The differential equation for φ as a function of u=M/r for a null geodesic with impact parameter b M is:
dφ/du = ±1/√[1/b^2 + (2u–1)u^2]
This is derived as eqn (21) on this page:
https://www.gregegan.net/PLANCK/Tour/TourNotes.html
but it is equivalent to Misner, Thorne and Wheeler’s eqn (25.27).
For b>3√3 light can’t enter the black hole, but for any b<3√3 this describes a possible null geodesic.
I haven’t been able to find an analytic expression for Δφ as u goes from 1/2 (i.e. the event horizon) to ∞ (the singularity) for the limit as b approaches 3√3 from below, but numerical integration seems to put it at around 2.63 radians.
Greg Egan
npub12c…gd38p
2025-10-27 15:03:00 UTC
in reply to nevent1q…0g2q
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"content": "The differential equation for φ as a function of u=M/r for a null geodesic with impact parameter b M is:\n\ndφ/du\t=\t±1/√[1/b^2 + (2u–1)u^2]\n\nThis is derived as eqn (21) on this page:\n\nhttps://www.gregegan.net/PLANCK/Tour/TourNotes.html\n\nbut it is equivalent to Misner, Thorne and Wheeler’s eqn (25.27).\n\nFor b\u003e3√3 light can’t enter the black hole, but for any b\u003c3√3 this describes a possible null geodesic.\n\nI haven’t been able to find an analytic expression for Δφ as u goes from 1/2 (i.e. the event horizon) to ∞ (the singularity) for the limit as b approaches 3√3 from below, but numerical integration seems to put it at around 2.63 radians.",
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