Surface gravity
The surface gravity of a Killing horizon is the acceleration, as exerted at infinity, needed to keep an object at the horizon. Mathematically, if is a suitably normalized Killing vector, then the surface gravity is defined by - ,
where the equation is evaluated at the horizon. For a static and asymptotically flat spacetime, the normalization should be chosen so that as , and so that . For the Schwarzschild solution, we take to be the time translation Killing vector , and more generally for the Kerr-Newman solution we take , the linear combination of the time translation and axisymmetry Killing vectors which is null at the horizon, where is the angular velocity.
Examples
The Schwarzschild solution The surface gravity for the Schwarzschild solution with mass is - .
The Kerr-Newman solution
The surface gravity for the Kerr-Newman solution is - ,
where is the electric charge, is the angular velocity, we define to be the locations of the two horizons and .
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