- #1

- 334

- 47

##\partial_\mu F^{\mu \nu} = 4\pi J^\nu##

##\partial_\mu F_{\nu \rho} + \partial_\nu F_{\rho \mu} + \partial_\rho F_{\mu \nu} = 0##

where the E.M. tensor is

##

F^{\mu \nu} = \begin{pmatrix}

0 & -B_3 & B_2 & E_1\\

B_3 & 0 & -B_1 & E_2\\

-B_2 & B_1 & 0 & E_3\\

-E_1 & -E_2 & -E_3 & 0

\end{pmatrix}##

and ##J^\nu = (\mathbf J ,\rho)##,

##\partial_\mu = (\nabla, \frac{\partial}{\partial t} )##

(allowing the notation with the equal sign between the component and the 4-vector.)

Now from this I'm trying to recreate our classic Maxwell's equations.

If I set ##\nu = 4## in the first equation we get the L.H.S. as ##4\pi \rho##

The R.H.S. becomes

##\partial_\mu F^{\mu 4} = -\frac{\partial}{\partial x}E_1-\frac{\partial}{\partial y}E_2-\frac{\partial}{\partial z}E_3 = -\nabla \cdot E##.

Now this is a sign error since we should get ##\nabla \cdot E = 4\pi \rho##.

So I guess maybe I shouldn't put a minus sign before those but that goes against what I learned by using the Minkowski metric, ##g_{\mu \nu} = diag(-1,-1,-1,1)##. Similarly I get the same sign error in the continuity equation. Should I always get + signs?