A novel hypothesis for explaining the orbiting speeds of stars in spiral galaxies without requiring dark matter or modified gravity.  Suggests that the speeds are consistent with Kepler’s Law if the correct reference frame for understanding their dynamics is influenced by the vast amount of rotating matter in the local galaxy.  The hypothesis has many interesting implications and readily lends itself to experimental and observational tests. 

Introduction

For stars in the disc of a spiral galaxy, the acceleration due to gravity at orbital radius r is GM/r² where M is the collective mass of the matter inside a sphere of radius r, and G is the gravitational constant.  

Using a suitable reference frame, Kepler’s Law says that for the stars to be in their observed orbits they should have a centrifugal acceleration equal to v²/r, where v is their tangential speed.  Hence

GM/r²  = v²/r     (1)

In the late 1960s and early 1970s Vera Rubin, a young astronomer at the Department of Terrestrial Magnetism at the Carnegie Institution of Washington, worked with a new sensitive spectrograph that could measure spectral red shifts in the light from stars in the discs of spiral galaxies to a high degree of accuracy.  At a 1975 meeting of the American Astronomical Society Rubin announced that the stars in the discs of a spiral galaxy all travel at roughly the same speed as each other, instead of decreasing in proportion to the inverse square root of r as predicted by Kepler’s Law.  A few years later Rubin presented her results in an influential paper (Rubin, Thonnard, & Ford, 1980). 

Although initially met with skepticism, the results have been confirmed by data gathered from tens of thousands of stars in thousands of galaxies, including those able to be observed in detail by space based telescopes e.g. (M. Persic, 1996).

But why were stars in the discs of spiral galaxies travelling so fast?  Kepler’s Law and Newtonian dynamics work exquisitely well within the Solar System.  No-one expected the dynamical relationship a galactic scale to be any different.  

Refer to equation (1).  Rubin’s evidence showed that v was increasing too fast as r increased.  The implication was that M must increase at a faster rate in order for the equation to remain true. This soon developed into a hypothesis that there must be an enormous amount of hitherto undetected dark matter in the halo of spiral galaxies.  

The hypothesis had some precedents.  A few decades earlier Ziggy had suggested that there was a lot more dust and gas in the Universe than could be easily seen and this was subsequently confirmed.  This is now called ‘ordinary’ dark matter.  In addition, physicists were discovering new types of subatomic particle every few years.  It seemed plausible that galactic halos were hiding huge amounts of some exotic new particle or other hitherto unknown dark matter.

A large amount of theoretical and experimental work has been undertaken since 1980 to put parameters around this hypothesis and into inventing exotic types of matter that might fit such parameters.  It was calculated that most of the mass in galaxies must consist of dark matter over and above normal dark matter such as dust, cold gases and brown dwarf stars, and that the new dark matter has to be ‘cold’ and it has to interact very weakly with normal matter.  

An extensive range of experiments have been undertaken over the last 45 years in attempts to find direct evidence for any constituents of the conjectured cold dark matter.  

Our own galaxy is a spiral galaxy and our own sun is orbiting the centre of the Milky Way several times faster than it ‘ought to be’ so cold dark matter should be close at hand.

However, to date (2024) there has been no conclusive success in identifying or discovering any cold dark matter.  In spite of this, the cold dark matter hypothesis has become a mainstream feature of the most generally accepted model of the Universe. 

The fact that equation (1) comes from the fundamentals of classical dynamics yet is violated on a grand scale by direct observational evidence from countless galaxies is a marvelous opportunity for physics to learn something new.  As such, every part of equation (1) and every assumption and derivation on which it rests ought to be given careful scrutiny. 

The observed relationship between the speed of orbiting stars at orbiting radius r and radius r is similar but different for each galaxy.  It is called the rotation curve.  It is calculated by observing the broadening of spectral line as a function of r after correcting for the angle of galaxy relative to us and any relative motion.

Although cold dark matter is the most accepted explanation of the rotation curve problem, other proposals have been suggested and supported.  For example, in 1983 Israeli physicist Mordehai Milgrom proposed that modifications to Newtonian dynamics might be able to explain the fact that the velocities of stars in galaxies are observed to be much larger than expected.  

Milgrom noted that the discrepancy could be resolved if the gravitational force experienced by a star in the outer regions of a galaxy was proportional to a modified form of the centripetal acceleration (as opposed to the centripetal acceleration itself, as in Newton’s Second Law), or alternatively if very weak gravity varies inversely with radius (as opposed to the inverse square of the radius in Newton’s Law of Gravity).  In Milgrom’s modified Newtonian Mechanics (MOND), violation of Newton’s Laws occurs at extremely small accelerations found in in the far reaches galaxies.  These are far below anything typically encountered in the Solar System or on Earth.

