Editing Gravitational lens (section)

==Approximate Newtonian description==
Newton wondered whether light, in the form of corpuscles, would be bent due to gravity. The Newtonian prediction for light deflection refers to the amount of deflection a corpuscle would feel under the effect of gravity, and therefore one should read "Newtonian" in this context as the referring to the following calculations and not a belief that Newton held in the validity of these calculations.<ref name="Meneghetti">{{cite book |last1=Meneghetti |first1=Massimo |title=Introduction to Gravitational Lensing With Python Examples |series=Lecture Notes in Physics |date=2021 |volume=956 |publisher=Springer |doi=10.1007/978-3-030-73582-1 |isbn=978-3-030-73582-1 |s2cid=243826707 |url=https://doi.org/10.1007/978-3-030-73582-1}}</ref> 

For a gravitational point-mass lens of mass <math>M</math>, a corpuscle of mass <math>m</math> feels a [[Newton's law of universal gravitation|force]] 

: <math>\vec F = -\frac{GMm}{r^2} \hat r,</math>

where <math>r</math> is the lens-corpuscle separation. If we equate this force with [[Newton's second law]], we can solve for the acceleration that the light undergoes:

: <math>\vec a = -\frac{GM}{r^2} \hat r.</math>

The light interacts with the lens from initial time <math>t = 0</math> to <math>t</math>, and the velocity boost the corpuscle receives is

: <math>\Delta \vec v = -\int_0^t dt'\, \frac{GM}{r(t')^2} \hat r(t').</math>

If one assumes that initially the light is far enough from the lens to neglect gravity, the perpendicular distance between the light's initial trajectory and the lens is ''b'' (the [[impact parameter]]), and the parallel distance is <math>r_\parallel</math>, such that <math>r^2 = b^2 + r_\parallel^2</math>. We additionally assume a constant speed of light along the parallel direction, <math>dr_\parallel \approx c\,dt</math>, and that the light is only being deflected a small amount. After plugging these assumptions into the above equation and further simplifying, one can solve for the velocity boost in the perpendicular direction. The angle of deflection between the corpuscle’s initial and final trajectories is therefore (see, e.g., M.&nbsp;Meneghetti 2021)<ref name="Meneghetti"/>

: <math>\theta = \frac{2GM}{c^2 r}.</math>
  
Although this result appears to be half the prediction from general relativity, classical physics predicts that the speed of light <math>c</math> is observer-dependent (see, e.g., L.&nbsp;Susskind and A.&nbsp;Friedman 2018)<ref>{{cite book |last1=Susskind |first1=Leonard |last2=Friedman |first2=Art |title=Special Relativity and Classical Field Theory |date=2018 |publisher=Penguin Books |isbn=9780141985015 |url=https://www.penguin.co.uk/books/303699/special-relativity-and-classical-field-theory-by-leonard-susskind-and-art-friedman/9780141985015}}</ref> which was superseded by a universal speed of light in [[special relativity]].