《时间简史―从大爆炸到黑洞》 ――史蒂芬?霍金

Figure 8:1
The history of the universe in real time, however, would look very different. At about ten or twenty thousand million
years ago, it would have a minimum size, which was equal to the maximum radius of the history in imaginary time. At
later real times, the universe would expand like the chaotic inflationary model proposed by Linde (but one would not
now have to assume that the universe was created somehow in the right sort of state). The universe would expand to
a very large size Figure 8:1 and eventually it would collapse again into what looks like a singularity in real time. Thus,
in a sense, we are still all doomed, even if we keep away from black holes. Only if we could picture the universe in
terms of imaginary time would there be no singularities.
If the universe really is in such a quantum state, there would be no singularities in the history of the universe in
imaginary time. It might seem therefore that my more recent work had completely undone the results of my earlier
work on singularities. But, as indicated above, the real importance of the singularity theorems was that they showed
that the gravitational field must become so strong that quantum gravitational effects could not be ignored. This in turn
led to the idea that the universe could be finite in imaginary time but without boundaries or singularities. When one
goes back to the real time in which we live, however, there will still appear to be singularities. The poor astronaut
who falls into a black hole will still come to a sticky end; only if he lived in imaginary time would he encounter no
singularities.
This might suggest that the so-called imaginary time is really the real time, and that what we call real time is just a
figment of our imaginations. In real time, the universe has a beginning and an end at singularities that form a
boundary to space-time and at which the laws of science break down. But in imaginary time, there are no
singularities or boundaries. So maybe what we call imaginary time is really more basic, and what we call real is just
an idea that we invent to help us describe what we think the universe is like. But according to the approach I
described in Chapter 1, a scientific theory is just a mathematical model we make to describe our observations: it
exists only in our minds. So it is meaningless to ask: which is real, “real” or “imaginary” time? It is simply a matter of
which is the more useful description.
One can also use the sum over histories, along with the no boundary proposal, to find which properties of the
universe are likely to occur together. For example, one can calculate the probability that the universe is expanding at
nearly the same rate in all different directions at a time when the density of the universe has its present value. In the
simplified models that have been examined so far, this probability turns out to be high; that is, the proposed no
boundary condition leads to the prediction that it is extremely probable that the present rate of expansion of the
universe is almost the same in each direction. This is consistent with the observations of the microwave background
radiation, which show that it has almost exactly the same intensity in any direction. If the universe were expanding
faster in some directions than in others, the intensity of the radiation in those directions would be reduced by an

A Brief History of Time - Stephen Hawking... Chapter 8
additional red shift.
Further predictions of the no boundary condition are currently being worked out. A particularly interesting problem is
the size of the small departures from uniform density in the early universe that caused the formation first of the
galaxies, then of stars, and finally of us. The uncertainty principle implies that the early universe cannot have been
completely uniform because there must have been some uncertainties or fluctuations in the positions and velocities
of the particles. Using the no boundary condition, we find that the universe must in fact have started off with just the
minimum possible non-uniformity allowed by the uncertainty principle. The universe would have then undergone a
period of rapid expansion, as in the inflationary models. During this period, the initial non-uniformities would have
been amplified until they were big enough to explain the origin of the structures we observe around us. In 1992 the
Cosmic Background Explorer satellite (COBE) first detected very slight variations in the intensity of the microwave
background with direction. The way these non-uniformities depend on direction seems to agree with the predictions
of the inflationary model and the no boundary proposal. Thus the no boundary proposal is a good scientific theory in
the sense of Karl Popper: it could have been falsified by observations but instead its predictions have been
confirmed. In an expanding universe in which the density of matter varied slightly from place to place, gravity would
have caused the denser regions to slow down their expansion and start contracting. This would lead to the formation
of galaxies, stars, and eventually even insignificant creatures like ourselves. Thus all the complicated structures that
we see in the universe might be explained by the no boundary condition for the universe together with the uncertainty
principle of quantum mechanics.
The idea that space and time may form a closed surface without boundary also has profound implications for the role
of God in the affairs of the universe. With the success of scientific theories in describing events, most people have
come to believe that God allows the universe to evolve according to a set of laws and does not intervene in the
universe to break these laws. However, the laws do not tell us what the universe should have looked like when it
started

A Brief History of Time - Stephen Hawking... Chapter 9
CHAPTER 9
THE ARROW OF TIME
In previous chapters we have seen how our views of the nature of time have changed over the years. Up to the
beginning of this century people believed in an absolute time. That is, each event could be labeled by a number
called “time” in a unique way, and all good clocks would agree on the time interval between two events.