Both approaches have issues.  The cold dark matter hypothesis struggles to explain the features of spiral galaxies without considerable fine tuning (see for example the ‘cuspy halo’ problem).  More significantly, nearly forty five years of searching have gone by without finding conclusive direct evidence for cold dark matter in any of its predicted manifestations (e.g. massive compact halo objects, or weakly interacting massive particles).  Success has been claimed several times, e.g. using interpretations of what is going on in the Bullet Cluster of colliding galaxies, but never conclusively.

The MOND approach and variations along similar lines have enjoyed greater success in explaining the observed features of spiral galaxies, see e.g. (S. McGaugh, 2013).  However, most astrophysicists seem reluctant to accept modifications to the basic laws of physics.  The MOND approach works well at a galactic level but has not yet been able to be used as the basis of a satisfactory cosmological model.

In spite of numerous papers claiming to disprove the MOND hypothesis, Milgrom and supporters of MOND have been able to fend off those assertions and have achieved good results in fitting their model to the ever increasing amounts of observational evidence. (For an extensive discussion of the data and its fit to MOND see (Milgrom, 2007) and more recently). 

The two approaches have argued with each other for forty years in the spirit of the scientific method and using astronomical evidence as it became available.  For a longtime the Bullet Cluster was argued to show direct evidence of dark matter at work, but this was disputed.  In 2023 a study of widely separated binary stars was argued as supporting the MOND hypothesis but a later paper by another group claimed the same date disproved the MOND hypothesis.  In mid 2024 a paper by Tobias Mistele, Stacy McGaugh and others (Mistele, 2024) used weak gravitational lensing as evidence that the flat rotation curve phenomena continues for very large values of  r, well beyond what could be explained by the dark matter hypothesis.

Mach’s Principle

Mach’s Principle was the name given by Einstein to part of the work of the German physicist and philosopher Ernst Mach.  Mach was a forerunner of Einstein in realizing that basic physical properties in nature do not have an absolute nature, but can only be described relative to each other.  Mach noted that whenever rotational effects were present there was also a rotation relative to a reference frame provided by the “fixed stars” and he postulated that this was not a coincidence.

There have been scores of papers written involving Mach’s Principle, and there are many versions of it.  One interpretation of Mach’s Principle is that distant matter is responsible for local inertia.  However, this raises the question of how could stars at enormous distances contribute to inertia here and now? 

Einstein tried to include Mach’s Principle into General Relativity.  Initially he felt he had included it without requiring action at a distance by making matter responsible for the geometry of spacetime and applying boundary conditions on the initial–value equations of his geodynamical model that gave rise to inertial reference frames consistent with Mach’s Principle.  Opinions amongst astrophysicists on whether Einstein was able to include Mach’s Principle are divided (Barbour & and Pfister, (1995).  Einstein himself eventually felt he had not succeeded in doing so. 

A discussion of how rotational inertia is treated in General Relativity is outside the scope of this paper.  However, as an aside, it appears to this author that while the Principle of Equivalence is straightforward for gravity and linear (translational) accelerations, it is not so straightforward for rotations.  A local observer can easily, always and everywhere tell when rotations are taking place and there is no gravitational field that easily produces basic rotational effects, such as the curved surface of the water in Newton’s spinning bucket.  

The meaning of “fixed stars” in the early 20th century was generally taken to be the stars plainly visible in the heavens above.  It was not until 1923-29 that Edwin Hubble was able to demonstrate that many of these objects were in fact other galaxies and that our Milky Way is just one galaxy amongst countless others.  Mach, and even Einstein at first, did not know about the other galaxies.  Hence Mach did not discuss whether he meant that rotations were always relative to the stars in our own galaxy or the full extent of matter in the universe more generally.  As far as Mach was aware, the stars in the Milky Way constituted the whole cosmos.  And without the existence of other galaxies there are no points of reference that make it clear that the Milky Way is itself rotating.  (Our solar system lies within the disc of the Milky Way at a distance of about 28,000 light years from its centre.  Our sun is travelling at around 828,000 kilometers per hour.  At this speed it takes about 240 million years to complete a full orbit.)

Newton’s laws of motion and gravity work superbly well at explaining the observations and model of the solar system put forward by Kepler.  The stars of the Milky Way provide such a convenient reference frame for rotational effects that this is invariably taken for granted.  Rotations under study typically occur in the range of microseconds out to hundreds of years.  However, on a galactic scale the period of rotation relative to the distant galaxies takes hundreds of millions of years, an increase of six orders of magnitude.  On this timescale the usefulness of the “fixed stars” for determining rotations becomes questionable, particularly as many of the stars will no longer be fixed at all, but will be moving in a variety of different directions and rates.  

