Equation 64 · Part 10 · The Bit Comes Back Before the Bearing

=

t_* \;\sim\; \frac{\beta}{2\pi}\,\ln S, \qquad \beta \equiv \frac{\hbar}{k_BT_H} = \frac{8\pi GM}{c^3},

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.

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The passage around this formula

The natural timescale for $t_L$ is the scrambling time, and it is worth deriving in real units rather than only in the abstract. Sekino and Susskind conjectured that black holes are the fastest scramblers nature permits, mixing information across all their degrees of freedom in a time growing only logarithmically with entropy [ 2 ] . Maldacena, Shenker, and Stanford later proved the sharp version of the bound this conjecture leans on: the exponential growth rate of an out-of-time-order correlator, the operational signature of chaos, cannot exceed $\lambda_L$ $\le$ 2 $\pi$ / $\beta$ for a thermal system at inverse temperature $\beta$ , with a black hole horizon saturating the bound [ 3 ] . Combining a…

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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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