AA geometric alternative to dark matter C OLIN R OURKE
The existence of “dark matter”, inferred from the observed rotation curves ofgalaxies, is a hypothesis which is widely regarded as problematic. This paperproposes an alternative hypothesis based on the space-time geometry near a rotatingbody and formulated in terms of the dragging of inertial frames. This hypothesisis true in a certain linear approximation to General Relativity (Sciama [9]) and isjustified in general by Mach’s principle. Dark matter corrects the rotation curvebut does not predict the ubiquitous spiral structure of galaxies. The geometricalternative suggested here deals with both problems and allows the construction ofa simple model for the dynamics of spiral galaxies which fits observations well.85A05; 85A15, 83C57, 83F05
One of the major problems with the current consensus model for the universe is theproblem of dark matter which has no observations to support its existence and noagreement as to its nature. The purpose of this paper is to propose an alternative basedon the geometry of space-time, which, like dark matter, explains observed rotationcurves. There is a dual problem to the dark matter problem, namely the spiral structureproblem. In the current consensus model for the universe, with dark matter, there isno satisfactory model for galactic dynamics that explains the ubiquitous spiral form ofmany galaxies. By contrast the geometric postulate made here fits observed rotationcurves and also provides a good model for galactic dynamics fitting observed spiralstructure.Dragging of inertial frames, henceforth abbreviated to “inertial drag” or ID, is arelativistic effect whereby the non-inertial motion of a body (acceleration and / orrotation) causes the inertial frames at other points to be dragged. In his thesis [9]Sciama proposed that this effect should embody Mach’s principle (that inertial framesdepend on the total distribution of matter in the universe) and that this should specifythe full dynamics of the universe. He illustrated this with a special case (a certain linearapproximation to general relativity) where it all works perfectly. His formula for the IDeffect of a rotating body is this: a r X i v : . [ phy s i c s . g e n - ph ] F e b Colin Rourke
Weak Sciama Principle (WSP)
A mass M at distance r from a point P , rotating withangular velocity ω , contributes a rotation of k M ω/ r to the inertial frame at P where k is constant. It is called the “Sciama principle” to distinguish it from the more general “Machprinciple” and “weak” because it specifies only the effect of one rotating body, insteadof all non-inertial motions. The “Full Sciama Principle” (for rotation) states that therotation of a local inertial frame is obtained by adding the effects for all rotating bodiesin the accessible universe (ie not regressing faster than c ).The factor k M / r must be regarded as a weighting factor and the sum must be dividedby the sum of the weights. For an example see the derivation of (1) below.This principle (the WSP) is the proposed geometric hypothesis. The key factor for therotation curve is 1 / r which has several pieces of evidence in its favour. It fits Sciama’sapproximation. It fits the behaviour of apparent acceleration (see Mach [6, Ch II.VI.7(page 286)]). There is a simple dimensional argument that supports it (see the discussionof the precession of the Foucault pendulum in Misner, Thorne and Wheeler [7] startingon page 547 para 3 with the margin note The dragging of the inertial frame ).The hypothesis is obviously related to Mach’s principle though it is much weaker. Butit has the advantage that it is capable of direct verification by observation. Althoughtrue in an approximation to general relativity (GR), it is not true in general in GR: inthe Kerr metric, ID drops off with 1 / r not 1 / r . Therefore to accomodate it withinGR it is necessary to assume that a rotating body has an effect on the local field thatcauses the hypothesised ID effect and stops it being a vacuum. This effect embedsMach’s principle (for rotation) within GR and avoids the paradoxes that arise in a naiveformulation of the principle: the ID field is a form of gravitational disturbance thatpropagates, like all gravitational disturbances, at the speed of light. The basic assumption is that the centre of every galaxy contains a heavy rotating mass(presumably a black hole). It is the ID effects from this mass that cause equatorialorbits to exhibit the characteristic rotation curve. However, the analysis applies to anyaxially-symmetric rotating body, which does not need to be assumed to be heavy.To fix notation, consider a central mass M at the origin in 3–space which is rotating inthe right-hand sense about the z –axis (ie counter-clockwise when viewed from above) geometric alternative to dark matter with angular velocity ω . Assume a flat background space-time, away from M , withsufficient fixed masses at large distances to establish a non-rotating inertial frame nearthe origin, if the effect of M is ignored. Let P be a point in the equatorial plane (the( x , y )–plane) at distance r from the origin. The rotation of the inertial frame at P isgiven by adding the contribution from M to the contribution from the distant masses.The inertial frame at P is rotating coherently with the rotation of M by the average of ω weighted kM / r and zero (for the distant fixed masses) weighted C say. Normalise theweighting so that C = k / C by k ) which leaves justone constant k to be determined by experiment or theory. The nett effect is a rotation of(1) ( kM / r ) × ω + × kM / r ) + = Ar + K where K = kM and A = K ω . Note
If the full Sciama principle is assumed then k = K = M and A = M ω .However the choice of k = K and M .When constructing models for galaxies in Section 3 the value k = P (at distance r from the origin) is rotatingwith respect to the background with angular velocity ω ( r ) counter-clockwise. Whencomputing rotation curves, the formula for ω ( r ) just found (1) will be used but for thepresent discussion it is just as easy to assume a general function. The inertial frame at P can be identified with the background space, but it is important to remember that it isrotating. There is no sensible meaning to the centre of rotation for an inertial frame.Two rotations which have the same angular velocity but different centres differ by auniform linear motion and inertial frames are only defined up to uniform linear motion.Thus it can be assumed for simplicity that all the rotations have centre at the origin.Then the inertial frames can be pictured as layered transparent sheets, each comprisingthe same point-set but with each one rotating with a different angular velocity about theorigin. Each sheet corresponds to a particuar value of r . It is necessary to be very clearabout the nature of motion in one of these frames. A particle moving with a frame (ie Colin Rourke one stationary in that frame) has no inertial velocity and its velocity is called rotational .In general if a particle has velocity v (measured in the background space) then v = v rot + v inert where its rotational velocity v rot is the velocity due to rotation of the local inertial frameand v inert is its inertial velocity which is the same as its velocity measured in the localinertial frame. Note that v rot = r ω ( r ) directed along the tangent.Inertial velocity correlates with the usual Newtonian concepts of centrifugal force andconservation of angular momentum. The fundamental relation
As a particle moves in the equatorial plane it moves between the sheets so that a rotationabout the origin which is rotational in one sheet becomes partly inertial in a nearby sheet.For definiteness, suppose that ω ( r ) is a decreasing function of r and consider a particlemoving away from the origin and at the same time rotating counter-clockwise about theorigin. The particle will appear to be being rotated by the sheet that it is in and thiscauses a tangential acceleration. This acceleration is called the slingshot effect becauseof the analogy with the familiar effect of releasing an object swinging on a string. Butat the same time the particle is moving to a sheet where the rotation due to inertial dragis decreased and hence part of the tangential velocity becomes inertial and is affectedby conservation of angular momentum which tends to decrease the angular velocity.These two effects balance each other out in the limit and this explains the flat asymptoticbehaviour. More precisely, let v be the tangential velocity of the particle in the directionof ω then the slingshot effect causes an acceleration dv / dr = ω ( r ) but conservation ofangular momentum that operates on the inertial part of v , namely v − r ω ( r ) causes adeceleration in v of ( v − r ω ( r )) / r or an acceleration dv / dr = ω ( r ) − v / r and addingthe effects we have the fundamental relation between v and r :(2) dvdr = ω − vr Given ω as a function of r , (2) can be solved to give v as a function of r . Rewrite it as r dvdr + v = ω r . The LHS is d / dr ( rv ) and the general solution is(3) v = r (cid:16)(cid:90) ω r dr + const (cid:17) . geometric alternative to dark matter It is now clear that any prescribed differentiable rotation curve can be obtained bymaking a suitable choice of continuous ω . Remark
It is worth remarking that the fundamental relation does not depend on theWSP. It is a purely geometric result depending only on axial symmetry and the factthat the dynamic is controlled by a central force. It is true in any metrical theory and inparticular in general relativity without the WSP. However for the models constructedhere it will be used with the formula for ω found in (1) which does depend on the WSP.Of interest here are solutions which, like observed rotation curves, are asymptoticallyconstant, and, inspecting (3), this happens precisely when (cid:82) ω r dr is asymptoticallyequal to Qr for some constant Q . But this happens precisely when 2 ω is asymptoticallyequal to Q / r . This proves the following result. Theorem
The equatorial geodesics have tangential velocity asymptotically equal toconstant Q if and only if ω is asymptotically equal to A / r where Q = A . The basic model
Now specialise to the case ω = A / ( r + K ) which gives the value of inertial dragformulated in (1).From (3) v = r (cid:18)(cid:90) Arr + K dr + C (cid:19) = Ar (cid:18)(cid:90) − Kr + K dr (cid:19) + Cr = A − AKr log (cid:16) rK + (cid:17) + Cr (4)where C is a constant depending on initial conditions. For a particle ejected from thecentre with v = r ω for r small, C =
0, and for general initial conditions there is acontribution C / r to v which does not affect the behaviour for large r . For the solutionwith C = r small, v ≈ r ω and the curve is roughly astraight line through the origin. And for r large the curve approaches the horizontal line v = A . A rough graph is given in Figure 1 (left) where K = A =
1. The similaritywith a typical rotation curve, Figure 1 (right), is obvious. Note that no attempt has beenmade here to use meaningful units on the left. See Figure 3 below for curves from themodel using sensible units.But notice that every equatorial orbit has the salient feature of observed rotation curves,namely a horizontal asymptote. This asymptote is the same for all equatorial orbits and
Colin Rourke hence any average over many orbits will also have this asymptote and this explains theobserved rotation curve.
