- #1
pellman
- 684
- 5
I know I really need to understand partial diff eqs better for this, but I don't know what subtopic to look for in a pde text.
Maxwell eqs are 2 scalar eqs plus 2 vector eqs (3 components) giving eight total equations, coupled in the 6 components of E(x) and B(x). Why not just 6 equations?
This is easier to see in the covariant form
[tex]\partial_{\mu}F^{\mu\nu}=j^\nu[/tex] , [tex]\nu=0,1,2,3[/tex]
[tex]\epsilon^{\alpha\beta\gamma\delta}\partial_\gamma F_{\alpha\beta} =0[/tex] , [tex]\delta=0,1,2,3[/tex]
F is a 4x4 anti-symmetric tensor, so it is made of six independent functions.
The equations are first order with respect to space and time derivatives. Contrast for example the Dirac equation which is also first order in space and time derivatives but is really only 4 equations for 4 unknown functions.
Do the Maxwell equations contain a hidden redundancy?
Maxwell eqs are 2 scalar eqs plus 2 vector eqs (3 components) giving eight total equations, coupled in the 6 components of E(x) and B(x). Why not just 6 equations?
This is easier to see in the covariant form
[tex]\partial_{\mu}F^{\mu\nu}=j^\nu[/tex] , [tex]\nu=0,1,2,3[/tex]
[tex]\epsilon^{\alpha\beta\gamma\delta}\partial_\gamma F_{\alpha\beta} =0[/tex] , [tex]\delta=0,1,2,3[/tex]
F is a 4x4 anti-symmetric tensor, so it is made of six independent functions.
The equations are first order with respect to space and time derivatives. Contrast for example the Dirac equation which is also first order in space and time derivatives but is really only 4 equations for 4 unknown functions.
Do the Maxwell equations contain a hidden redundancy?