Equation 9 · Part 1 · A Horizon Is a Toll Booth, Not a Loophole

Symbol tau_perp

\tau_\perp \geq \frac{\pi\hbar}{2\left(\langle H\rangle - E_0\right)},

\tau_\perp

What this part means

a time, the integral has action units with subscript perp (dimensionless).

Its job in the formula

ta $u_p$ erp is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Full expressionSymbol tau_perpArticle meaning

Where the article explains it

$N_\perp$ [ $\Gamma$ ] is a real-valued, non-negative functional of a timelike worldline in a fixed background spacetime together with a stated local energy convention; because the integrand is an energy and d $\tau$ is a time, the integral has action units and $N_\perp$ is dimensionless.

The passage around this formula

The baseline is the quantum speed limit. For a system with a fixed, time-independent Hamiltonian H evolving from a state $\lvert$ $\psi$ $\rangle$ toward any state orthogonal to it, the minimum time required is bounded below by the tight, unified Margolus-Levitin/Mandelstam-Tamm result of Levitin and Toffoli: $\tau_\perp \geq \frac{\pi\hbar}{2\left(\langle H\rangle - E_0\right)}$ . where $\langle$ H $\rangle$ is the mean energy in the evolving state and $E_0$ is the ground-state energy of H [ 2 , 1 ] . Inverting this gives a maximum rate of orthogonal state changes per unit proper time, 2( $\langle$ H $\rangle$ - $E_0$ )/ $\pi$ $\hbar$ , and Giovannetti, Lloyd, and Maccone showed that for a periodic evolution this rate integrates cleanly: a system with mean energy $\langle$ …

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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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Sources cited in the article section

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