math lessons - 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 $k a$ is a suitably normalized Killing vector, then the surface gravity is defined by

$k^a \nabla_a k^b = \kappa k^b$ ,

where the equation is evaluated at the horizon. For a static and asymptotically flat spacetime, the normalization should be chosen so that $k^a k_a \rightarrow -1$ as $r\rightarrow\infty$ , and so that $\kappa \geq 0$ . For the Schwarzschild solution, we take $k a$ to be the time translation Killing vector $k^a\partial_a = \frac{\partial}{\partial t}$ , and more generally for the Kerr-Newman solution we take $k^a\partial_a = \frac{\partial}{\partial t}+\Omega\frac{\partial}{\partial\phi}$ , 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

$\kappa = \frac{1}{4M}$ .

The Kerr-Newman solution

The surface gravity for the Kerr-Newman solution is

$\kappa = \frac{r_+-r_-}{2(r_+^2+a^2)} = \frac{\sqrt{M^2-Q^2-J^2/M^2}}{2M^2-Q^2+2M\sqrt{M^2-Q^2-J^2/M^2}}$ ,

where $Q$ is the electric charge, $J$ is the angular velocity, we define $r_\pm := M \pm \sqrt{M^2-Q^2-J^2/M^2}$ to be the locations of the two horizons and $a : = J / M$ .

Mathematics Encyclopedia and Lessons

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Surface gravity

Examples

The Schwarzschild solution

The Kerr-Newman solution

Mathematics Encyclopedia and Lessons

Lessons

Popular Subjects

References

Surface gravity

Examples

The Schwarzschild solution

The Kerr-Newman solution

Popular
Subjects