Newtonian limit to schwarschild radial geodesic

[vaibhavtewari]

Hello Everyone,

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

$$\frac{d^2r}{d\tau^2}=\frac{GM}{r^2}$$

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

regards,
Vaibhav

 

[haushofer]

Rewrite the derivative as

$$\frac{d^2r}{d\tau^2}= \Bigl(\frac{dt}{d\tau}\Bigr)^2 \frac{d^2 r}{dt^2}$$

and see what the Newtonian limit says about

$$\frac{dt}{d\tau} = \gamma$$

 

[vaibhavtewari]

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

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

any suggestions after this ? What do you think ?