10
1 A Brief Stroll in Special Relativity
Fig. 1.3 Rocket seen at two different times. The tail and nose are at x = 0 and x = L
moving rocket enter the eye at the same time, but are emitted at different times,
separated by
The visual length of the rocket is clearly given by
L v = c
(1.23)
During the time while the T photon moves from its tail to its nose, the
rocket moves a distance vvt, so the visual length may also be expressed as
L v = L + vvt.
(1.24)
From these two equations we may solve for the visual length in terms of L and
also in terms of the proper length from (1.21), giving
L v =
L
1 − β
=
L p
γ (1 − β)
=
√
1 + β
√
1 − β
L p .
(1.25)
The approaching rocket thus appears to the eye to be longer than its proper
length, due to the finite velocity of light which counteracts the length contraction effect. Similar effects occur if the rocket does not approach the observer
head-on, and in fact one finds that it also appears to rotate (Taylor 1963).
The Doppler effect is the observed change in the period or wavelength of light
emitted by a body in motion relative to the observer, and was known long before
relativity; however there is a modification of the effect due to relativity. The relativistic
expression for the Doppler effect can be obtained by reasoning very similar to that
in the example above. We consider a source of light moving at velocity v directly
toward us, as in Fig. 1.4. Wave front number 1 is emitted at t = 0 from the source at
x = 0.
Wave front number 2 is emitted at t = T with the source at x = vt, at which
time wave front number 1 has reached x = cT . From the figure it is clear that the
wavelength observed is given by
λ ob = cT − vT = (1 − β)cT.
(1.26)
1 A Brief Stroll in Special Relativity
Fig. 1.3 Rocket seen at two different times. The tail and nose are at x = 0 and x = L
moving rocket enter the eye at the same time, but are emitted at different times,
separated by
The visual length of the rocket is clearly given by
L v = c
(1.23)
During the time while the T photon moves from its tail to its nose, the
rocket moves a distance vvt, so the visual length may also be expressed as
L v = L + vvt.
(1.24)
From these two equations we may solve for the visual length in terms of L and
also in terms of the proper length from (1.21), giving
L v =
L
1 − β
=
L p
γ (1 − β)
=
√
1 + β
√
1 − β
L p .
(1.25)
The approaching rocket thus appears to the eye to be longer than its proper
length, due to the finite velocity of light which counteracts the length contraction effect. Similar effects occur if the rocket does not approach the observer
head-on, and in fact one finds that it also appears to rotate (Taylor 1963).
The Doppler effect is the observed change in the period or wavelength of light
emitted by a body in motion relative to the observer, and was known long before
relativity; however there is a modification of the effect due to relativity. The relativistic
expression for the Doppler effect can be obtained by reasoning very similar to that
in the example above. We consider a source of light moving at velocity v directly
toward us, as in Fig. 1.4. Wave front number 1 is emitted at t = 0 from the source at
x = 0.
Wave front number 2 is emitted at t = T with the source at x = vt, at which
time wave front number 1 has reached x = cT . From the figure it is clear that the
wavelength observed is given by
λ ob = cT − vT = (1 − β)cT.
(1.26)