D3 - tensors for EM

rank

• it denotes the number of spacetime indices
• rank 0: scalars $c^2=u^{\alpha }{u}_{\alpha }$, $d{s}^2=d{x}^{\alpha }d{x}_{\alpha }$
• rank 1: 4-vectors $x^{\alpha },p^{\alpha }$
• rank 2: Faraday tensor $F^{\alpha }{}_{\beta }$, Lorentz transform ${\mathrm{\Lambda }}^{\mu }{}_{\nu }$
• rank 3: differentiate Faraday tensor ${\partial }_{\lambda }{F}_{\mu \nu }$
• rank 4: Riemann tensor $R^{\alpha \beta \gamma \delta }$, antisymmetric tensor $E_{\alpha \beta \gamma \delta }$

indices

• the first one always represents the row (row-major)
• up = contravariant
• down = covariant

• LTs can change frames of reference:

• $A^{\alpha \alpha }$ represents the diagonal elements of $A^{\alpha }{}_{\beta }$

• $A^{\alpha }{}_{\alpha }$ represents the trace of $A^{\alpha }{}_{\beta }$

• index lowering and raising:

• in special relativity, for flat spacetime: ${\eta }_{\mu \nu }={\eta }^{\mu \nu }=\text{diag}(+1,-1,-1,-1)$, so:

• considering the Faradary tensor
• the covariant Faraday tensor:

• and the covariant:

• $F^{\alpha \beta }$ and $F_{\alpha \beta }$ are fully antisymmetric tensors with $F^{\alpha \alpha }=F_{\alpha \alpha }=0$, and $F^{\alpha \beta }=-F^{\beta \alpha }$