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The Motion of Massive Test Particles in Dark Matter

arXiv:hep-ph/9304300v2 16 Jul 1993IP-ASTP-11-93revised July, 1993The Motion of Massive Test Particles in Dark Matterwith an a0/r2 Energy DensityAchilles D. SpeliotopoulosInstitute of PhysicsAcademia SinicaNankang, Taipei, Taiwan 11529AbstractThe motion of massive test particles in dark matter is studied. It is shownthat if the energy density of the dark matter making up a galactic halo has alarge r behavior of 1/r2, then contrary to intuition the motion of these testparticles are not govern by Newtonian gravity, but rather by the equationsof geodesic motion from Einstein’s theory of general relativity.

Moreover,the rotational velocity curves of orbiting massive test particles in this energydensity do not approach a constant value at large r but will instead alwaysincrease with the radius of the orbit rc.Bitnet address: PHADS@TWNAS886

§1. IntroductionOne of outstanding problems in astrophysics today is to explain themotion of bodies with orbits in the galactic halo.

From Newtonian dynamicsone would expect that for a body in a circular orbitv2φrc= GMgr2cwhere G is the gravitational constant, Mg is the luminous mass of the galaxy,rc is the radius of the circular orbit and vφ is its rotational velocity. If onewere then to plot vφ verses r for various bodies orbiting in a galactic haloone would expect to obtain a rotational velocity curve which decreases as1/√rc.

This does not, in fact, happen. Experimentally, it is instead foundthat the rotational velocity curve approaches a constant value at large r. Toexplain this result, the presence of “dark matter”, matter which has yet tobe detected, has traditionally been proposed.

(See [1]−[3] and the referencescontained therein.) Namely, it has been postulated that the galactic halo isfilled with a gas of weekly interacting particles with a mass density ρ. Withtheir presence the total mass contained within the radius of the orbitingbody changes and the Newtonian equation of motion now becomesv2φ = GMgrc+ 4πrcZ rc0ρr2dr.If we now take ρ ≈a0c2/(Gr2) for large r where a0 is a dimensionlessconstant, then as long asa0 ≫MgG4πRgc2 ,(1)the Mg term may be neglected and a constant vφ with valuev2φc2 ≈4πa0 ,2

can be obtained. (Rg is the point at which the velocity curves become aconstant and is identified as the radius of the luminous galaxy.) Explana-tion of the rotational velocity curves then reduces to finding and detectingcandidates for this dark matter which will have the correct large r behavior.The basic premise of this argument is that Newtonian dynamics andgravity will still be valid even after the introduction of a 1/r2 energy den-sity into the system.

This need not be true. One should remember thatNewtonian gravity is only an approximation of Einstein’s theory of generalrelativity and that the very act of introducing a mass density ρ ∼1/r2 fordark matter introduces an unconfined energy density into the system.

Itspresence cannot help but have an affect on the geometry of the spacetimein the halo, and, consequently, on the motion of bodies orbiting there.In this paper we shall show that the standard argument using New-tonian dynamics and gravity for the existence of dark matter with a 1/r2energy density is inconsistent. The introduction of an energy density whichbehaves as 1/r2 for large r changes the geometry of the spacetime so dras-tically that the motion of bodies in the galactic halo is necessarily non-Newtonian.

The basic premise that Newtonian gravity is still valid evenafter the introduction of this dark matter is incorrect. Furthermore, afterusing the full geodesic equation from Einstein’s theory of general relativityto analyze circular orbits in the 1/r2 energy density, we find that v2φ ∼qrqcfor 0 < q ≤1.

The rotational velocity always increases with rc and will notapproach a constant value. In fact, for a static, spherical geometry thereare only a very narrow range of essentially unphysical choices for the energydensity which will give a constant vφ for r > Rg.§2.

Geometry of the a0/r2 energy densityWe begin by modeling the galaxy as a sphere of mass Mg and radius3

Rg which is surrounded by a galactic halo made up of dark matter. Themost general static, spherically symmetric metric is known to be [4]ds2 = −f(r)dt2 + h(r)dr2 + r2(dθ2 + sin2 θdφ2) ,where f and h are unknown functions of r only which need to be determined.As usual, we write the energy momentum tensor for the system asTµν = ρuµuν + p(gµν + uµuν) ,where ρ is the energy density of the particles in the halo, p is their pressure,and uµ is an unit velocity vector in the direction of the timelike Killingvector for the system.

