Mimicking Dark Matter

0.35 MB·

by  Lluís Bel

Mimicking Dark Matter
Dedicated to Vera Rubin
Ll. Bel∗
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1
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2 Abstract

n I show that a very simple model in the context of Newtonian physics
a promoted to a first approximation of general relativity can mimic Dark
J matter and explain most of its intriguing properties. Namely: i) Dark

2 matter is a halo associated to ordinary matter; ii) Dark matter does not
2 interactwithordinarymatternorwithitself; iii)Itsinfluencegrowswith
the size of the aggregate of ordinary matter that is considered, and iv)
]
Dark matter influences the propagation of light.
h
p
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n 1 Isolated Point particles
e
g
Let us consider two point particles, one of mass m and the other of mass unity,
.
s at a location xi apart, and assume that Newton’s law of attraction is modified
c
i so that :
s
y (cid:18) (cid:19)
Gm 1 1 1
h Fi =− xi−Gm + + xi (1)
p r3 a1r2 a22r a33
[
wherexi areCartesiancoordinates,GisNewton’sgravitationalconstantanda
i
2 are three parameters, with dimension length, supposedly to be determined by
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experiments or observations. It should be understood that at least one of these
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parameters is not infinite. This formula is a variant of proposals first made in
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0 Refs. ([2]), ([3]) and ([4]). 1
3 Defining the potential V, as usual up to an additive constant by:
0
.
1 Fi =−∂iV (2)
0
we get:
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1
:
v m m 3m
∆V =4πGρ, ρ=mδ(r)+σ(r), σ(r)= + + (3)
Xi 4πa1r2 2πa22r 2πa33
r
a From where it follows that the potential V would be:
Gm (cid:18) 1 r r2(cid:19)
V =− +Gm ln(r)+ + (4)
r a a2 a3
1 2 3
∗e-mail: [email protected];www.lluisbel.com
1InthreeofmypreviousarXivpapers,[5],[6]and[7],Iassumedthata2 anda3 were∞
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From the definition of σ above it follows that to calculate the force that a
particle of mass m exerts on a particle of mass unit it is equivalent to use (1)
or to calculate:
G (cid:18) (cid:90) u (cid:19)
Fi =− m+4π σ(u)u2du xi (5)
r2
0
Equivalently we can say that the gravitational mass of the source particle has
increased by a fictitious mass, or Dark mass, whose density is σ.
Thus, dark mass comes to live as halos surrounding every single particle of
ordinary mass and we can not expect these halos to interact otherwise with
ordinary mass or among themselves if several particles are considered.
Obviously the credibility of this interpretation should be checked against
possible contradictions derived from observations.
Several qualitative considerations can be made already. There are in the
Universe different aggregates of ordinary matter; e.g., globular clusters (size
scale say of the order of 102 ly) , small spheroidal galaxies (size scale say of the
order of 103 ly) , large galaxies of different types (size scale say of the order of
105 ly), and aggregates or clusters of them (size scale say of the order of 106
ly). The value of any one of the parameters a’s will affect only those structures
with a scale larger than the corresponding size scale.If it is large compared to
the scale of the parameters a(cid:48)s they will be very much affected by Dark mass,
bothinthedynamicsofitsownstructureandinthestrengthofitsgravitational
interaction with external objects. If it is small they will be little affected. This
would explain, for instance, that if all three parameters a’s are much greater
than the size scale of globular clusters, dark matter effects are not important
for them, while they are for larger structures. This is nice because it is one of
the deep mysteries that everybody has in mente.
2 Point particles aggregates
