and outgoing fields of radiation,
1
fab 2 ( ab ab (2-13)
2 (i +outj (2
2.1.2 The Equations of Motion of an Electron
The interaction between an electron and the electromagnetic field can be
examined from the equations of motion for the electron, i.e., the equations to
determine the world-line of the particle in motion. The laws of conservation of
energy and momentum are used to get information on this question. First, one
surrounds the singular world-line of the particle by a thin world-tube, whose radius
is much smaller than the range of interaction between the particle and the field in
consideration. Then, one calculates the flow of energy and momentum across the
surface of this world-tube, using the stress tensor Tac of Maxwell's theory, which is
calculated from the actual field Fab via
1 bd
47rTac FabFcb + 4gacFbd bd. (2-14)
4
By the conservation laws, the total flow of energy (or momentum) out from the
surface of any finite length of world-tube must be equal to the difference in the
energy (or momentum) residing within the tube at the two ends of this length:
depending only on conditions at the two ends of this length, the rate of flow
of energy (or momentum) out from the surface of the tube must be a perfect
differential.
The information obtained in this manner is independent of shape and size
of the world-tube provided that it is much smaller than the realm of the Taylor
expansions used in the calculations. If we take two world-tubes surrounding
the singular world-line, the divergence of the stress tensor aTac/ x will vanish
everywhere in the region of spacetime between them, since there are no singularities