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Equation 73 · Part 3 · A Horizon Is a Toll Booth, Not a Loophole

Symbol T_H

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

What this part means

THT_H is an input to the expression that computes the quantity on the left.

Its job in the formula

THT_H is an input to the expression that computes 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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Sources cited in the surrounding passage

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