Equation 64 · Part 16 · The Bit Comes Back Before the Bearing

Denominator: k_BT_H

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

k_BT_H

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

$k_B$ $T_H$ occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Full expressionDenominator: k_BT_HArticle meaning

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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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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