Killing Vector Field & Surface Gravity in Kerr BH Explained

Killing Vector Field

The Killing vector field is a vector field on a differentiable manifold that preserves the spacetime metric. In other words, it generates an isometry of the metric. Although the Killing vector that corresponds to time translation is timelike (c^2 dt^2 > dr^2) at spatial infinity, it need not be timelike everywhere outside a black hole (BH) horizon. For the Kerr metric (and in the non-rotating limit the Schwarzschild metric) the relevant metric component is

$$\tag{1} \kappa^2=g_{tt}=-\frac{\Delta-a^2\sin^2\theta}{\rho^2}$$

Kerr metric definitions

Here

$$\Delta= r^{2}+a^{2}-2Mr$$

$$\rho^2=r^2+a^2\cos^2\theta$$

and

M = G m / c^2 (the gravitational radius, i.e. mass expressed as a length) and a = J / m c (the angular momentum per unit mass, with dimensions of length).

Equation (1) can be written as

$$\kappa^2=-\left(1-\frac{2Mr}{r^{2}+a^{2}\cos^{2}\theta}\right)$$

From this expression, the Killing vector norm (g_tt) vanishes where coordinate time becomes null; for a static Schwarzschild black hole this occurs at the event horizon, while for a rotating Kerr black hole it vanishes at the ergosurface. Outside of that surface the coordinate time direction is spacelike (c^2 dt^2 < dr^2).

Killing Surface Gravity

Surface gravity (often denoted κ) has slightly different expressions in different contexts and unit conventions. In geometric units (G = c = 1) the common formulae are:

For a static Schwarzschild black hole:

$$\tag{2}\kappa=\frac{c^4}{4GM}$$

For a rotating Kerr black hole (outer/inner horizons):

$$\tag{3}\kappa_{\pm}=\frac{r_{\pm}-r_{\mp}}{2\left(r_{\pm}^2+a^2\right)}$$

where

$$\tag{4} r_+=M+\sqrt{M^2-a^2}$$

$$\tag{5} r_-=M-\sqrt{M^2-a^2}$$

Note: M here is the gravitational radius M = G m / c^2 (mass expressed in metres). The parameter a is the spin parameter (in metres) a = J / m c.

Numerical example and unit clarification

Using your example of a rotating 3-solar-mass black hole and spin parameter a = 0.95 (in geometrized units), you quoted:

κ_+ = 2.6855e-5

κ_- = -5.1241e-5

These numerical values are consistent with geometric units where κ has dimensions of inverse length (1/m). To convert to SI acceleration units (m/s^2) multiply κ by c^2. That is, κ_SI ≈ κ_geom × c^2. This matches the alternative source you found that multiplies by c^2.

Killing Horizon and Angular Velocity

The horizon angular velocity (the angular velocity of the Killing horizon) is

$$\tag{6}\Omega_H=\frac{a}{\left(r_{\pm}^2+a^2\right)}=\frac{1}{2} \frac{a}{M} \frac{1}{r_{\pm}}$$

You reported the numeric results:

Ω_+ = 8.1699e-5

Ω_- = 1.5589e-4

In geometric units these Ω values have dimensions of inverse length. To obtain angular frequency in radians per second (as measured at infinity) multiply Ω by c. In other words, Ω_SI (rad/s) ≈ Ω_geom × c.

Interpretation

• κ in geometric units is an inverse length; κ × c^2 gives a surface gravity in m/s^2 (acceleration).
• Ω_H in geometric units is an inverse length; Ω_H × c gives an angular velocity in rad/s (frame-dragging rate as measured from infinity, typically evaluated in the equatorial plane).

Updates and notes

UPDATE — on the reduced circumference: if you compute the reduced circumference R at r_+ and r_- using the usual Kerr expressions, you may find results that coincide with the Schwarzschild radius in the special limits; this can occur depending on which coordinate slice (equatorial plane, Boyer–Lindquist coordinates, etc.) is used. Be explicit about the coordinate definitions when comparing radii.

UPDATE — regarding the inner horizon (r_-): equation (3) yields κ_- negative in geometric units. This sign difference is a well-known feature of the inner (Cauchy) horizon; converting to SI units with c^2 preserves the sign but changes the units to acceleration.

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References

Steve

References cited in the original post:

• Max Camenzind, “Compact Objects in Astrophysics” — pages referenced by the original author: (1) page 256, (3)(4)(5)(6) page 253.
• Max Camenzind & A. Müller, “General Relativity, The Kerr Black Hole” — (7) page 211.
• Wikipedia — Surface gravity