On the Geometrical Conditions to Determine the Flat Behaviour of the Rotational Curves in Galaxies

Transcription

1 On the Geometrial Conditions to Determine the Flat Behaviour of the Rotational Curves in Galaxies Departamento de Físia, Universidade Estadual de Londrina, Londrina, PR, Brazil M.E.X. Guimarães Departamento de Matemátia, Universidade de Brasília, Brasília, DF, Brazil M. Leineker Costa Departamento de Físia, Universidade de Brasília, Brasília, DF, Brazil Observational data establish that in large samples of disks galaxies the tangent veloity of tests partiles in a oplanar orbit is radii independent. With this knowledge, we onstrut a theorem that furnishes the geometrial onditions on the metri oeffiients of an axisymmetri stationary spae-time in order to tests partiles to obey these data. After this, we apply this theorem to a dilatoni urrent-arrying osmi string and arrive to a onstraint on the mirosopi gauge model. Fifth International Conferene on Mathematial Methods in Physis April 2006 Centro Brasileiro de Pesquisas Fisias (CBPF/MCT), Rio de Janeiro, Brazil Speaker. Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommerial-ShareAlike Liene.

2 1. Introdution Observational data measuring the rotational urves in some galaxies show that oplanar orbital motion of gas in the outer part of galaxies maintains a onstant veloity up to several luminous radii [6, 7, 8, 9, 10]. The most aepted explanation for this effet is that there exists a spherial halo of dark matter whih surrounds the galaxy and aounts for the missing mass needed to produe the flat behavior of the rotational urves. It is reasonable to suppose that the halo of the dark matter is symmetri with respet to the rotation axis of the galaxy, so we onsider here an axisymmetri spaetime. In previous works a osmi string in salar-tensor gravities were onsidered [1, 2, 3]. This kind of soure is an example of axisymmetri spaetime. This work is organized as follows. In setion 2 we impose the trajetory of the test partile in this stati axisymmetri spae-time to be oplanar and radii independent and then obtain its angular veloity in terms of the oeffiients of the metri. In setion 3 we re-write the line element of this region using the Chandrasekhar form and we alulate the tangential veloity of these test partiles. Suh alulation leads to a theorem that gives a neessary and suffiient ondition on the metri oeffiients in order to have tangential veloities of equatorial objets irling the galaxy and whose magnitude is radii independent. In setion 4 we apply this theorem to the ase of a spae-time generated by a dilatoni urrent arrying osmi string [1, 2]. Finally, in setion 5 we present some onlusions. 2. The Line Element The line element of an axially symmetri spae-time is given in the form [11]: ds 2 = e 2ψ ( + ωdϕ)+e 2ψ [ e 2γ ( d 2 + dz 2) + µ 2 dϕ 2] (2.1) where ψ, ω, γ and µ are funtions of (,z). The Lagrangean for a test partile travelling on the stati spae-time (ω = 0) desribed by (2.1) is given by: 2L = e 2ψ ṫ 2 + e 2ψ [ e 2γ ( 2 + ż 2) + µ 2 ϕ 2], (2.2) thus, the assoiated anonial momenta, p x a = L ẋ a, are p t = E = e 2ψ ṫ, p ϕ = L = µ 2 e 2ψ ϕ, p = e 2(ψ γ), p z = e 2(ψ γ) ż, (2.3) where E and L are onstants of motion for eah geodesi, a fat that omes from the symmetries of the spae-time analyzed. As there is no expliit dependene on time t, the Hamiltonian, H = p a ẋ a L is another onserved quantity, whih we normalize to be equal minus one half for timelike geodesis. Also, we restrit the motion to be at the equatorial plane, thus ż = 0. In this way, we obtain the following equation for the radial geodesi motion: 2 e 2(ψ γ) [Eṫ L ϕ 1] = 0. (2.4) 2

