What Does Up Indice Paired with Up Indice Mean in Relativity Notation?

[Joe D]

Homework Statement

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.

Homework Equations

m(d²x^α/dτ²)=(e/c)(∂^αA^β-∂^βA^α)(dx_β/dτ)

The Attempt at a Solution

Also, the 4 vector potential that I've already calculated is:

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

or A^α=(-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.

 

[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

 

[nrqed]

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

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

 

[Brian T]

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

Ohh thanks. Got them flipped xD