Equation 71 · Part 10 · A Horizon Is a Toll Booth, Not a Loophole

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T_H=\hbar c^3/(8\pi GMk_B)\approx1.5\times10^{-14}\,\mathrm K

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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

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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

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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