Time loops

I read an article in a Sunday paper recently with the headline
‘Could Einstein have been wrong after all?’. “Oh no,” I thought,
“another crazy idea trying to disprove special relativity.” For me,
to read that special relativity has been proven wrong would be
equivalent to reading that it has been discovered that the Earth is
flat after all. Both would be quite preposterous given everything
we know. But it was general relativity that was being discussed in
the article and, far from being under threat, was alive and well. It
was just the headline that was misleading.
Many physicists regard general relativity to be the most
beautiful scientific theory ever discovered. Its beauty lies in the
simplicity, elegance and richness of its mathematical equations.
I admit that only a tiny fraction of the human population can
appreciate this beauty because they have had years of training.
For most people it is just a bunch of Greek symbols. But then I
have never been able to appreciate or understand cubism as an art
form. Anyway, as well as being pleasing to theoretical physicists,
general relativity has been confirmed by experimental evidence
time and time again. However, in almost all these cases it has only
been in what is called the ‘weak field limit’ (i.e. weak gravity).
It has yet to really prove its mettle in situations where it departs
radically from Newtonian gravity.
The newspaper article I mentioned described a new type
of experiment that, it is anticipated, will confirm yet another
prediction of general relativity known as gravity waves. The first
paragraph in the newspaper article was in fact stating that if such
gravity waves were not found then general relativity would be in
trouble. However, such is physicists’ faith in the theory that those
working on these new experiments fully expect to find what they
are looking for very soon. Unfortunately, a headline proclaiming:
‘More experimental proof that Einstein was right is just round the
corner’ is simply not as newsworthy.
Gravity waves are disturbances, or ripples, in the fabric of
space which are caused by the motion of a massive object. Think
of the trampoline model of space. When you stand in the centre
your weight makes a dent in the canvas. This is the simple view of how mass affects the space around it. If you then jump up and
down you will make the canvas vibrate and, provided it has a very
large area, these vibrations would travel outwards in the same way
that ripples spread out on the surface of a pond when a stone is
thrown in. Likewise, the motion of massive objects such as the
collapse of a massive star into a black hole will send out ripples
not through space but of space that will affect any objects in their
path. The hope is that experiments on Earth will be able to detect
the effect these gravity waves have on sensitive equipment which
will be ever so slightly stretched and squeezed as the waves pass
through.
Of course gravity waves have nothing to do with time
machines. I mention them as an example of an unambiguous
prediction of general relativity that has yet to be confirmed
experimentally. However, general relativity is so rich that it also
allows (theoretically of course) other, more exotic spacetime shapes
to exist which we are not nearly so confident about. One of these,
of relevance to this chapter, is called a closed timelike curve. This is
a circular path, or route, through warped spacetime in which time
itself is bent round in a circle. If you were to follow such a path
it would seem to you that you were travelling through ordinary
space. If you were to check your watch at any time during this
roundtripyouwouldsee it running forwards as normal. However,
you will, after some time had elapsed for you, eventually arrive
back at the same place and time that you started from according to
a clock that had been left there. Such a path would require you to
be travelling into the past for part of your journey. Of course if you
are travelling back in time you might as well get back to where you
started from before you set off , otherwise you will not have gained
anything from the trip. Any further warping of spacetime will
cause the time loop to take you back into the past.
Thus ‘closed timelike curve’ is Jargonese for ‘time machine’.
I shall refer to them here simply as ‘time loops’. It has long been
known that general relativity allows for the existence of time loops,
but whenever one popped up in the mathematics it was usually
disregarded on the grounds that the initial assumptions that were
fed into the equations were unreasonable. What party-poopers physicists are. Unfortunately, this attitude is justified and there
are plenty of other examples to illustrate why. Take the following
simple one. If you are told that a square has an area of 9 square
metres then you deduce that it has sides of length 3 metres, since
the area of the square comes form 3 × 3. However −3 × −3
also gives 9 (remember that a negative times a negative gives a
positive). But, we would never talk about the side of a square
having a length of −3 metres and so we ignore this option because
it is unphysical. Mathematical equations that describe the real
world often give, along with the correct answer, such unphysical,
or nonsensical answers, which should be ignored. For the vast
majority of physicists working on general relativity, time loops
fall into this category. They are considered unphysical because of
all the problems associated with time travel into the past.
In recent years, however, some physicists have been more
reluctant to dismiss time loops so quickly and they have become
a fashionable field of study. As we shall see, this is in part due
to Kip Thorne’s work on wormholes. However, despite the ease
with which time loops can be produced as solutions of Einstein’s
equations, physicists are still undecided whether they can really
exist in our Universe.
The first solution of Einstein’s field equations of general
relativity that described a spacetime containing time loops was
due to W J van Stockum in 1937. However, a connection between
this strange mathematical solution and the possibility of using it
to describe time travel was not appreciated until much later. The
van Stockum solution required an infinitely long cylinder of very
densely packed material spinning rapidly in empty space; not the
sort of thing you are likely to come across by accident unless you
are on board the Starship Enterprise. General relativity predicted
that the region of spacetime surrounding the cylinder would be
twisted around it and could contain a time loop. But the infinitely
long cylinder was understandably dismissed as too unreasonable
to be taken seriously. What is more, the mathematics predicted
that even spacetime very far from the cylinder would have strange
properties, proving that such a cylinder of matter could not exist
in our Universe or we would be able to see its effects locally even
if it were on the other side of the Universe. Time loops really hit the scientific headlines with the work of
KurtG¨odel in 1949. In a classic paper, he described mathematically
an abstract universe that would contain time loops. However,
G¨odel’s universe differed from the one we inhabit in the way it
maintained its stability against the inward gravitational pull of
its matter. Instead of expanding, as ours does, his universe was
rotating. If a space traveller in such a universe were to follow a
large enough circular path then she would get back to her starting
point before she set off. Time travel!
Although Einstein, who worked in the same building at
Princeton’s Advanced Study Institute as G¨odel, was initially
disturbed by this result, he (and most other physicists) soon
dismissed the result as being of little relevance to the real Universe
which we know is not rotating. Even G¨odel himself ignored the
possibility of time travel because it was so unachievable in practice,
not just because his model universe was unlike the real one, but
because of the unrealistic speeds required and distance that would
have to be covered by a rocket in order to complete a time loop in
such a universe. The fact remains, however, that G¨odel had come
up with a scenario (albeit an unrealistic one) in which no laws
of physics were violated and which was entirely consistent with
general relativity, but which contained time loops with all the time
travel paradoxes they implied. Most physicists believed, and still
do believe, that the loopholes in the physics that allow this sort of
solution to exist will eventually be plugged up through a better
understanding. Until then, G¨odel’s universe has been relegated to
the status of a mathematical curiosity.