aa r X i v : . [ phy s i c s . g e n - ph ] D ec Varying G dynamics
B.G.Sidharth ∗ International Institute of Applicable Mathematicsand Information SciencesAdarsh Nagar, Hyderabad&B.S.Lakshmi † Department of Mathematics, JNTUCEHKukatpally, Hyderabad
Abstract
In this paper it is shown that dynamics based on a variation of thegravitational constant G with time solves several puzzling and anoma-lous features observed, for example the rotation curves of galaxies (at-tributed to as yet undetected Dark matter). It is also pointed outthat this provides an explanation for the anomalous acceleration ofthe Pioneer space crafts observed by J.D.Anderson and co-workers. The Milky Way to which our sun belongs, contains about 100 billion stars.On even larger scales, individual galaxies are concentrated into clusters ofgalaxies. These clusters consist of the galaxies and any material which is inthe space between the galaxies. The force that holds the cluster together isgravity. The space between galaxies in clusters is filled with a hot gas. Thecluster includes the galaxies and any material which is in the space between ∗ Email:[email protected] † Email:[email protected] · · · · · Orbitalspeed (a) Expected Keplerian curves (b) Observed Rotation Curves
Figure 1: Galactic Rotation Curves2
Varying G Dynamics we would first like to observe that even after all these decades there is neitherevidence for the hypothesized Dark matter, nor any clue to its exact nature,if it exists. Let us see how varying G dynamics removes the need for DarkMatter thus providing an alternative explanation. Cosmologies with timevarying G have been considered in the past, for example in the Brans-Dicketheory or in the Dirac large number theory or in the model of Hoyle [1, 16,15, 3, 17]. In the case of the Dirac cosmology, the motivation was Dirac’sobservation that the supposedly large number coincidences involving N , thenumber of elementary particles in the universe had an underlying message ifit is recognized that √ N ∝ T (1)where T is the age of the universe. Equation (1) leads to a G decreasinginversely with time in Dirac’s hypothetical development.The Brans-Dicke cosmology arose from the work of Jordan who was moti-vated by Dirac’s ideas to try and modify General Relativity suitably. In thisscheme the variation of G could be obtained from a scalar field φ which wouldsatisfy a conservation law. This scalar tensor gravity theory was further de-veloped by Brans and Dicke, in which G was inversely proportional to thevariable field φ . (It may be mentioned that more recently the ideas of Bransand Dicke have been further generalized.)In the Hoyle-Narlikar steady state model, it was assumed that in the Machiansense the inertia of a particle originates from the rest of the matter present inthe universe. This again has been shown to lead to a variable G . The abovereferences give further details of these various schemes and their shortcom-ings which have lead to their falling out of favour.Then there is fluctuational cosmology in which particles are fluctuationallycreated from a background dark energy, in an inflationary type phase tran-sition and this leads to a scenario of an accelerating universe with a smallcosmological constant. This 1997 work [22] was observationally confirmed ayear later due to the work of Perlmutter and others [20]. Moreover in thiscosmology, the various supposedly miraculous large number coincidences asalso the otherwise inexplicable Weinberg formula which gives the mass of anelementary particle in terms of the gravitational constant and the Hubbleconstant are also deduced from the underlying theory rather than being ad3oc.To quote the main result, the gravitational constant is given by G = G T (2)where T is time (the age of the universe) and G is a constant. Furthermore,other routine effects like the precession of the perihelion of Mercury and thebending of light, and so on have also been explained in this model. Moreoverin this model, the cosmological contant Λ is given by Λ ≤ H ) and showsan inverse dependence 1 /T on time. We will see that there is observationalevidence for (2).With this background, we now give some tests for equation (2). Let us first see the correct gravitational bending of light. In fact in Newtoniantheory too we obtain the bending of light, though the amount is half thatpredicted by General Relativity[14, 5, 23, 4]. In the Newtonian theory wecan obtain the bending from the well known orbital equations (Cf.[6]),1 r = GML (1 + ecos Θ) (3)where M is the mass of the central object, L is the angular momentumper unit mass, which in our case is bc , b being the impact parameter orminimum approach distance of light to the object, and e the eccentricity ofthe trajectory is given by e = 1 + c L G M (4)For the deflection of light α , if we substitute r = ±∞ , and then use (4) weget α = 2 GMbc (5)This is half the General Relativistic value.We now observe that in this case we have, G = G o (1 − tt o ) (6) r = r o (cid:18) t o t o + t (cid:19) (7)4e now observe that the effect of time variation of r is given by equation(7)(cf.ref.