Equation 73 · Part 7 · A Horizon Is a Toll Booth, Not a Loophole

subscript

T_{\rm loc}(r)=T_H/\sqrt{f(r)}

subscript

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

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The passage around this formula

Staying cold is the second cost, and here the numbers run the other way. The Hawking temperature of Sagittarius A* at infinity is $T_H$ = $\hbar$ $c^3$ /(8 $\pi$ GM $k_B$ ) $\approx$ 1.5 $\times$ 10^{-14}\, $\mathrm K$ [ 10 ] . A static observer at radius r sees this blueshifted by the same Tolman factor that governs any locally measured temperature in a static spacetime, $T_{\rm loc}(r)$ = $T_H$ / $\sqrt{f(r)}$ [ 11 ] . Setting $T_{\rm loc}$ equal to a demanding but realistic dilution-refrigerator base temperature of 10\, $\mathrm{mK}$ and solving for the required clearance gives $\Delta$ r $\approx$ 2.8 $\times$ 10^{-14}\, $\mathrm m$ — a few nuclear radii from the horizon. For a supermassive hole, in other words, the ambient Hawking bath…

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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 surrounding passage

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