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Flatness in disc galaxy dynamics without MOND or dark matter

In previous work (astro-ph/0306228) we have established that a particular quasi-classical theoretical viewpoint about the nature of gravitation in galaxy discs provides extremely high concordance (comparable with MOND) with the observed dynamics in optical discs. In this paper, we consider the possibility that the optical disc and the flat radio disc represent distinct dynamical regions which are inherently described by distinct phases of the same over-arching theory of which the quasi-classical theory is but one phase. This is a defining moment for the theory under discussion since the only natural generalization available is that from the existing quasi-classical (one-clock) model into a quasi-relativistic (two-clock) model. The quasi-relativistic disc turns out to be a stationary hyperbolic disc within which information propagates along characteristics and within which the rotation curve is necessarily exactly flat. The transition between the two components of the disc is managed by physically determined jump conditions which are represented by a set of non-linear algebraic equations. These equations predict the existence of a forbidden region within the parameter space which coincides exactly with that part of the parameter space in which there is an already observed highly significant underpopulation of galaxy discs (astro-ph/0107300).


A globally inertial material vacuum
This is the third paper in a series which has as its purpose a reformulation of gravitation theory on the basis of what might be called an 'ultra-strong Machian worldview'. The initial paper (Roscoe [1], astro-ph/0107397) asked the question:

Is it possible to conceive a globally inertial space & time with is irreducibly associated with a non-trivial global mass distribution and, if so, what are the properties of this distribution?
This question received a positive answer subject to the conditions that: • the global mass distribution concerned is fractal, D = 2; • the constituent particles of this distribution are photon-like in the sense that they have a velocity property, v, associated with them satisfying |v| = const; • the velocity property is not a true velocity but is, rather, a conversion factor between times scales and length scales exactly like Bondi's interpretation of light velocity, c.
For obvious reasons, the foregoing was considered to be a description of a rudimentary material vacuum.

Gravitation as perturbations in the material vacuum
The second paper (Roscoe [2], astro-ph/0306228) was concerned with the question of how gravitational processes could arise in this material vacuum. It began by showing how pointwise perturbations of the D = 2 distribution gave rise to classical Newtonian gravitational processes for test particles placed in the region of the perturbation and went on to derive the equations of motion for an idealized spiral galaxy. This theory was then applied to model the rotation curves for a small sample of low surface brightness (LSB) galaxies giving results that were directly comparable in quality to those obtained from the MOND algorithm -and far superior to anything achieved by the multi-parameter dark matter models.

Flat rotation curves and an unsought-for bonus
However, whilst the theory of the second paper, which we shall refer to as the quasi-classical (one-clock) theory, works extremely well within the optical disc, it is not able to generate the flat extensions to rotation curves usually associated with the radio disc. The purpose of this third paper is to show how a natural generalization of the quasi-classical one-clock theory into a quasi-relativistic 1 two-clock theory solves this problem in a direct and simple way. In essence, we find that the rotation curve of the quasi-relativistic disc is necessarily everywhere flat. Thus, in the presented worldview, galaxy discs exist in two phases -one, the optical disc, which is described by the quasi-classical theory and the other, the radio disc, described by the quasi-relativistic theory. The transition from the quasi-classical disc to the quasi-relativistic disc, which can be considered analogous to the transition between a subsonic and a supersonic flow in a compressible fluid, is managed by a set of physically derived jump conditions which are represented by a set of non-linear algebraic equations.
As an unsought-for bonus, it transpires that the jump-condition equations become degenerate in exactly that part of the disc galaxy parameter space which we already know to be very significantly underpopulated in such objects in the four large samples that we have analyzed (Roscoe [30]). In other words, the theory predicts the difficulty of forming galaxy discs in exactly that part of the parameter space which is already known to be significantly underpopulated by disc galaxies.

