Surface gravity

The surface gravity κ of a black hole is the acceleration, as measured by a distant observer, of a test particle held static just outside the event horizon. Because the proper acceleration needed to hover diverges at the horizon while the gravitational redshift factor vanishes there, the product — the force per unit mass measured at infinity — remains finite, and that finite quantity is κ. For a non-rotating Schwarzschild black hole of mass M it equals κ = c⁴/4GM, inversely proportional to the mass: smaller holes have larger surface gravity.

Surface gravity is the temperature variable of black-hole thermodynamics. The 1973 Bardeen–Carter–Hawking theorems established that κ is constant everywhere over the horizon of any stationary black hole — even a rapidly spinning Kerr hole with strong frame-dragging — which is the exact analogue of the zeroth law of thermodynamics, that temperature is uniform throughout a body in equilibrium. The first law of black-hole mechanics, dM = (κ/8πG)dA + ΩdJ + ΦdQ, identifies the term (κ/8πG)dA with the heat term T dS once Hawking's 1974 result supplies the temperature T = ħκ/2πk_Bc.

Surface gravity also encodes the third law: κ falls to zero only in the extremal limit (maximal spin or charge), and that state cannot be reached in any finite sequence of physical operations, mirroring the unattainability of absolute zero. The same κ sets the Unruh-like temperature of the horizon and, through the formula T = ħκ/2πk_Bc, determines the spectrum of Hawking radiation — making surface gravity the single quantity that ties together the hole's mechanics, its thermodynamics, and its quantum emission.

Shown to be constant over a stationary horizon (the zeroth law) by Bardeen, Carter, and Hawking in 1973; promoted from a mechanical quantity to a genuine temperature by Hawking's 1974 radiation calculation via T = ħκ/2πk_Bc.