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30
Вестник Челябинского государственного университета. 2012. № 31 (285). Физика. Вып. 15. С. 32-33.
АСТРОФИЗИКА
A. R. Khaybullina, J. A. Arslanova, S. N. Mikhailov,
I. R. Khamidullin, D. I. Fazlyev, R. N. Izmailov
INFLUENCE OF CONICAL DEFECT ON LIGHT BENDING IN THE GALACTIC HALO
Observed flat rotation curves in the galactic halo led the scientific community to hypothesize the existence of copious amounts of invisible matter, called dark matter, in the halo region of any galaxy. However, current theoretical research indicates that Weyl conformal gravity can consistently predict flat rotation curves without requiring the existence of hypothetical dark matter. We investigate here the influence of a conical defect on light deflection in the galactic halo within the framework of conformal gravity. The observational limit of the defect is calculated.
Keywords: Weyl conformal gravity, flat rotation curves, dark matter.
The hypothesis of dark matter in the galactic halo is a well-known one. There have been numerous models in the literature for the dark matter, all of which of course have their own merits. Recent research however indicates that the astrophysical observations of flat rotation curves in the halo region can be predicted within the Weyl gravity without requiring the dark matter [1]. The model also allows determination of maximal size of individual galaxies [2]. The dark matter is hypothesized due to the following reason: Doppler emissions from stable circular orbits of neutral hydrogen clouds in the halo allow the measurement of tangential velocity vg (r) of the clouds treated as probe particles. According to Newton’s laws, centrifugal acceleration vg / r should balance the gravitational attraction GM(r)/ r2, which immediately gives v2g = GM(r) / r . That is, one would expect a fall-off of vg (r) with r. Observations indicate that this is not the case: vg approximately levels off with r in the distant halo region. The only way to reconcile this result of observation is to hypothesize that the mass M (r) increases linearly with distance r. Luminous mass distribution in the galaxy does not follow this behavior. Hence the conclusion that there must be huge amounts of nonluminous matter hidden in the halo. This unseen matter is given a technical name dark matter”. Gravitational lensing measurements have further confirmed the presence of dark matter. Current estimates suggest that about 23 % of matter in the whole universe consists of dark matter residing in the galactic haloes.
The relevant metric in the conformal theory is given by Mannheim and Kazanas [3] which shows modifications to Schwarzschild metric by two parameters Y and k . Small departures from
exact spherical symmetry are often needed to accommodate the practical situation, for instance, situations like solar oblateness etc. In this paper, we shall introduce yet another small departure from the Mannheim—Kazanas metric in terms of a conical defect b (= 1 - s) and calculate its effect on light deflection when the rays pass through the edge of a galaxy.
The metric exterior to a static spherically symmetric distribution in Weyl conformal gravity has been obtained by Mannheim and Kazanas [3], which reads (G = c = 1):
di2 =-B(r)dt2 + ^^ dr2 + r2(d02 + sin2 0d^2),(1)
2M 2
B(r) = a-----\-jr -кг ,
r
(2)
where a = (1 - 6My)1/2 is a constant, M is the luminous mass, y = 3,03 x 10-30 cm -1 and k = 9,54 x10-54 cm-2 that fit the rotation curve data quite satisfactorily [1]. For distances neither too small nor too large, one may take a = 1, as we do here. We introduce the conical defect parameter b as follows
d T2 =-B(r )dt2 +dr2 + B(r )
+ r2 d 02 + b2 r2 sin2 0d ф2,
B(r) = 1 -2M + jr -кг2,
r
(3)
(4)
with u =1, the photon trajectory from (1) is given by
,,l2
^4 = -ub2 +3MU2b2 -Yb d ф2 2
V- (5)
As evident, k has disappeared from the above equation. Solving perturbatively to first order in M, we have
and taking y ~ 10 30 cm [1], we have
1 sin bti Y M
u =- =----------- —- + -
R
2 4R2
•[6 + 3R2y2 - 3RY(n - 2b^) cos b^ + (6)
+ 2cos2b^- 6Ry sin b^],
where the parameter R is related to the closest approach distance r0 at ^ = n /2. Note that the familiar Schwarzschild orbit equation u = ^sin^ + m(3 + cos2^) is obtained at Y = 0, b = 1. 2R
The angle between the two asymptotes gives the total angle of deflection 89, which works out, for small s, to
5^ = -b
( 4M \ (4M \
1 R - yr — ——yR
) 1 R )
(1 + £). (7)
Thus, the difference in deflection between the case s 4 0 and s = 0 is
4M
R
- YR
£.
Thus the effect on deflection is a tiny portion of the deflection under purely spherical symmetry. An upper limit on s by order of magnitude can be obtained from the error involved in the measurement of deflection. Typically, for galaxies
5^ U'
10-
and so at the galactic scale length, if the measurement involves, say, an error of 10 %, the conical defect parameter will be bounded by
s < 10-3.
This bound coincides with the bound obtained from the measurement of light deflection by the Sun
[4].
References
1. Mannheim, P. D. Impact of a global quadratic potential on galactic rotation curves / P. D. Mannheim, J. G. O’Brien // Phys. Rev. Lett. 2011. Vol. 106. P. 121101-121105.
2. Nandi, K. K. Impact of a global quadratic potential on galactic rotation curves / K. K. Nandi, A. Bhadra // Phys. Rev. Lett. 2012. Vol. 109. P. 079001-079002.
3. Mannheim, P. D. Exact vacuum solution to conformal Weyl gravity and galactic rotation curves / P. D. Mannheim, D. Kazanas // Astrophys. J. 1989. Vol. 342. P. 635-640.
4. Freire, W. H. C. Cosmological constant, conical defect and classical tests of general relativity / W. H. C. Freire, V. B. Bezerra, J. A. S. Lima [preprint]: gr-qc /020103.
M
1015cm, R
1022 cm
