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To wrap-up this mini-series on rotating black holes, I analyze the strong frame dragging near a rotating black hole. The rather formidable math is simplified somewhat by considering the equatorial plane only, thereby sacrificing generality, but not losing accuracy. The emphasis is on using the equations for calculations in order to get a feel for the magnitudes.
Figure 1 below shows, for ease of reference, a slightly modified diagram of the Kerr black hole picture from the original Blog article of this mini-series. Extreme frame dragging occurs near and inside the dotted circle representing the Schwarzschild radius of an equivalent non-rotating black hole.
Figure 1:
On the equatorial plane of a spinning black hole, the frame dragging can be expressed by the angular velocity (ω) of the spacetime as a function of the radial distance parameter r, the mass M and the spin parameter a of the hole. Put differently, ω is the angular rate, as observed from afar, by which a local observer, stationary relative to the spacetime in the vicinity of the hole, is being dragged around the hole.
Summary of the meanings and units of the parameters used:
Radial parameter2 (Δ) converts local distances to coordinate distances. Also see end-notes(1,3).
Figure 2 below shows how the spacetime is dragged around a spinning black hole in its equatorial plane. The spiraling radial lines are the paths of photons on the equatorial plane, aimed directly at the hole from afar. Photons move at the same local velocity in all directions (i.e., isotropic), hence the spiraling paths indicate that the local space is being dragged around the hole.
Figure 2: Credit:(1)
In order to make the equations more palatable, lets put in values and get some tangible answers. For our Sun, with M ≈ 1.99 x 1030 kg and J ≈ 1.63 x 1041 kg m2 s-1, lets first check how large the frame dragging will be on Earth's orbit, with r ≈ 1.5 x 1011 m. This gives:
a = J/(Mc) ≈ 273 m, Mbar ≈ 1470 m, Δ ≈ 2.32 x 1022 m2, giving
ω ≈ 7.16 x 10-20 radians per second, or ≈ 10-5 arcseconds per century.
As expected, this is negligible since we are still in the weak gravitational field environment and frame dragging falls off with roughly the cube of distance. To find out what will happen in a strong field, lets visit a relatively small black hole of 1.5 solar masses with the same angular momentum than our Sun. We will first check the frame dragging rate at rtest = 4Mbar ≈ 8800 m. This gives a ≈ 182 m, Δ ≈ 3.92 x 107 m2 and ω ≈ 350 radians per second, or a frequency of 56 Hz.
That's one whopping angular rate! Anything that hasn't got some means of propulsion will be dragged around the hole at this rate. However, this is not very fast as far as black holes are concerned. The fastest a black hole can spin is when a = Mbar (the extreme Kerr black hole), or a ≈ 2200 m for this case. At rtest = 8800 m in the equatorial plane, the frame dragging will be: ω ≈ 3876 radians per second, or 617 Hz.
It gets more severe as one goes down to the static limit, rstatic = 2Mbar ≈ 4400 m at the equator, giving a frame drag rate of ω ≈ 22600 radians per second, or 3.6 KHz. Inside rstatic, i.e., in the ergosphere, no amount of propulsion can prevent a spacecraft from being dragged around the black hole, because to do so would require a local speed through spacetime exceeding c.
It doesn't stop there. Frame dragging can theoretically be observed down to the outer horizon of this black hole, at router= Mbar ≈ 2200 m, yielding ω ≈ 67800 radians per second, or 10.8 KHz. Figure 3 shows the main frame dragging rates for a 1.5 solar mass black hole, spinning at the maximum allowable rate, a =Mbar.
Figure 3:
Interestingly enough, due to the gravitational redshift,(2) a distant observer would not observe an unpowered spacecraft at the static limit to whiz around the hole at c, but rather at about 0.4c. You can check this roughly by calculating ωr, but see end-note(3).
Finally, we will check the frame dragging magnitude experienced by that "GIGO-city"(4) with the clean energy. For this, we must find the radial distance of the city from the hole for the required "1g" of local gravity. When talking a low 1g gravity, we can easily find the distance with Newton's dynamics: r = √[GM/9.81] ≈ 4.5 x 109 m. This is less than one tenth of Mercury's distance from the Sun.
Even for an extreme Kerr hole, with all the other variables remaining the same, the frame dragging is ≈ 3 x 10-14 radians/sec, or some 20 arc-seconds per century. Not too bad at all - Earth "wobbles" more than that over the course of a century!
This concludes this mini-series on spinning black holes - and some heads are probably spinning too (including mine)! Now it's time for rants, raves or rotten tomatoes. I'll be grateful if someone would check my calculations - it's all too easy to goof at these sums!
Regards, Jorrie
(1) Equations are adapted for "engineering use" (i.e., straight SI units) from http://www.astro.ku.dk/~cramer/RelViz/text/exhib4/exhib4.html. The referenced website tends to use SI units and geometric units (where c=G=1) interchangeably and this can be very confusing, but it has great computer-generated graphics of frame dragging.
(2) The gravitational redshift factor (the ratio: frequency observed at r1 -› infinity, to frequency emitted at r) in the plane of the equator of a Kerr black hole is given by:
The other parameters are as defined in the text above. For an extreme Kerr black hole, the redshift factor approaches zero as r approaches Mbar. It's hard to see in the equation, but when spin parameter a = 0, it reduces to the Schwarzschild redshift factor: dτ/dt = √[1-2GM/(rc2)]. (Edit: Changed equation to be compatible with MTW, i.e., using dτ/dt in place of α (alpha), which has a different meaning there.)
(3) The radial parameter r is not quite a circumferential radius, where the circumference is 2Πr. Parameter r includes a spin or rotational component and is used to generalize the Schwarzschild radial distance for rotating space-times. It is defined in Boyd-Lindquist coordinates.
(4) The "GIGO-city" is discussed in this Blog entry.
-J
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