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What is the connection between Planck's constant and gravity? Why is the Planck scale the natural scale for quantum gravity? I would have though the scale would be related to G, not h.

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There is a difference between Planck's constant $h$ and the "Planck scale" as it is most often used colloquially, the latter referring to the Planck length, Planck time, and Planck mass. Planck's constant, as you note, has no direct connection to gravity and can be derived experimentally by studying black body spectra, as one example. The "Planck scale" involves gravity by design: it refers to the physical scales at which both general relativity and quanutum mechanics are relevant, and so involves both the gravitational constant $G$ and the Planck constant $h$.

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Can you specify the mathematical expression that combines h,g – Richardbernstein May 13 '12 at 2:23
Thank you; can you specify the mathematical expression that combines G h and c to give the Planck length, time, or mass? – Richardbernstein May 13 '12 at 2:24
Sure, the Plank length is $\sqrt{\hbar G / c^3}$, the Planck time is $\sqrt{\hbar G/c^5}$, and the Planck mass is $\sqrt{\hbar c / G}$. Note that these are the simplest ways to combine the fundamental constants $c$, $h$, and $G$ to give the units of length, time, and mass. Wikipedia offers some nice interpretations for what these scales might mean physically. By the way, $\hbar \equiv h/(2 \pi)$ – kleingordon May 13 '12 at 2:27

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