the Planck scale
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Take the three constants that anchor the three great pillars of physics, the speed of light c (relativity), Planck's constant hbar (quantum mechanics), and Newton's gravitational constant G (gravity), and ask: can you combine them into a length, a time, a mass, an energy? You can, and the answer is essentially unique. Those combinations define the Planck scale, the natural yardstick nature itself seems to pick out, and the frontier where all three theories must be used at once.
The Planck length is l_P = sqrt(hbar G / c^3), about 1.6 x 10^-35 metres; the Planck time is t_P = l_P / c, about 5.4 x 10^-44 seconds; the Planck mass is m_P = sqrt(hbar c / G), about 2.2 x 10^-8 kilograms; and the Planck energy is E_P = m_P c^2, about 1.2 x 10^19 GeV. These are built from constants alone, with no reference to any particular particle or force, which is why they are considered fundamental. At these scales the quantum fluctuations of spacetime geometry become as large as the geometry itself, so a classical smooth spacetime ceases to make sense.
The Planck scale matters because it marks where quantum gravity becomes unavoidable: below the Planck length, the very notions of distance and duration are expected to lose their ordinary meaning, and both general relativity and quantum field theory are expected to fail. It is staggeringly far from experiment: the Planck energy is about a quadrillion (10^15) times higher than the energies reached at the Large Hadron Collider, which is why the physics of this scale is probed by thought, mathematics, and cosmology rather than by accelerators. Note that the Planck mass, unlike the length, is not tiny, roughly the mass of a flea's egg; it is huge for a particle, which is part of the hierarchy problem.
The Large Hadron Collider reaches about 10^4 GeV; the Planck energy is about 10^19 GeV. To reach Planck-scale collisions with the same technology you would need an accelerator larger than the solar system, which is why we cannot probe quantum gravity directly.
The Planck energy sits a quadrillion times beyond our biggest collider.
Planck units are natural in that they use only c, hbar, and G, but the factors of order one (and the choice of hbar versus h, or 8 pi G) are conventions; the Planck scale marks roughly, not exactly, where quantum gravity takes over. The Planck mass is large for a particle, not small, which is itself a puzzle.