The surface gravity κ of a Killing horizon is the acceleration, as exerted at infinity, needed to keep an object at the horizon. Mathematically, if *k*^{a} is a suitably normalized In mathematics, a Killing vector field is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. Killing fields are named for Wilhelm Killing. Specifically, a vector field...
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 *k*^{a} to be the time translation In mathematics, a Killing vector field is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. Killing fields are named for Wilhelm Killing. Specifically, a vector field...
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 Ω is the angular velocity.
## Examples ### The Schwarzschild solution The surface gravity for the Schwarzschild solution with mass *M* is - .
### The Kerr-Newman solution The surface gravity for the Kerr-Newman solution is - ,
where *Q* is the electric charge, *J* is the angular velocity, we define to be the locations of the two horizons and *a*: = *J* / *M*. |