However, the discovery that the speed of light appeared the same to every observer, no matter how he was
moving, led to the theory of relativity

A Brief History of Time - Stephen Hawking... Chapter 9
and that these two arrows necessarily always point in the same direction. If one assumes the no boundary
condition for the universe, we shall see that there must be well-defined thermodynamic and cosmological
arrows of time, but they will not point in the same direction for the whole history of the universe. However, I
shall argue that it is only when they do point in the same direction that conditions are suitable for the
development of intelligent beings who can ask the question: why does disorder increase in the same direction
of time as that in which the universe expands?
I shall discuss first the thermodynamic arrow of time. The second law of thermodynamics results from the fact
that there are always many more disordered states than there are ordered ones. For example, consider the
pieces of a jigsaw in a box. There is one, and. only one, arrangement in which the pieces make a complete
picture. On the other hand, there are a very large number of arrangements in which the pieces are disordered
and don’t make a picture.
Suppose a system starts out in one of the small number of ordered states. As time goes by, the system will
evolve according to the laws of science and its state will change. At a later time, it is more probable that the
system will be in a disordered state than in an ordered one because there are more disordered states. Thus
disorder will tend to increase with time if the system obeys an initial condition of high order.
Suppose the pieces of the jigsaw start off in a box in the ordered arrangement in which they form a picture. If
you shake the box, the pieces will take up another arrangement. This will probably be a disordered
arrangement in which the pieces don’t form a proper picture, simply because there are so many more
disordered arrangements. Some groups of pieces may still form parts of the picture, but the more you shake
the box, the more likely it is that these groups will get broken up and the pieces will be in a completely jumbled
state in which they don’t form any sort of picture. So the disorder of the pieces will probably increase with time if
the pieces obey the initial condition that they start off in a condition of high order.
Suppose, however, that God decided that the universe should finish up in a state of high order but that it didn’t
matter what state it started in. At early times the universe would probably be in a disordered state. This would
mean that disorder would decrease with time. You would see broken cups gathering themselves together and
jumping back onto the table. However, any human beings who were observing the cups would be living in a
universe in which disorder decreased with time. I shall argue that such beings would have a psychological
arrow of time that was backward. That is, they would remember events in the future, and not remember events
in their past. When the cup was broken, they would remember it being on the table, but when it was on the
table, they would not remember it being on the floor.
It is rather difficult to talk about human memory because we don’t know how the brain works in detail. We do,
however, know all about how computer memories work. I shall therefore discuss the psychological arrow of
time for computers. I think it is reasonable to assume that the arrow for computers is the same as that for
humans. If it were not, one could make a killing on the stock exchange by having a computer that would
remember tomorrow’s prices! A computer memory is basically a device containing elements that can exist in
either of two states. A simple example is an abacus. In its simplest form, this consists of a number of wires; on
each wire there are a number of beads that can be put in one of two positions. Before an item is recorded in a
computer’s memory, the memory is in a disordered state, with equal probabilities for the two possible states.
(The abacus beads are scattered randomly on the wires of the abacus.) After the memory interacts with the
system to be remembered, it will definitely be in one state or the other, according to the state of the system.
(Each abacus bead will be at either the left or the right of the abacus wire.) So the memory has passed from a
disordered state to an ordered one. However, in order to make sure that the memory is in the right state, it is
necessary to use a certain amount of energy (to move the bead or to power the computer, for example). This
energy is dissipated as heat, and increases the amount of disorder in the universe. One can show that this
increase in disorder is always greater than the increase in the order of the memory itself. Thus the heat
expelled by the computer’s cooling fan means that when a computer records an item in memory, the total
amount of disorder in the universe still goes up. The direction of time in which a computer remembers the past
is the same as that in which disorder increases.
Our subjective sense of the direction of time, the psychological arrow of time, is therefore determined within our
brain by the thermodynamic arrow of time. Just like a computer, we must remember things in the order in which
entropy increases. This makes the second law of thermodynamics almost trivial. Disorder increases with time

A Brief History of Time - Stephen Hawking... Chapter 9
because we measure time in the direction in which disorder increases You can’t have a safer bet than that!