Why bother? Some difficulties faced by the standard model
Although not yet widely accepted as problematical, it is a fact that there exist various wellfounded observations and difficulties which, if taken at their face value, would cast reasonable doubt about the long-term viability of the standard cosmological model. These fall into two broad categories which we can classify as redshift phenomena and as non-redshift phenomena. The point can be illustrated by reference to just two examples which can be considered to serve as 'straws in the wind'.
For more than two decades, Tifft (see [33] for an early reference) has been making claims about the existence of what he terms as redshift periodicities. The claims have become increasingly elaborate over the years but, at their most simple, they relate to the supposed existence of a periodicity at around 72km/sec between galaxies in clusters and one at around 36km/sec between our own galaxy and galaxies in the local group, out to about 2000km/sec. Unfortunately, Tifft was never able to support his hypotheses with statistical analyzes rigorous enough to match the extreme nature of the claims. However, sometime in the early 1990s Napier brought scepticism and a long experience in dealing with noisy periodic data in the geophysical and cometary record (variously [?, 4, 5, 6]) to the problem . There followed a short series of little-known papers by Napier & Guthrie ([23,24,14]). These papers were remarkable on two counts: firstly, for their statistical rigour and secondly for their conclusion -that the Tifft hypotheses relating to the 72km/sec and the 36km/sec periodicities were supported on the (non-Tifft) samples used by these authors at the level of virtual statistical certainty (see specifically [14]). The basic technical problems associated with these analyzes revolve around the difficulties involved in finding line-profiles that are symmetric enough to allow HI determinations of galaxy redshifts to within better than 5km/sec -it can be shown that anything much worse than this makes it impossible to reliably detect any periodicities at the 36km/sec level which might exist. So far as non-redshift phenomena are concerned, we need only cite the problems associated with modelling rotation curves in disc galaxies in general -and in low surface brightness galaxies in particular -using the multi-parameter dark matter models, and then compare this with the simplicity and effectiveness with which the one-parameter MOND recipe (see Milgrom [20,21,22] for early predictions) succeeds in this task. Of course, MOND is not the answer for it is not even a theory -but the details of its successes over a wide variety of applications (see variously McGaugh, de Blok and Sanders [18,19,32]) should be raising serious questions about the standard model's failures in this respect -notwithstanding well-known problems experienced with MOND when modelling galaxy clusters.

An historic perspective on the proposed alternative view
Given that the future of the standard model is by no means secure, then there is room to consider alternatives, such as the one being considered here. This alternative view was originally driven by what could be termed as an 'ultra-strong Machian worldview' and this remains as the core philosophy. But, interestingly, the theory which has emerged -which employs all the machinery more usually associated with curved spacetime theories -falls directly into the category of LeSage-type theories. See the introduction of Roscoe [2] (astro-ph/0306228) for a brief discussion of this. LeSage [15] was interested in the causal mechanisms which he believed must underly Newtonian gravitation theory and he proposed the existence of an isotropic and homogeneous substratum (in modern terms, a material vacuum) consisting of microscopic particles in states of agitated motion. These particles were envisaged as colliding with, and transferring momentum to, ordinary material bodies. For one body alone in this agitated substratum, the overall effect of the transferred momentum was seen to be neutral. However, when two such ordinary bodies were near each other, then they provided mutual 'shade' to each other from the bombardment of the substratum particles thereby creating an asymmetric imbalance in the rate of momentum transfer from the substratum to the two bodies -the net effect being that the two bodies experienced net forces which would appear as mutual attraction. When the details of schemes such as this are worked through, it is a simple matter to recover a quantitatively correct 'Newtonian gravitational force'.
The LeSage theory, and many others of a similar kind that have appeared over the years (see for example, Edwards [13] for a good review) envisage the substratum particles in a deeply classical way -that is, as essentially microscopic versions of macroscopic classical bodies existing within a pre-determined inertial space & time. By contrast, the 'substratum particles' associated with the theory of the inertial frame developed by Roscoe [1] (astro-ph/0107397) are the very antithesis of the classical momentum-bearing particle: specifically, they do not exist within space & time -rather, what we call space & time can be considered as a metaphor for the relationships which exist between these vacuum particles -in effect, they define what we call space & time. Thus, according to this theory, a globally inertial space & time is irreducibly associated with a material vacuum within which the vacuum particles are fractally distributed with D = 2. Conventional gravitational processes arise in a given locality when this distribution is perturbed in that locality. Consequently, an intrinsic assumption of our approach is that conventional bodies act as sinks for the vacuum particles and thereby act as local perturbers of the D = 2 equilibrium distribution.
3 Review of the middle-disc data analysis

Disc galaxy rotation curves
When one studies rotation curves 'in the whole' one is typically confronted by various complexities which would appear to defy any chance of a simple description. These vary from non-monotonic behaviour (typically) near the central regions, through generally monotonically rising sections through, finally, abrupt transitions into almost perfect flatness somewhere in the boundary between the optical and the radio parts of the disc. Astronomers have, traditionally, fitted very complicated functional forms to rotation curves in attempts to provide global classifications of their shapes. (cf Courteau [7], Rix et al [26]). In this paper, we describe the second step of a quite different strategy whereby we consider the possibility that galaxy discs might reasonably be partitioned into three distinct dynamical regions, each to be considered separately within the context of the same general theory and merged into a whole at their joining boundaries by appropriate jump conditions: • an innermost region dominated by the central bulge where the dynamics frequently appear messy and complicated; • a mid-region comprising most of the optical disc where the dynamics appear to be reasonably regular; • an outer region comprising the radio disc where the dynamics gives rise to the very peculiar constancy of rotational velocity.
The phenomenology of the optical mid-regions has already been comprehensively analyzed in Roscoe [27,30] whilst the related theory has been developed in Roscoe [1,2] (astro-ph/0107397, astro-ph/0306228). Correspondingly, our main task here is to show how the flat radio disc can be understood from the same general theoretical perspective and finally merged seamlessly with the optical mid-region. The process by which this is done also makes clear how the overall task, that of including the three dynamical regions within a unified theoretical perspective, can finally be completed.

