Coordinates and turning points

Coordinates and conventions follow Expiscating the black hole singularity in holography.

Here d is the boundary dimension; d = 4 and 6 correspond to bulk AdS₅ and AdS₇.

Quantity	Notation	Meaning
Coordinates	$r = radius squared / rplus squared; u = 2/(1+r)$	B: u = 0; outer horizon H₊: u = 1; origin S (neutral) or O (RN): u = 2.
Frequency phase	$vartheta = arg fraktur w$	Frequency rotates at fixed physical ζ. The finite phase offsets illustrate the lateral resolutions.
Primary turning points	$r_j$	Compact sphere tags j₀, j₁, …; fixed during frequency rotation.
Core turning points	$b_i$	Compact tags b₀, b₁, …; the additional boundary turning point is separate.
Neutral core variables	$neutral core scaling relations$	Resolve the core cluster near S.

Neutral core

The limiting core equation and its zero-momentum root labels are

$neutral core polynomial$

$rhat_a(0) = exp[-pi i (2a+1)/(d-1)]$

For the displayed neutral examples, |𝔴| = 12, ν = 3/4 and ζ̂₀² = 0.1. At fixed physical ζ,

$zetahat squared = 0.1 exp[i d vartheta/(d-1)]$

RNAdS₅ core

At zero phase b₀ is the real core root, while b₁ and b₂ are the upper and lower roots in the squared-radius coordinate. These labels continue across the displayed phases. O is ordinary for the charged differential. The parameters are |𝔴| = 24, ν = 1.3, ϱ = 0.4 and ζ = 0.1085844617.

Primary roots and horizons

Planar primary labels follow the root formula in Expiscating the black hole singularity in holography. For a global complex pair j₀ is the upper member in the squared-radius plane; for an exterior real pair it is the larger root. Global d = 6 uses j₁ for the negative real root. RN uses j₀,j₁ for the upper/lower complex pair and j₂ for the real root.

H₊ is the outer horizon and H₋ the RN inner horizon. H_c is the negative squared-radius RN horizon. In the neutral views H₁ (d = 4) and H₁,H₂ (d = 6) denote the other horizons; the latter form a complex-conjugate pair. For d = 4, H₁ lies at coordinate infinity.

Spheres and continuation

The spheres compactify the base coordinate, rather than the two-sheeted WKB spectral curve. The full and magnified views show the same finite-frequency core graph. Solid and faint dashed segments distinguish front and rear hemispheres.

The red contour Γ(t) = t − 0.025 i sin(πt) runs from B to H₊; it is a continuation contour, not a Stokes trajectory. Crossings C₁,C₂,… are ordered along it. General graphs use phases −0.08,0,+0.08; neutral core graphs additionally include ±0.02.