From Einstein’s equations [4] we then obtain8πρ = 1r2ddrr1 −1h,8πp = f ′rhf −1r21 −1h,8πp =12f ′′fh −14f ′fhf ′f + h′h+f ′2rfh −h′2rh2 ,(2)where we are using units in which G = c = 1 and the primes denote deriva-tives with respect to r.Suppose now that the energy density of dark matter is ρ ≈a0/r2 forr > Rg where a0 > 0 is a (dimensionless) constant. Then the first equationin (2) is trivial to integrate givingh−1 = 1 −8πa0 −Kr ,where K is an integration constant.

Since in the absence of dark matterwe would expect to have obtained the Schwarzchild solution, we identify Kwith 2Mg. Next, notice that as long as the Newtonian bound (1) on a0holdsa0 ≫Mg4πr ,4

and we can neglect this term in h and can approximate h−1 ≈1 −8πa0 asa constant for r > Rg. We shall justify this approximation later.The difference of the second two equations gives0 = 12f ′′f −14f ′ff ′f + h′h−12rf ′f + h′h+ (h −1)r2.

(3)Since h is a constant for r > Rg, it is straightforward to solve,f(r) ≈k+rq+/2 + k−rq−/22,whereq± = 2 ± 21 −16πa01 −8πa01/2,and k± are integration constants. Because f > 0, a0 ≤1/16π otherwisef will contain oscillatory solutions.Since we are working in the large rlimit, and since q+ > q−, only one of the two solutions to (2) will survive.Linear combinations of the two will not.

We can thus consider each solutionindependently and for convenience we shall write f± = k±rq±. Then8πp± ≈q± + 1 −hhr2,and we see that the pressure also varies as 1/r2.

Moreover,p =q±4 −q±ρ ,and since q+ ≥2, p+ > ρ. Because p ≤ρ/3, we must therefore excludethe (+) solutions as being unphysical.

The only physical solutions are the(−) solutions, and we thus set q = q−, p = p−and f = krq. Furthermore,0 < q ≤1 while Mg/(4πRg) < a0 ≤3/(56π).

Consequently, 1 < h ≤7/4,and, since f ∼rq, we can see explicitly that spacetime in the presence ofthe 1/r2 energy density is not flat, but is instead quite curved.5

We next consider the motion of a massive test particle in a static,spherically symmetric geometry with a velocity vµ such that −1 = vµvµ.This constraint gives−1 = −f dtdτ2+ hdrdτ2+ r2dθdτ2+ r2 sin2 θdφdτ2,where τ is the proper time of the particle. Working in the equatorial θ = π/2plane, and using energy and angular momentum conservation,E = f dtdτ,L = r2 dφdτ ,(4)where E and L are the energy and orbital angular momentum per unit mass,respectively, of the particle, we obtain0 =drdt2+ V (r) ,whereV (r) = f 2E2h1 + L2r2−fh ,is an effective potential energy.

For circular motion, r = rc, the radius ofthe circular orbit which is a constant in time. Consequently, V (rc) = 0 andV ′(rc) = 0, givingE2 =2f 2(rc)2f(rc) −rcf ′(rc),L2r2c=rcf ′(rc)2f(rc) −rcf ′(rc).

(5)Defining the rotational velocity asvφ = r dφdtthen from (4) and (5) we find that for circular motion,v2φ = 12rcf ′(rc) . (6)6

This equation holds for any f. In particular, if we are dealing with a spher-ical mass M in free space, then f = (1 −2M/r) and (6) reduces to whatone obtains from Newtonian gravity. If, on the other hand, ρ ∼1/r2, thenfrom the above f = krq, so thatv2φ = 12qkrqc.For q ̸= 0, vφ always increases with the radius of the orbit and never ap-proaches a constant value as one would naively expect from Newtoniangravity.In the above solution of (3) for f we have neglected the contributionof h′/h in comparison to f ′/f.

We shall now justify this approximation.First, we note that for the galaxy, 2M/r ≪1 and, due to the bound (1)on a0, this term is very small in comparison to 1 −8πa0. Next, note thatwhile h′/h ∼2M/r2, the solution we obtained by taking h as a constantgives f ′/f ∼1/r.

