4. The Transformation of Four-Vectors, Four-Forms and the Matrices F and M
How do four-vectors and our matrices F, M etc. transform if we switch from one inertial frame S to another S' ? We assume that S and S' are (as usual) oriented in a special way. So the transformation matrices correspond to a Lorentz-boost, and they are given with L und L-1 .
By definition, ordinary four-vectors as Vi or Ji , written in a single column, i.e. a 4x1-matrix, transform as follows:
(Vi) ' = L · Vi
To each four-vector Vi belongs a linear form Vi (written in a single row, i.e. a 1x4-matrix) defined by
Vi := ( v0 , –v1 , –v2 , –v3 ) , where v0, v1, v2 resp. v3 are the components of Vi
Note the position of the index i and the sign-changes in Vi compared to Vi !
How do this corresponding (or 'dual') four-forms transform?
It is easy to demonstrate (and will be done later) that the following holds:
(Vi) ' = Vi · L-1
Every linear form transforming like this under a Lorentz boost is called a four-form .
Now let us suppose that the electromagnetic force f = q ( E + v x B ) in its matrix representation Ki = (q/c ) · F · Vi is form-invariant under Lorentz-transformations. Then we must have F' = L · F · L-1 .
From this postulate we can easily derive the following special transformations for the components of the field vectors E and B
(see Freund [1-218f], e.g.) :
What about our matrix M ? The answer is: M transforms in the very same way as F, i.e. M' = L · M · L-1 !!
( Otherwise M would not be very useful ... ). You can proof this statement directly just by doing some annoying matrix multiplications and simplifying the result, using the above transformations of the coponents of E and B . A modern pocket calculator as the TI-89 can do as well, or you hand it all over to a computer algebra system, e.g. Mathematica® . In section 6.4 we will give a nice proof using matrix algebra.
Finally, let's take a look at our nabla-operator. How does it transform under a Lorentz boost ? You can check by direct calculation that the following rule holds:
Our nabla-operator is a four-form, it behaves as any four-form does ! Compare with [1-232], especially with formula (36.5): Just multiply from the left side with G and then transpose the results on both sides !
Now we are ready to show the Lorentz-invariance of Maxwell's equations on a few lines.