Morris-Thorne throat profile r(ℓ) — the real funnel cross-section, plus light-ray bending
Throat
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Throat radius b₀
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Circumference 2πb₀
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Proper dist. ℓ (throat→edge)
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Through-throat length 2ℓ
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Embedding diagram: the shape function b(r) = b₀·(b₀/r)2n−1
sets how fast the throat flares outward. n = 1 gives the classic Morris-Thorne
solution r(ℓ) = √(b₀²+ℓ²), z(ℓ) = b₀·asinh(ℓ/b₀) — the
two-funnel “hourglass” joining two universes at the throat. Proper
distance ℓ is measured radially outward from the throat along the surface, not
straight through 3D embedding space.