Gravitational times

Now that we have seen how time is described in special relativity
I will discuss briefly how it behaves in general relativity. Einstein
showed that gravity provides an alternative way of slowing time
down to travelling at very high speeds. We have already seen
how general relativity describes the way massive objects cause
spacetime in their vicinity to curve (notice how I can finally talk
about spacetime now rather than just space). Just as space gets
stretched inside gravitational fields, so does time. Consider the
effect on time near the event horizon of a black hole. An observer
watching, from a safe distance, someone holding a clock while
falling into the hole will see the clock run more slowly. This is
why we see objects that fall into black holes appear as though they
are frozen at the horizon. To us, time at the horizon is standing
still. However, this is not simply an optical illusion. We have
seen that the time dilation in special relativity is itself relative.
Two observers moving at high speed relative to each other will
each see the other’s clock run slower. But in the case of the two
observers near the black hole, the one falling in will see the clock
of the distant observer running faster!
You are entitled to feel less than convinced by this discussion.
After all, no one has come face to face with a black hole for such an
effect to be tested. So how can we be sure that time would really
slow down? The answer is that we can test it here on Earth. The
gravitational field of the Earth is nowhere near as strong as a black
hole’s but we can still measure the tiny effect it has on time.
The dilation of time due to the Earth’s gravity was confirmed
in a famous experiment carried out by two Americans in 1960.
Robert Pound and Glen Rebka made use of the recently discovered
M¨ossbauer effect, which states that an atom of a particular type will
emit light of a specific wavelength when pumped with energy.
And because this wavelength is compatible with other similar
atoms, they will readily absorb the light. If the wavelength is
changed ever so slightly, say by a Doppler shift, then the other
atoms will not be able to absorb it. Pound and Rebka placed some
‘emitting’ atoms of iron at the bottom of a 23 metre high tower
and identical atoms at the top. They found that the light emitted by the atoms at the bottom was not absorbed by those at the top,
and showed that the reason for this was that the wavelength of
the light was redshifted. This ‘gravitational redshift’ is a direct
result of the slowing down of time at the base of the tower. You
see the top of the tower is further away from the Earth and gravity
is therefore weaker there (not by much, of course, but enough to
alter the wavelength of the light sufficiently.
To understand this redshift as a slowing down of time,
consider what a wavelength actually means. You can think of
the atoms of iron as clocks with each crest of a light wave they
emit as a ‘tick’. If we see longer wavelengths, it will be because
more time has elapsed between successive ticks and we say the
atomic clock is running slower. To measure how much time
was slowing down, Pound and Rebka did something which I
remember thinking, when I first learnt about it, was an absolute
masterstroke. They made the atoms at the top of the tower move
down with a specific speed towards the ones at the bottom. The
moving atoms now saw the wavelength of the light travelling up
to meet them slightly squashed due to the Doppler shift. This
shortening of the wavelength could be adjusted, by controlling the
speed of the downward-moving atoms, to restore the wavelength
of the light to its correct value and the falling atoms were thus able
to absorb the light.
In certain situations, the two time dilation effects (due to
special and general relativity) can act against each other. Consider
two atomic clocks, one on the ground and one in a satellite in
orbit. Which will be running slower? To the clock on the ground,
the high speed motion of the one in orbit should be making it run
slower, while the fact that it is orbiting the Earth in zero gravity
should be making it run faster. Which effect wins? The answer
is that it depends on how high up the satellite is. Scientists need
to know this sort of thing when analysing information sent down
by navigational satellites which have their own atomic clocks. As
an example, if a satellite is orbiting at an altitude that is more
than the diameter of the Earth, it will be sufficiently far away for
the gravitational time dilation to win. Its clock will be running
faster than the clocks on Earth, which are slowed down by Earth’s gravity, by a few millionths of a second each day (an unforgivable
inaccuracy where atomic clocks are concerned).
So just remember, if your watch is running slow, hold it
above your head! It will speed up now that it is feeling a weaker
gravitational force. Of course you would never be able to measure
such a tiny effect however long you hold you arm aloft.