21 10 20 30 40Figure 1: The rotation curve from the model (left) and for the galaxy NGC3198 (right) takenfrom Begeman [4]
Units
The models for the rotation curve and for galactic dynamics given in this paper are fully quantitative using natural units, in which G (Newton’s gravitational constant) and c (the velocity of light) are both 1. Time is measured in years, distances in light-years,velocities in fractions of c and mass converted into distance using the Schwarzschildradius: a mass of 1 means a mass with Schwarzschild radius 1 (light-year). Thus avelocity of .001 is 300km / s, a distance of 45,000 is 15Kpc and a mass of 1 is 3 × solar masses all approximately. When using natural units, pure numbers are used. Theycan be converted into more familiar units as indicated here.There are other shapes for rotation curves; see [10] for a survey. All agree on thecharacteristic horizontal straight line. Figure 2 is reproduced from [10] and gives a goodselection of rotation curves superimposed. In Figure 3 is a selection of rotation curvesagain superimposed, sketched using Mathematica and the model given here. Thecentral masses for the curves in Figure 3 vary from 3 × to 10 solar masses. Thedifferent curves correspond to choices of A , K and C . The similarity is again striking. The notebook
Rots.nb used to draw this figure can be collected from [2] and the values ofthe parameters used read off. geometric alternative to dark matter Figure 2: A collection of rotation curves from [10]
Figure 3: A selection of rotation curves from the model
It is worth commenting that the observed rotation curve for a galaxy is not the sameas the rotation curve for one particle, which is what has been modelled here. Whenobserving a galaxy, many particles are observed at once and what is seen is a rotationcurve made from several different rotation curves for particles, which may be close butnot identical. So it is expected that the observed rotation curves have variations fromthe modelled rotation curve for one particle, which is exactly what is seen in Figure 1(right) and Figure 2.
Postscript
As remarked earlier, the effect described in this section is independent of mass. Howeverfor rotating bodies of small mass the effect is unobservably small. For example the
Colin Rourke sun has K ≈ K = M , and ω Sun = π/
25 days. Thus the asymptotictangential velocity 2 A = K ω Sun is 6km per 4 days or 1m / s approximately. The analysis given in Section 2 will now be extended to find equations for orbits ingeneral (not just for the tangential velocity) and, using a hypothesised central generator,the spiral arm structure will be modelled as well. The basic idea is that the central massaccretes a belt of matter which develops instability and explodes feeding the roots of thearms. Stars are formed by condension in the arms and move outwards as they develop.Thus a typical star is on a long outward orbit and the rotation curve observed for stars ina spiral arm is formed of many such similar orbits. But this full picture is not necessaryto explain the observed rotation curves, since the tangential velocity for all orbits hasthe same horizontal asymptote.
Note
The assumption of a hypermassive central black hole in a spiral galaxy directlycontradicts current beliefs of the nature of Sagittarius A ∗ and this problem togetherwith other observational matters are dealt with in Chapter 6 of the book [8] of whichthis paper is a fragment. Very briefly, SgrA ∗ and the stars in close orbit around it forman old globular cluster near the end of its life with most of the matter condensed intothe central black hole. It is not at the centre of the galaxy but merely roughly on lineto the centre and it is about half-way from the sun to the real galactic centre which isinvisible to us. Plotting orbits
Equation (4) gave a formula for the tangential velocity in an orbit. What is needed is aformula for the radial velocity (again in terms of r ) and these two will describe the fulldynamic in the equatorial plane, which can then be used to plot orbits.There are two radial “forces” on a particle: a centripetal force because of the attractionof the massive centre and a centrifugal force caused by rotation in excess of that due toinertial drag. Thus radial acceleration ¨r is given by(5) ¨r = v r − F ( r )where v inert = v − ω r and F ( r ) is the effective central “force” at radius r , per unit mass,which, since we are using a Newtonian approximation, is M / r . The same notation as geometric alternative to dark matter in the last section is used here and in particular ω = ω ( r ) is the inertial drag at radius r .Now specialise to the case ω = A / ( r + K ), equation (1), which was the formula forinertial drag coming from the Weak Sciama Principle. Here A = K ω and K = kM ,where M , ω are the mass and angular velocity of the central mass, and k is a weightingconstant which can be taken to be 1 for purposes of exposition. The following formulafor v , equation (4), was found:(6) v = A − AKr log (cid:16) rK + (cid:17) + Cr where C is a constant which can be read from the tangential velocity for small