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Newton's universal law of gravitation

  1. Feb 7, 2010 #1

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

    Non-relativistic Schwarzschild metric:
    [tex]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[/tex]

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

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

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

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

    metric identity:
    [tex]e^{\lambda} - 1 = \frac{r_s}{r - r_s}[/tex]

    Gravitational acceleration:
    [tex]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}[/tex]

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

    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. jcsd
  3. Feb 7, 2010 #2
    How did you get [tex]g = - \frac{d^2 s}{dt^2} = - g_{00} {\nu}'[/tex]?

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

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

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

    Gravitational acceleration:
    [tex]g = - \frac{d s^2}{dt^2} \cdot \frac{d}{ds} = - \frac{d^2 s}{dt^2}[/tex]
     
    Last edited: Feb 7, 2010
  5. Feb 7, 2010 #4
    This is not true! If we assume that the motion is radial, then

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

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

    AB
     
    Last edited: Feb 7, 2010
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