Equation 71 · Part 8 · A Horizon Is a Toll Booth, Not a Loophole

=

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

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

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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 equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

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