Surface gravity ( $κ$ )

The surface gravity

κ

measures the failure of a horizon-generating Killing vector field to be affinely parametrized. For a static metric

d s^{2} = - f (r) d t^{2} + f (r)^{- 1} d r^{2} + \dots

with a simple horizon at

r = r_{+}

,

κ = ∣ f^{'} (r_{+}) ∣/2

.

Table of contents

Signed surface gravity of a de Sitter horizon ( $κ_{future} = 1/ R, κ_{past} = - 1/ R$ )

For the future-directed central-time Killing vector field in a static patch of de Sitter spacetime, the outgoing null chart gives the future branch coefficient

+ 1/ R

in

\nabla_{k} k = κk

. The ingoing chart gives

- 1/ R

on the past branch. Both follow from

d (k^{2}) = - 2 κ k^{♭}

; the cross-term sign changes the horizon covector. The physical Hawking temperature uses

∣ κ ∣

, not a signed negative temperature.

Surface gravity of a static spherical horizon ( $κ = \frac{1}{2} A^{'} (r_{+}) B^{'} (r_{+})$ )

For a static spacetime with

g_{tt} = - A (r)

,

g_{rr} = 1/ B (r)

and simple simultaneous horizon zeros, the squared surface gravity of

\partial_{t}

is

lim B A^{'2} / (4 A) = A^{'} (r_{+}) B^{'} (r_{+}) /4

. Choose the positive value for the outer horizon. A rescaling of the time generator rescales its surface gravity.

Surface gravity of a five-dimensional charged black hole ( $κ_{\pm} = \pm 2 M^{2} - Q^{2} / r_{\pm}^{3}$ )

The time translation is normalized at infinity. For the roots

r_{\pm}^{2} = M \pm M^{2} - Q^{2}

, evaluate

V^{'} (r) /2

to get the signed horizon values. The outer Hawking temperature in natural units is

M^{2} - Q^{2} / (π r_{+}^{3})

. The extremal limit has zero surface gravity; the neutral case has only the outer positive-radius horizon and temperature

1/ (2 π 2 M)

.

Surface gravity from the Killing derivative ( $κ^{2} = - \frac{1}{2} (\nabla_{a} K_{b}) (\nabla^{a} K^{b})$ )

At a regular Killing horizon, the generator is hypersurface orthogonal and satisfies

\nabla_{K} K = κ K

. The Killing derivative contraction identity gives the displayed squared surface gravity, including through a bifurcation surface by continuity. The generator's normalization is part of this formula.

Affine horizon-generator scaling ( $ξ = κλk$ )

If a Killing horizon generator

ξ

has constant nonzero surface gravity

κ

, write

ξ = F (λ) k

with

k = d / d λ

affine. The equation

\nabla_{ξ} ξ = κ ξ

implies

d F / d λ = κ

, so shifting the affine origin gives

ξ = κλk

.

Schwarzschild surface gravity ( $κ = 1/ (4 M)$ )

For a Schwarzschild black hole of mass

M

in geometrized units,

f = 1 - 2 M / r

gives

κ = f' (2 M) /2 = 1/ (4 M)

. The Killing vector field is normalized at infinity; changing that normalization changes

κ

.

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Surface gravity by

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Surface gravity refers to the gravitational acceleration experienced at the surface of a celestial body, such as a planet, moon, or star. It is a measure of how strongly the body attracts objects toward its center due to gravity. Surface gravity can be calculated using the formula: \[ g = \frac{G \cdot M}{r^2} \] Where: - \( g \) is the surface gravity, - \( G \) is the gravitational constant (\(6.

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