Implicitly assuming the mean rest frame of the Universe or the frame afforded by the cosmic microwave background is the correct the reference frame for Equation (1) is ‘heroic’.  It warrants  much more careful consideration.  

It is not unreasonable to imagine that when the scale of a dynamic system under study is a billion times bigger than the solar system, rotations are taking place as slowly as one every 300 million years or so, the masses involved are about 50 billion times bigger than the mass of the Sun and the gravitational field strength is extremely weak, then some hitherto undetected effects might be taking place.  Perhaps the acceleration term on the right hand side of equation (1) is the issue.  Milgrom’s MOND approach explores this possibility in a particular way and with some success.  

This paper will also focus attention on the right hand side of Equation (1), but will do so using an approach in accordance with Mach’s thinking, and without having to violate Newtonian physics.  

This paper argues that if we consider the motion of the so-called “fixed stars” arising from their orbits around the centre of the Milky Way, it is clear that these stars and other celestial objects do not provide a simple and unambiguous frame of reference at all, at least not at the scale of the rotation curve problem.

Consider an extremely long observation of the ‘heavens’ from a star which is a galactic orbit but is not otherwise spinning.  Most of the stars in the ‘heavens’ appear to moving in arcs through the sky at variable rates relative to each other.  Furthermore, there are innumerable galaxies further away.  Some of these will have appreciable drift rates, some not at all.  The star itself might keep facing the galactic core as it orbits or it might not. 

In short, on a galactic time scale there are no ‘fixed stars’.  The observers on such a star are then faced with a dilemma – what exactly is the correct reference frame for understanding their own rotational motion around their galaxy?

Mixed Rotational Reference Frame effect hypothesis

In more detail, the Mixed Rotational Reference Frame effect (MiRRFe) hypothesis is:

1. The difference between the externally observed rates of orbit of stars in the discs of spiral galaxies and the rates predicted by Kepler’s Law is due to a dragging of the rotational reference frame for the stars in question.  
2. Localized observers will be able to establish that their local non-rotating reference frame is being dragged around and spun around at an extremely slow rate through an effect arising from, or coincident with, the gross movements of matter in their local galaxy.  
3. The smaller rate of rotation is consistent with Kepler’s Law.  
4. The orbiting stars experience centrifugal forces related to a smaller rate of rotation than is apparent to external observers.  
5. There is no need to assume that spiral galaxies contain large amounts of cold dark matter.   
6. The Newtonian equation for gravity and Kepler’s Law do not need to be modified to account for galactic rotation curves.  
7. A better understanding of the interaction between matter, rotational inertia and rotational reference frames will help to explain the evolution and structures of spiral galaxies.
8. Matter, gravitational mass, gravity, inertial mass and rotational inertia all emerged in an inter-related way during the evolution of the universe.

It should come as no surprise that the effect is not readily noticeable – the angular velocity involved in this relative motion is of the order of 10^-8 radians/year and the inertial effects at a local scale are almost immeasurable.  It is only on the scale of the galaxy itself that they become noticeable. 

The hypothesis does not necessarily imply that rotating galactic matter is directly responsible for dragging the local inertial reference frame around with it.  Although this would be the most obvious explanation if the MiRRFe hypothesis turns out to have validity, it is too soon to rule out more exotic explanations.  For example, it might be that spiral galaxies form because of a ‘whirlpool in spacetime’ and not the other way around. 

The MiRRFe hypothesis is not at variance with the Principle of General Covariance.  In essence this principle states that physical outcomes should not change if the reference frame in which they are studied is moved or turned.  However physics is definitely not immune if the whole system starts to rotate or changes its rate of rotation.  All sorts of effects arise. 

(Note – the English language is quite confusing when it comes to revolutions, one-off rotations, changes from static to rotating states, rotational accelerations, spins about an internal axis, orbits around an external axis and so on.) 

MiRRFe applied to Spiral Galaxies

Typical spiral galaxies can be thought of as large concentrations of matter in their bulge, surrounded by one or two pairs of spiraling arms lying in a broad flat disc extending well beyond the luminous region, all enclosed in a spherical halo and quite often featuring a bar of stars across much of the disc.  For purposes of analysis, the disc can be thought of as concentric thin rings of matter each rotating at particular rates as r increases.  These rings contain luminous stars, luminous gases and normal dark matter but as r increases they consist mainly of hydrogen, helium and dust.  The spherical halo surrounding the budge and disc is relatively free of stars.

