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Encyclopedia > Surface gravity

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 ka 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 ka 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.


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