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Equation 79 · Part 5 · A Horizon Is a Toll Booth, Not a Loophole

Symbol r^3

atidal(r)≈2GMr3 ℓ,a_{\rm tidal}(r) \approx \frac{2GM}{r^3}\,\ell,
r3r^3

What this part means

r3r^3 occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Its job in the formula

r3r^3 occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

The bill here is tidal, not electromagnetic or mechanical, and it is the one place this construction is bounded directly by what has actually been observed rather than by an idealization. A rigid processor of size ℓ\ell falling radially experiences a stretching tidal acceleration between its two ends of approximately atidal(r)≈2GMr3 ℓa_{\rm tidal}(r) \approx \frac{2GM}{r^3}\,\ell. 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…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the surrounding passage

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