Redshift data provides evidence of tangential velocities.  External observers and observers travelling with the stars in question can agree on these velocities, and also on the radius r to the galactic centre.  So both sets of observers can agree that the stars at radius r have angular velocity ω (.

For stars in the disc of a galaxy, the acceleration due to gravity at orbital radius r is GM/r² where M is the collective mass of the matter inside a sphere of radius r.  Most of this mass is inside the bulge or core of the galaxy.

If the external observers use Kepler’s Law and a reference frame aligned to the distant universe, they will calculate that for the stars to be in their observed orbits they should have an angular velocity equal to √(GM/r)/r.

However this value is much lower than the angular velocities observed in practice.

This is the conundrum of the observed galactic rotation curves.  GM/r² is supposed to equal ω²r but the latter is observed to be anything up to an order of magnitude too large.

The cold dark matter hypothesis responds by boosting M with dark matter.  The MOND hypothesis responds by replacing G/r² with something stronger, or by weakening the effect on stellar matter of the acceleration term ω²r.

The MiRRFe hypothesis suggested here is that the issue is resolved by a better understanding of the rotational reference frames that apply for the dynamics of spiral galaxies.

MiRRFe applied to galactic rotation curves

The diagram below is for the Milky Way and is fairly typical of the galactic rotation curve issue.  In practice it is easier to generate this curve for nearby galaxies such as Andromeda or M33 than it is for our own Milky Way.  This is due to large clouds of dust, the masking effect of the galactic bulge and the need to correct for all the relatively large motions of the Sun and Earth for telescopes based in Earth orbit.  One feature that is more apparent here than on most other rotation curves is the wavy nature of the velocity curve as r increases.  The rotation curves for several thousand spiral galaxies can be found on-line.  

[Data for several million galaxies is available from huge online databases e.g. Hypercat (Lyon/Meudon Extragalactic Database http://www-obs.univ-lyon1.fr/hypercat/) classifies references to spatially resolved kinematics (radio/optical/1-dimension/2-dim/velocity dispersion, and more) for 2724 of its over 1 million galaxies.  NED (NASA/IPAC Extragalactic Database /index.html) contains velocities for 144,000 galaxies.]

Fig 1: The blue line is the measured rotation curve (in this case for the Milky Way). The red dotted line is the rate expected if Kepler’s Law is applied in a frame at rest to the distant universe.  (The diagram is taken from a Wikipedia article on galactic rotation curves.)  In similar diagrams for other galaxies the turning point on the observed velocity curve is closer to the crossover with the calculated Keplerian line.  The more or less flat part of the velocity curve can be higher or lower for different galaxies. It sometimes has a gentle general slope up or down.

The solid line shows the tangential velocities of stars in the rim.  The dashed line shows the rotation curve derived from Kepler’s Law under the assumption that the appropriate reference frame for such an analysis is the universe at large.  

The MiRRFe hypothesis is that the observed angular velocities arise from a combination of rotational velocities speeds consistent with Kepler’s Law plus an effect arising from a very slow rotation of the inertial frame of reference at each value of r.  Accordingly there is a need to explain why the hypothesized effect becomes bigger as r increases through the luminous part of the galactic rim and out as far as observations permit.

MiRRFe might arise in various conceptual ways.  One avenue to explore is that the enormous mass of a galaxy (~10⁴¹ kg) has, over billions of years of evolution, managed to create a kind of whirlpool in spacetime that contributes to the determination of the correct non-rotating reference frame and hence rotational inertia effects affecting stars in the disc.  Another, more exotic possibility is that a whirlpool in spacetime is responsible for the formation of the galaxy.

The section below will give an equation for the frame dragging effect across the full rotation curve, along with a suggested physical interpretation based on Mach’s Principle.

Components of frame dragging effect

Consider a test star orbiting at radius r around a galactic centre.  Denote the angular velocity with respect to the universal reference frame as observed by external observers by ω_g(r).  However, observers on the star think that they have an angular velocity of ω_K(r) in accord with Kepler’s Law.  Denote the difference by ω_g(r).

With an appropriate sign convention ω_g(r)  =  ω_K(r) + ω_g(r) (2)

I.e. the observed angular velocity relative to the distant galaxies is the sum of the angular velocity according to Kepler’s Law, plus angular velocity ω_g(r) that can be interpreted as the rate at which local reference frames are being rotated at radius r, as viewed by external observers.  (Conversely, observers on the star in question are entitled to think that ω_g(r) is the rate the rate at which the distant galaxies, their own galactic core and everything else is rotating around them.)  

