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Relativity Mathematical Notation

  1. Dec 15, 2015 #1
    1. The problem statement, all variables and given/known data

    Find t(τ), x(τ), y(τ), z(τ) for constant electric field E=E(sinθx+cosθz) and B=Bz, and constant magnetic field where E,B, and θ are all constants.

    I haven't seen this notation before, and I really just want to know what it means specifically. I know and up indice paired with a down indice essentially gives an inner product, but I don't really know what an up indice paired with an up indice gives me. So this is equation 12.32 in Jackson Electrodynamics that I'm trying to understand.
    2. Relevant equations
    m(d2xα/dτ2)=(e/c)(∂αAβ-∂βAα)(dxβ/dτ)


    3. The attempt at a solution
    Also, the 4 vector potential that I've already calculated is:

    Aα=(-E(xsinθ+zcosθ),-By,0,0) where E, B and θ are all constants.

    or Aα=(-E(xsinθ+zcosθ),0,Bx,0) I can't decide which one is more convenient.

    I've attached an image of my best guess for what that equation means.
     

    Attached Files:

    Last edited: Dec 15, 2015
  2. jcsd
  3. Dec 15, 2015 #2
    I think its best to think of this way:

    Since beta is a free index, you get one equation for each value of beta (so you get 4 equations). In each of those equations, alpha is summed over (it is up for A and down for x). So each one of those equations can now be solved for the four functions you need.

    By the way for n free indicides in an equation in dimension d, you get d^n equations. In this case, d=4 and n=1
     
    Last edited: Dec 15, 2015
  4. Dec 16, 2015 #3

    nrqed

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    Just a small correction (I am sure it is a typo): it is alpha which is a free index and beta is summed over.
     
  5. Dec 17, 2015 #4
    Ohh thanks. Got them flipped xD
     
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