But why should the thermodynamic arrow of time exist at all? Or, in other words, why should the universe be in
a state of high order at one end of time, the end that we call the past? Why is it not in a state of complete
disorder at all times? After all, this might seem more probable. And why is the direction of time in which
disorder increases the same as that in which the universe expands?
In the classical theory of general relativity one cannot predict how the universe would have begun because all
the known laws of science would have broken down at the big bang singularity. The universe could have
started out in a very smooth and ordered state. This would have led to well-defined thermodynamic and
cosmological arrows of time, as we observe. But it could equally well have started out in a very lumpy and
disordered state. In that case, the universe would already be in a state of complete disorder, so disorder could
not increase with time. It would either stay constant, in which case there would be no well-defined
thermodynamic arrow of time, or it would decrease, in which case the thermodynamic arrow of time would point
in the opposite direction to the cosmological arrow. Neither of these possibilities agrees with what we observe.
However, as we have seen, classical general relativity predicts its own downfall. When the curvature of
space-time becomes large, quantum gravitational effects will become important and the classical theory will
cease to be a good description of the universe. One has to use a quantum theory of gravity to understand how
the universe began.
In a quantum theory of gravity, as we saw in the last chapter, in order to specify the state of the universe one
would still have to say how the possible histories of the universe would behave at the boundary of space-time in
the past. One could avoid this difficulty of having to describe what we do not and cannot know only if the
histories satisfy the no boundary condition: they are finite in extent but have no boundaries, edges, or
singularities. In that case, the beginning of time would be a regular, smooth point of space-time and the
universe would have begun its expansion in a very smooth and ordered state. It could not have been
completely uniform, because that would violate the uncertainty principle of quantum theory. There had to be
small fluctuations in the density and velocities of particles. The no boundary condition, however, implied that
these fluctuations were as small as they could be, consistent with the uncertainty principle.
The universe would have started off with a period of exponential or “inflationary” expansion in which it would
have increased its size by a very large factor. During this expansion, the density fluctuations would have
remained small at first, but later would have started to grow. Regions in which the density was slightly higher
than average would have had their expansion slowed down by the gravitational attraction of the extra mass.
Eventually, such regions would stop expanding and collapse to form galaxies, stars, and beings like us. The
universe would have started in a smooth and ordered state, and would become lumpy and disordered as tim
went on. This would explain the existence of the thermodynamic arrow of time.
But what would happen if and when the universe stopped expanding and began to contract? Would the
thermodynamic arrow reverse and disorder begin to decrease with time? This would lead to all sorts of
science-fiction-like possibilities for people who survived from the expanding to the contracting phase. Would
they see broken cups gathering themselves together off the floor and jumping back onto the table? Would they
be able to remember tomorrow’s prices and make a fortune on the stock market? It might seem a bit academic
to worry about what will happen when the universe collapses again, as it will not start to contract for at least
another ten thousand million years. But there is a quicker way to find out what will happen: jump into a black
hole. The collapse of a star to form a black hole is rather like the later stages of the collapse of the whole
universe. So if disorder were to decrease in the contracting phase of the universe, one might also expect it to
decrease inside a black hole. So perhaps an astronaut who fell into a black hole would be able to make money
at roulette by remembering where the ball went before he placed his bet. (Unfortunately, however, he would not
have long to play before he was turned to spaghetti. Nor would he be able to let us know about the reversal of
the thermodynamic arrow, or even bank his winnings, because he would be trapped behind the event horizon
of the black hole.)
At first, I believed that disorder would decrease when the universe recollapsed. This was because I thought that
the universe had to return to a smooth and ordered state when it became small again. This would mean that
the contracting phase would be like the time reverse of the expanding phase. People in the contracting phase
would live their lives backward: they would die before they were born and get younger as the universe

A Brief History of Time - Stephen Hawking... Chapter 9
contracted.
This idea is attractive because it would mean a nice symmetry between the expanding and contracting phases.