From equations (1) and (2)   ω_g(r)  =  v_e/r – √(GM) / (r√r) (3)

From observations we know that v_e is approximately constant.  Choose to express this as v_e = k√(GM)    where k is a value derived from observation for each galaxy under consideration.  (Detailed analysis of the large number of galaxies for which data has been gathered might enable k to be replaced by a general expression that applies to all spiral galaxies.  Note that Milgrom’s MOND theory derives v_e ⁴ = (a₀GM) where a₀ is a new constant.)

Hence ω_g(r)  has the form ω_g(r)  =  k√(GM) / r – √(GM) / (r√r) (4)

This is not profound.  We have simply subtracted the Keplerian angular velocity from the externally observed angular velocity.  The equation holds for r well outside of the core and as far out as observations have been made so far.  M is often treated as being constant, since most of the mass of a galaxy lies in its inner regions, but strictly speaking it increases slightly with r until there is no more matter to be found.  

If the equation holds for every test star at radius r then it holds for the whole ring of stars at radius r.

What follows is an attempt to explain the hypothesized frame dragging effect by reference to Mach’s Principle.  It is interesting that expression (4) has both a positive and a negative component.  

It may be helpful to think of a gyroscope in a circular galactic orbit at radius r with the axis of the gyroscope pointed towards the galactic centre at time t = 0 and completing a complete orbit (as seen by external observers) in time T = 2πr/v_e.  

The question is – what happens to the axis of the gyroscope as it orbits the galactic core?

If the relevant reference frame is provided by distant galaxies then the axis of the gyroscope should remain aligned to such distant galaxies.

The MiRRFe hypothesis is that a gyroscope would be rotated clockwise by a full circle during the orbit, less a counter rotation that depends on its radius r.  And since the gyroscope is in free fall around the galactic core, its behavior is a good indication of what is happening to the local reference frame.  

In more detail the conjecture is that there are two main effects that contribute to the behaviour of the gyroscope.  For ease of reference, the two effects will be called the Core Effect and the Distant Galaxies Effect.  There may also be some second order effects.  

Core Effect: Mach’s Principle associates rotation with movement relative to the “fixed stars” and non-rotation with lack of movement relative to the “fixed stars”.  However Mach was not aware that the Milky Way is just one galaxy among innumerable others.  Hence what is meant by “fixed stars” is wide open to question.  

Consider a star that is not revolving as it orbits.  Observers on such as star would see all the stars in the core as more or less fixed, apart from their local motions.  They would also see the stars above and below them, and in front and behind, to be more or less fixed as well.  However all the distant galaxies would be moving, apart from those directly above and below the plane of the orbit.

The Core Effect is a conjecture that a gyroscope pointed at the galactic centre will try to remain pointed at the centre as it completes its orbit.  External observers would see the gyroscope rotated by 2π radians every orbit and in the same direction as the orbit itself (which we have called anti-clockwise).  The core effect tries to make a local rotational reference frame roll around a complete circle every orbit.  The rate at which this happens is 2π/T = v_e / r = k√(GM) / r.

Distant Galaxies Effect:  With the core effect in place a gyroscope at radius r has to travel at the Keplerian rate, relative to this rotating reference frame, in order to stay in stable orbit.  While it is completing its very slow galactic orbit, observers traveling with it would perceive the distant galaxies to be drifting in a clockwise direction at a relative angular velocity of  ω_k(r) = √(GM)/(r√r).  

The traditional interpretation of such an observation is that the observers would deduce that they themselves must be rotating at this very low rate.  

However, our observers are not so sure.  They are getting a conflicting Machian message from the galactic core and most of the rest of the stars in their local galaxy.  This message is  telling them that they are non-rotating when they remain oriented to their galaxy’s centre.

This gives rise to a kind of tussle between the two effects.  The core effect is trying to keep the gyroscope oriented towards its initial position relative to the core while the distant galaxies effect is trying to keep the gyroscope oriented towards its initial position relative to the distant galaxies.  

Net Effect:  A physical interpretation for the MiRRFe hypothesis is that the reference frame with the least evidence of local rotational effects (i.e. the best local inertial reference frame) will be determined the sum of the Core Effect and the Distant Galaxies Effect.  

External observers will see this to be rotating at the rate given by the core effect less the rate given by the distant galaxies effect i.e.

ω_g(r)  =  k√(GM) / r – √(GM) / (r√r) (5)

which is the same as equation (4), as required.  

An orbiting star is moving at ω_k(r) over an above that ω_g(r), making up the balance of the angular rate of rotation as seen by an external observer.

Second order effects:  Since we have argued that local non-rotating reference frames are influenced by large amounts of matter within the spiral galaxy, we need to acknowledge that there might be some second order effects as well.  

