Frame dragging around a rotating black hole.

[Imax]

Hi:

I’m a new member to this forum.

I’m looking at the behavior of frame dragging around a rotating black hole. It seems to me that frame dragging should be zero about a rotating black hole’s axis of rotation, but should be at a maximum at the equator (i.e. 90 degrees from its axis of rotation). I’m having problems with r, the distance from a reference center, the black hole’s singularity.

Imax

 

[stevebd1]

Oddly enough, the frame dragging rate at the event horizon (as observed from infinity) is constant over the entire event horizon, even at the poles-

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

The above equation applying to the event horizon only where $r_\pm=M\pm\sqrt{(M^2-a^2)}$, M=Gm/c and a=J/mc (multiply $\Omega_H$ by c to get SI units of rad/s).

What reduces (observed from infinity) relative to moving from the equator to the pole is the centripetal acceleration brought on by the frame dragging where $a_c=\Omega_H^2 R$ where R is the reduced circumference ($R=\Sigma \sin \theta/\rho$).

As you're already probably aware, the equation for the frame dragging rate where $r\neq r_\pm$ is-

$$\omega=\frac{2Mra}{\Sigma^2}$$

where-

$$\] \Sigma^2=(r^2+a^2)^2-a^2\Delta \sin^2\theta\\ \\ \Delta= r^{2}+a^{2}-2Mr\\ \\ \rho^2=r^2+a^2 \cos^2\theta\\ \[$$

All the above applies for as observed from infinity, if you want the local rates, then you need to divide by the redshift- $\alpha=\rho \sqrt{(\Delta)}/\Sigma$.Regarding the actual location of the singularity in a rotating black hole seems to be an area of contention (though it seems accepted that it exists at r=0 and $\theta=\pi/2$), the diagrams in the following papers might shed some light-

http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.0622v3.pdf page 24

http://arxiv.org/PS_cache/gr-qc/pdf/0411/0411060v2.pdf page 11

http://casa.colorado.edu/~ajsh/phys5770_08/bh.pdf page 35

 

[stevebd1]

A number of papers also use http://en.wikipedia.org/wiki/Elliptic_coordinate_system" which give some coordinate properties for the ring singularity. Using a standard Cartesian coordinate system, x and y can be defined using the following-

$$x=\sqrt{r^2+a^2}\sin \theta$$

$$y=r \cos \theta$$

where the outer and inner event horizon would be $r_\pm=M\pm\sqrt{(M^2-a^2)}$ and the outer and inner ergospheres would be $r_{e\pm}=M\pm\sqrt{(M^2-a^2\cos^2 \theta)}$. Using these coordinates, the ring singularity has a coordinate radius of $a$ while the singularity still resides at r=0 and $\theta=\pi/2$. This implies there are r<0, a new region of space (or extended Kerr geometry) as suggested in these links-

http://jila.colorado.edu/~ajsh/insidebh/waterfall.html (scroll down)

http://arxiv.org/PS_cache/gr-qc/pdf/9702/9702060v1.pdf

Attached are two images for a black hole with a spin parameter of a/M=0.95. The top image uses regular spherical coordinates while the bottom image uses elliptical coordinates and defines the ring singularity.

 

[Imax]

Hi stevebd1:

Thanks for the Matt Visser info.

I’m looking at frame dragging between the outer ergosurface and outer event horizon (r2), and frame dragging occurring outside the outer ergosurface (r1). Each point in these regions can be a function of r and θ. I’m trying to make a model consisting of several surfaces, where frame dragging is constant along each surface.

Imax

 

[stevebd1]

On a side note, it's often stated that space rotates at c at the ergosphere, this can be proved for various $\theta$ using the above equations-

tangential velocity brought on by frame dragging (observed from infinity)-

$$v_t=\omega R$$

local tangential velocity brought on by frame dragging (divide by redshift)-

$$v_t=\frac{\omega R}{\alpha}$$

which can be rewritten-

$$v_t=\frac{2Mra}{\rho^2 \sqrt{\Delta}}\sin \theta$$

if r is substituted with $r_{e\pm}=M\pm\sqrt{(M^2-a^2\cos^2 \theta)}$, the ergosphere boundary, then v_t will always equal 1 (i.e. speed of light), irrespective of $\theta$.

 

[Imax]

What happens around a sphere with radius r = 2 r+ = 2 M + 2 √(M^2 - a^2) ?

 

[AEM]

Stevebd1, do you have a good reference for the derivation of the equations that describe frame dragging? Is this the same as the Lense-Thirring effect?

 

[Chronos]

Gravity Probe B meticulously examined frame dragging effects. See
http://www.nasa.gov/mission_pages/gpb/index.html
And yes, it is related to the lense-thirring effect:
http://www.nasa.gov/vision/earth/lookingatearth/earth_drag.html
Gravity Probe B was a more exacting study.