General comments
The mid-1990's was an optimal time for our partitioning approach since Persic & Salucci [25] had just made available a sample of 900 folded optical rotation curves (ORCs) which originated from the Southern Sky survey of Mathewson et al [16]. This sample was notable for three things: • being purely optical it contained no radio disc component; • it was sufficiently large that it could support a rigorous statistical analysis; • the individual ORCs were very accurately folded -this quality being driven by Persic & Salucci's intention of using them for studying the interior dynamics of individual discs.
The availability of this large high-quality sample meant that there was a reasonable chance of gaining a genuine insight into the generic behaviour of dynamics in the middle-disc regions of galaxies. Furthermore, the quality of Persic & Salucci's folding process turned out to be extremely important for it added crucial precision to the statistical study we had in mind.

Summary of previous results
The most simple solution provided by the quasi-classical theory when applied to modelling an idealized disc stated that rotation velocities followed power laws V = AR α where (A, α) are parameters that define individual idealized discs. This solution turns out to be merely the most simple in a class of much more complex solutions, but it defined the basic question which has driven most of this author's data analyzes: specifically, to what extent can rotation velocities in real galaxy discs be said to be described by simple power laws? This question turned out to be an immensely productive one. The results arising from pursuing it -both directly and indirectly -are listed below.
• Over large samples of late-type galaxies (Sb..Sd), and to an extremely high statistical precision, rotation velocities in the outer regions of optical discs can be represented by power-laws of the type where the scaling constants, (V 0 , R 0 ) are very strong functions of galaxy luminosity properties (Roscoe [27,30]); • The power-law structure in the outer regions of optical discs gives automatic rise to the classical Tully-Fisher relations which are properly calibrated for I-band luminosity data and R-band luminosity data according to whether the sample being analyzed uses I-band or R-band photometry (Roscoe [30]); • The pursuit of the power-law question led directly to the discovery of the discrete dynamical states phenomenology in disc galaxies according to which the distribution of the parameter A in the law V = AR α is multi-peaked with four strong peaks satisfying ln A ≈3.90, 4.17, 4.72 and 5.11 being identified. This translates into the statement that rotational velocities at 1kpc tend to cluster around the values of 49km/sec, 65km/sec, 112km/sec and 166km/sec (Roscoe [28,30]); • The statistical success of the simple power-law formulation for outer-disc optical rotation curves prompted a much deeper study of the underlying quasi-classical theory. Subsequently, we were able to show that it could be applied to model the dynamics of low surface brightness galaxies (LSBs) giving results that hitherto have only been matched by MOND (Roscoe [2], astro-ph/0306228). It is the general success of the quasi-classical model, as outlined above, which has encouraged the development of the work which forms the main body of this paper.

Summary of new results
The current paper provides three additional significant results which can be stated as: • A fresh, but simple, insight into the quasi-classical model provides a qualitative understanding of why the power-law model V = AR α fits the generality of middle-disc dynamics as well as it does -even though it is a simple matter to find examples of individual discs which are in clear departure from any power-law description.
• the rotation curve flatness problem is resolved within the theory: specifically, the flat exterior sections of rotation curves arise as the only solutions of the two-clock quasirelativistic generalization of the one-clock quasi-classical theory; • the jump conditions between the quasi-classical and the quasi-relativistic parts of the disc form a set of non-linear algebraic equations which become degenerate in that part of the parameter space which corresponds exactly to the deep valley between the ln A = 4.17 and the ln A = 4.72 peaks of the ln A frequency diagram. That is, the jump conditions predict the existence of a forbidden zone in the parameter space which appears to exist on the data.

Review of the quasi-classical disc theory
A brief review of the theory underlying the quasi-classical middle-disc model (Roscoe [1,2] astro-ph/0107397, astro-ph/0306228) provides a useful precursor for the main analysis of this paper and provides the first new result.

