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Equation 76 · Part 4 · A Horizon Is a Toll Booth, Not a Loophole

superscript

Δr≈2.8×10−14 m\Delta r\approx2.8\times10^{-14}\,\mathrm m
superscript

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.

The passage around this formula

​ [ 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 stays colder than any laboratory cryostat until the processor is closer to the horizon than an atomic nucleus, twenty-odd orders of magnitude closer than the point at which the support-acceleration bill already becomes unpayable. Cooling is a genuine, non-negotiable line item — it is what ties this construction to horizon thermodynamics at all — but for this mass scale it is not…

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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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Sources cited in the surrounding passage

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