664
M. Dohlus et al.
the Poynting vector
− →
S =
− →
E ×
− →
B
/μ 0 is parallel to − → n . It is thus immediately
clear that the radiation detected by the observer always seems to be originating from
the retarded position of the particle, i.e. the retarded position is the apparent origin
of the radiation, as it is intuitively expected when the speed of light is taken into
account.
Equation (11.5) suggests that one just has to inspect the acceleration of the charge
transverse to its apparent line of sight in order to understand the behaviour of the
electric field component. For example, it is thus easily seen that radiation from an
electron moving on a circular orbit is linearly polarized in the plane of the circle if
the observer is located in the same plane. If, however, the radiation is observed from
a position elevated out of this (say, horizontal) plane, one expects also a vertical
field component.
Since Eqs. (11.2) and (11.4) look so differently, it is useful to sketch how Eq.
(11.2) can be derived from Eq. (11.4): a key point is that Eq. (11.2) requires all
derivations to be taken with respect to the retarded time t = t − R/c while Eq.
(11.5) contains derivatives with respect to the time of observation t. Explicitly, this
reads
− →
β = −
d
− →
R
cdt = −
d
R − → n
cdt = −
dR
cdt
− → n − R
d − → n
cdt .
(11.6)
From t = t − R/c it follows: dt /dt = 1 − (dR(t)/cdt )(dt /dt). Solving for dt /dt
yields: dt /dt = 1/(1 + dR/cdt ).
Due to − → n ⊥ d − → n /dt we get from Eq. (11.6): dR/cdt = − − → n ·
− →
β and thus
dt
dt
=
1
1 − − → n ·
− →
β
.
(11.7)
For ultrarelativistic motion, and in particular in forward direction − → n
− →
β , dt’
and dt differ by a really large factor
1
1 − β
=
1 + β
1 − β 2 ≈ 2γ
2
1
(11.8)
thus resulting in a huge compression of the time scale at which radiation properties
are observed compared to the one at which electron motion takes place. It should
be noted that this has nothing to do with any Lorentz transform but is rather a
property of relativistic Doppler shift. The particle position at the retarded time is
just a different point in space-time.
M. Dohlus et al.
the Poynting vector
− →
S =
− →
E ×
− →
B
/μ 0 is parallel to − → n . It is thus immediately
clear that the radiation detected by the observer always seems to be originating from
the retarded position of the particle, i.e. the retarded position is the apparent origin
of the radiation, as it is intuitively expected when the speed of light is taken into
account.
Equation (11.5) suggests that one just has to inspect the acceleration of the charge
transverse to its apparent line of sight in order to understand the behaviour of the
electric field component. For example, it is thus easily seen that radiation from an
electron moving on a circular orbit is linearly polarized in the plane of the circle if
the observer is located in the same plane. If, however, the radiation is observed from
a position elevated out of this (say, horizontal) plane, one expects also a vertical
field component.
Since Eqs. (11.2) and (11.4) look so differently, it is useful to sketch how Eq.
(11.2) can be derived from Eq. (11.4): a key point is that Eq. (11.2) requires all
derivations to be taken with respect to the retarded time t = t − R/c while Eq.
(11.5) contains derivatives with respect to the time of observation t. Explicitly, this
reads
− →
β = −
d
− →
R
cdt = −
d
R − → n
cdt = −
dR
cdt
− → n − R
d − → n
cdt .
(11.6)
From t = t − R/c it follows: dt /dt = 1 − (dR(t)/cdt )(dt /dt). Solving for dt /dt
yields: dt /dt = 1/(1 + dR/cdt ).
Due to − → n ⊥ d − → n /dt we get from Eq. (11.6): dR/cdt = − − → n ·
− →
β and thus
dt
dt
=
1
1 − − → n ·
− →
β
.
(11.7)
For ultrarelativistic motion, and in particular in forward direction − → n
− →
β , dt’
and dt differ by a really large factor
1
1 − β
=
1 + β
1 − β 2 ≈ 2γ
2
1
(11.8)
thus resulting in a huge compression of the time scale at which radiation properties
are observed compared to the one at which electron motion takes place. It should
be noted that this has nothing to do with any Lorentz transform but is rather a
property of relativistic Doppler shift. The particle position at the retarded time is
just a different point in space-time.