Newton's universal law of gravitation

[Orion1]

Derivation of Newton's law of universal gravitation...

Non-relativistic Schwarzschild metric:
c^2 {d \tau}^{2} = e^{\nu} c^2 dt^2 - e^{\lambda} dr^2 - r^2 d\theta^2 - r^2 \sin^2 \theta d\phi^2

metric identity:
g_{00} = e^{\nu} c^2 = \frac{ds^2}{dt^2}

Non-relativistic Einstein tensor:
G_{11} = \frac{- r \nu' + e^{\lambda} - 1}{r^2} = 0

Einstein tensor metric differential:
\nu' = \frac{e^{\lambda} - 1}{r} = \frac{d}{ds}

Non-relativistic metric identity:
e^{\nu} = \frac{1}{2} \left(1 - \frac{r_s}{r} \right)

metric identity:
e^{\lambda} - 1 = \frac{r_s}{r - r_s}

Gravitational acceleration:
g = - \frac{d s^2}{dt^2} \cdot \frac{d}{ds} = - \frac{d^2 s}{dt^2} = - g_{00} \nu' = - \frac{c^2 e^{\nu} (e^{\lambda} - 1)}{r} = - \frac{c^2}{2r} \left(1 - \frac{r_s}{r} \right) \left(\frac{r_s}{r - r_s} \right) = - \frac{c^2 r_s}{2 r^2} = - \frac{G M_1}{r^2}

Newton's second law and universal law of gravitation:
F_g = M_2 g = - M_2 g_{00} \nu' = - \frac{G M_1 M_2}{r^2}

Are these equations correct?
[/Color]
Reference:
http://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation"

 

[Altabeh]

Derivation of Newton's law of universal gravitation...

Non-relativistic Newton-Schwarzschild metric:
c^2 {d \tau}^{2} = \frac{1}{2} \left(1 - \frac{r_s}{r} \right) c^2 dt^2 - \frac{dr^2}{1 - \frac{r_s}{r}} - r^2 d\theta^2 - r^2 \sin^2 \theta d\phi^2

metric identity:
g_{00} = e^{\nu} c^2

Non-relativistic Einstein tensor:
G_{11} = \frac{- r \nu' + e^{\lambda} - 1}{r^2} = 0

Einstein tensor metric differential:
\nu' = \frac{e^{\lambda} - 1}{r}

metric identity:
e^{\nu} = \frac{1}{2} \left(1 - \frac{r_s}{r} \right)

metric identity:
e^{\lambda} - 1 = \frac{r_s}{r - r_s}

Gravitational acceleration:
g = - \frac{d^2 s}{dt^2} = - g_{00} \nu' = - \frac{c^2 e^{\nu} (e^{\lambda} - 1)}{2r} = - \frac{c^2}{2r} \left(1 - \frac{r_s}{r} \right) \left(\frac{r_s}{r - r_s} \right) = - \frac{c^2 r_s}{2 r^2} = - \frac{G M_1}{r^2}

Newton's second law and universal law of gravitation:
F_g = M_2 g = - M_2 g_{00} \nu' = - \frac{G M_1 M_2}{r^2}

Are these equations correct?
[/Color]
Reference:
http://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation"

How did you get g = - \frac{d^2 s}{dt^2} = - g_{00} {\nu}'?

AB

 

[Orion1]

How did you get g = - \frac{d^2s}{dt^2} = - g_{00} \nu'?

metric identity:
g_{00} = e^{\nu} c^2 = \frac{ds^2}{dt^2}

Einstein tensor metric differential:
\nu' = \frac{e^{\lambda} - 1}{r} = \frac{d}{ds}

Gravitational acceleration:
g = - \frac{d s^2}{dt^2} \cdot \frac{d}{ds} = - \frac{d^2 s}{dt^2}
[/Color]

 

[Altabeh]

metric identity:
g_{00} = e^{\nu} c^2 = \frac{ds^2}{dt^2}

Einstein tensor metric differential:
\nu' = \frac{e^{\lambda} - 1}{r} = \frac{d}{ds}

Gravitational acceleration:
g = - \frac{d s^2}{dt^2} \cdot \frac{d}{ds} = - \frac{d^2 s}{dt^2}
[/Color]

This is not true! If we assume that the motion is radial, then

g=\frac{-e^\lambda \dot{r}\dot{{\dot{r}}}}{(e^{\nu}c^2-e^{\lambda}{\dot{r}}^2)^{1/2}},

where a dot over r refers to the derivative wrt time.

AB