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Equation 71 · Part 6 · A Horizon Is a Toll Booth, Not a Loophole

Symbol k_B

TH=ℏc3/(8πGMkB)≈1.5×10−14 KT_H=\hbar c^3/(8\pi GMk_B)\approx1.5\times10^{-14}\,\mathrm K
kBk_B

What this part means

kBk_B is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

kBk_B is one of the signed contributions combined to compute the quantity on the left.

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 THT_H=ℏ\hbar c3c^3/(8π\pi GMkBk_B)≈\approx1.5×\times10^{-14}\,K\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, Tloc(r)T_{\rm loc}(r)=THT_H/f(r)\sqrt{f(r)} [ 11 ] . Setting TlocT_{\rm loc} equal to a demanding but realistic dilution-refrigerator base temperature of 10\,mK\mathrm{mK} and solving for the required clearance gives Δ\Delta r≈\approx2.8×\times10^{-14}\,m\mathrm m — a few nuclear radii from the horizon. For a supermassive hole, in other words, the ambient Hawking bath…

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