<p>While researching online I've found two expressions for the angle of deflection by gravitational lensing:</p>
<ul>
<li><span class="math-container">$\dfrac{2GM}{rv^2},$</span> Classical(Newtonian) prediction</li>
<li><span class="math-container">$\dfrac{4GM}{rc^2},$</span> Relativistic prediction</li>
</ul>
<p>I know the Classical definition is accurate for small <span class="math-container">$v$</span>, and the Relativistic prediction is accurate for light(and particles traveling close to the speed of light, presumably). However, what's in the middle ground? For instance, what would the deflection be for a particle traveling at <span class="math-container">$0.5c$</span>?</p>
<p>Sources:<br />
<a href="https://en.wikipedia.org/wiki/Gravitational_lens#Explanation_in_terms_of_spacetime_curvature" rel="nofollow noreferrer">The Wiki</a> (Relativity)<br />
<a href="https://web.mit.edu/6.055/old/S2009/notes/bending-of-light.pdf" rel="nofollow noreferrer">Web article</a> (Relativity/Newtonian)<br />
<a href="https://www.researchgate.net/publication/2173573_Newtonian_gravitational_deflection_of_light_revisited" rel="nofollow noreferrer"> Publication</a> (Relativity/Newtonian)<br />
And more</p>

CC BY-SA 4.0
2022-06-02T19:30:41.607Z
It would be best to include references to where you found each.
Greg Miller
CC BY-SA 4.0
2022-06-03T00:05:00.213Z
BTW, that relativistic equation is an approximation. The Schwarzschild radius is $r_s=\frac{2GM}{c^2}$, so that expression can be written as $\frac{2r_s}r$. Actually, the $r$ in the denominator should be $b$, the [impact parameter](https://en.wikipedia.org/wiki/Impact_parameter), the perpendicular distance to the path that the light would take if there were no deflection. (Note that the Wikipedia diagram is for a repulsion interaction, not attraction). The approximation is ok for large $b$, eg $b>10r_s$, but it's too low for small $b$.
PM 2Ring
CC BY-SA 4.0
2022-06-03T00:15:46.217Z
You may enjoy my post on photon trajectories on our sister site https://physics.stackexchange.com/a/680961/123208 Here's a graph of deflection angles (in degrees), with $b$ in units where $r_s=1$. The green curve is the simple $2/b$ equation, the red dots are (hopefully) correct values, computed using an integral. https://i.sstatic.net/vG2f2.png
PM 2Ring
CC BY-SA 4.0
2022-06-03T19:16:51.443Z
If you know some calculus, this may be helpful: https://hepweb.ucsd.edu/ph110b/110b_notes/node72.html
PM 2Ring
<general-relativity><newtonian-gravity>
In between Classical Gravitational Deflection (2GM/rv^2) and General Relativity (4GM/rc^2)
Inigo Montoya
# In between Classical Gravitational Deflection (2GM/rv^2) and General Relativity (4GM/rc^2)

(Asked by: username_0 on 2022-06-02)

While researching online I've found two expressions for the angle of deflection by gravitational lensing:

* $\dfrac{2GM}{rv^2},$ Classical(Newtonian) prediction
* $\dfrac{4GM}{rc^2},$ Relativistic prediction

I know the Classical definition is accurate for small $v$, and the Relativistic prediction is accurate for light(and particles traveling close to the speed of light, presumably). However, what's in the middle ground? For instance, what would the deflection be for a particle traveling at $0.5c$?

Sources:  
[The Wiki](https://en.wikipedia.org/wiki/Gravitational_lens#Explanation_in_terms_of_spacetime_curvature) (Relativity)  
[Web article](https://web.mit.edu/6.055/old/S2009/notes/bending-of-light.pdf) (Relativity/Newtonian)  
 [Publication](https://www.researchgate.net/publication/2173573_Newtonian_gravitational_deflection_of_light_revisited) (Relativity/Newtonian)  
And more

**Tags:** general-relativity, newtonian-gravity
> **Comments:**
> **username_1**: It would be best to include references to where you found each.  
> **username_2**: BTW, that relativistic equation is an approximation. The Schwarzschild radius is $r_s=\frac{2GM}{c^2}$, so that expression can be written as $\frac{2r_s}r$. Actually, the $r$ in the denominator should be $b$, the [impact parameter](https://en.wikipedia.org/wiki/Impact_parameter), the perpendicular distance to the path that the light would take if there were no deflection. (Note that the Wikipedia diagram is for a repulsion interaction, not attraction). The approximation is ok for large $b$, eg $b>10r_s$, but it's too low for small $b$.  
> **username_2**: You may enjoy my post on photon trajectories on our sister site https://physics.stackexchange.com/a/680961/123208 Here's a graph of deflection angles (in degrees), with $b$ in units where $r_s=1$. The green curve is the simple $2/b$ equation, the red dots are (hopefully) correct values, computed using an integral. https://i.sstatic.net/vG2f2.png  
> **username_2**: If you know some calculus, this may be helpful: https://hepweb.ucsd.edu/ph110b/110b_notes/node72.html  

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## 0 answers