3 In order to have stable irular motion, whih is the motion we are interested in, we have to satisfy three onditions: i) = 0 ii) V() = 0, where V() = e 2(ψ γ) [Eṫ L ϕ 1], iii) 2 V() 2 extr > 0, in order to have a minimum. With these onditions, from (2.4), we obtain a set of two equations onstraining the motion to be irular extrema in the equatorial plane: Eṫ L ϕ 1 = 0, (2.5) ( ) e 2(ψ γ) [Eṫ L ϕ 1] = 0. (2.6) From (2.3), we an express ṫ and ϕ in terms of E and L, and the metri oeffiients as: ṫ = e 2ψ E, (2.7) ϕ = e2ψ L. (2.8) µ 2 Using these equations in the onstraints ones and realling that E and L are onstants for eah irular orbit, after some rearranging, we arrive at the following equations: µ 2 e 2ψ ( 1 e 2ψ E 2) + L 2 = 0, (2.9) ( e 2ψ) ( ) e 2ψ E2 + = 0, (2.10) where the subindex stands for derivative with respet to. Solving for E and L, we obtain: E = e ψ µ ψ µ 2ψ, L = µe ψ ψ µ 2ψ. (2.11) µ 2 The seond derivative of the potential V() evaluated at the values of E and L whih onstraint the motion to be irular and extrema, is given by: ( V extr = 2e2(ψ γ) µ µ 2ψ µ ψ µ ψ + 4ψ 3 6 µ ( ) ) 2 µ µ ψ2 + 3 ψ. (2.12) µ We an now obtain an expression for the angular veloity of a test partile, Ω, moving in a irular motion in the orbital plane, in terms of the metri oeffiients, realling that: Ω = dϕ = ϕ ṫ, (2.13) thus, using Eqs. (2.8) and (2.11) in this last equation for the angular veloity, we obtain that: Ω = e2ψ ψ µ µ ψ. (2.14) 3

4 3. The Tangential Veloity We now want to express the tangential veloity of the test partiles in irular motion in the equatorial plane, in terms of the metri oeffiients, following [5], we rewrite the line element (2.1) as: ds 2 = e 2ψ 2 + e 2ψ µ 2 dϕ 2 + e 2(ψ γ) d 2 (3.1) thus, in terms of the proper time, dτ 2 = ds 2, we have that [ ( ) dϕ 2 ( ) ] d 2 dτ 2 = e 2ψ 2 1 e 4ψ µ 2 e 2γ e ψ. (3.2) from whih we an write 1 = e 2ψ u 02 [ 1 v 2 ], (3.3) where u 0 = dτ is the usual time omponent of the four veloity, and a definition of the spatial veloity, v 2, omes out naturally in this way. This spatial veloity is the 3-veloity of a partile measured with respet to an orthonormal referene system, thus has omponents: ( ) dϕ 2 ( ) d 2 v 2 = e 4ψ µ 2 + e 2γ e 4ψ. (3.4) The orthogonal veloity is the 3-veloity of a partile measured with respet to an orthonormal referene system, thus has omponents: v 2 = v (ϕ)2 + v ()2. (3.5) From these last two expressions we obtain for the ϕ-omponent the spatial veloity: v (ϕ) = e 2ψ µω, (3.6) and replaing Ω from Eq. (2.14), we finally obtain an expression for the tangential veloity of a test partile in stable irular motion, in terms of the metri oeffiients of the general line element given by Eq. (2.1), suh tangential veloity has the form: v (ϕ) = ψ µ ψ. (3.7) It was our goal to obtain this expression for the tangential veloity for a general axisymmetri stati spae-time, and to be able to desribe it in terms of the metri oeffiients alone, beause now we an impose onditions on this tangential veloity, and dedue a onstraint equation among the metri oeffiients, whih has to be satisfied in order to fulfill the ondition imposed on the veloity. In partiular, the tangential veloity for a trajetories in eah orbit is onstant, that is v (ϕ) = 0, thus v () = v (ϕ) (3.7), we have that:, with v (ϕ) a onstant, representing the value of the veloity, from Eq. 2 µ = 1+v(ϕ) ψ 2 (3.8) v (ϕ) 4

