12 An introduction to the physics of cosmology
The most well-known example of the power of the equivalence principle
is the thought experiment that leads to gravitational time dilation. Consider an
accelerating frame, which is conventionally a rocket of heighth, with a clock
mounted on the roof that regularly disgorges photons towards the floor. If the
rocket accelerates upwards atg, the floor acquires a speedv=gh/cin the time
taken for a photon to travel from roof to floor. There will thus be a blueshift in the
frequency of received photons, given byν/ν=gh/c^2 , and it is easy to see that
the rate of reception of photons will increase by the same factor.
Now, since the rocket can be kept accelerating for as long as we like, and
since photons cannot be stockpiled anywhere, the conclusion of an observer on
the floor of the rocket is that in a real sense the clock on the roof is running
fast. When the rocket stops accelerating, the clock on the roof will have gained
a timetby comparison with an identical clock kept on the floor. Finally, the
equivalence principle can be brought in to conclude that gravity must cause the
same effect. Noting thatφ=ghis the difference in potential between roof and
floor, it is simple to generalize this to
t
t=
φ
c^2The same thought experiment can also be used to show that light must be deflected
in a gravitational field: consider a ray that crosses the rocket cabin horizontally
when stationary. This track will appear curved when the rocket accelerates.
2.1.2 Applications of gravitational time dilation
For many purposes, the effects of weak gravitational fields can be dealt with by
bolting gravitational time dilation onto Newtonian physics. One good example is
in resolving the twin paradox (see p 8 of Peacock 1999).
Another nice paradox is the following: Why do distant stars suffer no time
dilation due to their apparently high transverse velocities as viewed from the
frame of the rotating Earth? At cylindrical radiusr, a star appears to move at
v=rω, implying time dilation by a factor 1 +r^2 ω^2 / 2 c^2 ; this is not observed.
However, in order to maintain the stars in circular orbits, a centripetal acceleration
a=v^2 /ris needed. This is supplied by an apparent gravitational acceleration in
the rotating frame (a ‘non-inertial’ force). The necessary potential is=r^2 ω^2 /2,
so gravitational blueshift of the radiation cancels the kinematic redshift (at least
to orderr^2 ). This example captures very well the main philosophy of general
relativity: correct laws of physics should allow us to explain what we see,
whatever our viewpoint.
For a more important practical application of gravitational time dilation,
consider theSachs–Wolfe effect. This is the dominant source of large-scale
anisotropies in the cosmic microwave background (CMB), which arise from
potential perturbations at last scattering. These have two effects: