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Fixed-angular-momentum stability at 2s, 3s and 4s

Physical scope

Worked parameter values for the conditional Schwarzschild fixed-angular-momentum radial model; not full orbital stability or an observation.

Example

Use $s > 0$ and the reduced potential $W_{ℓ} (r) = (1 - s / r) (c^{2} + ℓ^{2} / r^{2})$ of Lyapunov stability of Schwarzschild circular orbits in the fixed-angular-momentum radial system. At a circular orbit, $ℓ^{2} = c^{2} s r_{0}^{2} / (2 r_{0} - 3 s)$ .

Stable radial equilibrium at $r_{0} = 4 s$ . The circular value is $ℓ^{2} = 16 c^{2} s^{2} / 5$ , and

$W_{ℓ}^{''} (4 s) = \frac{2 c^{2} s (4 s - 3 s)}{(4 s)^{3} (8 s - 3 s)} = \frac{c^{2}}{160 s^{2}} > 0.$

The fixed- $ℓ$ radial equilibrium is Lyapunov stable.
Unstable radial equilibrium at $r_{0} = 2 s$ . Here $ℓ^{2} = 4 c^{2} s^{2}$ and

$W_{ℓ}^{''} (2 s) = \frac{2 c^{2} s (2 s - 3 s)}{(2 s)^{3} (4 s - 3 s)} = - \frac{c^{2}}{4 s^{2}} < 0.$

The fixed- $ℓ$ equilibrium is unstable.
Degenerate endpoint at $r_{0} = 3 s$ . Here $ℓ^{2} = 3 c^{2} s^{2}$ and $W_{ℓ}^{'} (r) = c^{2} s (r - 3 s)^{2} / r^{4}$ . It has a stationary inflection rather than a strict extremum, and the radial equilibrium is not Lyapunov stable by the one-sided perturbation argument in the theorem.

Facts & Assumptions

Given: $s, c > 0$ , the Schwarzschild radial potential, and the fixed-angular-momentum stability theorem.

[F1]

At a timelike circular orbit $r_{0} > 3 s / 2$ , $ℓ^{2} = c^{2} s r_{0}^{2} / (2 r_{0} - 3 s)$ and $V_{ℓ}^{''} (r_{0}) = c^{2} s (r_{0} - 3 s) / (r_{0}^{3} (2 r_{0} - 3 s))$ for $V_{ℓ} = W_{ℓ} / 2$ (Circular timelike geodesics, their radial-potential stability threshold, and the photon sphere at r = 3s/2 clause (c)).

[F2]

For the fixed- $ℓ$ radial system, radii $r_{0} > 3 s$ are Lyapunov stable and radii $3 s / 2 < r_{0} \leq 3 s$ are not (Lyapunov stability of Schwarzschild circular orbits in the fixed-angular-momentum radial system).

Verification

technique · substitute each radius in the exact circular angular-momentum and potential-curvature formulas, then apply the radial stability theorem

1.1givenF1F2algebra

At $r_{0} = 4 s$ , substitution gives $ℓ^{2} = c^{2} s (16 s^{2}) / (8 s - 3 s) = 16 c^{2} s^{2} / 5$ and [F1] gives $W_{ℓ}^{''} = 2 V_{ℓ}^{''} = c^{2} / (160 s^{2}) > 0$ . Since $4 s > 3 s$ , [F2] classifies the equilibrium as Lyapunov stable in the fixed- $ℓ$ radial state.

1.2givenF1F2algebra

At $r_{0} = 2 s$ , substitution gives $ℓ^{2} = 4 c^{2} s^{2}$ and $W_{ℓ}^{''} = 2 V_{ℓ}^{''} = - c^{2} / (4 s^{2}) < 0$ . Since $3 s / 2 < 2 s < 3 s$ , [F2] classifies this equilibrium as unstable.

2.1givenF1F2algebra∎

At $r_{0} = 3 s$ , substitution gives $ℓ^{2} = 3 c^{2} s^{2}$ . Direct differentiation of $W_{ℓ} (r) = (1 - s / r) (c^{2} + ℓ^{2} / r^{2})$ at this fixed angular momentum gives $W_{ℓ}^{'} (r) = c^{2} s (r - 3 s)^{2} / r^{4}$ , positive away from $3 s$ and zero at $3 s$ . The equilibrium is a stationary inflection; [F2] classifies it as not Lyapunov stable. All three results are restricted radial-model classifications.

Fixed-angular-momentum stability at 2s, 3s and 4s

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources