Frame Dragging Model
This model equates to the description in the Rouke-Mackay papers that has been referenced on several occasions throughout this overview. In many respect, the use of these papers has mainly been to provide a framework for some of the many and varied hypotheses within this branch of astrophysics, rather than by way of any implicit support of this model.
In fact, understanding the premise on which the Rouke model is built is not an easy task as some of the fundamental equations do not appear to be anchored in any obvious physics and possibly compounded by the abstraction of geometric or natural units, e.g.
The equation above is said to represent the net rotation of a local frame of reference, i.e. a galaxy, and is the starting point for an explanation of the observed galactic rotation curves. The form of the equation is said to be anchored to Mach’s principle and the inertial drag caused by a ‘heavy’ rotating central mass, i.e. a super massive black hole. However, as indicated, the physical arguments underpinning  are not necessarily obvious and therefore this discussion will question some of these ideas. The introduction of  is explained in the following terms:
|“The concept of inertial frame near the origin is correlated with the rotation of the central mass and frames will tend to rotate in the same sense. The ‘simple dimensional considerations’ show that this frame dragging effect, which from now on we refer to as inertial drag, is proportional to [M ω/r], where [M, ω] are mass and angular velocity of the central mass and [r] is distance from the origin. The effects for all the masses in the universe need to be added with appropriate weighting.”|
As such, we are left to resolve how this description of frame dragging is being linked to the proportionality [Mω/r]. Based on an earlier discussion of ‘Mach’s Principle and Frame Dragging’, the following weak-field formulation was derived:
The appearance of the [G/c2] factor in  can be explained in terms of the use of SI units, rather than geometric units, and if we make [r=R] in  then we might have the rationale for the [Mω/r] proportionality being suggested in  and the extract above, i.e.
However, if we return to , it would seem that [r] and [R] are two different measures of distance. The general implication is that [r] is simply some arbitrary radius from the centre of mass for which the frame dragged angular velocity [ωD] can be calculated. In contrast, [R] is the radius of mass [M], which is rotating at angular velocity [ω]. As such, the radius [R] is a fixed value for some given mass [M], while radius [r] is a variable, which implies that [R] and [r] in the numerator and denominator of  cannot simply be cancelled, as seems to be implied in . At face value, this appears to put one of the fundamental assumptions in  into some doubt and we still need to understand why the form of  is so different to , if they are both thought to quantify the frame dragging angular velocity [ωD] of spacetime caused by some central rotating mass [M]. So, at this point, we shall turn our attention to the (1*0) expression in , which is explained in the following terms:
|“The local inertial frame rotates with respect to distant galaxies by the weighted sum of ω weighted kM/r and 0, for all the distant ‘stationary’ galaxies, weighted Q say. We can normalise the weighting so that Q=1, which is the same as replacing k=Q by k, which leaves just one constant k to be determined by experiment or theory.”|
In the current context, it is believed the main idea of Mach’s principle might be interpreted in the following basic form, where the frame dragging angular velocity [ωD] is the sum of a local rotating system [mS] and the net total of all the other stars in the visible universe [mN]:
The reason for presenting the equation above in the form of a hybrid of  and  is because it highlights an interesting aspect of the mass-to-radius ratio in , which is cited in the following extract:
|“There is some evidence that k is in fact 1. This follows from the observation that the sum [mN/rN] over all masses [m] in the universe at distance [r] is approximately 1, which makes the choice of normalised weighting [Q=1], the same as weighting purely by mass. This suggests a deep property of spacetime, namely that a third concept of ‘mass’, i.e. the inertial drag mass, is the same as the other two, i.e. (gravitational mass and ordinary inertial mass.”|
We might test this idea by substituting an estimate for the mass of the universe based on the critical energy density [ρC] multiplied by the volume of the visible universe, as defined by the Hubble radius [c/H]. Based on these assumptions, the result below would appear suggest that the unity approximation holds true:
Now in the case of the Rouke model, the local rotating system is a black hole, sized in the order of 1012 solar masses, where the radius is linked to its Schwarzschild radius [Rs]. If we now repeat  with these black hole assumptions:
Clearly, there is some aspect of both the visible universe and a black hole that leads to this unity result. However, we can get an initial reality check by repeating the process one last time, but now substituting the value of our own star, i.e. the sun.
