$$\langle H(\tau)\rangle$$

Equation 28 · Evolutionary Physics

A Horizon Is a Toll Booth, Not a Loophole

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

$N_\perp$[$\Gamma$] 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…

Meanings in this article

• $\tau$: a time, the integral has action units.

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