5 Theorem: The tangential veloity of irular stable equatorial orbits is onstant iff the oeffiient metri are related as ( ) µ l e ψ = (3.9) with l = te. We an see that this is a neessary and suffiient ondition for the veloity v (ϕ) to be the same for ( ) ( 2/ ( ) ) 2 two orbits at different radii at the equatorial plane, provided that l = 1+. In order to have tangential veloities of equatorial objets irling the galaxy, and whose magnitude is radii independent, the form of the line element in the equatorial plane has to be µ 0 v (ϕ) v (ϕ) ( ) µ 2l ( ) µ 2l ds 2 = 2 [ + e 2γ d 2 + µ 2 dϕ 2]. (3.10) 4. Stable Cirular Geodesis Around a Dilatoni Eletrially Charged Cosmi String where The metri of a eletrially harged osmi string is [1, 2]: ( r ds 2 E = ( r ) 2l 2 2n ( r W 2 (r)(dr 2 + dz 2 )+ ) 2n W 2 (r)b 2 r 2 dθ 2 ) 2n 1 W 2 (r) 2 (4.1) W(r) ( r ) 2n + k. 1+k The onstants m, n, k and B will be determined after the inlusion of matter fields. Our objetive in this setion will be to derive the geodesi equations in the equatorial plane (ż = 0), where dot stands for derivative with respet to the proper time τ". First of all, let us re-write the metri (4.1) in a more ompat way: with ds 2 = A(r) [ dr 2 + dz 2] + B(r)dθ 2 C(r) 2, (4.2) ( r A(r) = ) 2m 2 2n W 2 (r), ( ) r 2n B(r) = W 2 (r)β 2 (r), ( ) r 2n C(r) = W 2 (r) 5

6 The Lagrangian for a test partile moving in this spae-time is given by: 2L = A(r) [ ṙ 2 + ż 2] + B(r) θ 2 C(r)ṫ 2 (4.3) The assoiated anonial momenta, p α = L ẋ α, are: p t = E = C(t)ṫ, p θ = L = B(r) θ, p r = A(r)ṙ, p z = A(r)ż. (4.4) Beause of the symmetries of this partiular spae-time, the quantities E and L are onstants for eah geodesi and, beause this spae-time is stati, the Hamiltonian, H = p α ẋ a L, is a onstant. Combining this information with the restrition of a motion in an equatorial plane, we arrive to the following equation for the radial geodesi: ṙ 2 A 1[ Eṫ L θ 1 ] = 0. (4.5) In this work, we will onentrate on stable irular motion. Therefore, we have to satisfy three onditions simultaneously. Namely: ṙ = 0 ; V(r) r = 0, where V(r) = A 1[ Eṫ L θ 1 ] ; 2 V(r) r 2 ext > 0, in order to have a minimum. Consequently, we have: Eṫ L θ 1 = 0 (4.6) { A 1 [ Eṫ L θ 1 ]} = 0 r Expressing ṫ and θ in terms of the onstant quantities E and L respetively, we an re-write the above equations as: ( ) ( ) 1 1 E 2 L 2 1 = 0 C ( 1 C B ) E 2 ( ) 1 L 2 = 0 (4.7) B where prime means derivative with respet to the oordinate r", whih finally gives us expressions for E and L: B E = C B C BC, C L = B B C BC (4.8) 6