Physical time and mathematical closure
The basic relation of the quasi-classical theory is given by where the indices take values 1..3 and M provides a scalar representation of the mass distribution concerned -but see Roscoe [2] (astro-ph/0306228) for a complete discussion. In the present case, we are concerned with modelling disc galaxies and here M(r) has the straightforward interpretation as the amount of mass contained within a radius r in a disc of unit thickness. Furthermore, for disc galaxies, the assumption of cylindrical symmetry is appropriate implying Γ k ab ≡ 0. Using cylindrical polars, we then find that the line element can be written as Introducing the quantities R = r/r 0 and T = t/t 0 where (r 0 , t 0 ) are scaling constants, we get the dimensionless form The equations of motion can now be written down in the usual way using the variational principle However, because I is homogeneous degree zero in the temporal ordering parameter, τ , then the resulting equations of motion are invariant wrt arbitrary transformations of this parameter. This has two related consequences: firstly, it implies that τ does not, in general, represent physical time and, secondly, the resulting equations of motion form an incomplete set. In fact, because the analysis is two-dimensional, the variational principal only gives one independent equation of motion, namely which corresponds to the conservation of angular momentum in purely classical theory. The situation is, of course, identical for General Relativity -there, the equations of motion arising from the variational principle are completed by introducing the concept of particle proper time. In the case under discussion (see Roscoe [2] or astro-ph/0306228 for a complete discussion) this was not an option, and a different solution had to be sought: Certain symmetry arguments led to the conclusions that, within the context of the application of the theory to disc galaxies, • the forces acting within the disc are not necessarily central; • the ratio of the magnitudes of the transverse forces to the radial forces along any radius within the disc is constant.
This led to the closure of the system in the following way: the line element is given by where A and B are defined in (2). Since the closure information is derived from the ratio of force components, it is necessary to define what we mean by force in a disc with line element given by (3). To this end, we factorize this latter relation as where S 1 = ±1, S 2 = ±1 and where, in the standard notation,R = (cos θ, sin θ) T and θ = (− sin θ, cos θ) T . Hence, introducing an arbitrary temporal ordering parameter, τ , we can define a generalized velocity as from which a generalized acceleration vector can be derived aṡ However, we also require that the ratio of the magnitude of the transverse force component to the radial force component is to be constant, so that we get the additional equation for some constant K. Since this equation is not invariant under arbitrary transformations of the parameter τ then, effectively, it selects a specific τ from all the possibilities and defines it as physical time -that is, as that definition of time which is consistent with our constraint on disc forces. Thus, at face value, we now have two equations for the system -one from the variational principle (not shown) and (5), above. However, it transpires that if one assumes that light traces mass in a disc galaxy (that is, no significant dark matter) then, typically for light-inferred mass distributions, one finds A ≡ ∂ 2 M ∂R 2 < 0 which corresponds to density distributions which fall off more quickly than 1/R but no quicker than 1/R 2 . Since B, defined at (2), is necessarily positive for any M, then equation (5) is necessarily complex which, in turn, implies K ≡ q + pj for real parameters (p, q). Thus, in the final analysis we end up with three equations -which implies that the theory determines the dynamics and the mass distribution for a disc galaxy.

The power-law as a limiting velocity distribution
The same theory that gave almost perfect modelling for the sample of eight LSBs considered in Roscoe [2] (astro-ph/0306228) (data provided by McGaugh 2 ) also provides an understanding of why the simple power-law model provides such a high-fidelity statistical resolution of middle-disc ORC data over large data sets. In effect, we find that the power-law solution, V = AR α , is the only solution which is bounded over finite discs and that all other possible solutions become rapidly unbounded and therefore, by definition, cannot persist over substantial radial ranges within stable discs. However, according to the theory, these non-power-law solutions can persist over small radial ranges and are therefore, in terms of the theory, responsible for much of the complexity commonly observed within optical discs. However, since, by definition, stable discs have bounded velocity distributions, then it is the power-law which is the dominant underlying form. To obtain a quantitative understanding of these remarks, we must consider the basic theory: the density-distribution equation derived in Roscoe [2] (astro-ph/0306228) can be written as

∂M ∂R
where ρ is surface density, M is the basic mass function of the theory, m 1 is a scaling parameter having dimensions of mass and (p, q) are free dimensionless parameters. The corresponding rotational velocity equation can be written as It is easily shown that if where m 0 has dimensions of mass and d 0 is dimensionless, then ρ ∼ 1/R 2 , ǫ 2 = 1 and Ψ = const etc with the final consequence that the velocity equation gives V = AR α exactly. But also, by the considerations of §4.1, we know that ρ cannot fall of more rapidly than 1/R 2otherwise equation (5) becomes real, and the whole analysis becomes invalidated. We are therefore left to consider solutions for which ρ ∼ 1/R k , k < 2: from (6) it is easy to see that, for such solutions, Ψ 2 → 0 as R increases. But, by (7), dV /dR → ∞ as Ψ → 0 so that, as stated, all other possible solutions are unbounded over any finite disc.
To conclude, the power-law solution is the only solution which is bounded over a finite disc and therefore, according to the theory, provides the underlying dynamical 'shape' of any stable disc.

