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The electromagnetic 3force acting on a charge moving with 3velocity
is given by the wellknown formula

(1556) 
When written in component form this expression becomes

(1557) 
or

(1558) 
where use has been made of Eq. (1469).
Recall that the components of the and
fields can be written
in terms of an antisymmetric electromagnetic field tensor via
Equation (1558) can be written

(1561) 
where
is the particle's 4velocity.
It is easily demonstrated that

(1562) 
Thus, the 4force acting on the particle,

(1563) 
can be written in the form

(1564) 
The skew symmetry of the electromagnetic field tensor ensures that

(1565) 
This is an important result, since it ensures that electromagnetic fields
do not change the rest mass of charged particles. In order to appreciate this,
let us assume that the rest mass is not a constant.
Since

(1566) 
we can use the standard results
and
to give

(1567) 
Thus, if rest mass is to remain an invariant, it is imperative that
all laws of physics predict 4forces acting on particles which
are orthogonal to the particles' 4velocities. The laws of electromagnetism
pass this test.
Next: The electromagnetic energy tensor
Up: Relativity and electromagnetism
Previous: Relativistic particle dynamics
Richard Fitzpatrick
20060202