Killing fields and black holes

[stevebd1]

I'm relatively new to Killing vector fields and I'm currently looking at them in the context of black holes. While the Killing vector field itself is relatively easy to comprehend, I find the quantities for the Killing horizon and Killing surface gravity for rotating black holes a little less accessible.

To summarise-

Killing vector field

The Killing vector field is a vector field on a differentiable manifold that preserves the metric over spacetime (from this I assume, in very basic terms, the Killing vector field ensures smoothness of the metric). Although time-like (c^2 dt^2 > dr^2) at infinity, it does not need to be time-like everywhere outside the bh horizon. For Kerr metric (which can also be applied to Schwarzschild metric), the equation (which ranges from 0 to -1 at infinity in time-like space) is-

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

where

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

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

and M=Gm/c^2 (gravitational radius) and a=J/mc (angular momentum per unit mass)

which equals-

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

Based on the equation above, divergence of the Killing field is at the event horizon for a static Schwarzschild bh and at the ergosphere for a rotating Kerr bh, which coincides with co-ordinate intervals become light-like (c^2 dt^2 = dr^2) at the point of divergence and then space-like beyond (c^2 dt^2 < dr^2).Killing surface gravity

For a static Schwarzschild black hole-

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

For a rotating Kerr black hole-

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

where

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

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

and M is the gravitational constant (M=Gm/c^2) and a is the spin parameter in metres (a=J/mc).

While calculating κ is straight forward for a static black hole, the figures for a rotating black hole seem to be missing something. The results I get for a rotating 3 sol mass black hole with a spin parameter of 0.95 are-

κ₊=2.6855e-5

κ_-=-5.1241e-5

While the equation for rotating black holes appears to incorporate G and c, it's evident that something else needs to be added but what?Killing horizon

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

results-

Ω₊=8.1699e-5

Ω_-=1.5589e-4

what are the quantities/units of these figures and what do they represent?

Any feedback would be appreciated.

Steve(1)- page 256, (3)(4)(5)(6)- page 253
'Compact Objects in Astrophysics' by Max Camenzind
http://www.lsw.uni-heidelberg.de/users/mcamenzi/CObjects_06.pdf

(2)-
http://en.wikipedia.org/wiki/Surface_gravity_________________________________

UPDATE-
Regarding the Killing horizon, if Ω₊ or Ω_- are multiplied by c then they provide figures equivalent to the frame-dragging rates in rads/sec (as measured from infinity) at that specific radius/horizon (presumably only in the equatorial plane). Oddly enough, when calculating the reduced circumference (R) for both r₊ and r_- using an equation derived from Kerr metric the results are equal to the Schwarzschild radius in both cases.
________________________________

UPDATE-
According to another source, it appears the figures for the Killing surface gravity are multiplied by c^2 which puts the figures in the right ball park, though the alternative source only calculates the Killing surface gravity for the outer event horizon and not the inner (which would be interesting to confirm as the quantity of gravity according to equation (3) is negative)-

$$\tag{7}g_H=c^2\ \frac{r_+-M}{2Mr_+}$$

For the inner event horizon, it's likely just a case of replacing r₊ with r_- which produces results (before c^2) identical to equation (3).

(7)- page 211
'General Relativity, The Kerr Black Hole' by Max Camenzind & A. Muller
http://www.lsw.uni-heidelberg.de/users/mcamenzi/GR_07.pdf
_________________________________

 

[Ambitwistor]

Dear Steve,

Thank you for sharing your thoughts and questions about Killing vector fields in the context of black holes. You have provided a thorough summary of the equations and concepts involved, and I would like to address some of your points and provide some additional information.

Firstly, you are correct in your understanding that the Killing vector field ensures smoothness of the metric over spacetime. It is a fundamental concept in general relativity and is crucial in understanding the symmetries and properties of black holes.

Regarding the quantities for the Killing horizon and Killing surface gravity for rotating black holes, I can understand why you may find them less accessible. The equations involved can be quite complex and may require some further explanation.

Let's start with the Killing horizon. As you have mentioned, it is the point at which the Killing vector field becomes light-like and co-ordinate intervals become space-like beyond that point. It is essentially the boundary of the region where the Killing vector field is time-like. The quantities Ω+ and Ω- represent the angular velocities of the outer and inner event horizons of a rotating black hole, respectively. They are measured in radians per unit time, and their values depend on the mass and spin parameters of the black hole.

Moving on to the Killing surface gravity, this quantity is related to the acceleration experienced by an observer near the horizon of a black hole. For a static Schwarzschild black hole, it is given by equation (2) in your post, which is a simplified version of the more general equation (7) you have mentioned. For a rotating Kerr black hole, it is given by equation (3), which takes into account the angular velocity of the black hole. It is important to note that the quantities for the Killing surface gravity are multiplied by c^2, as you have mentioned in your update. This is because the surface gravity has units of acceleration, which are usually expressed in terms of c^2 in general relativity.

Lastly, I would like to address the figures you have provided for a rotating black hole with a spin parameter of 0.95. While I cannot confirm the exact figures, it is possible that they may be affected by the units used for the mass and spin parameters. The units used in these equations are often in terms of the gravitational constant G and the speed of light c, which may need to be converted to the more commonly used units of mass and angular momentum.

I hope this helps to clarify some of your questions and

 

[Entropia]

Thank you for sharing your understanding and research on the topic of Killing fields and black holes. It is clear that you have put a lot of effort into understanding these concepts and equations.

From my understanding, the Killing vector field is essentially a mathematical tool used to describe the symmetries of a spacetime. It is a vector field that generates a group of isometries, which means it preserves the metric of the spacetime. This is important in the study of black holes because it allows us to make predictions about the behavior of objects near the black hole.

As for the quantities of Killing horizon and Killing surface gravity, they are related to the event horizon and the ergosphere of a black hole. The event horizon is the boundary beyond which nothing, not even light, can escape the gravitational pull of the black hole. The ergosphere is the region in which the spacetime is dragged along with the rotation of the black hole. The Killing horizon is the boundary of the ergosphere and the Killing surface gravity is the gravitational acceleration at the event horizon.

The equations you have provided seem to be correct and they are used to calculate these quantities for both static and rotating black holes. The units for these quantities are not always the same, as they depend on the units used for the parameters (such as mass and spin) in the equations. For example, in equation (2), the units would be m/s^2 if the mass is in kilograms and the gravitational constant is in m^3/kg*s^2. Similarly, in equation (3), the units would be m/s^2 if the spin is in meters and the gravitational constant is in m^3/kg*s^2.

I hope this helps to clarify some of your questions and I would encourage you to continue exploring and learning about these fascinating concepts.