However, one cannot adopt it on its own, independent of other ideas about the universe. The question is: is it
implied by the no boundary condition, or is it inconsistent with that condition? As I said, I thought at first that the
no boundary condition did indeed imply that disorder would decrease in the contracting phase. I was misled
partly by the analogy with the surface of the earth. If one took the beginning of the universe to correspond to
the North Pole, then the end of the universe should be similar to the beginning, just as the South Pole is similar
to the North. However, the North and South Poles correspond to the beginning and end of the universe in
imaginary time. The beginning and end in real time can be very different from each other. I was also misled by
work I had done on a simple model of the universe in which the collapsing phase looked like the time reverse of
the expanding phase. However, a colleague of mine, Don Page, of Penn State University, pointed out that the
no boundary condition did not require the contracting phase necessarily to be the time reverse of the expanding
phase. Further, one of my students, Raymond Laflamme, found that in a slightly more complicated model, the
collapse of the universe was very different from the expansion. I realized that I had made a mistake: the no
boundary condition implied that disorder would in fact continue to increase during the contraction. The
thermodynamic and psychological arrows of time would not reverse when the universe begins to recontract, or
inside black holes.
What should you do when you find you have made a mistake like that? Some people never admit that they are
wrong and continue to find new, and often mutually inconsistent, arguments to support their case

A Brief History of Time - Stephen Hawking... Chapter 9
agree with the cosmological arrow is that intelligent beings can exist only in the expanding phase. The
contracting phase will be unsuitable because it has no strong thermodynamic arrow of time.
The progress of the human race in understanding the universe has established a small corner of order in an
increasingly disordered universe. If you remember every word in this book, your memory will have recorded
about two million pieces of information: the order in your brain will have increased by about two million units.
However, while you have been reading the book, you will have converted at least a thousand calories of
ordered energy, in the form of food, into disordered energy, in the form of heat that you lose to the air around
you by convection and sweat. This will increase the disorder of the universe by about twenty million million
million million units

A Brief History of Time - Stephen Hawking... Chapter 10
CHAPTER 10
WORMHOLES AND TIME TRAVEL
The last chapter discussed why we see time go forward: why disorder increases and why we remember the
past but not the future. Time was treated as if it were a straight railway line on which one could only go one way
or the other.
But what if the railway line had loops and branches so that a train could keep going forward but come back to a
station it had already passed? In other words, might it be possible for someone to travel into the future or the
past?
H. G. Wells in The Time Machine explored these possibilities as have countless other writers of science fiction.
Yet many of the ideas of science fiction, like submarines and travel to the moon, have become matters of
science fact. So what are the prospects for time travel?
The first indication that the laws of physics might really allow people to travel in time came in 1949 when Kurt
Godel discovered a new space-time allowed by general relativity. Godel was a mathematician who was famous
for proving that it is impossible to prove all true statements, even if you limit yourself to trying to prove all the
true statements in a subject as apparently cut and dried as arithmetic. Like the uncertainty principle, Godel’s
incompleteness theorem may be a fundamental limitation on our ability to understand and predict the universe,
but so far at least it hasn’t seemed to be an obstacle in our search for a complete unified theory.
Godel got to know about general relativity when he and Einstein spent their later years at the Institute for
Advanced Study in Princeton. His space-time had the curious property that the whole universe was rotating.
One might ask: “Rotating with respect to what?” The answer is that distant matter would be rotating with
respect to directions that little tops or gyroscopes point in.
This had the side effect that it would be possible for someone to go off in a rocket ship and return to earth
before he set out. This property really upset Einstein, who had thought that general relativity wouldn’t allow time
travel. However, given Einstein’s record of ill-founded opposition to gravitational collapse and the uncertainty
principle, maybe this was an encouraging sign. The solution Godel found doesn’t correspond to the universe
we live in because we can show that the universe is not rotating. It also had a non-zero value of the
cosmological constant that Einstein introduced when he thought the universe was unchanging. After Hubble
discovered the expansion of the universe, there was no need for a cosmological constant and it is now
generally believed to be zero. However, other more reasonable space-times that are allowed by general
relativity and which permit travel into the past have since been found. One is in the interior of a rotating black
hole. Another is a space-time that contains two cosmic strings moving past each other at high speed. As their
name suggests, cosmic strings are objects that are like string in that they have length but a tiny cross section.
Actually, they are more like rubber bands because they are under enormous tension, something like a million
million million million tons. A cosmic string attached to the earth could accelerate it from 0 to 60 mph in 1/30th
of a second. Cosmic strings may sound like pure science fiction but there are reasons to believe they could
have formed in the early universe as a result of symmetry-breaking of the kind discussed in Chapter 5.
