Special Theory of Relativity
23
frequencies of the radiation of fairly well-defined except for the uncertainty due
to the natural lifetime τ of the excited atom. The radiation has a frequency
distribution
2
2
0
2
1
( ) ~
4 (
)
1
v
h
I
ρ
ν − ν
+
′
(1.78)
where ν 0 is the central value, and Γ′ is the uncertainty in the energy related to
the lifetime of the excited state by
τΓ =
(1.79)
34
,
6.67 10 Js
2
−
=
=
×
π
h h
being the Planck’s constant. In an experiment
performed by Hay et al. (1960), photons are emitted by excited
57
Fe atoms
embedded in the crystal, which have energy centred around hv 0 = 14.4 keV and
a linewidth Γ = 4.7 × 10
–9
eV. The emitter is placed at the centre of a centrifuge.
The photons are observed by
57
Fe atoms in the ground state, embedded in a
crystal and kept at the edege of the centrifuge. When the centrifuge is not
rotating, the photons from the emitter are absorbed by the absorber, since the
photons have just the right energy for exciting the
57
Fe atoms. However, once
the centrifuge starts rotating, the absorber sees the photons with shifted frequency
given by Eq. (1.77) (the speed is ν = ω r, ω being the angular speed and r being
the radial distance of the absorber from the centre) and the rate of absorption
goes down. The experimental observations for the shifts agree with the shifts
given by Eq. (1.77) within experimental errors, thus confirming the predictions
of the special theory of relativity for the transverse Doppler shift.
1.13 EXAMPLES
A few examples are now discussed to illustrate some applications, and elaborate
the ideas that have been analysed.
Example 1
This example shows that Galilean transformations are not consistent with
Maxwell’s equations.
Consider an infinitely long, stationary line-charge and a positively charged
particle P with charge q, moving away from the line charge with velocity u. The
only force acting on P is the repulsive force due to the electric field, and it acts
in a direction prependicular to the line charge. Now, an observer in a frame
moving parallel to the line charge with velocity v sees both electric and magnetic
23
frequencies of the radiation of fairly well-defined except for the uncertainty due
to the natural lifetime τ of the excited atom. The radiation has a frequency
distribution
2
2
0
2
1
( ) ~
4 (
)
1
v
h
I
ρ
ν − ν
+
′
(1.78)
where ν 0 is the central value, and Γ′ is the uncertainty in the energy related to
the lifetime of the excited state by
τΓ =
(1.79)
34
,
6.67 10 Js
2
−
=
=
×
π
h h
being the Planck’s constant. In an experiment
performed by Hay et al. (1960), photons are emitted by excited
57
Fe atoms
embedded in the crystal, which have energy centred around hv 0 = 14.4 keV and
a linewidth Γ = 4.7 × 10
–9
eV. The emitter is placed at the centre of a centrifuge.
The photons are observed by
57
Fe atoms in the ground state, embedded in a
crystal and kept at the edege of the centrifuge. When the centrifuge is not
rotating, the photons from the emitter are absorbed by the absorber, since the
photons have just the right energy for exciting the
57
Fe atoms. However, once
the centrifuge starts rotating, the absorber sees the photons with shifted frequency
given by Eq. (1.77) (the speed is ν = ω r, ω being the angular speed and r being
the radial distance of the absorber from the centre) and the rate of absorption
goes down. The experimental observations for the shifts agree with the shifts
given by Eq. (1.77) within experimental errors, thus confirming the predictions
of the special theory of relativity for the transverse Doppler shift.
1.13 EXAMPLES
A few examples are now discussed to illustrate some applications, and elaborate
the ideas that have been analysed.
Example 1
This example shows that Galilean transformations are not consistent with
Maxwell’s equations.
Consider an infinitely long, stationary line-charge and a positively charged
particle P with charge q, moving away from the line charge with velocity u. The
only force acting on P is the repulsive force due to the electric field, and it acts
in a direction prependicular to the line charge. Now, an observer in a frame
moving parallel to the line charge with velocity v sees both electric and magnetic