Since r is large, we would expect h′/h to have a verysmall affect on the krq solution of (3). Consequently, we can take h′/h asa small perturbation and solve (3) perturbatively about the krq solution.After doing so, we find that to first order in 2M/r,f = krq1 −2Mh0(3 −q)rq2 + 2h0 + 1,where h−10≡1−8πa0.

The inclusion of the 2M/r term in h modifies the krqsolution very slightly since 2M/r ≪1, and 1 < h0 ≤7/4. Consequently, wewere justified in neglecting this contribution to f.§3.

Constant vφ energy densityWe now ask whether or not it is possible for any physically reasonableρ to result in a vφ which will be constant for r > Rg. Using (6), we findthat for vφ is to be a constant outside of the galaxy, f must then have the7

approximate form of fv ≈2v2φ log(r/r0) for r greater than some r0. (Thesubscript v will denote the fact that we are looking for solutions of Einstein’sequations which will result in a constant vφ.) Because fv > 0, r > r0, andsince vφ is a constant only outside of the galaxy, we shall identify r0 with Rg.Since fv is now given and hv unknown, (3) becomes a differential equationfor hv which may be written as0 = ddy 1hv−1 + 2yy 1hv+4y1 + 2y ,where y = log(r/Rg) > 0.

Its solution is straightforward,1hv= ye2ych + 2Z ∞2ye−t1 + tdt,where ch is an integration constant. Then from (2) we find that:8πr2ρv =1 +4y1 + 2y −(1 + 3y)e2ych + 2Z ∞2ye−t1 + tdt,8πr2pv =(1 + y)e2ych + 2Z ∞2ye−t1 + tdt−1.

(7)Physically, ρv ≥3pv. This gives an upper bound of 0.0013 for ch.

As h > 0for all y > 0, ch ≥0. Consequently, 0 ≤ch ≤0.0013 and there are only avery narrow range of values for ch which will result in a physically reasonableρv and pv.

Moreover, if ch > 0, then it is only when y is between some yminand ymax that ρv ≥3pv. Outside of these two bounds the energy densitymust have a different form and vφ cannot be a constant.

If, on the otherhand, ch = 0 then ρv ≥3pv for all y > 0.627. As long as r > e0.627Rg theenergy density given in (7) for ch = 0 will result in a constant rotationalvelocity curve outside of the galaxy.

For r < e0.627Rg the energy densitywill have a different form and vφ will not be a constant, as expected.§4. Concluding Remarks8

We have thus shown that if the energy density ρ of dark matter behavesas 1/r2 for large r, then f ≈krq for some constant k while h ≈1 −8πa0.Consequently, the premise that ρ ∼1/r2 for large r contradicts the premisethat Newtonian gravity is valid in the halo. The introduction of this energydensity, which is not confined but spread over a large area, alters the geom-etry of spacetime so drastically that Newtonian dynamics is no longer validin the halo.

The assumption that Newtonian gravity is valid even after theintroduction of dark matter with a 1/r2 energy density is incorrect. In fact,contrary to what is expected from Newtonian gravity, circular orbits for thisenergy density have a v2φ = qkrqc/2 which always increases with rc.

It neverapproaches a constant, although for very small q it increases very slowly.From a physicist point of view, however, the major problem with using a1/r2 energy density for dark matter is not that there was an inconsistency inthe argument for its introduction since this is straightforwardly resolved byusing general relativity instead of Newtonian gravity to analyze the system;nor is it that vφ always increases with rc since q may be taken to be quitesmall so that any increase in vφ occurs very gradually (with one cavet;see [5]).It is rather that the spacetime in a 1/r2 energy density is socurved that Newtonian dynamics is no longer valid. If the energy density ofthe dark matter in the galactic halo truly does behave as 1/r2 for large r,this will present great difficulties in interpreting extragalactic astronomicalobservations.As most galaxies have a halo, including presumeably ourown, light from a distant galaxy would then first have to pass through itsown halo, a region of curved spacetime, and then our through own galactichalo, another region of curved spacetime, before we can observe it.