The Development of the quasi-relativistic disc-model
We show, in appendix A, that it is impossible to have non-trivial power-law solutions merging into flat solutions within the context of the quasi-classical one-clock theory. The solution to this problem turns out to lie in the quasi-relativistic generalization of the quasi-classical model: Specifically, it transpires that the flat component of rotation curves is defined in a hyperbolic space which arises directly from the quasi-relativistic generalization whilst the optical mid-region component remains associated with the quasi-classical model. The boundary between the two components can be considered analogous to a shock front, with the transition across this boundary being managed by appropriately defined jump conditions.

The Euler-Lagrange component
The basic definition for the metric tensor, given at (1) remains unchanged -except that now, the indices vary over 1..4 where x 4 ≡ ct and t represents conventional clock time and c is a scaling parameter with the dimensions of velocity. In cylindrical polars, the line element becomes which, in dimensionless form, is given by where R = r/r 0 and T = t/t 0 where (r 0 , t 0 ) are scaling constants, as before. Now introduce noting that (A, B, C, D) all have dimensions of mass, so that from which we get the variational principle I = Ldτ, L = g ijẋ iẋj ≡ AṘ 2 + BR 2θ2 + DṘṪ + EṪ 2 1/2 .
As before, this is homogeneous degree zero in the parameter τ , so that the equations of motion form an incomplete set given by where m 1 is a scaling constant with dimensions of mass, together with d dτ In a classical context, the equation corresponding to (12) would give angular momentum conservation. Clearly, angular momentum is not conserved here (the forces are not assumed central) -but something is. For convenience, we shall refer to (12) as the equation for the conservation of generalized angular momentum.

Physical time and mathematical closure
As before, the problem is that the system is invariant wrt to arbitrary (monotonic) transformations of the temporal ordering parameter, τ , which implies that physical time is not defined -some physical condition is missing from the system. To progress, we could adopt the GR solution of introducing the notion of particle proper time -but that would entail ignoring the considerable success of the earlier approach (above and Roscoe [2], astro-ph/0306228) which depended critically upon the idea that the ratio of the magnitudes of transverse to radial force components was constant within any given disc. We therefore seek to generalize the approach of §4.1 to the present case using the line element (10). In the following, we sketch out the analysis, giving the details in appendix B. The only factorization of the line element (10) similar to that of (4) turns out to be whereR andθ are unit vectors as before, subject to the condition on M that We can now use (14) to define a generalized velocity, ds/dτ ≡ V, and a corresponding generalized acceleration, dV/dτ ≡V, in the space in a way which is directly analogous to the development of §4.1. The arbitrariness associated with the temporal ordering parameter, τ , is removed by applying the physical condition that the ratio of radial to transverse generalized accelerations is to be constant within the disc. Adopting the conventions that then this physical condition gives rise to the additional equations whilst (15) becomes In the above, V θ ≡ Rθ and V R ≡Ṙ are the rotational and radial components of the generalized velocity, V, and (p, q) are dimensionless parameters directly analogous to those of §4.1.
To summarize, our equations of motion are given by (12), (13), (16) and (17) together with the condition (18) which must be also satisfied. Thus, we now have five equations to be solved for four unknowns, (Ṙ, Rθ,Ṫ , M) so that the system appears to be overdetermined -unless there is redundancy somewhere. We shall show that this is, in fact, the case.

Linear dependencies with the system
We show, in appendix C, how (12), (16) and (17) lead to the equations where The structure of (19) and (20) is fundamental to the system. By removing the time-dependent terms, it is easily seen that the quasi-classical form of this latter pair is given by from which it is immediate that non-trivial solutions for v θ and v R can only exist if This equation is, in fact, a rearranged form of the quasi-classical mass-equation, (6), which immediately suggests the possibility that (19) and (20) are similarly linearly dependent. If so, we must necessarily have the conditions simultaneously satisfied. We immediately see that this is only possible if the additional mass condition holds -remember that A, B, D, E, Ψ are all defined in terms of M. But M must also satisfy (18), so we must now consider whether or not such a thing is possible.

