# Relativity Mathematical Notation

1. Dec 15, 2015

### Joe D

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.

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Last edited: Dec 15, 2015
2. Dec 15, 2015

### Brian T

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
3. Dec 16, 2015

### nrqed

Just a small correction (I am sure it is a typo): it is alpha which is a free index and beta is summed over.

4. Dec 17, 2015

### Brian T

Ohh thanks. Got them flipped xD