Measure of gravitational acceleration at a horizon, determining Hawking temperature.Surface gravity is the gravitational acceleration measured at a horizon, such as the event horizon of a black hole. In the context of black hole thermodynamics, it plays a crucial role: the surface gravity κ is directly proportional to the Hawking temperature, T = κ / (2π), meaning that a black hole with higher surface gravity radiates more intensely. This quantity bridges geometry and thermodynamics, showing that black holes are not just gravitational objects but also thermal systems with entropy and temperature.
The concept of surface gravity in black hole physics was formalized in the 1970s by Stephen Hawking and Jacob Bekenstein, who established the analogy between black hole mechanics and thermodynamics, with κ playing the role of temperature.
For a Schwarzschild black hole, surface gravity is κ = c⁴ / (4 G M), inversely proportional to mass. It represents the acceleration a stationary observer at the horizon would need to resist the pull of the black hole. In more general spacetimes, surface gravity is defined via the Killing horizon, and it determines the rate at which quantum vacuum fluctuations near the horizon become real particles, leading to Hawking radiation.
Surface gravity is used in theoretical astrophysics to predict black hole evaporation rates and in numerical relativity to simulate black hole mergers and their gravitational wave signatures.
For a black hole of one solar mass, the surface gravity is about 10¹² m/s², yet the Hawking temperature is only 60 nanokelvin, far colder than the cosmic microwave background.
| δM | variable | Infinitesimal change in black hole mass — mass kilogram |
| κ | variable | Surface gravity of the black hole — surface gravity metre per second squared |
| δA | variable | Infinitesimal change in horizon area — area square metre |
| Ω_H | variable | Angular velocity of the horizon — angular velocity radian per second |
| δJ | variable | Infinitesimal change in angular momentum — angular momentum kilogram square metre per second |
| δ | operator | Infinitesimal variation of a quantity — variation |
| T_H | variable | Hawking temperature of the black hole — temperature kelvin |
| κ | variable | surface gravity of the black hole — surface gravity metre per second squared |
| ħ | constant | fundamental constant equal to h/(2π), relating angular frequency to energy — reduced planck constant = 1.054571817e-34 joule second |
| k_B | constant | fundamental constant relating temperature to energy — boltzmann constant = 1.380649e-23 joule per kelvin |