The Mass Equation
Equation (18) has two basic classes of solution, given by a logarithmic class where (m 0 , d 0 ) are constants, and a variables separable class. The constant d 0 is irrelevant here since only derivatives of M appear in the theory; we ignore it from now on. The variables separable class, which is considered in appendix D for completeness, turns out to be associated with degeneracy in the system and so plays no part in the main body of this paper. The logarithmic class concerns us directly and we ask under what conditions (if any) the solution (24) can satisfy condition (23). Using the definitions (9) and (21) we easily find that the logarithmic class can only satisfy (23) if the condition is satisfied. Noting that R = −(1 + T ) is excluded by the structure of (24), then we must have With this latter condition, we find that M(R, T ) = m 0 ln(1 + R + T ) is a solution subject to R = 1 + T and the conditions (22) which now reduce to the single algebraic equation where M ≡ m 0 /m 1 , S 1 = ±1 and S 2 = ±1. To summarize, the solution can be expressed as on the characteristic R = 1+T subject to the condition that the dimensionless mass parameter M satisfies (25).

Rotational velocity = constant in the hyperbolic disc
In this section, we obtain the extremely interesting result that, when the mass distribution is as described in §5.4, then the rotational velocity is constant with a value determined by the parameters (p, q) and the algebraic equation (25). The four basic (but rearranged) equations of motion are: The first two equations arise directly from the variational principle (11), whilst the third and fourth equations arise from the conditions (16) and (17). In §5.3 we showed the subsidiary mass condition (18) could be satisfied by making (27) and (30) linearly dependent so that, in effect, we can consider the latter of these to be redundant. Consequently, we have three equations (27), (28) and (29) say, for the determination of (V θ , V R ,Ṫ ).
To progress, (28) must be expanded explicitly and this is done in appendix E, where we show how it reduces to an algebraic equation in the dimensionless velocity v θ with the solution where M ≡ m 0 /m 1 must satisfy (25). That is, the only possibility for rotation velocities in the disc of the quasi-relativistic theory is that of constancy -the rotation curve is flat.

Jump Conditions
We are concerned with the transition from the quasi-classical rising part of the rotation curve to the flat quasi-relativistic disc and, in all that follows, we use the superfix * to denote quantities in the quasi-relativistic disc. It transpires that everything reduces to deriving the relationships between the quasi-classical parameters (m 0 , m 1 ) and their quasi-relativistic counterparts (m * 0 , m * 1 ) and, to this end, we use jump conditions which quantify • conservation of generalized angular momentum -cf equation (12); • conservation of mass-flow across the transition boundary; • no-slip for the rotational flow at the transition boundary.
The first of these requires some discussion: within the context of the present model, forces within discs are not central forces -consequently, angular momentum is not conserved. However, if we consider (12) we see that the quantity R 2θ B/L (which plays the part here of classical angular momentum) is conserved in both the quasi-classical and hyperbolic discs. Accordingly, it is this quantity which must be conserved across the transition boundary.

Conservation of Generalized Angular Momentum
Reference to (12) shows that the quantity R 2θ B/L must be conserved across the radial transition boundary, R = R T say. Using the no-slip condition v * θ = v θ at R = R T then this requirement reduces to B * L * = B L across R = R T . Hence, following appendix F, we obtain the condition for generalized angular momentum across the transition boundary as m * 1 = m 1 .

Conservation of mass flow
The mass-flow conservation across the transition boundary, R = R T , can be stated as: Hence, following appendix G, we obtain the condition for mass conservation across the transition boundary as At this stage, we can note that Ψ = 0 necessarily otherwise, as is easily seen by reference to (39) and (40), dynamics in the quasi-classical disc becomes indeterminate. It follows immediately, from (32) above that

Summary of jump relations
For convenience, we collect the jump conditions in the form we use them, below: Using (34) then (32) and (33) give, after squaring, where we remember that M * ≡ m * 0 /m * 1 . Use of the generalized angular momentum jump condition m * 1 = m 1 now gives, after a bit of algebra, which we refer to as our first jump condition. Use of (34) in (32) gives which we refer to as our second jump condition whilst our third is simply

Forbidden zones in the parameter space
In the following, where the details of merging the rising part of a rotation curve (typically, the optical rotation curve) with its flat radio extension are considered, we remember that (p, q) are dimensionless disposable parameters which appeared firstly after (5) via K = q + pj and similarly in appendix B. We assume that they are fixed across the whole disc.
To simplify matters, we represent the rising part of the rotation curve purely in terms of the power law solution V = AR α . The theory says nothing about A since this would generally be determined by initial conditions, but is explicit about the structure of α in terms of the free parameters in the theory. Specifically, by (7) and the sentence which follows it, we have and we shall consider the constraints placed upon this quantity by the theory. Note that, in practice, we are only concerned with 0 < α < 1 since there are essentially no optical rotation curves outside of this range (Roscoe [27,30]). The constraining equations are, from (6) in the case of a power-law solution on the quasiclassical disc, the condition from (25) on the quasi-relativistic disc. Together with (35) and (36), these represent four constraints on the five unknowns (p, q, M * , m 0 /m * 0 , Ψ).

