General-momentum Stokes graphs

Interactive companion to Expiscating the black hole singularity in holography.

Primary WKB turning points r subscript j · fixed ζ during frequency rotation vartheta = arg fraktur w

H₊ destination H₋ destination (RN) Other horizon Root / endpoint connection Γ: boundary → H₊ ○ j₀, j₁, …: primary WKB turning points · ◇ horizons · □ B: boundary · S: neutral origin · O: RN origin

ϑ = −0.08

Lower frequency tilt

Route detail in u

ϑ = 0

Real-frequency graph

Route detail in u

ϑ = +0.08

Upper frequency tilt

Route detail in u

Drag any sphere to rotate all three views. Rear hemispheres are faint and dashed. The u-chart menu switches between the transport region and all finite landmarks.

Turning-point names and coordinates
Pointru

Crossings along Γ

CrossingEmitter / armDestinationRe uIm u

Reading these graphs. Each primary WKB turning point, tagged j₀, j₁, …, emits three critical trajectories. B is a simple pole of the principal differential and also emits one trajectory. In neutral d = 4, S emits one. The neutral d = 6 origin S is ordinary for the principal differential; the RN origin O is ordinary for the charged differential.

Coordinates and frequency. r = r squared / rplus squared; u = 2/(1+r). Thus B is at u = 0, H₊ at u = 1, and the origin S (neutral) or O (RN) at u = 2. These are spheres of the base coordinate. The primary roots remain fixed as vartheta = arg fraktur w changes; frequency magnitude and scalar mass do not enter the principal graph. The ±0.08 tilts illustrate lateral resolutions at finite angles.

Continuation route. The same contour Γ(t) = t − 0.025 i sin(πt) is used on every lateral, from the boundary to the outer horizon. It bypasses real exterior turning points below and is not a Stokes trajectory. C₁, C₂, … are ordered along Γ. Selecting a crossing highlights the corresponding critical trajectory.

Momentum samples. The momentum samples are ζ = 0.35, 0.65 and 0.96. Global holes have r₊ = 1; RN has ϱ = r₋²/r₊² = 0.5. The global ζ = 0.96 samples have an exterior turning-point pair. The tags j₀, j₁, … identify the primary turning points. Planar indices follow the analytic root formula. The global and RN indexing convention is given on the notation page.

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