Because they would be under enormous tension and could start in any configuration, they might accelerate to
very high speeds when they straighten out.
The Godel solution and the cosmic string space-time start out so distorted that travel into the past was always
possible. God might have created such a warped universe but we have no reason to believe he did.
Observations of the microwave background and of the abundances of the light elements indicate that the early
universe did not have the kind of curvature required to allow time travel. The same conclusion follows on
theoretical grounds if the no boundary proposal is correct. So the question is: if the universe starts out without
the kind of curvature required for time travel, can we subsequently warp local regions of space-time sufficiently
to allow it?
A closely related problem that is also of concern to writers of science fiction is rapid interstellar or intergalactic

A Brief History of Time - Stephen Hawking... Chapter 10
travel. According to relativity, nothing can travel faster than light. If we therefore sent a spaceship to our nearest
neighboring star, Alpha Centauri, which is about four light-years away, it would take at least eight years before
we could expect the travelers to return and tell us what they had found. If the expedition were to the center of
our galaxy, it would be at least a hundred thousand years before it came back. The theory of relativity does
allow one consolation. This is the so-called twins paradox mentioned in Chapter 2.
Because there is no unique standard of time, but rather observers each have their own time as measured by
clocks that they carry with them, it is possible for the journey to seem to be much shorter for the space travelers
than for those who remain on earth. But there would not be much joy in returning from a space voyage a few
years older to find that everyone you had left behind was dead and gone thousands of years ago. So in order to
have any human interest in their stories, science fiction writers had to suppose that we would one day discover
how to travel faster than light. What most of thee authors don’t seem to have realized is that if you can travel
faster than light, the theory of relativity implies you can also travel back in the, as the following limerick says:
There was a young lady of Wight
Who traveled much faster than light.
She departed one day,
In a relative way,
And arrived on the previous night
The point is that the theory of relativity says hat there is no unique measure of time that all observers will agree
on Rather, each observer has his or her own measure of time. If it is possible for a rocket traveling below the
speed of light to get from event A (say, the final of the 100-meter race of the Olympic Games in 202) to event B
(say, the opening of the 100,004th meeting of the Congress of Alpha Centauri), then all observers will agree
that event A happened before event B according to their times. Suppose, however, that the spaceship would
have to travel faster than light to carry the news of the race to the Congress. Then observers moving at
different speeds can disagree about whether event A occurred before B or vice versa. According to the time of
an observer who is at rest with respect to the earth, it may be that the Congress opened after the race. Thus
this observer would think that a spaceship could get from A to B in time if only it could ignore the speed-of-light
speed limit. However, to an observer at Alpha Centauri moving away from the earth at nearly the speed of light,
it would appear that event B, the opening of the Congress, would occur before event A, the 100-meter race.
The theory of relativity says that the laws of physics appear the same to observers moving at different speeds.
This has been well tested by experiment and is likely to remain a feature even if we find a more advanced
theory to replace relativity Thus the moving observer would say that if faster-than-light travel is possible, it
should be possible to get from event B, the opening of the Congress, to event A, the 100-meter race. If one
went slightly faster, one could even get back before the race and place a bet on it in the sure knowledge that
one would win.
There is a problem with breaking the speed-of-light barrier. The theory of relativity says that the rocket power
needed to accelerate a spaceship gets greater and greater the nearer it gets to the speed of light. We have
experimental evidence for this, not with spaceships but with elementary particles in particle accelerators like
those at Fermilab or CERN (European Centre for Nuclear Research). We can accelerate particles to 99.99
percent of the speed of light, but however much power we feed in, we can’t get them beyond the speed-of-light
barrier. Similarly with spaceships: no matter how much rocket power they have, they can’t accelerate beyond
the speed of light.
That might seem to rule out both rapid space travel and travel back in time. However, there is a possible way
out. It might be that one could warp space-time so that there was a shortcut between A and B One way of doing
this would be to create a wormhole between A and B. As its name suggests, a wormhole is a thin tube of
space-time which can connect two nearly flat regions far apart.
