Stephen Hawking gets it wrong

Soon after I started my PhD in 1987 I was in my university library
carrying out what is known as a literature search. I was working
on a problem in physics which involved a lengthy mathematical
calculation describing what happens when two atomic nuclei
collide, and I was looking up some references in scientific journals
related to my work. Not having been very successful in locating a
particular paper and becoming a little bored I decided to look for
any recent scientific papers by Stephen Hawking, for no reason
other than that I felt his work on cosmology might provide a
welcome break from mine. I found a paper of his dating back
a couple of years to 1985 in which he discussed how the direction
of time might get switched round if the Universe ever began to
contract. This sounded promising. I made a photocopy of the
article and read it on the train home. I followed the arguments of the first few pages but very soon
became stuck in the mathematics. Nevertheless, that evening I
decided that he had to be wrong, but since I could not follow the
mathematical details I did not feel I was on safe enough ground.
After all, he was a world famous scientist and I had just started as a
research student in a different field of physics altogether. Although
I did not know it at the time, Hawking had already realized that
his conclusions in that paper, which had attracted considerable
attention, had been wrong. I wish, nonetheless, to discuss the
ideas involved mainly to show just how confusing and illusory
time can be if someone of the stature of Stephen Hawking could
get things wrong. In fact, it has been fascinating for me to see
how so many other renowned scientists and world experts can still
hold completely opposite views about something as fundamental
as this3. It is all down to the confusion many have with the concept
of entropy. I shall first briefly describe why Hawking reached his
controversial conclusion.
The second law of thermodynamics should apply without
discrimination everywhere in the Universe stating that the entropy
of any isolated system cannot decrease. So why shouldn’t it apply
to the whole Universe? After all, the Universe in its entirety is
by definition an isolated system since there is nothing outside
it. In fact, the entropy of the Universe is indeed increasing and
implies that it must have been more ordered in the past. In fact, it
must have had a minimum entropy at the Big Bang and has been
unwinding, or running down, ever since.
Of course you may consider it rather ambitious, if not
arrogant, of us to talk about the entropy of the whole Universe,
but since we are trying to figure out its size, shape and age,
why not its entropy too? To begin with, I will consider a simple
‘model’ universe that has little to do with reality but will help us
understand what role the second law might play in the evolution So gravity increases entropy, but this still doesn’t explain how
entropy in the box could increase if it was at a maximum already.
The answer is that all the time the molecules are evenly separated,
gravity will be pulling in all directions and cancelling out, and
entropy is at a maximum. If, by chance (and anything is possible),
the molecules in a certain region find themselves closer together
than average then this represents a temporary departure from the
maximum entropy (equilibrium) state. In order for the second law
to rectify the situation, these molecules have two choices: they can
either drift back apart again to their original equilibrium state, or
they can gravitate together to form a clump. Either way entropy
increases back to a maximum again. Both scenarios can be seen as
a running down of the system, but now we have two alternative
pictures of the maximum entropy state.
Now we are ready to tackle the real Universe. Hawking
began his argument by stating that the Universe must have had
a minimum entropy at the Big Bang and, since it must obey the
second law of thermodynamics, has been unwinding ever since,
moving towards a state of maximum entropy. He had developed a
theory of the Universe which required it to be closed and believed
that it contained enough matter to one day halt the expansion and
cause it to collapse to a Big Crunch.In Hawking’s model, the Big Bang and Big Crunch
singularities were identical. After all, in both cases all the matter
and energy in the Universe would be crushed to infinite density
and zero size. Thus if the Big Bang singularity was in a state of low
entropy, then so should the Big Crunch singularity be. It therefore
followed that, as the Universe contracted, its entropy would have
to decrease again and the second law of thermodynamics would
be violated during this phase. Hawking believed that the state
of maximum expansion also represented the state of maximum
entropy. Thus the contracting phase of the Universe would be the
time reverse of the expanding phase. In terms of arrows of time, if entropy starts to decrease
during the contracting phase then the thermodynamic arrow
must get flipped over (since it is defined to always point in
the direction of increasing entropy), and if our own subjective
(psychological) arrowis always aligned in the same direction as the
thermodynamic one, our time will also be running in reverse. This
would mean that rather than the Big Crunch being an event in our
future, it would be an event in our past. Of course I am assuming
that humans will survive for the billions of years necessary to
put this to the test, but if we did we would not actually see the
Universe contracting. Since our time would be running in reverse
wewould think it was still expanding. We would also therefore not
see any violation of the second law of thermodynamics. After all,
according to us entropy would be increasing as normal. The most
fascinating conclusion to draw from this weird situation is that the
Universe may in fact be collapsing at the moment, and it is only
because we have an arrow of time that points in the direction of
increasing entropy that we mistakenly believe it to be expanding!
I did not realize it at the time, but this idea of the reversal of
the direction of time during a collapsing universe was actually due
to Thomas Gold in the 1960s. Hawking tried to put the idea on
a firmer theoretical footing by appealing to the quantum nature
of the two singularities. In fact, the behaviour of the Universe
when it is nearmaximumexpansion would have to be very strange
in Hawking’s original picture. Let’s say that a human survives
from the expansion phase through to the contracting phase while
enclosed inside a spaceship. Her arrowof time would have flipped
over suddenly and she would not remember the time of maximum
expansion since that would now be in her future.