 

[Imax]

I’m trying to build an isobar analogue, a contour map that describes constant frame dragging.

 

[stevebd1]

Stevebd1, do you have a good reference for the derivation of the equations that describe frame dragging? Is this the same as the Lense-Thirring effect?

Hello AEM

The frame dragging equations are derived from Kerr metric so you would probably need to look at how Kerr metric is derived, though the papers linked to in post #2 and the following paper might be of some use-

http://pisces.as.utexas.edu/GenRel/KerrMetric.pdf

 

[George Jones]

Stevebd1, do you have a good reference for the derivation of the equations that describe frame dragging?

Interesting discussion of this: Chapter 13, Kerr Geometry, from General Relativity: An Introduction for Physicists, by Hobson, Efstathiou, and Lasenby; and Chapter 3, Rotating Black Holes, from Black Holes: An Introduction, by Raine and Thomas. Both books have interesting and useful exercises.

Is this the same as the Lense-Thirring effect?

Yes.

 

[AEM]

Thanks to all for the references on frame dragging.

I have another question. Consider the very massive black holes believed to be at the center of most spiral galaxies. If they were rotating rapidly would the frame dragging effect alter the orbital dynamics of stars in orbit around the black hole?

 

[George Jones]

Consider the very massive black holes believed to be at the center of most spiral galaxies. If they were rotating rapidly would the frame dragging effect alter the orbital dynamics of stars in orbit around the black hole?

I don't think observable stars are close enough for orbital effects of frame dragging to be observed. There is, however, some evidence for frame dragging of the accretion disk around the black hole at the centre of our galaxy,

http://arxiv.org/abs/astro-ph/0310821.

The article's penultimate paragraph explains how frame dragging (possibly) explains the observations.

There is a good chance that, within a decade or so, we should be able use microwaves to "image" the black hole at the centre of our galaxy. See

http://www.scientificamerican.com/article.cfm?id=portrait-of-a-black-hole

http://arxiv.org/abs/astro-ph/0607279.

These images should show strong frame dragging effects. Very exciting.

 

[stevebd1]

To get some idea of the effect that frame dragging has on orbits, you can look at the Kepler equation for an object in stable orbit in Kerr metric-

$$\Omega_s=\frac{\pm\sqrt{M}}{r^{3/2}\pm a\sqrt{M}}$$

where $\Omega_s$ is the angular velocity, $\pm$ denotes prograde and retrograde orbits, $M=Gm/c^2$ and $a=J/mc$.

Though it's likely than any star in close orbit around a rotating black hole will have a wildly elliptical orbit (see the video below), the above shows that an object rotating with the black hole can orbit much closer than one that is in retrograde orbit. For the record, the supermassive black hole at the core of our galaxy is predicted to have a spin of a/M=0.44-

http://arxiv.org/abs/0906.5423

video of stars orbiting SMBH at our galactic centre- http://astro.uchicago.edu/cosmus/projects/UCLA_GCG/

 

[AEM]

Thank you, Steve and George.

 

[Imax]

Hi Stevedb1 :

Thanks for the link:

http://pisces.as.utexas.edu/GenRel/KerrMetric.pdf

It’s a good read. I find the math problematic. Whenever I try solving anything, I always get a quartic of the form:

A*R^4 + B*R^3 + C*R^2 + D*R + E = 0

Messy stuff. Can these equations be simplified by assuming that some parameters are those of our Milky Ways supermassive black hole:

a/M = 0.44±0.08.

M =4.2 ± 0.4 ×10^6 the mass of our Sun?

Imax

 

[stevebd1]

Hi Imax

You would work in geometric units where $M=Gm/c^2$ and $a=J/mc$ where G is the gravitational constant, m is mass, c is the speed of light and J is angular momentum. In the case of a/M=0.44 you can simply write a=0.44M to obtain a. You can then put these quantities into the equation in post #14 which will provide you with the angular velocity in geometric units, multiply by c to get SI units of rad/s.

For the record, what were you looking to work out with your first equation.

regards
Steve

 

[Imax]

Hi Steve :

With my first equation I was trying to describe frame dragging, but I was trying to describe it as a scalar quantity. Frame dragging depends on tangent velocity v, and an inward force along gravitational lines, and on upwards movement of a particles. It's math in 3 D. Messy

Imax
.

 

[Imax]

The reason why I’m asking about frame dragging is because I’m trying to make a video of what it would be like to travel from Earth to a quasar. Imagine leaving Earth and traveling outside our Milky Way galaxy. Go trough inter galactic space to a nearby quasar, another galaxy with an active galactic center. Travel through stars and molecular clouds and get close to the suppermassive black hole driving the quasar. This is what I have near the black hole, minus background stellar field.



It’s not realistic, because I’ve neglected frame dragging.

Imax