Equation 24 · Part 2 · A Horizon Is a Toll Booth, Not a Loophole

Symbol τ

d\tau

\tau

What this part means

a time, the integral has action units.

Its job in the formula

τ is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Full expressionSymbol τArticle 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

$N_\perp$ [ $\Gamma$ ] 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. It is invariant under reparametrizing $\Gamma$ ’s own curve parameter, since only proper time enters, and invariant under any passive coordinate change of the ambient spacetime, since both d $\tau$ and a comoving-frame energy are scalars. It is not invariant under boosting the local tetrad used to evaluate $\langle$ H( $\tau$ ) $\rangle$ — a different local observer at the same event assigns a different local energy — which is why…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the article section

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