[19]). Using this, the well known equation for the trajectory is givenby, u ′′ + u = GML + u tt + 0 (cid:18) tt (cid:19) (8)where u = r and primes denote differentiation with respect to Θ.The first term on the right hand side represents the Newtonian contributionwhile the remaining terms are the contributions due to (7). The solution of(8) is given by u = GML (cid:20) ecos (cid:26)(cid:18) − t t (cid:19) Θ + ω (cid:27)(cid:21) (9)where ω is a constant of integration. Corresponding to −∞ < r < ∞ in theNewtonian case we have in the present case, − t < t < t , where t is largeand infinite for practical purposes. Accordingly the analogue of the receptionof light for the observer, viz., r = + ∞ in the Newtonian case is obtained bytaking t = t in (9) which gives u = GML + ecos (cid:18) Θ2 + ω (cid:19) (10)Comparison of (10) with the Newtonian solution obtained by neglecting terms ∼ t/t in equations (8) and (9) shows that the Newtonian Θ is replaced by Θ2 , whence the deflection obtained by equating the left side of (10) to zero, is cos Θ (cid:18) − t t (cid:19) = − e (11)where e is given by (4). The value of the deflection from (11) is twice theNewtonian deflection given by (5). That is the deflection α is now given notby (5) but by the formula, α = 4 GMbc , (12)The relation (12) is the correct observed value and is the same as the GeneralRelativistic formula which however is obtained by a different route [4, 2, 7].5 Galactic Rotation Curves and Dark Matter
We now come to the problem of galactic rotational curves mentioned earlier(cf.ref.[14]). We would expect, on the basis of straightforward dynamics thatthe rotational velocities at the edges of galaxies would fall off according to v ≈ GMr (13)However as seen in Section (1), it is found that the velocities tend to aconstant value, v ∼ km/sec (14)as we approach the edges of the galaxies. This, as noted, has lead to the pos-tulation of the as yet undetected additional matter alluded to, the so calleddark matter.(However for an alternative view point Cf.[24]). We observe thatfrom (7) it can be easily deduced that [20, 18] a ≡ (¨ r o − ¨ r ) ≈ t o ( t ¨ r o + 2 ˙ r o ) ≈ − r o t o (15)as we are considering infinitesimal intervals t and nearly circular orbits.Equation (15) shows (Cf.ref[19] also) that there is an anomalous inward ac-celeration, as if there is an extra attractive force, or an additional centralmass.So, we now have GM mr + 2 mrt o ≈ mv r (16)From (16) it follows that v ≈ (cid:18) r t o + GMr (cid:19) / (17)From (17) it is easily seen that at distances within the edge of a typicalgalaxy, that is r < cms the equation (13) holds but as we reach the edgeand beyond, that is for r ≥ cms we have v ∼ cms per second, inagreement with (14). In fact as can be seen from (17), the first term in thesquare root has an extra contribution (due to the varying G ) which is roughlysome three to four times the second term, as if there is an extra mass, roughlythat much more. In fact the velocity at the edge of the galaxies as calculated6rom equation (15) are tabulated in the following table, where the radius isin units of 10 cm. The table shows that equation (15) is in agreement withthe observed velocity given in equation (12).7 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Velocity Radius . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . × . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G explains observation without invoking dark mat-ter.It may be added that this also explains the latest studies by Metz,Kroupaand others of the satellite galaxies of the Milky Way galaxy , which also throwup the faster than expected rotational velocities, ruling out however, in thiscase dark matter, in addition. There could be other explanations, too. One of the authors and A.D. Popovahave argued that if the three dimensionality of space asymptotically falls off,then the above can be explained [21].Yet another prescription was given by Milgrom [8] who approached the prob-lem by modifying Newtonian dynamics at large distances. It must be men-tioned that this approach is purely phenomenological.The idea was that perhaps standard Newtonian dynamics works at the scaleof the solar system but at galactic scales involving much larger distances per-haps the situation is different. However a simple modification of the distancedependence in the gravitation law, as pointed by Milgrom would not do, evenif it produced the asymptotically flat rotation curves of galaxies. Such a lawwould predict the wrong form of the mass velocity relation. So Milgrom sug-gested the following modification to Newtonian dynamics: A test particle ata distance r from a large mass M is subject to the acceleration a given by a /a = M Gr − , (18)where a is an acceleration such that standard Newtonian dynamics is agood approximation only for accelerations much larger than a . The aboveequation however would be true when a is much less than a . Both thestatements in (18) can be combined in the heuristic relation µ ( a/a ) a = M Gr − (19)In (19) µ ( x ) ≈ x >> , and µ ( x ) ≈ x when x <<
1. It mustbe stressed that (18) or (19) are not deduced from any theory, but ratherare an ad hoc prescription to explain observations. Interestingly it must bementioned that most of the implications of Modified Newtonian Dynamicsor MOND do not depend strongly on the exact form of µ .It can then be shown that the problem of galactic velocities is now solved99, 10, 11, 12, 13].Finally it maybe mentioned that the above varying G dynamics explains thepuzzling anomalous acceleration of the Pioneer spacecrafts of the order of10 − − cm/sec observed by J.D.Anderson of the JPL Pasadena, andco-workers [25]. References [1] Barrow, J.D. and Parsons, P. (1997).
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