# 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.

[1]

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
[1] are not necessarily obvious and therefore this discussion will question
some of these ideas. The introduction of [1] 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:

[2]

The appearance of the [G/c^{2}] factor in [2] can be explained
in terms of the use of SI units, rather than geometric units, and if
we make [r=R] in [2] then we might have the rationale for the [Mω/r]
proportionality being suggested in [1] and the extract above, i.e.

[3]

However, if we return to [2], 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 [ω_{D}] 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 [2] cannot simply be cancelled,
as seems to be implied in [3]. At face value, this appears to put one of the fundamental
assumptions in [1] into some doubt and we still need to understand why
the form of [1] is so different to [2], if they are both thought to
quantify the frame dragging angular velocity [ω_{D}] of
spacetime caused by some central rotating mass [M]. So, at this point,
we shall turn our attention to the (1*0) expression in [1], 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 [ω_{D}] is the sum of a local rotating
system [m_{S}] and the net total of all the other stars in the
visible universe [m_{N}]:

[4]

The reason for presenting the equation above in the form of a hybrid of [2] and [3] is because it highlights an interesting aspect of the mass-to-radius ratio in [4], 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 [m
_{N}/r_{N}]
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 [ρ_{C}]
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:

[5]

Now in the case of the Rouke model, the local rotating system is
a black hole, sized in the order of 10^{12} solar masses, where
the radius is linked to its Schwarzschild radius [Rs]. If we now repeat
[5] with these black hole assumptions:

[6]

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.

[7]

Of course, it is not really a coincidence why [5] and [6] equate to unity, while [7] does not, although it is possibly easier to see in [6]. The mass of a black hole is directly proportional to its Schwarzschild radius [Rs]:

[8]

Based on [8], 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 [6]. While this relationship is not so obvious in [5], 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 [7] reflects a much lower value. However, there are a couple of points in [6] and [8] that might be questionable in terms of the mass-radius of a black hole, when considered in terms of geometric units, e.g.

[9]

The idea of geometric mass [Μ] is measured in metres, as defined in [9], 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 [2] and [3], which may undermine a key assumption in [1]. In order to better understand the physical premise of [1], 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:

[10]

Based on the analysis above, we might now re-write [4] in the following form:

[11]

At this point, we might recognise that [11] is similar to the numerator in [1], such that it not clear where the denominator included in [1] is sourced, i.e.

[12]

The implied numeric solution in [12] is based on the figures used
in [6], while [10] 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 [ω_{N}], in [11],
is assumed to be zero, which aligns to the (1*0) expression in [1].
So based on all the comments above, we might re-write [1] in the following
equivalent form of [11], where the leading [1/2] reflects the numeric
weighting shown in [12]:

[13]

However, while we have attempted to rationalise the form of [1] in terms of MKS units, the problem cited in [2] and [3] still remains, i.e.

[14]

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 [ω_{D}] is to be determined. If we
consider [14] in terms of a frame dragging velocity [v_{D}= ω_{D*}r], then [14a] would appear to suggest that the velocity [v_{D}]
remains constant, irrespective of the radial distance [r], while [14b]
suggests that the velocity [v_{D}] will fall as an inverse square
of distance.

While the rotation velocity [v_{D}] 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 [v_{D}] 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.

Footnotes:

*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 [v*_{O}] 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 [v_{D}] 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 form.*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.*