Brief computational details
There are five unknowns and four constraints together with various unknown signatures. A simple (inefficient with lots of redundancy!) approach is as follows: • choose q on the range (−0.5 < q < +0.5) in a uniform random way; • choose S * 1 = ±1 in a uniform random way, and then set S * 2 = −S * 1 ; • choose S 0 = ±1 and S * 0 = ±1 in uniform random ways; • solve the four constraining equations for (p, M * , m 0 /m * 0 , Ψ); • calculate α from (38) for each signature in turn and choose the value that satisfies 0 < α < 1, if one exists, and record it; • repeat a large number of times. • on branch I, the Jacobian is singular at about α = 0.43;

Results
• on branch II, the Jacobian is singular at about α = 0.40.
Thus, if the theory does describe aspects of reality for disc galaxies, then we would expect there to be a relative under-abundance of disc galaxies in that part of the parameter space which is in the neigbourhood of these singularities. However, there is a problem associated with considering α-data directly: whilst we know that α and ln A are extremely strongly correlated ( [27,30]) we also know that estimates of α from the data are an order of magnitude less accurate than those of ln A (Roscoe [28]). Consequently, rather than consider the α data directly, we consider it through the proxy of ln A. Figure 1b shows what we find: • The figure primarily shows the strong four-peak structure in the ln A distribution discussed in detail in Roscoe [30] in the context of the discrete dynamical states phenomenology; • Low values of ln A in the figure correspond to dim objects whilst high values of ln A correspond to bright objects. Thus, there are virtually no objects in the ranges ln A < 3.5 or ln A > 5.5; • The mean α-values corresponding to each of the four major peaks are indicated in the diagram; • The vertical dotted lines of figure 1b indicate the estimated ln A positions of the two singularities shown in α-space in figure 1a; • We see that, whilst they occupy the middle of the ln A range where we might expect to see the highest density of objects, they in fact lie in a region where there is a very significant under-abundance of objects.
We can conclude that the theory's predictions about the existence of singularities in the parameter space is in strong concordance with that which is observed.

Summary and conclusions
The quasi-classical one-clock model has enjoyed a great deal of success within the context of dynamics in the optical disc -but it fails to admit the possibility of a transition to flatness within discs. We then considered the possibility that a generalization of the quasi-classical model into a quasi-relativistic form might provide a resolution to this problem and found this to be the case. However, in order to merge solutions between the quasi-classical and quasirelativistic parts of the disc, it then became necessary to consider jump-conditions relating the two components of the disc. It transpired that the equations representing these jumpconditions became degenerate in exactly that part of the parameter space within which disc galaxies were already know to be significantly under-abundant -that is, in the valley between the two peaks corresponding to ln ≈ 4.17 and ln A ≈ 4.72 in figure 1.
Although the valley between 4.17 < ln A < 4.72 is the deepest of the valleys in figure 1, it is not unique -if the theory is able to provide an explanation for this deepest valley, we should also expect it to provide an explanation for the valleys between 3.90 < ln A < 4.17 and 4.72 < ln A < 5.10 -either in explicit terms of why these regions are underpopulated in discs or in explicit terms of why discs appear to prefer the peaks. As things stand, the twocomponent disc model does not do this. However, there are two major ways in which this latter model is significantly over-simplified: • Real disc galaxies have a central bulge region. In modelling terms, this could be represented by an interior rotating spherical distribution so that the over-all model would be a three-component disc model which would have two sets of jump-conditions placed upon the dynamics -one from interior sphere to the middle disc and one from the middle disc to the exterior flat disc. We might then expect these extra jump-conditions to provide additional explanations for observed features; • Our model assumes perfect rotational symmetry within discs whereas real discs have a very strong tendancy for a 180 o symmetry -that is, to have two symmetrically opposed spiral arms. We might expect the additional complexity arising from accounting for this two-arm structure to give rise to significant improvement in the explanatory power of the theory.
To conclude, the theory, which has the underlying structure of a LeSage-type vacuum gravitation theory, has 'clocked up' sufficiently many significant successes that one is inclined to believe that, at some level, it is reflecting reality. This alone is enough to justify a continued effort in its development.