At time t = 0 consider a line of stars either side of the gyroscope.  This line points towards the core on one side and out through the rim on the other.  The gyroscope is in the middle.  The gyroscope and the stars are all lined up like runners in a race around a circular track.  As observed and agreed by all, all the stars have the same speed.  However, stars nearer the centre have the advantage of a smaller orbit and so by the time the gyroscope completes a lap they will be further advanced.  Similarly the stars on the outside will have fallen behind.  For every light year that inner stars are closer to the centre they will be 2π light years ahead, and for every light year that outer stars are further from the centre they will be 2π light years behind.  Hence the starting line will now not be straight but will be angled by θ = arctan (2π).  This will have taken place in the orbital period of the gyroscope = T = 2πr/v_e .

So the gyroscope will note that the starting line of stars around it is being rotated at a rate equal to   θ/T = arctan(2π) v_e / 2πr = 0.225 k√(GM) / r (6)

This is a rotation in the clockwise direction.  It is similar to an effect that can be seen in the spinning teacups in a traditional fairground roundabout.  It may or may not have an influence on the behaviour of the gyroscope.  But since the number of stars in the local neighborhood is much smaller than the number of stars in the core, it is reasonable to assume that any effect is of the second order.  Another small effect might arise in conjunction with proximity to a galactic bar. 

Discussion of Galaxy Rotation Curves with MiRRFe

The MiRRFe hypothesis is that the stars are actually orbiting at their correct Keplerian rates relative to their local reference frames and that the local reference frames are turning at a rate equal to the sum of a core effect and a distant galaxies effect.  The combined result is what is  observed.

Returning to Figure 1.  Note that there is generally a point at which the observed tangential velocity equals the Keplerian velocity i.e. the point where the observed velocity curve crosses the expected Keplerian velocity curve.    

In our suggested explanation for the hypothesized MiRRFe, this occurs when the distant galaxies effect exactly matches and counterbalances the core effect so that the ω_g(r) is zero.  The Keplerian orbit is so fast that its period equals the observed period (fast might be a misleading term as the period may still be in the order of several hundred million years).  From equation (5) we have that this occurs when k√r = 1.

For values of r a bit smaller than this crossover radius, the calculated Keplerian rotation is actually bigger than the observed rate of rotation.  Figure 1 suggests that this may be the case for the Sun within the Milky Way.  Not all calculations of Keplerian curves in galaxies show a region of negative discrepancy, but many do.  (It is not clear to this author how adding extra mass in the form of cold dark matter helps to resolve a negative rotation curve discrepancy.)

Under the MiRRFe hypothesis the explanation is simply that the distant galaxies effect has become bigger than the core effect and so the stars are still at their correct Keplerian rates relative to their correct local rotational reference frames.

For r bigger than the cross over radius the observed tangential velocities tend to be constant, so our hypothesized rotational reference frame effect has to show an increase.  This is demonstrated in Equation (5) due to the extra √r term in the denominator of the second term.  In Machian terms the core effect weakens but the distant galaxy effect weakens faster, in accord with the slowing down in the observed drift rate of the distant galaxies.  Thus stars have to “hurry up” to stay in stable orbits and this gives rise to the flat rotation curves first reported by Vera Rubin in 1975. 

Finally a comment about what happens as r becomes very large.  As the local galaxy recedes into the very far distance it eventually starts to look like any other galaxy.  One would expect any that any effects from the rotating galactic mass would fade away.  This is suggested in expression (4) and (7) by the fact that ω_g(r) tends to zero as r becomes large.

Potential Tests, Explanations and Predictions

A hypothesis can become a useful theory if it meets three criteria:

1. it provides a plausible explanation for a range of observed effects
2. it is not contradicted by the results of reliable relevant experiments, and
3. it makes predictions which turn out to be true.

 The MiRRFe hypothesis is fundamental to the dynamical structure of spiral galaxies and hence to a lot of other astrophysics and cosmology.  Because it is a basic large-scale effect, it lends itself to a wide range of tests under all three criteria.  It may also be helpful to a range of other issues in the dynamics of spiral galaxies. 

Virial Theorem:  The virial theorem in dynamics implies that the total kinetic energy of an n-body system should be half of its total energy.  Applied to galaxies this suggests that their total kinetic energy should be half of the total gravitational binding energy.  However, observationally, the total kinetic energy appears to be much greater.  If the apparent rotational velocities of the orbiting stars are adjusted by a rotational frame dragging effect back to their Keplerian levels, the total kinetic energy of the galaxies would be reduced to levels more consistent with the virial theorem prediction.

