Newtonian limit to schwarschild radial geodesic

Hello Everyone,

While trying to find the newtonian limit to radial geodesic I was able to find that

[tex]\frac{d^2r}{d\tau^2}=\frac{GM}{r^2}[/tex]

In the weak field limit we can naively replace [tex]\tau[/tex] by "t" and recover Newtons Law, this though does not sound very rigorous. Can some-one suggest a much systematic way.

regards,
Vaibhav
 

haushofer

Science Advisor
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Rewrite the derivative as

[tex]
\frac{d^2r}{d\tau^2}= \Bigl(\frac{dt}{d\tau}\Bigr)^2 \frac{d^2 r}{dt^2}
[/tex]

and see what the Newtonian limit says about

[tex]
\frac{dt}{d\tau} = \gamma
[/tex]
 
Thankyou for replying, I was thinking about that and was writing [tex]\gamma=1-(\frac{dr}{dt})^2[/tex] that leads to a second order non-linear differential equation

[tex]\frac{d^2r}{dt^2}=[1-(\frac{dr}{dt})^2]\frac{Gm}{r^2}[/tex]

any suggestions after this ? What do you think ?
 

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