46 Special Relativity
Variable speed of light is in conflict with Lorentz invariance and with Einstein’s
time dilation formula in Equation (2.5). Although there have been many attempts
to test this by measuring the transverse second-order Doppler shift by Mössbauer
spectroscopy, the results are claimed to be either wrong or doubtful, and in need of
improved technology.
On the theoretical side there are interesting generalizations of the linear Lorentz
transformations to uniformly accelerated or rotating frames. Some generalized trans-
formations predict acceleration-dependent Doppler shift and time dilation, as well as
a maximal acceleration [6, 7].
Problems
- Starting from the postulates (i) and (ii) in Section 2.1 and the requirement that
d푠^2 in Equation (2.1) should have the same form in the primed and unprimed
coordinates, derive the linear Lorentz transformation in Equation (2.2) and the
expression in Equation (2.3). - The radius of the Galaxy is 3× 1020 m. How fast would a spaceship have to travel
to cross it in 300yr as measured on board? Express your result in terms of
훾= 1 ∕
√
1 −푣^2 ∕푐^2 [7].
- An observer sees a spaceship coming from the west at a speed of 0. 6 푐and a
spaceship coming from the east at a speed 0. 8 푐. The western spaceship sends a
signal with a frequency of 10^4 Hz in its rest frame. What is the frequency of the
signal as perceived by the observer? If the observer sends on the signal imme-
diately upon reception, what is the frequency with which the eastern spaceship
receives the signal [7]? - If the eastern spaceship in the previous problem were to interpret the signal as
one that is Doppler shifted because of the relative velocity between the western
and eastern spaceships, what would the eastern spaceship conclude about the
relative velocity? Show that the relative velocity must be(푣 1 +푣 2 )∕( 1 +푣 1 푣 2 ∕푐^2 ),
where푣 1 and푣 2 are the velocities as seen by an outside observer [7]. - A source flashes with a frequency of 10^15 Hz. The signal is reflected by a mirror
moving away from the source with speed 10kms−^1. What is the frequency of the
reflected radiation as observed at the source [7]? - Suppose that the evolution of the Universe is described by a constant deceler-
ating parameter푞=^12. We observe two galaxies located in opposite directions,
both at proper distance푑P. What is the maximum separation between the galax-
ies at which they are still causally connected? Express your result as a fraction
of distance to푑P. What is the observer’s particle horizon? - Show that the Hubble distance푟H=푐∕퐻recedes with radial velocity
푟̇H=푐( 1 +푞). (2.69)
8.IsthespheredefinedbytheHubbleradius푟Hinside or outside the particle
horizon?