A modification of Newton’s law of force would modify also the dynamics of
Galaxy clusters in dynamical equilibrium through an application of the Virial
theorem at an approximation where such clusters could be compared to an
aggregate of point particles. Theorem that in this case tells us that:
−G(cid:88)(cid:18) mimj +m m (cid:18) 1 + |ri−rj|2 + |ri−rj|3(cid:19)(cid:19)+(cid:88)m (cid:18)dri(cid:19)=0
2 |r −r | i j a a2 a3 i dt2
i(cid:54)=j i j 1 2 3 i
(6)
contributing thus to explain the anomaly pointed out by Zwicky, ([1]), who
checked that the preceding formula without the terms depending on the a’s did
not fit the data.
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3 Continuous spherical distributions
Continuoussphericaldistributionsofordinarymattercanbedealtwithasusual
in Newtonian theory. For example if the density of ordinary matter is a contin-
uous function µ(u) this will require to replace the function σ(r) in (3) by:
(cid:90) R (cid:90) π µ(u)u2sinθ
σ =2π du (7)
r2+u2−2rucosθ
0 0
where R is the radius, eventually ∞, of the configuration. Considering as an
example the cuspidal Hernquist profile:
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µ(u)= (8)
u(1+u)3
The figures below show the graphics of µ(u), and σ (u) corresponding to the
a
parameters a = 1, a = a = ∞ in the first case and a = 1, a = a = ∞ in
1 2 3 2 1 3
the second.
Hernquistvs.Darkmatter:(a(2)=1)
Hernquistvs.Darkmatter
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3
2
2
1 1
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
r r
Hernquist Dark Hernquist Dark
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The familiar Newtonian law for the rotational velocity curves beyond the finite
radius of galaxies becomes:
(cid:115)
Gm(cid:18) r r2 r3(cid:19)
v = 1+ + + (9)
r a a2 a3
1 2 3
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5 Light bending
ThereremainstounderstandhowamodificationofNewton’slawcouldproduce
light bending. A problem that I already considered promoting the framework
of Newtonian physics to the framework of Einsten’s theory to first order of
approximation in a simplified case: a =a =∞ ([6]).
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I consider here two possible promotions of the Newtonian-like formalism: i)
either introducing the line-element:
ds2 =−(1+2V(r))dt2+(1−2V(r))(dr2+r2(dθ2+sin2θdφ2)) (10)
where V(r) is the potential considered in (4). Or ii) the line-element
ds2 =−(1+2V(r))dt2+(1−2V(r))dr2+r2(dθ2+sin2θdφ2) (11)
Assumingthata =a =a =∞,theline-element(10)isthefamiliarlinear
1 2 3
approximation of Schwarzschild solution in harmonic or isotropic coordinates,
whiletheline-element(11)isthefamiliarlinearapproximationofSchwarzschild
solution in Drosde’s coordinates. However when the parameters a(cid:48)s have finite
values the two line-elements have quite different interpretations that can be
made explicit calculating the corresponding Einstein’s tensor. In the first case
we get:
(cid:18) (cid:19)
2 4 12
S =−Gm + + (12)
00 a r2 a2r a3
1 2 3
the remaining components being zero.
In the second case the non zero components are:
(cid:18) (cid:19)
2 4 6
S = Gm (1+ln(r)+ + (13)
00 r2a ra2 a3
1 2 3
(cid:18) (cid:19)
2 4 6
S = Gm − (1+ln(r)− − (14)
11 r2a rb2 a3
1 3
(cid:18) 1 2r 6r2(cid:19)
S = Gm − − − (15)
22 a a2 a3
1 2 3
S = S sin2θ (16)
33 22
Notice that (10) can be very nicely understood as a solution of Einstein’s
equation with Dark matter density but no pressure and noteworthy is that
assuminga =a =∞whatwegetisthat(12)becomesthefirstapproximation
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of the Kottler solution ([8]).
On the other hand the interpretation of the line-element (11) would require
to make sense of the pressure terms.
Now comes a second delicate point. I have used the same letters r,θ,φ as
if they were polar coordinates of Euclidean space in Newtonian physics. But is
this justified on both theories based on the line-elements (10) and (11). I think
