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The Beginning of Time
S.D.Hawking
In this lecture, I would like to discuss whether time itself has a beginning, and whether it will
have an end. All the evidence seems to indicate, that the universe has not existed forever, but
that it had a beginning, about 15 billion years ago. This is probably the most remarkable
discovery of modern cosmology. Yet it is now taken for granted. We are not yet certain
whether the universe will have an end. When I gave a lecture in Japan, I was asked not to
mention the possible re-collapse of the universe, because it might affect the stock market.
However, I can re-assure anyone who is nervous about their investments that it is a bit early
to sell: even if the universe does come to an end, it won't be for at least twenty billion years.
By that time, maybe the GATT trade agreement will have come into effect.
The time scale of the universe is very long compared to that for human life. It was therefore
not surprising that until recently, the universe was thought to be essentially static, and
unchanging in time. On the other hand, it must have been obvious, that society is evolving in
culture and technology. This indicates that the present phase of human history can not have
been going for more than a few thousand years. Otherwise, we would be more advanced than
we are. It was therefore natural to believe that the human race, and maybe the whole
universe, had a beginning in the fairly recent past. However, many people were unhappy with
the idea that the universe had a beginning, because it seemed to imply the existence of a
supernatural being who created the universe. They preferred to believe that the universe, and
the human race, had existed forever. Their explanation for human progress was that there had
been periodic floods, or other natural disasters, which repeatedly set back the human race to a
primitive state.
This argument about whether or not the universe had a beginning, persisted into the 19th and
20th centuries. It was conducted mainly on the basis of theology and philosophy, with little
consideration of observational evidence. This may have been reasonable, given the notoriously
unreliable character of cosmological observations, until fairly recently. The cosmologist, Sir
Arthur Eddington, once said, 'Don't worry if your theory doesn't agree with the observations,
because they are probably wrong.' But if your theory disagrees with the Second Law of
Thermodynamics, it is in bad trouble. In fact, the theory that the universe has existed forever
is in serious difficulty with the Second Law of Thermodynamics. The Second Law, states that
disorder always increases with time. Like the argument about human progress, it indicates that
there must have been a beginning. Otherwise, the universe would be in a state of complete
disorder by now, and everything would be at the same temperature. In an infinite and
everlasting universe, every line of sight would end on the surface of a star. This would mean
that the night sky would have been as bright as the surface of the Sun. The only way of
avoiding this problem would be if, for some reason, the stars did not shine before a certain
time.
In a universe that was essentially static, there would not have been any dynamical reason,
why the stars should have suddenly turned on, at some time. Any such "lighting up time" would
have to be imposed by an intervention from outside the universe. The situation was different,
however, when it was realised that the universe is not static, but expanding. Galaxies are
moving steadily apart from each other. This means that they were closer together in the past.
One can plot the separation of two galaxies, as a function of time. If there were no
acceleration due to gravity, the graph would be a straight line. It would go down to zero
separation, about twenty billion years ago. One would expect gravity, to cause the galaxies to
accelerate towards each other. This will mean that the graph of the separation of two galaxies
will bend downwards, below the straight line. So the time of zero separation, would have been
less than twenty billion years ago.
At this time, the Big Bang, all the matter in the universe, would have been on top of itself. The
density would have been infinite. It would have been what is called, a singularity. At a
singularity, all the laws of physics would have broken down. This means that the state of the
universe, after the Big Bang, will not depend on anything that may have happened before,
because the deterministic laws that govern the universe will break down in the Big Bang. The
universe will evolve from the Big Bang, completely independently of what it was like before.
Even the amount of matter in the universe, can be different to what it was before the Big Bang,
as the Law of Conservation of Matter, will break down at the Big Bang.
Since events before the Big Bang have no observational consequences, one may as well cut
them out of the theory, and say that time began at the Big Bang. Events before the Big Bang,
are simply not defined, because there's no way one could measure what happened at them.
This kind of beginning to the universe, and of time itself, is very different to the beginnings that
had been considered earlier. These had to be imposed on the universe by some external
agency. There is no dynamical reason why the motion of bodies in the solar system can not be
extrapolated back in time, far beyond four thousand and four BC, the date for the creation of
the universe, according to the book of Genesis. Thus it would require the direct intervention of
God, if the universe began at that date. By contrast, the Big Bang is a beginning that is
required by the dynamical laws that govern the universe. It is therefore intrinsic to the
universe, and is not imposed on it from outside.
