Point-like source solutions in modified gravity with a critical acceleration
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DELING IN NUCLEAR TECHNOLOGIES
Point-like Source Solutions in Modified Gravity with a Critical Acceleration1 Ja. V. Balitskya, b, * and V. V. Kiseleva, b, ** aMoscow
Institute of Physics and Technology (State University), Institutsky 9, Dolgoprudny, Moscow oblast, 141701 Russia Russian State Research Center Institute for High Energy Physics (National Research Centre Kurchatov Institute), Nauki 1, Protvino, Moscow oblast, 142281 Russia e-mail: *[email protected]; **[email protected]
b
Received November 6, 2015
Abstract—We consider equations of modified gravity with critical accelerations and find its solutions for the point-like source by suggesting the appropriate symmetry of metrics in empty space-time. Key words: modified Newtonian dynamics, critical acceleration, metrics DOI: 10.1134/S1063778816100021
Modified Newtonian dynamics [1] is a successful phenomenological setting for the description of “dark matter effects” at galactic scales [2]. Its concept is based on the introduction of a universal critical acceleration g 0 as the manifestation of empirical regularities in dark matter halos, so that the gravitation law is crucially modified at accelerations less than g 0 with the ordinary visible or baryonic sources in the following way
⎛g⎞ (1) g ζ ⎜ ⎟ = −∇ φ M , ⎝ g0 ⎠ where the interpolating function ζ is originally set to be equal to 1 ⎞2
⎛ y2 (2) ζ( y) = ⎜ , 2⎟ ⎝1 + y ⎠ while ϕ M is the gravitational potential of visible matter with density ρ M : Δφ M = 4π G ρ M . This law reproduces flat rotation curves in dark galactic halos of disc galaxies without introduction of any dark matter. We can rewrite the law of modified gravity in (1) with interpolating function (2) in terms of Ricci curvature tensor
∫ ∫
⎛ d 3rR00 ⎞ ⎟ = d 3rR00, (3) d 3 0 ⎜ d rK 0 ⎟ ⎝ ⎠ where the bar tensor is defined in terms of energymomentum tensor Tμν for the visible matter,
∫
3
rR00ζ ⎜
∫
1 The article is published in the original.
(
)
ν μ 1 ν ν R μ = 8πG Tμ − δ μT , T = Tμ , 2 and the external curvature is given by
(4)
(5) K 00 = g 0 2 , r Nevertheless, for instance, we can get the relativistic static solution in the modified gravity with the critical acceleration in the case of point-like source, that conserve both the spherical symmetry and stationarity. This symmetry in general relativity provides the metrics of the form 2 2 2 2 2 2 2 ds = f (r )dt − 1 dr − r (d θ + sin θ d φ ), (6) f (r ) due to the symmetry of source energy-momentum tensor: T00 = Trr . It is natural to extend this symmetry of metrics to the case of modified gravity with the point-like source to find the function f (r ). It is spectacular that (3) can be extrapolated in cosmology with a little modification in order to conserve the spatial homogeneity at extra-galactic scales in large scale structure of Universe as it was done in [3]: one can simply replace the external Ricci curvature by de Sitter one, K 00 K 00 = 3 g '0 , while the integration inside the sphere is canceled, that gives
⎛R0 ⎞ (7) R00ζ ⎜ 00 ⎟ = R 00. ⎝K 0⎠ In this paper
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