Of course, it is not really a coincidence why  and  equate to unity, while  does not, although it is possibly easier to see in . The mass of a black hole is directly proportional to its Schwarzschild radius [Rs]:
Based on , we can see that the Schwarzschild radius [Rs] is directly proportional to the mass [M] of a black hole, which leads to the unity result in . While this relationship is not so obvious in , in principle, the same effect exists within the parameters used, because the Hubble radius [c/H] has some attributes of a Schwarzschild radius [Rs]. In the case of the sun, the Schwarzschild radius [Rs] is much smaller than its physical radius, which is why  reflects a much lower value. However, there are a couple of points in  and  that might be questionable in terms of the mass-radius of a black hole, when considered in terms of geometric units, e.g.
The idea of geometric mass [Μ] is measured in metres, as defined in , and leads to the idea that the physical mass of a black hole might be linked to [Rs/2], not [Rs]. However, having highlighted this issues, we need to return to a more fundamental issue rooted in the logic of  and , which may undermine a key assumption in . In order to better understand the physical premise of , it might be useful to point out that the expressions in [5,6,7] reduce to a number with no units, when the geometric units are replaced by normal MKS units, such that we can show the apparent equivalence of the Rouke expression as follows:
Based on the analysis above, we might now re-write  in the following form:
At this point, we might recognise that  is similar to the numerator in , such that it not clear where the denominator included in  is sourced, i.e.
The implied numeric solution in  is based on the figures used in , while  implies that the result [1/2] aligns to a numeric weighting without any associated units. It is also highlighted that the rotation of the visible universe [ωN], in , is assumed to be zero, which aligns to the (1*0) expression in . So based on all the comments above, we might re-write  in the following equivalent form of , where the leading [1/2] reflects the numeric weighting shown in :
However, while we have attempted to rationalise the form of  in terms of MKS units, the problem cited in  and  still remains, i.e.
As already outlined, [14a] does not equate to [14b], as the parameter [R] corresponds to radial size of the black hole, which is directly proportional to its mass [M] and therefore a fixed value. In contrast, the radial distance [r] is a coordinate variable at which the frame drag angular velocity [ωD] is to be determined. If we consider  in terms of a frame dragging velocity [vD= ωD*r], then [14a] would appear to suggest that the velocity [vD] remains constant, irrespective of the radial distance [r], while [14b] suggests that the velocity [vD] will fall as an inverse square of distance.
|While the rotation velocity [vD] of a black hole might approach the speed of light [c] as the Schwarzschild radius [Rs] is approached, the standard equation [14b] would suggest that the frame dragging velocity [vD] would quickly fall as an inverse square function of larger radii, such that it would seem to become negligible effect outside the most inner region of the galaxy, .|
Although, frame dragging may not extend far into the galaxy, it may still be the source the initial rotational velocity. In this respect, the mass distribution model might then explain the rotation curve without any excessive frame dragging effects in the outer regions. However, given that some aspects of the Rouke model may have been simply misunderstood, further analysis will be put on hold with the footnotes below reflecting on some of the complexities surrounding this subject.
- Irrespective of the actual formulation implied
by the mass distribution within a galaxy, it is still true to say that
a mass [m] within the galaxy will require a rotational velocity [vO]
to counteract the inward pull of gravity. Observational measurement
suggest that this orbital velocity is in the order of 250,000 m/s, which
although much, much higher than the implied frame dragging velocity
[vD] in the outer disk, this velocity still has to be acquired
from within the rotational dynamics of the galaxy itself. In this respect,
the Rouke model may provide a possible explanation of the spin of the
accretion layers around the central hole, which then imparts the acquired
angular momentum into the bulge and spiral arms regions in which stars then subsequently
- In order to agree with observations, it is normally assumed that accretion disk models have to support a mechanism by which angular momentum is redistributed. As such, if matter is to fall inwards, it must lose not only gravitational energy, but also angular momentum. Since the total angular momentum of the disk, as a whole, has to be conserved, the angular momentum lost by any mass falling into the centre has to be balanced by the angular momentum gained by the effective mass further from the centre, i.e. angular momentum has to be transported outwards for matter to accrete. However, the actual details of this transport mechanism still seem to be an issue of much debate, which extend far beyond the scope of this review.