Although the laws of science seemed to predict the universe had a beginning, they also
seemed to predict that they could not determine how the universe would have begun. This was
obviously very unsatisfactory. So there were a number of attempts to get round the
conclusion, that there was a singularity of infinite density in the past. One suggestion was to
modify the law of gravity, so that it became repulsive. This could lead to the graph of the
separation between two galaxies, being a curve that approached zero, but didn't actually pass
through it, at any finite time in the past. Instead, the idea was that, as the galaxies moved
apart, new galaxies were formed in between, from matter that was supposed to be continually
created. This was the Steady State theory, proposed by Bondi, Gold, and Hoyle.
The Steady State theory, was what Karl Popper would call, a good scientific theory: it made
definite predictions, which could be tested by observation, and possibly falsified. Unfortunately
for the theory, they were falsified. The first trouble came with the Cambridge observations, of
the number of radio sources of different strengths. On average, one would expect that the
fainter sources would also be the more distant. One would therefore expect them to be more
numerous than bright sources, which would tend to be near to us. However, the graph of the
number of radio sources, against there strength, went up much more sharply at low source
strengths, than the Steady State theory predicted.
There were attempts to explain away this number count graph, by claiming that some of the
faint radio sources, were within our own galaxy, and so did not tell us anything about
cosmology. This argument didn't really stand up to further observations. But the final nail in
the coffin of the Steady State theory came with the discovery of the microwave background
radiation, in 1965. This radiation is the same in all directions. It has the spectrum of radiation
in thermal equilibrium at a temperature of 2 point 7 degrees above the Absolute Zero of
temperature. There doesn't seem any way to explain this radiation in the Steady State theory.
Another attempt to avoid a beginning to time, was the suggestion, that maybe all the galaxies
didn't meet up at a single point in the past. Although on average, the galaxies are moving
apart from each other at a steady rate, they also have small ad***ional velocities, relative to
the uniform expansion. These so-called "peculiar velocities" of the galaxies, may be directed
sideways to the main expansion. It was argued, that as you plotted the position of the galaxies
back in time, the sideways peculiar velocities, would have meant that the galaxies wouldn't
have all met up. Instead, there could have been a previous contracting phase of the universe,
in which galaxies were moving towards each other. The sideways velocities could have meant
that the galaxies didn't collide, but rushed past each other, and then started to move apart.
There wouldn't have been any singularity of infinite density, or any breakdown of the laws of
physics. Thus there would be no necessity for the universe, and time itself, to have a
beginning. Indeed, one might suppose that the universe had oscillated, though that still
wouldn't solve the problem with the Second Law of Thermodynamics: one would expect that
the universe would become more disordered each oscillation. It is therefore difficult to see how
the universe could have been oscillating for an infinite time.
This possibility, that the galaxies would have missed each other, was supported by a paper by
two Russians. They claimed that there would be no singularities in a solution of the field
equations of general relativity, which was fully general, in the sense that it didn't have any
exact symmetry. However, their claim was proved wrong, by a number of theorems by Roger
Penrose and myself. These showed that general relativity predicted singularities, whenever
more than a certain amount of mass was present in a region. The first theorems were
designed to show that time came to an end, inside a black hole, formed by the collapse of a
star. However, the expansion of the universe, is like the time reverse of the collapse of a star.
I therefore want to show you, that observational evidence indicates the universe contains
sufficient matter, that it is like the time reverse of a black hole, and so contains a singularity.
In order to discuss observations in cosmology, it is helpful to draw a diagram of events in
space and time, with time going upward, and the space directions horizontal. To show this
diagram properly, I would really need a four dimensional screen. However, because of
government cuts, we could manage to provide only a two dimensional screen. I shall therefore
be able to show only one of the space directions.
As we look out at the universe, we are looking back in time, because light had to leave distant
objects a long time ago, to reach us at the present time. This means that the events we
observe lie on what is called our past light cone. The point of the cone is at our position, at the
present time. As one goes back in time on the diagram, the light cone spreads out to greater
distances, and its area increases. However, if there is sufficient matter on our past light cone,
it will bend the rays of light towards each other. This will mean that, as one goes back into the
past, the area of our past light cone will reach a maximum, and then start to decrease. It is
this focussing of our past light cone, by the gravitational effect of the matter in the universe,
that is the signal that the universe is within its horizon, like the time reverse of a black hole.

Hayumiko

Continue:
If
one can determine that there is enough matter in the universe, to focus our past light cone,
one can then apply the singularity theorems, to show that time must have a beginning.
