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Nordström's second gravitational theory

  1. Jan 6, 2010 #1
    A sort of simple question (equation of motion)

    1. The problem statement, all variables and given/known data

    Hello everybody
    I'm new to these forums, so wish to be clear enough in my first post to not ask again. :biggrin:

    In the Nordström's second theory of gravitation, the field equation is [tex]\varphi \,\square \left( \varphi \right) =4\,\pi { \it GT}_{{m}}[/tex] where [tex]\square[/tex] is the D'Alembertian operator defined in the Minkowskian spacetime with metric (+,-,-,-), T_m is the trace of the material contribution to the total stress-energy-momentum tensor [tex]T_{{\mu \nu }}[/tex] and finally we have [tex] \varphi [/tex] implying the potential.

    This field is said to have the following Lagrangian proposed by Einstein: [tex]L={\frac {{\eta}^{\mu \nu }\partial _{{\mu}} \left( \varphi \right) \partial _{{\nu}} \left( \varphi \right) }{8\pi }}-\rho \varphi [/tex] where [tex] \rho=\varphi \,T_{{m}} [/tex] is the density of matter.

    Now my question is that how can one proceed to use the above Lagrangian to show that the equation of motion of a test particle moving in the field under discussion is [tex] \varphi \,d_{{\tau}} \left( u_{{\mu}} \right) =-\partial _{{\mu}} \left( \varphi \right) -d_{{\tau}} \left( \varphi \right) u_{{\mu}} [/tex] where [tex] \tau [/tex] is the proper time, [tex]u_{{\mu}}[/tex] is the 4-velocity of the moving particle and [tex]d_{{\tau}}(..)[/tex] refers to the derivative of (..) with respect to [tex] \tau [/tex]??

    Thanks in advance

    AB
     
    Last edited: Jan 7, 2010
  2. jcsd
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