not. Ithinkthatitisjustifiedin(10)anditisnotin(11)apointthatIpresent
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more explicitly in the Appendix, and therefore I consider (10) to be the proper
promotion of the Newtonian generalization that I have been considering.
NowwecandealwithlightraysasusualinGeneralrelativityconsideringthe
equations of null auto-parallels. Because of the spherical symmetry of the line-
elementthespace-trajectoriesoflightrayswilllieonaplaneanditisconvenient
to assume that this plane is the plane φ=0
The explicit equations are then, s being an arbitrary parameter along the
rays:
d2t dV dr dt dt
−2 −b =0 (17)
ds2 dr dsds ds
d2r dV (cid:18)dr(cid:19)2
+
ds2 dr ds
(cid:18)dV (cid:19)(cid:32)(cid:18)dθ(cid:19)2 (cid:18)dφ(cid:19)2(cid:33) dV (cid:18)dt(cid:19)2 dr
−r r+1 + − −b =0 (18)
dr ds ds dr ds ds
d2θ 2(cid:18)dV (cid:19)drdθ
+ r+1 =0 (19)
ds2 r dr dsds
d2φ 2(cid:18)dV (cid:19)drdφ dθ
+ r+1 −b (20)
ds2 r dr ds ds ds
b being a function of s, that is by definition an affine parameter if b=0.
Choosing as parameter s=t reduces the preceding system to the following
two equations:
d2r dV (cid:32) (cid:18)dr(cid:19)2 (cid:18)dθ(cid:19)2 (cid:33) (cid:18)dθ2(cid:19)
+ 3 −r2 −1 −r = 0 (21)
dt2 dr dt dt dt
d2θ dθdr (cid:18) dV 2(cid:19)
+ 4 + = 0 (22)
dt2 dt dt dr r
That it is equivalent to the system of equations of the geodesics of the three
dimensional elliptic metric:
dt2 =(1−4V)(dr2+r2(dθ2+sin2θdθ2) (23)
These equations have to be integrated using initial conditions:
(cid:18) (cid:19) (cid:18) (cid:19)
dr dθ
r , , θ , (24)
0 dt 0 dt
0 0
satisfying the following condition:
(cid:32)(cid:18)dr(cid:19)2 (cid:18)dθ(cid:19)2(cid:33)
−(1+2V(r ))+(1−2V(r )) +r2 =0. (25)
0 0 dt 0 dt
0 0
Let us now calculate the general relativistic bending of light by the Sun
assuming that a = a = a = ∞. Using units such that G = c = r = 1,
1 2 3 0
r being the radius of the Sun it turns out that the mass of the Sun is m =
0
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0.000002136477017meters. Integratingthesystemofdifferentialequations(21)
and (22) with initial conditions:
(cid:18) (cid:19) (cid:18) (cid:19)
dr dθ 2m
r =1, θ =0, =0, r =1+ (26)
0 0 dt 0 dt r
0 0 0
from t=−t to t=+t , t being a sufficiently large value of t. we get:
∞ ∞ ∞
∆θ =0.00000875rad (27)
thatcorrespondto1.8arcsec,whichisthewellknownresult2. Itiseasytocheck
repeatingtheintegrationprocesswiththreesetofparameters: i)a =100,a =
1 2
∞,a = ∞, ii) a = ∞,a = 100,a = ∞) and iii) a = ∞,a = ∞,a =
3 1 2 3 1 2 3
1000 the corresponding bending results are equal to the observed observed one.
Thereforetheseparametersorgreateronescannotbeexcludedtomimickdark
matter beyond the solar system to much larger scales.
Appendix
Let us consider a general relativistic metric:
ds2 =−A2(t,xk)(−dt2+f (t,xk)dxi)2+A−2g¯ (t,xk)dxidxj, i,j,k =1,2,3
i ij
(28)
Call the time-like world-lines congruence t variable the reference system, and
the 3-dimensional metric g¯ the metric of space. Consider now the Euclidean
ij
metric (or any other 3-dimensional constant curvature metric):
ds˜2 =g˜ (t,xk)dxidxj (29)
ij
where xk are whatever coordinates suits you. Then its meaning in (29) and in
(28) will be the same if and only if:
(Γ¯i −Γ˜i )g¯jk =0 (30)
jk jk
where the Γ(cid:48)s are the Christoffel connection symbols of g¯ and g˜
ij ij
Acknowledgments
JuanMariAguirregabiriaandLuisAcedohelpedmetoimprovethismanuscript.
References
[1] F. Zwicky, Helv.Phys.Acta, 6, 1933
[2] A. Finzi, MNRAS 127, 1963
[3] , R. H. Sanders, Astron. Astrophys, 136, 1984
[4] J. R. Kuhn & L. Kruglyak, Ap. J., 1987
2thereference[9]givesalistofhistoricalresults
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[5] Ll. Bel, arXiv:1308.0249v2 [physics.gen-ph]
[6] Ll. Bel, arXiv:1311.6891v1[gr-qc]
[7] Ll. Bel, arXiv:1404.0225v1[gr-qc]
[8] F. Kottler, Annalen Physic, 410 (1918)
[9] K. Brown Reflections on Relativity, page 422, 2016
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