The gravitational deflection of light in MOND

Abstract

The deflection angle $\Delta\phi$ of light rays by the gravitational field of a spherical system $M(r)$ is calculated using the MOdified Newtonian Dynamics (MOND). It is shown that $\Delta\phi$ with an impact parameter $r_0$ can be expressed by the measured rotation velocity $v(r)$ as $\Delta\phi(r_0)=2\int_{r_0}^{\infty} \frac{v^2(r)}{c^2}\frac{r_0dr}{r\sqrt{r^2-r_0^2}}$, where $v(r)=\left\{\left(Ga_0M(r)\right)^{1/4}, & r_0>r_c;\left(\frac{GM(r)}{r}\right)^{1/2},& r_0\leq r_c,\right$, and $r_c$ is the critical radius that is determined by the critical acceleration $a_0$. In the Newtonian limit of the gravitational acceleration $a\gg a_0$, $\Delta\phi$ approaches $\Delta\phi=2Gm(r_0)/c^2r_0$ with the projected surface mass $m(r_0)$. Whilst the asymptotic value of $\Delta\phi$ reaches a constant $\pi(v_{\infty}/c)^2$ in the low-acceleration limit of $a\ll a_0$. Taking the empirical correction of a factor of 2 from the theory of general relativity into account and utilizing the relation between rotation velocity $v$ and velocity dispersion $\sigma$, MOND results naturally in a constant deflection angle, $4\pi(\sigma/c)^2$, which has been widely used in the present-day study of gravitational lensing by galaxies and clusters of galaxies, implying that without introducing the massive halos acting as $r^{-2}$ for dark matter MOND has no difficulty in reproducing the known cases of gravitational lensing associated with galaxies and clusters of galaxies.