Winding Problem:  The winding problem is that since matter nearer to the centre of a spiral galaxy rotates faster than the matter at the edge of the galaxy, the arms would become indistinguishable from the rest of the galaxy after only a few orbits.  However, spiral arms in spiral galaxies are clearly quite persistent.  MiRRFe may provide a partial remedy to this issue.  It suggests that, viewed in the correct reference frame, the stars in the outer parts of the arms are actually rotating at much lower angular velocities that are consistent with Kepler’s Law.  (The rest of their apparent angular velocities are due to a dragging of their rotational reference frames.)

Relationship between Luminosity and Rotational Velocity:  The Tully–Fisher relation is derived from observations on a statistical basis and shows that for spiral galaxies the rotational velocity is well related to its total luminosity.  A consistent way to predict the rotational velocity of a spiral galaxy is to measure its bolometric luminosity and then read its rotation rate from its location on the Tully–Fisher diagram.  Conversely, knowing the rotational velocity of a spiral galaxy gives its luminosity.  Thus the magnitude of the galaxy rotation is related to the galaxy’s visible mass.  However, there is not yet any straightforward explanation as to exactly how and why the observed scaling relationship exists.  MiRRFe may be able to contribute to such an explanation.

The cuspy halo problem:  The cuspy halo problem is that cold dark matter (CDM) simulation models predict halos have a core which are too dense, or have an inner profile that is too steep, compared to calculations of the dark matter densities required by the CDM hypothesis as applied to low mass galaxies.  Nearly all simulations form dark matter halos which have “cuspy” dark matter distributions, with density increasing steeply at small radii, while the rotation curves of most observed dwarf galaxies suggest that they would need to have a flat central dark matter density profile.  MiRRFe does not require CDM and hence does not suffer from this problem.  

Cosmological models:  A criticism of the MOND approach is that it has not proved to be helpful in constructing full cosmological models of the universe.  MiRRFe does not modify classical dynamics and MiRRFe itself disappears well outside of galaxies.  However, the implications of the wider MiRRFe hypothesis would extend beyond galactic models and would provide a Machian influence or constraint upon cosmological models more generally.  The general idea is that inertia might arise from an interaction between matter and some sort of universal background.  There is a correspondence between inertia and the “fixed stars” but not because of any mysterious magical action-at-a-distance, but because the local matter and the distant matter are embedded in the same field.  

This author favours the idea that there is never any action at a distance – all physics is local.  Even gravity.  Bodies moving under the gravitational influence of other bodies are doing so in response to the local environment and that local environment has been influenced by the presence of nearly matter.  Many people will immediately say “of course – that’s spacetime curvature as described by general relativity.”  And the author will agree that curved spacetime is a marvellous way of describing the situation.  But is cannot be a full and complete description of the underlying physics because if it was, physics would not be faced with the current catastrophe in which it cannot explain the gross motion of stars in spiral galaxies, including our Milky Way.

Computational Tests:  Powerful computers and simulation models have been used for several decades to try to model the evolution of galaxies in an attempt to replicate their observed features.  A relatively easy test for MiRRFe is to use it in such models in place of cold dark matter and examine whether reasonable outcomes can be achieved.  MiRRFe predicts substantive effects, e.g. that the angular momentum of spiral galaxies is less than it seems, so the outcomes ought to be noticeable.  If not MIRRFe exactly as proposed here, then perhaps variations of the basic idea.

Note however that the MiRRFe would only emerge as the galaxy itself emerges.

In addition to the mainstream family of spiral galaxy types, there are several hundred non-standard spiral galaxies that have been studied in detail with modern high-resolution instruments.  MiRRFe may or may not prove useful in helping to understand how their peculiarities came about.  

Dynamics of the Core: If the MiRRFe hypothesis is valid it has some predictions for stars inside the core.  A star in the core which is non-rotating with respect to the distant galaxies would still be adjudged to be rotating by local observers due to the rotation of the core and rim local galaxy.  Similarly a cloud of stars in the core would feel itself to be rotating on account of MiRRFe and would either rotate in sympathy with the aggregate effect from the rest of the galaxy, or it would feel itself pulled apart slightly.  If the later case applies then this might be help to stop the cloud from collapsing. 

Test Experiments: In principle, observers could resort to experimental determinations.  For example, an incredibly sensitive Foucault pendulum or gyroscope could reveal how fast the local reference frame is rotating in response to galactic orbital motion.  A satellite launched at very high speed towards the rim or centre of the Milky Way would tend to be swept around slightly by the frame dragging effect.  

Comets:  A comet orbiting a star (e.g. the Sun) with a long elliptical orbit in the plane of the galactic disc would demonstrate a MiRRFe in the precession of its perihelion.  

