Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Newton's universal law of gravitation
  1. Universal Gravitation (Replies: 7)

Loading...