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I am interested in the issue of frame dragging used in a number of galactic rotation models. However, I wanted to first make sure that I have a better understanding the relativistic implications of frame dragging. While the issue of galactic rotation is not the subject of this thread, it is possibly useful to mention that the standard galactic model appears to assume a central black surrounded by a accretion layer, which is often collectively described as the active galactic nucleus (AGN) from which the typical spiral arms extend. Now the only point of this detail is to highlight that AGN appear to extend out many 1000’s of times beyond the Schwarzschild radius [Rs] of the black hole (BH). If so, then it is assumed that the relativistic effects in terms of both velocity and gravitation within the spiral arms may be minimal. Therefore, in the current context, the following PF discussions are only provided as a general cross reference:

PF frame dragging reference

black holes, frame dragging and the effect on gravity

While many discussion of frame dragging are anchored in the Kerr metric of a rotating black hole, it seems that the angular velocity of the dragged frame can be presented in the following form, although this assumption may be incorrect:

[1] [itex] \omega_D = k \left( \frac{G}{c^2} \right) \left( \frac{M \omega_{(BH)}}{R}

\right) [/itex]

The parameter [k] is often described in terms of some density distribution, which I think Lense-Thirring originally quantified as 4/3 for a spherical distribution, which might not be applicable for a disk galaxy. The [G/c^2] is the gravitation constant and speed of light that essentially act as a conversion factor for MKS units. The parameter [M] is assumed to be the mass of the central black hole rotating at an angular velocity [w]. For the purpose of approximate calculations, the mass [M] of a galactic black hole can be estimate in millions of solar masses, which then determines the radius [Rs]:

[2] [itex]Rs = \frac{2GM}{c^2}[/itex]

Again, the angular velocity [w] of the black hole itself can be estimated by calculating the circumference and dividing by a maximum tangential velocity up to [0.99c]; although the actual figure is not directly relevant at this stage. Therefore, it would seem that the frame dragging angular velocity is inversely proportional to the distance [R] from the BH. However, in order to convert the frame drag angular velocity into a linear tangential velocity, it is assumed that the relationship [w=v/r] applies, such that [1] would appear to become:

[3] [itex] v_D = k \left( \frac{G}{c^2} \right) \left(M \omega_{(BH)} \right)[/itex]

While [3] seem right and aligns to the correct units of velocity, i.e. m/s, the implication appears to be that this tangential drag velocity will remain constant irrespective of distance!

So my first question is whether this is right?

My second issue is really predicated on the answer to the question above, but on the assumption that the extreme relativistic effects of the BH do not extend out beyond the AGN, then a stable orbit would only require a classical rotational velocity to counteract the pull of gravity, i.e.

[4] [itex]F= \frac{mv_O^2}{r}= \frac{GMm}{r^2}[/itex]

Re-arranging, this velocity falls with the inverse square-root of distance:

[5] [itex]v_O=\sqrt{ \frac{GM}{r}}[/itex]

The implication of [3] and [5], as they stand, seems to suggest that the constancy of the frame dragging velocity in [3] must eventually exceed the orbital velocity in [5], such that it would sweep stars within the spiral arm to ever greater distance. Something seems intuitively wrong with this description, but I am not sure where and would therefore appreciate any clarifications or insights. Thanks.

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# Frame Dragging

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