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Weak field Geodesic equation

  1. Nov 15, 2013 #1
    Geodesic equation:

    [itex]m_{0}\frac{du^{\alpha}}{d\tau}+\Gamma^{\alpha}_{\mu\nu}u^{\mu}u^{\nu}= qF^{\alpha\beta}u_{\beta}[/itex]

    Weak-field:

    [itex]ds^{2}= - (1+2\phi)dt^{2}+(1-2\phi)(dx^{2}+dy^{2}+dz^{2})[/itex]

    Magnetic field, [itex]B[/itex] is set to be zero.

    I want to find electric field, [itex]E[/itex], but don't know where to start, so could someone give me a procedure to find it?

    Thank you
     
    Last edited: Nov 16, 2013
  2. jcsd
  3. Nov 15, 2013 #2

    WannabeNewton

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    ##F_{\mu\nu}## is a covariant object but ##E^{\mu}## and ##B^{\mu}## are not i.e. they are frame dependent quantities. They only have meaning if you have a background observer with 4-velocity ##u^{\mu}## who can decompose ##F_{\mu\nu}## into ##E^{\mu}## and ##B^{\mu}## relative to ##u^{\mu}##.

    Once you have such a ##u^{\mu}##, then ##E^{\mu} = F^{\mu}{}{}_{\nu}u^{\nu}## and ##B^{\mu} = -\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}u_{\nu}##.
     
  4. Nov 16, 2013 #3

    vanhees71

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    That's not entirely true. You can build a formalism of classical electromagnetics with socalled bivectors by representing the electromagnetic field with [itex]\mathbb{C}^3[/itex] vector fields,
    [tex]\vec{\mathfrak{E}}=\vec{E} + \mathrm{i} \vec{B}.[/tex]
    This makes use of the group-isomorphism [itex]\mathrm{SO}(1,3)^{\uparrow} \simeq \mathrm{SO}(3,\mathbb{C}).[/itex]

    Thus the product
    [tex]\vec{\mathfrak{E}}^2=\vec{E}^2-\vec{B}^2 + 2\mathrm{i} \vec{E} \cdot \vec{B}[/tex]
    is Lorentz invariant wrt. to proper orthochronous Lorentz transformations, and indeed also the four-vector formalism shows that the corresponding invariants given by the real and imaginary part of the bivector scalar product are proportional to the invariants [itex]F_{\mu \nu} F^{\mu \nu}[/itex] and [itex]\epsilon_{\mu \nu \rho \sigma} F^{\mu \nu} F^{\rho \sigma}=F^{\mu \nu} F^{\dagger}_{\mu \nu}[/itex].

    A nice review on this funny bivector formalism can be found here:

    http://arxiv.org/abs/1211.1218
     
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