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Equation 81 · Part 1 · A Horizon Is a Toll Booth, Not a Loophole

=

ℓ=1 cm\ell=1\,\mathrm{cm}
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

which at the horizon itself scales as ℓ\ell c6c^6/(4G2G^2M2M^2) — falling as the inverse square of the mass. For a processor of laboratory size ℓ\ell=1\,cm\mathrm{cm} at the horizon of a black hole with the mass measured for the progenitors of the first LIGO gravitational-wave detection, around 30\,M⊙M_\odot , this tidal stress is already about 1.1×\times10^5\,m s−2\mathrm{m\,s^{-2}} , on the order of ten thousand Earth gravities across a single centimeter — enough to disassemble ordinary hardware well before the horizon is reached [ 16 ] . For the Sagittarius A* mass used above, the same formula gives roughly 6×\times10^{-6}\,m s−2\mathrm{m\,s^{-2}} across the same centimeter: negligible, and consistent with the…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

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