r . Thisimplies:(7) v inert = A − AKr log (cid:16) rK + (cid:17) + Cr − ArK + r Thus: ¨r = v r − Mr = r (cid:20) A − AKr log (cid:16) rK + (cid:17) + Cr − ArK + r (cid:21) − Mr Multiplying by ˙ r and integrating with respect to t gives ˙ r = (cid:90) ¨rdr = − C r + M − ACr + A KK + r + A log( K + r )(8) + AK ( C + Ar ) log(1 + r / K ) − (2 AK log(1 + r / K )) r + E where E is another constant determined by the overall energy of the orbit. From thisequation ˙ r can be read off (in terms of r ). Moreover since there is a formula for v , thereis also a formula for ˙ θ = v / r , where polar coordinates ( r , θ ) are used in the equatorialplane. From this it is possible to express θ and t in terms of r as integrals. Theseintegrals are not easy to express in terms of elementary functions but Mathematica canbe used to integrate them numerically and this can be used to plot the orbits of particlesejected from the centre. Now use the hypothesis that the centre of a normal galaxycontains a belt structure, which emits jets of gas / plasma, which condense into stars.The orbits of these stars can be modelled and a “snapshot” of all the orbits taken at aninstant of time, in other words a picture of the galaxy can be given, Figure 4.This compares well with classic observed spiral galaxies, Figure 5.To end this section it is worth remarking that it may be thought that motion along thearms is a new hypothesis and that observations should be able to verify this. However the observations aleady exist . The rotation curve shows that there is a general outwardmotion and, since the arms are close to tangential, as seen in the above figures, this Colin Rourke - - - - Figure 4: Mathematica plot using equations (8) and (6)Figure 5: M101 (left) and M51 (right): images from the Hubble site [1] motion is roughly along the arms, which is therefore already observed. The newhypothesis is that this motion is responsible for the long-term maintenance of the spiralstructure. The model demonstrates all of this both graphically and quantitatively. geometric alternative to dark matter As noted earlier, this paper is an extract from the author’s book “A new paradigm forthe universe” [8], which covers all the topics discussed here in greater detail. Thejustification for the key factor 1 / r in the main geometric hypothesisis is given in detail in[8, Chapter 2]. The derivation of the fundamental relation (2) and the radial accelerationequation (5) are both proved in detail for a larger class of suitable metrics in [8, Chapters3 and 5]. The proposed accretion structure (the “generator”) which creates the jetswhich condense into the spiral arms is described in more detail in [8, Chapter 5]. TheMathematica notebook used for the plot, Figure 4, is typed out (and can be downloadedfrom [2] as Basic.nb ) and the data used spelt out as follows.2 A the asymptotic tangential velocity is .
001 which is 300km/s in MKS units. M hasbeen set to 10 solar masses. The minimum radius for the plot, rmin , has been set to5,000 and the maximum, rmax , to 50,000 light years (corresponding to a visible diameterof 100,000 light years). There is a precession constant B which arises because inertialdrag causes the frame at the origin to appear to rotate at A / rmin and B = A / rmin corrects this and causes the familiar spiral form. Time elapsed along the visible armsis 5 . × years. The nature of the visible spiral arms is discussed carefully in [8,Chapter 6]. Here merely note that the visible arms correspond to strong star-producingregions and bright short life stars, which burn out or explode in 10 to 10 years. Thusa total time elapsed of 5 . × years allows several generations of stars to be formedand to create the heavy elements necessary for planets such as the earth to be formed. Remark
The rate of rotation of the central mass ω can be read from the othervariables. Since A = K ω , from equation (1), and the default value K = M has beenchosen, ω = A / M = . /. = /
60. In other words the central mass is rotating atabout one radian every 60 years or one revolution every 360 year approx. This oughtto be observable, but since the central mass in a galaxy is obscured by the bulge, allthat can be observed are the velocities of stellar regions near the centre and these aredominated by the rotation curve. Since the correct rotation curve is built into the model,observations here are in agreement with the model.The program is intended for interactive use and the reader is recommended to downloada copy and investigate the output. Hints on using it can be found in [8, Section 5.6].The book also covers a wide range of other topics and in particular gives a new frameworkfor quasars, which fits with the observations of Halton Arp [3] and others, that quasarstypically exhibit intrinsic redshift, and explains the apparently paradoxical results ofHawkins [5]. Colin Rourke
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The Hubble site , http://hubblesite.org [2] Mathematica notebooks , available from http://msp.warwick.ac.uk/~cpr/paradigm/Nb [3]
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