There need be no relation between the distance through the wormhole and the separation of its ends in the
nearly Hat background. Thus one could imagine that one could create or find a wormhole that world lead from
the vicinity of the Solar System to Alpha Centauri. The distance through the wormhole might be only a few
million miles even though earth and Alpha Centauri are twenty million million miles apart in ordinary space. This
would allow news of the 100-meter race to reach the opening of the Congress. But then an observer moving
toward 6e earth should also be able to find another wormhole that would enable him to get from the opening of

A Brief History of Time - Stephen Hawking... Chapter 10
the Congress on Alpha Centauri back to earth before the start of the race. So wormholes, like any other
possible form of travel faster than light, would allow one to travel into the past.
The idea of wormholes between different regions of space-time was not an invention of science fiction writers
but came from a very respectable source.
In 1935, Einstein and Nathan Rosen wrote a paper in which they showed that general relativity allowed what
they called “bridges,” but which are now known as wormholes. The Einstein-Rosen bridges didn’t last long
enough for a spaceship to get through: the ship would run into a singularity as the wormhole pinched off.
However, it has been suggested that it might be possible for an advanced civilization to keep a wormhole open.
To do this, or to warp space-time in any other way so as to permit time travel, one can show that one needs a
region of space-time with negative curvature, like the surface of a saddle. Ordi-nary matter, which has a
positive energy density, gives space-time a positive curvature, like the surface of a sphere. So what one needs,
in order to warp space-time in a way that will allow travel into the past, is matter with negative energy density.
Energy is a bit like money: if you have a positive balance, you can distribute it in various ways, but according to
the classical laws that were believed at the beginning of the century, you weren’t allowed to be overdrawn. So
these classical laws would have ruled out any possibility of time travel. However, as has been described in
earlier chapters, the classical laws were superseded by quantum laws based on the uncertainty principle. The
quantum laws are more liberal and allow you to be overdrawn on one or two accounts provided the total
balance is positive. In other words, quantum theory allows the energy density to be negative in some places,
provided that this is made up for by positive energy densities in other places, so that the total energy re-mains
positive. An example of how quantum theory can allow negative energy densities is provided by what is called
the Casimir effect. As we saw in Chapter 7, even what we think of as “empty” space is filled with pairs of virtual
particles and antiparticles that appear together, move apart, and come back together and annihilate each other.
Now, suppose one has two parallel metal plates a short distance apart. The plates will act like mirrors for the
virtual photons or particles of light. In fact they will form a cavity between them, a bit like an organ pipe that will
resonate only at certain notes. This means that virtual photons can occur in the space between the plates only
if their wavelengths (the distance between the crest of one wave and the next) fit a whole number of times into
the gap between the plates. If the width of a cavity is a whole number of wavelengths plus a fraction of a
wave-length, then after some reflections backward and forward between the plates, the crests of one wave will
coincide with the troughs of another and the waves will cancel out.
Because the virtual photons between the plates can have only the resonant wavelengths, there will be slightly
fewer of them than in the region outside the plates where virtual photons can have any wavelength. Thus there
will be slightly fewer virtual photons hitting the inside surfaces of the plates than the outside surfaces. One
would therefore expect a force on the plates, pushing them toward each other. This force has actually been
detected and has the predicted value. Thus we have experimental evidence that virtual particles exist and have
real effects.
The fact that there are fewer virtual photons between the plates means that their energy density will be less
than elsewhere. But the total energy density in “empty” space far away from the plates must be zero, because
otherwise the energy density would warp the space and it would not be almost flat. So, if the energy density
between the plates is less than the energy density far away, it must be negative.
We thus have experimental evidence both that space-time can be warped (from the bending of light during
eclipses) and that it can be curved in the way necessary to allow time travel (from the Casimir effect). One
might hope therefore that as we advance in science and technology, we would eventually manage to build a
time machine. But if so, why hasn’t anyone come back from the future and told us how to do it? There might be
good reasons why it would be unwise to give us the secret of time travel at our present primitive state of
development, but unless human nature changes radically, it is difficult to believe that some visitor from the
future wouldn’t spill the beans. Of course, some people would claim that sightings of UFOs are evidence that
we are being visited either by aliens or by people from the future. (If the aliens were to get here in reasonable
time, they would need faster-than-light travel, so the two possibilities may be equivalent.)
However, I think that any visit by aliens or people from the future would be much more obvious and, probably,
much more unpleasant. If they are going to reveal themselves at all, why do so only to those who are not