A The possibility of transition to flatness within the quasiclassical model
It is sufficient to consider only equations (6) and (7) for which the parameters (p, q) are independent and, once chosen, fix the space of possible solutions. Suppose we consider the general power-law solution which necessarily arises whenever ρ = k/R 2 . In this case, (6) gives so that, for any choice of (p, q) then Ψ is defined to have one of two possible values, both constant. Thus, (7) becomes the simple power-law equation for some constant α ≡ α(p, q) which gives the exponent in the solution, V = AR α and, in Roscoe [30], we have seen that it is this class of solutions which defines the mean behaviour of optical rotation curves. Now, clearly, a flat rotation curve is a special case of the power-law for which α = 0. Thus, in this special case we have the additional relation where the ∓ arises because ǫ 2 = 1. Thus, (39) and (40) together define a relationship f (p, q) = 0 between the parameters (p, q) which must hold if α = 0. It follows that if the conditions for transition to flatness in the radio disc are satisfied within any given disc (that is, f (p, q) = 0), then the conditions for a non-zero power-law solution within the optical disc are excluded. But this is contrary to the conclusions the analyzes of Roscoe [27,30] for the generality of optical discs -that is, there are substantial reasons for believing that the transition to flatness does not occur within the context of the quasi-classical model.

B Physical time and mathematical closure
The omitted details for the analysis of §5.2 are given below. Using the line element (10) we find subject to the condition that M must satisfy We can now use (41) to define a generalized velocity in the space from which a generalized acceleration vector,V, can be defined aṡ Using the condition that the ratio of radial to transverse accelerations is to be constant within the disc, then, after multiplying through by S 2 , this latter equation leads to as the generalization of (5). Since, from the earlier analysis, we have that A < 0, it follows from (42) that E < 0 also. Since B > 0 necessarily, then (43) is also complex so that we must assume that K is complex, as before. Adopting the conventions that

C A Rearrangement of the equations of motion
The key to a successful resolution of the system lies in recognizing redundancy in a certain subset of the four basic equations of motion. This subset is obtained by processing equations (12), (16) and (17) in a certain way: From (12) with use of (18) we get directly We can now write equations (16) and (17) as We now eliminate d( √ BV θ )/dτ between these equations to get, after some work, Finally, using the identityΨ (there is no θ dependency) we find

∂Ψ ∂TṪ
Noting that (44) and this latter equation are homogeneous algebraic relations between V θ , V R andṪ and that the ordinary transverse and radial velocity components are given v θ = V θ /Ṫ and v R = V R /Ṫ , then we can write them as

∂Ψ ∂T
The structure of these two equations is fundamental to the system.
We ask under what conditions (if any) this solution (with β 0 = β 1 = 0 for simplicity) can satisfy condition (23). Using the definitions (9) and (21) we easily find that the variables separable class can only be a solution for the particular case M(R, T ) = m 0 √ RT , and S 0 = S 1 = ±1. If these conditions are satisfied, then we find that both of (22) reduce to a single algebraic condition having the general structure This corresponds to M = const on the characteristic surface R = const/T . As we shall show, it transpires that this class of solution is associated with a degeneracy in the system which removes them from the current discussion.

E Expansion of an Equation of Motion
Equation (28) is given by d dτ where L ≡ AṘ 2 + BR 2θ2 + DṘṪ + EṪ 2 1/2 =Ṫ Av 2 R + Bv 2 θ + Dv R + E 1/2 ≡Ṫ L and A, B, D and E are defined at (9). Thus (45) can be written as d dT where we have used the identity 1Ṫ d dτ ≡ d dT .
Equations (12) and (19) give, respectively Substituting these in (46), we find that the velocity derivatives, which all arise from the term, vanish identically so that (46) becomes purely algebraic in the velocity terms: in terms of L, v R and v θ it becomes using the definitions of A, B, D and E given at (9), and using Maple, we obtain the remarkable result that v θ = ± 1 √ 2 − M on the characteristic R = 1 + T where M ≡ m 0 /m 1 .

F Conservation of Generalized Angular Momentum
In the following, we give the details omitted in §7.1. Reference to (12) shows that the quantity R 2θ B/L must be conserved across the transition boundary. Using the no-slip condition v * θ = v θ at R = R T , this requirement reduces to B * L * = B L at the boundary so that, using (11) for L and the corresponding definition for L * we get Using

G Conservation of mass flow
In the following, we give the details omitted in §7.2. If we use R = R T to denote the radial transition boundary, the mass-flow conservation across this boundary can be stated as: on both parts of the disc, then we can write the mass-flow condition as Thus, we need appropriate expressions for (B * , v * R ) and (B, v R ). From §4.2 we have already noted that M * = m * 0 ln R so that using the definitions given at (9), then whilst in §5. 4 we have noted that M = m 0 ln(1 + R + T ) on R = 1 + T so that using the definitions given at (9), then also on R = 1 + T . Noting that the quasi-classical relations can be obtained from the quasirelativistic relations simply by dropping of all time-dependent terms then (27) gives, after using (48) and (21),