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Divide in space and time component

  1. Sep 15, 2011 #1
    For [tex] -\frac{1} {4} F_{\mu\nu} F^{\mu\nu} [/tex] We can write [tex] -\frac{1} {4} F_{i j} F^{ij} -\frac{1}{2}F_{0i} F^{0i} [/tex] Where [tex] F_{\mu\nu} \equiv \partial_\mu W_\nu-\partial_\nu W\mu [/tex]
    If there are 3 indices how can I separate them like this?
    I want to separate [tex] \frac{1} {12} G_{\mu\nu\rho} G^{\mu\nu\rho} [/tex] into time and space component . Where [tex] G_{\mu\nu\rho}\equiv\partial_{\mu}\phi_{\nu\rho}+ \partial_{\nu}\phi_{\rho\mu}+\partial_{\rho}\phi_{\mu\nu} [/tex]

    How can I do it?
     
    Last edited: Sep 15, 2011
  2. jcsd
  3. Sep 16, 2011 #2
  4. Sep 16, 2011 #3
    Hi.

    What is Φ with two indexes? I have not seen it before. Thank you in advance.
     
  5. Sep 16, 2011 #4
    [tex] \phi_{\nu\rho} [/tex] is a antisymmetric tensor field.
     
  6. Sep 16, 2011 #5
    Actually I want to separate this in space and time component. There is some hint in the "Classical Electrodynamics by Jackson" section 11.6

    I can separate the space and time component for two indices (like: [tex] F_{\mu\nu} [/tex] ) but I am not sure how to do it when there are three indices.
    can anybody help?
     
  7. Sep 16, 2011 #6
    Hi. aries.

    I see. so G is antisymmetric tensor. Exchange of any pair of indexes changes signature. Among 4^6 = 64 components, only four components are independent, i.e. 012, 013, 023 and 123. So the formula you are looking for is

    1/2 { G_012 G^012 + ( similar other three terms ) }

    Regards
     
  8. Sep 17, 2011 #7
    sweet springs
    many many Thanx :-)
     
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