I will now describe my objection to this idea. First of
all, Hawking used the words ‘expansion’ and ‘contraction’ and
‘surviving through the period of maximum expansion into the
contraction phase’. Such language implies that there must be a
separate, external, arrow of time that points from the Big Bang
to the Big Crunch. Otherwise there is nothing to distinguish the
two and we cannot say that one was ‘before’ the other. When it
is claimed that we might ‘mistakenly’ think that we are living in the expanding phase but are ‘really’ in the collapsing phase, we
would need such an external time to act as an adjudicator and
tell us what the Universe is really doing. We know of no such
arrow of time and to suggest that one might exist is reminiscent
of my earlier discussion of a hypothetical external time against
which we would need to measure the rate of flow of our time.
And if there is no preferred overall direction of time that would
label the expansion and contraction phases then the Big Crunch
really should be equivalent to the Big Bang and would also mark
a beginning of time. We would therefore have time flowing from
both singularities, in opposite directions, towards an ‘end of time’
at maximum expansion.
I will highlight this problem of an end to time by considering
the fate of the surviving human in a spaceship near the time of
maximum expansion. She calculates that the Universe will reach
maximum expansion at three o’clock that afternoon (let’s call it
T-max). She is aware that her arrow of time is about to flip over.
At one second to three everything is normal and she knows she
has a second to go. What will be happening two seconds later? It
is now one second past three and we are in the contracting phase.
If her arrow of time has now reversed and all processes inside the
spaceship are running backwards then her clock will now say one
second before three again. She will still think that the Universe has
another second’s worth of expansion.
Even at one millionth of a second to three on this side of T-max
there would be nothing unusual, but two millionths of a second
later she would still believe T-max to be a millionth of a second
away. We could get as close to T-max as we liked but there would
never be a time later than it. It would really mark an end to time.
The above objections do not prove that Hawking was wrong,
rather that the language used presupposed an extra arrow of time
that did not change directions at T-max, and to which no reference
had been made.
After discussing his theory with colleagues, Hawking soon
realized that the Universe need not return to a state of low entropy
at the Big Crunch and hence there would not have to be a reversal
in the direction of our arrow of time. The entropy of the Universe could carry on increasing from the expanding phase through to
the contracting phase. UnfortunatelyHawkingcaught pneumonia
and was unable to write a quick follow-up paper explaining his
mistake. I vividly remember reading his best seller A Brief History
of Time while on the train to work a year or two after it came out—a
friend had bought me the paperback edition at New Delhi airport
long before it was available in Britain—and I remember feeling
both surprised and full of admiration for Hawking’s honesty.
Above all, I remember being embarrassed that the stupid grin on
my face had attracted the attention of the other commuters.
So how can we understand the difference between the low
entropy Big Bang and the high entropy Big Crunch? One
explanation is that space near the two singularities has different
geometries. Current thinking is that black holes are reservoirs of
entropy. The bigger they are, the higher their entropy. Since the
Big Crunch can be considered as the ultimate black hole which has
swallowed the whole Universe, it should have an extremely high
entropy. The Big Bang, in contrast, is like a white hole and would
have very low entropy.
This is rather unsatisfying though. After all, where does
gravity come in? Where does expansion come in? And how did
the Universe get to be in such a low entropy state in the first place?
At first sight, it would appear that the Universe is in a state
of low entropy at the moment. Stars are hot spots in space
which are radiating their heat into their surroundings and causing
entropy to increase (remember the idea of heat transfer was one
way of defining entropy). When a star stops shining it will have
completely wound down and would be in a state of high entropy
(whether or not it ends up as a black hole). So there will be a time
in the distant future when all the stars will have burnt out and
their radiation would be spread out evenly in space (high entropy).
There is a serious problem here, however, which physicists have
attempted to wriggle their way out of with lesser and greater
degrees of success. Before stars and galaxies had formed in the
early Universe, before even matter had had a chance to form from
pure energy, the Universe would have been in a state of thermal
equilibrium, with its energy spread out evenly so that no one region of space was any hotter than any other. Surely this is a
state of maximum entropy! So what caused the stars to form in
the first place?
One proposal goes like this: It is true the Universe started
off in a state of maximum entropy, but it was also very small
then. The entropy it had was the maximum possible for that
sized universe. The Universe then went through a period of rapid
expansion (inflation) and the maximum amount of entropy it could
have increased dramatically. However, its actual entropy quickly
fell behind this maximum possible value creating an ‘entropy gap’.
In his book The Emperor’s New Mind, Roger Penrose criticizes
this view by claiming that the time reverse situation should also
apply if and when the Universe were to finally collapse to a Big
Crunch. As it shrinks, the entropy gap will decrease until it again
reaches a size in which the entropy is the maximum possible. Any
further shrinking would squeeze the entropy down further, in
violation of the second law.
How can we therefore understand this asymmetry between
the two singularities? Can gravity provide it? An obvious
difference between the expanding and contracting phases is that in
the former there would have been some initial conditions that set
the Universe expanding in the first place. The contracting phase on
the other hand is due entirely to the gravitational pull of the matter
within the Universe. Thus the physical origins of the expansion
and contraction are different. But it would be satisfying to be able
to explain the evolution of the Universe in terms of entropy.
Another oft-quoted difference is that a very old contracting
universe would no longer have any stars still burning. It would
consist entirely of cold background radiation, dead stars and black
holes. Clearly a high entropy landscape. But this is not the
only possible scenario. Let us for simplicity assume that the
contracting Universe contains only low energy light (photons)
and black holes. Hawking has shown that black holes evaporate
and we can therefore imagine a universe that is so old—one that
has just enough matter to close it means it would take gravity
a very long time to halt and reverse the expansion—that all the
black holes could have evaporated away. Whether they leave behind them empty, naked, singularities is unclear, but if they do not then the Universe will finally consist entirely of cold
radiation