Supernovae:  Assume the pattern of matter ejected from a supernova has a distinctive shape, such as a spherical shell.  If MiRRFe exists there will be some apparent distortion in the pattern of the ejected matter over time.  Material ejected towards the out rim will look like it is being “blown” in a pro-grade direction by some sort of cosmic wind.  In fact it is just try to conserve its angular momentum against a reference field that is itself rotating due to the local influence of all the other stars in the galaxy.  

The local reference frame will be undergoing bigger rotations at larger r and smaller rotations at smaller r.  This will affect the trajectory of matter flung towards or away from the galactic core.  Generally speaking there will be a tendency for an expanding shell to flatten and rotate a bit, in directions parallel to the galactic rim.  There are enough supernova remnants in the Milky Way for a check to be done to see if there are some detectable traces of this effect.

Light bending effects:  The way in which electromagnetic radiation from distant quasars is bent when it passes just above and across each side of the disc of an intervening galaxy might shed some light on the answer.  MiRRFe suggests that distant light travelling through or near to the disc in the same direction as the galaxy’s rotation (pro-grade) will arrive sooner than light passing through or above the other side of the disc against the direction of rotation (retrograde).  

So suppose light from a supernova in a far off galaxy passes through an intervening galaxy and is recorded in our telescopes.  If we watch that part of the sky closely we may see, some years later, the same event reoccur some distance away having been gravitationally lensed through another part of the same intervening galaxy.  If we can calculate the effects in accordance with General Relativity we can look to see if there is an additional time of arrival difference in favour of the pro-grade path.  A kind of ‘fly-by’ boost for the light itself.

If not a supernova, then perhaps a discrete event in another bright source of light, e.g. blip in the light from a quasar, possibly caused by an eclipse from a companion star.

Proof by Exception – Cold Dark Matter:  MiRRFe has been suggested here as an alternative to the hypothesis that cold dark matter makes up 27% of the Universe and over half of the mass of typical spiral galaxies, including the Milky Way.  It follows that if CDM is ever detected in an unambiguous physical experiment or observation, then there would not be a need for MiRRFe or any other alternative ideas to CDM.  Conversely, the longer that time goes on without CDM being conclusively identified and detected then the greater the need for a plausible alternative.

Objections to the MiRRFe hypothesis

It is understandable and natural that the MiRRFe hypothesis will not sit comfortably with anyone heavily invested in the dark matter hypothesis. The simplest rebuttal would be to find some dark matter.  It is supposed to make up most the matter in spiral galaxies, including the Milky Way.  But if dark matter cannot be found within another few decades (say) then it might turn out to have been wise to have some alternative ideas ready to be explored in full detail.

Proponents of the dark matter hypothesis often point to the Bullet Cluster of galaxies as providing firm evidence for the existence.  The Bullet cluster is a group of about 40 galaxies that have evidently passed through a larger cluster of galaxies with dramatic effects.  A later essay in this series will suggest an explanation of the effects that does not require dark matter. 

An expected criticism of any new idea is that it is ‘”ad hoc” which is a Latin way of saying that the explanation has been conjured up specifically to meet the circumstances of a particular problem and that it needs to be tested against the entirety of other relevant physical evidence before it deserves any consideration.  This is valid to the extent that testing new ideas is part of the scientific method.  It is not so valid if it is used just as an excuse for ignoring new ideas, killing them off in their infancy, rejecting them at the peer review stage or denying them support for experiments.  A variation of this attitude is to ignore new ideas because they have not attracted a lot of popular support.  However, the scientific method should rely on evidence, not popularity.  The balance of other people’s opinions is persuasive, but it needs to be recognised that new ideas will have very few supporters at first, even if the ideas have merit, or at east the seeds of something worthwhile. 

In relation to the MiRRFe hypothesis a start has been made on checking it against the rest of physics and this is discussed in later essays.  The hypothesis is different from theories that try to explain inertia as a side effect of gravity or which propose a static universal background field. 

Conclusion

The gross violation of classical physics in the orbits of stars in spiral galaxies has given rise to two main explanatory hypotheses, cold dark matter or modified Newtonian physics.  This paper suggests a third hypothesis based on Mach’s Principle.  It suggests that the rotational velocities are actually in accord with Kepler’s Law if the reference frames against which these should be measured are themselves rotating at rates of up to several degrees per million years.  The hypothesis may or may not be correct but it lends itself to observational and computational verification and may be worth further consideration. 

References

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Van de Vusse, Sjoerd B.A., 2024,  “Some ideas and experiments for issues affecting modern physics”.  https://hereticalphysics.com.au
Author contact:  SBAvan@utas.edu.au
Author’s location:  Hobart, Australia