Sincethe usual assumption is that light from other galaxies passes through aspacetime which is essentially flat before it reaches us, if the energy density9

of dark matter has a 1/r2 large r behavior, then all of the extragalacticobservational data would have to be re-evaluated and interpreted. This isjust one of the many problems that would arise from using a 1/r2 energydensity for dark matter which have yet to be addressed.The 1/r2 energy density is an unconfined energy density which, pre-sumeably, extends for large distances into the spacetime.

As the universeis known to be expanding and thus changing with time, one may questionthe validity of using a static solution of Einstein’s equations to analyze themotion of test particles in the dark matter as we have done. As, however,the galactic rotation curves only extend out to a few galactic radii, we areonly interested in the behavior of the motion of bodies relatively close tothe galaxy and in this region a static approximation is certainly valid.

Ofcourse, at some r ≫Rg the 1/r2 energy density must be cutoff-ed and ouranalysis will no longer be valid, but this will only happen outside the regionwe are interested in. We should also note that precisely the same static ap-proximation is made in the standard analysis of the motion test particles inthe galactic halo using Newtonian gravity.

The important point here is notwhether the use of a static, 1/r2 energy density to model the galactic halois a valid approximation or not, but rather that our analysis of the motionof test particles in a 1/r2 energy density using general relativity holds inprecisely the same regime in which the standard analysis using Newtoniangravity was presumed to be valid.As (3) is a non-linear second order differential equation in f, it maybe that we truly cannot neglect the h′/h term in (3) no matter how small2M/r is. In §3, however, we have shown that in a static, spherical geometrya rotational velocity curve which is truly a constant for r > Rg is obtainableonly for the very special choices of the ρv given in (7).

The only approx-10

imation made in obtaining (7) was once again that the system is static,and spherically symmetric. Notice, however, that if (7) is truly the energydensity of dark matter, this would mean that no matter what the internalproperties of the galaxy are, once one leaves it the energy density of theparticles making up its’ halo has to be determined within one part in 1000.This is extraordinarily and prohibitively restrictive.

The form that ρv takesis very particular and it is difficult to imagine a physical process which willnot only reproduce it, but also determine ρv to such a high degree of ac-curacy. On this basis alone we would tend to rule out ρv as a physicallyviable energy density for dark matter.

We also note, however, that becausef ∼log(r/Rg), once again the spacetime with this energy density is curvedand in using ρv as the energy density of dark matter we would once againbe faced with the problem of interpreting the observational data.The analysis done in this paper was done for a very special system un-der some very restrictive conditions. For example, although we have used aspherically symmetric geometry to model the galaxy, most observed galaxiesare axisymmetric and have a definite angular velocity.

It is, moreover, noteven clear whether the experimental data gives rotational velocity curveswhich are truly a constant or whether they are instead slightly increasing ordecreasing with rc. All that we are comfortable concluding from this analy-sis, therefore, is that one must be much more careful about introducing anyunconfined energy density for dark matter into the system.

Not only mustit be able to explain the experimental rotational velocity curves within theframework of general relativity, but one must also consider the subsequentaffects of this energy density on the geometry of the spacetime. As we haveseen, for the 1/r2 energy density these affects are considerable.11

AcknowledgementsADS would like to thank K.-W. Ng for many helpful discussions whilethis paper was being written. This work is supported by the National ScienceCouncil of the Republic of China under contract number NSC 82-0208-M-001-086.REFERENCES[1] V. C. Rubin, W. K. Ford, and N. Thonnard, Ap.

J. Letters, 225 L107(1978).

[2] V. Trimble, Ann. Rev.

Astron. and Astrophys., 25 425 (1987).

[3] E. W. Kolb and M. S. Turner, The Early Universe, Chapter 1 (Addison-Wesley Publishing Company, Inc., New York, 1990). E. W. Kolb and M.S.

Turner, The Early Universe: Reprints, Chapter 1 (Addison-WesleyPublishing Company, Inc., New York, 1988). [4] R. M. Wald, General Relativity, Chapter 6 (The University of ChicagoPress, Chicago, 1984).

[5] Since we have taken a0 ≫MgG/(4πRgc2), there is a lower bound belowwhich we would not be able to decrease q. For a typical galaxy withMg ≈1012M⊙and Rg ≈5 kpc, this bound turns out to be ∼2 × 10−5.Note also that because p = q/(4 −q)ρ, very small q would correspondto very “cold” dark matter which would have a temperature which isvery much smaller than its mass.12

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