# Newton's universal law of gravitation

1. Feb 7, 2010

### 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?

Reference:
http://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation" [Broken]

Last edited by a moderator: May 4, 2017
2. Feb 7, 2010

### Altabeh

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

AB

Last edited by a moderator: May 4, 2017
3. Feb 7, 2010

### Orion1

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:
$$g = - \frac{d s^2}{dt^2} \cdot \frac{d}{ds} = - \frac{d^2 s}{dt^2}$$

Last edited: Feb 7, 2010
4. Feb 7, 2010

### Altabeh

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

Last edited: Feb 7, 2010