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21. Radiation from relativistically moving systems
Reading: Shu, Vol.I, Ch. 16 and 17
21.1 Lorentz transformations
Let us recapitulate the Lorentz transformations of special relativity.
One frame of reference,
K
0
, may move with velocity
~v
=
~
βc
relative to another system,
K
. For the coordinates parallel
and perpendicular to the transformation direction the Lorentz transformations are
x
k
=
γ
±
x
0
k
+
βct
0
²
x
⊥
=
x
0
⊥
(21
.
1)
ct
=
γ
±
βx
0
k
+
ct
0
²
(21
.
2)
In astrophysics we often cannot measure where exactly a particle or an emitter is at a given time.
If the length of the lineofsight changes during a time interval
dt
, for example by relativistic
motion of the emitter that is at rest in system
K
0
, a distant observer at rest in system
K
would
measure a time interval that is changed by retardation, i.e. the relocation of the emitter during
the time of measurement.
dt
obs
= (1

β
cos
θ
obs
)
dt
=
γ
(1

β
cos
θ
obs
)
dt
0
=
dt
0
D
D
= Dopplerfactor
(21
.
3)
where
θ
obs
is the angle between the lineofsight (the propagation direction of the radiation)
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