasing on that interval. The travel-time integral to $3s-\eta$ is finite; near $r_0$ its integrand is bounded by a constant times $(r_0-r{)}^{-1\mathrm{/}2}$ because $W_{\mathrm{\ell }}^{\mathrm{\prime }}(r_0)>0$. The energy identity bounds $w$ while $r$ stays in the compact interval $[3s-\eta ,r_0]\subset (s,\mathrm{\infty })$, so [F5] extends the solution past any finite maximal endpoint before its finite travel time. It therefore reaches $3s-\eta$, leaving the fixed $d$-ball of dimensionless radius $\eta \mathrm{/}s$. Since $d((r_0,0),z_c)=\mathrm{\mid }{r}_0-r_c\mathrm{\mid }\mathrm{/}s$ can be made smaller than every proposed $\delta$, [F3] says the equilibrium is not Lyapunov stable.

The two subcritical cases in 2.2–2.3 are not Lyapunov stable, while the supercritical case in 2.1 is Lyapunov stable. These are deductions under the adopted Schwarzschild free-fall model [A1] and concern only its fixed-angular-momentum radial subsystem.