Equation 70 · A Horizon Is a Toll Booth, Not a Loophole
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Symbol g
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superscript
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To make this concrete rather than asymptotic, take the mass of Sagittarius A* as measured by the Event Horizon Telescope, M410^6\, , giving a Schwarzschild radius 1.1810^{10}\, and a horizon-scale acceleration factor GM/3.8010^{6}\, [ 14 ] . At a clearance of just one kilometer above the horizon — a distance smaller than a millionth of the horizon’s own radius — f8.4610^{-8} and the required proper acceleration is already a1.310^{10}\, , more than a billion Earth gravities. Even at a clearance of 1.810^6\, — about fifteen percent of the horizon’s own…
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To make this concrete rather than asymptotic, take the mass of Sagittarius A* as measured by the Event Horizon Telescope, M410^6\, , giving a Schwarzschild radius 1.1810^{10}\, and a horizon-scale acceleration factor GM/3.8010^{6}\, [ 14 ] . At a clearance of just one kilometer above the horizon — a distance smaller than a millionth of the horizon’s own radius — f8.4610^{-8} and the required proper acceleration is already a1.310^{10}\, , more than a billion Earth gravities. Even at a clearance of 1.810^6\, — about fifteen percent of the horizon’s own radius, a distance that sounds generous — the acceleration is still about 10^6\,g . No structure holds together under a sustained load anywhere near that scale; this is not an engineering inconvenience to be solved by better materials, it is the support-energy term diverging exactly where the naive argument wanted its subsidy.
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