7 Realling that the angular veloity of a test partile moving in a irular motion in an orbital plane is Ω = dθ = θṫ, we have: C Ω = B (4.9) We are now in position to ompute the tangential veloity of the motion in an orbit plane. From now on, we will follow the presription established by Chandrasekhar. Let us re-express the metri (4.2) in terms of the proper time τ, as dτ 2 = ds 2 : [ dτ 2 = C(r) 2 1 A ( ) dr 2 B ( ) ] dθ 2 (4.10) C C and omparing with the expression where u 0 = dτ, we an easily obtain the spatial veloity v2 : whose omponents are, respetively: ( A dr v (r) = C ( B dθ v (θ) = C 1 = C(r) ( u 0) 2[ 1 v 2 ], (4.11) v 2 = (v (r)) 2 ( + v (θ)) 2, (4.12) ), ) = B Ω. (4.13) C In order to have stable irular orbits, the tangential veloity v (θ) must be onstant at different radii at the equatorial plane. Therefore, we an impose: BC v (θ) = B C = v(θ) = onst. (4.14) Applying the theorem to this ase, we get provided l = v(θ) 1+v (θ) ( ) r l C 1/2 =, (4.15). This theorem implies that the line element in the equatorial plane must be: ( ) r 2l ( ) [ r 2l ( ) ] r 2m 2 ds 2 = 2 + dr 2 + B 2 r 2 dθ 2 (4.16) This form is learly not asymptotially flat and also does not desribe a spae-time orresponding to a entral blak hole. Therefore, we an infer that it desribes solely the region where the tangential veloity of the test partiles is onstant, being probably joined in the interior and exterior regions with other metris, suitably hosen in order to ensure regularity in the asymptoti limits. 7

8 Let us notie, however, that this metri has the form whih has been found previously [1, 2], after identifying l with the appropriate onstant parameters whih depend on the mirosopi details of the model. The alulations are straightforward but length. For this partiular onfiguration, onsisting of an eletrially harged dilatoni string, we have: l = 2G 0 α(φ 0 ) [ U + T + I 2], (4.17) where U, T and I 2 are the energy per unit length, the tension per unit length and the urrent of the string, respetively. α(φ 0 ) measures the oupling of the dilaton to the matter fields. 5. Conlusion We found the onditions on the metri oeffiients of a stati axisymmetri spae-time to admit a test partile with a oplanar irular orbit radii independent up to several luminous radii. A remarkably fat is that the results presented in setions 2 and 3 are independent of the type of the energy-matter tensor present in the spae-time and urving it. It is a purely geometri analysis. A possible example of this kind of spae-time is the one generated by a dilatoni eletrially harged osmi string. [ Considering osmi strings formed at GUT sales, G 0 U + T + I 2 ] , and for a oupling α(φ 0 ) whih is ompatible with present experimental data, α(φ 0 ) < 10 3, the parameter l (and thus the tangential veloity v θ ) seems to be too small. The observed magnitude of the tangential veloity being v (θ) > annot be explained by a single dilatoni urrent-arrying osmi string in this ase. As argued by Lee [4], if a bundle of N osmi strings formed at GUT sales seeded one galaxy, then the total magnitude of the tangential veloity would be Nv (θ). In our ase, to be ompatible with astronomial observations, one must have a bundle of N 10 5 strings seeding a galaxy. With suh a density, a osmi string network would be dominating the universe, and its dynamis would be ompletely different. The only situation where suh a huge number of strings ould be possible is at muh lower energy sales (eletroweak sale, say) but then of ourse the energy sale is far too low to have any relevane for struture formation. Aknowledgements M. Leineker Costa would like to thank CAPES for a support. and M. E. X. Guimarães would like to thank CNPq for a support. Referenes [1] A. L. N. Oliveira and M. E. X. Guimarães, Breakdown of the Mehanism of Forming Wakes by a Current-Carrying String, Phys. Lett. A (2003) [hep-th/ ]. [2] A. L. N. Oliveira and M. E. X. Guimarães, Wakes in Dilatoni Current-Carrying Cosmi Strings, Phys. Rev. D (2003) [hep-th/ ]. ed. G.W. Gibbons, S. W. Hawking and T. Vahaspati (Cabridge Univ. Press, Cambridge, 1990). [3] A.L. Naves de Oliveira, Dilatoni, Chiral Cosmi Strings, PoS WC2004 (2004) 34 [gr-q/ ]. 8

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