How can we tell from the observations, whether there is enough matter on our past light cone,
to focus it? We observe a number of galaxies, but we can not measure directly how much
matter they contain. Nor can we be sure that every line of sight from us will pass through a
galaxy. So I will give a different argument, to show that the universe contains enough matter,
to focus our past light cone. The argument is based on the spectrum of the microwave
background radiation. This is characteristic of radiation that has been in thermal equilibrium,
with matter at the same temperature. To achieve such an equilibrium, it is necessary for the
radiation to be scattered by matter, many times. For example, the light that we receive from
the Sun has a characteristically thermal spectrum. This is not because the nuclear reactions,
which go on in the centre of the Sun, produce radiation with a thermal spectrum. Rather, it is
because the radiation has been scattered, by the matter in the Sun, many times on its way
from the centre.
In the case of the universe, the fact that the microwave background has such an exactly
thermal spectrum indicates that it must have been scattered many times. The universe must
therefore contain enough matter, to make it opaque in every direction we look, because the
microwave background is the same, in every direction we look. Moreover, this opacity must
occur a long way away from us, because we can see galaxies and quasars, at great distances.
Thus there must be a lot of matter at a great distance from us. The greatest opacity over a
broad wave band, for a given density, comes from ionised hydrogen. It then follows that if
there is enough matter to make the universe opaque, there is also enough matter to focus our
past light cone. One can then apply the theorem of Penrose and myself, to show that time
must have a beginning.
The focussing of our past light cone implied that time must have a beginning, if the General
Theory of relativity is correct. But one might raise the question, of whether General Relativity
really is correct. It certainly agrees with all the observational tests that have been carried out.
However these test General Relativity, only over fairly large distances. We know that General
Relativity can not be quite correct on very small distances, because it is a classical theory. This
means, it doesn't take into account, the Uncertainty Principle of Quantum Mechanics, which
says that an object can not have both a well defined position, and a well defined speed: the
more accurately one measures the position, the less accurately one can measure the speed,
and vice versa. Therefore, to understand the very high-density stage, when the universe was
very small, one needs a quantum theory of gravity, which will combine General Relativity with
the Uncertainty Principle.
Many people hoped that quantum effects, would somehow smooth out the singularity of infinite
density, and allow the universe to bounce, and continue back to a previous contracting phase.
This would be rather like the earlier idea of galaxies missing each other, but the bounce would
occur at a much higher density. However, I think that this is not what happens: quantum
effects do not remove the singularity, and allow time to be continued back indefinitely. But it
seems that quantum effects can remove the most objectionable feature, of singularities in
classical General Relativity. This is that the classical theory, does not enable one to calculate
what would come out of a singularity, because all the Laws of Physics would break down there.
This would mean that science could not predict how the universe would have begun. Instead,
one would have to appeal to an agency outside the universe. This may be why many religious
leaders, were ready to accept the Big Bang, and the singularity theorems.
It seems that Quantum theory, on the other hand, can predict how the universe will begin.
Quantum theory introduces a new idea, that of imaginary time. Imaginary time may sound like
science fiction, and it has been brought into Doctor Who. But nevertheless, it is a genuine
scientific concept. One can picture it in the following way. One can think of ordinary, real, time
as a horizontal line. On the left, one has the past, and on the right, the future. But there's
another kind of time in the vertical direction. This is called imaginary time, because it is not the
kind of time we normally experience. But in a sense, it is just as real, as what we call real
time.
The three directions in space, and the one direction of imaginary time, make up what is called
a Euclidean space-time. I don't think anyone can picture a four dimensional curve space. But it
is not too difficult to visualise a two dimensional surface, like a saddle, or the surface of a
football.
In fact, James Hartle of the University of California Santa Barbara, and I have proposed that
space and imaginary time together, are indeed finite in extent, but without boundary. They
would be like the surface of the Earth, but with two more dimensions. The surface of the Earth
is finite in extent, but it doesn't have any boundaries or edges. I have been round the world,
and I didn't fall off.
If space and imaginary time are indeed like the surface of the Earth, there wouldn't be any
singularities in the imaginary time direction, at which the laws of physics would break down.
And there wouldn't be any boundaries, to the imaginary time space-time, just as there aren't
any boundaries to the surface of the Earth. This absence of boundaries means that the laws of
physics would determine the state of the universe uniquely, in imaginary time. But if one
knows the state of the universe in imaginary time, one can calculate the state of the universe
in real time. One would still expect some sort of Big Bang singularity in real time. So real time
would still have a beginning. But one wouldn't have to appeal to something outside the
universe, to determine how the universe began. Instead, the way the universe started out at
the Big Bang would be determined by the state of the universe in imaginary time. Thus, the
universe would be a completely self-contained system. It would not be determined by anything
outside the physical universe, that we observe.
The no boundary con***ion, is the statement that the laws of physics hold everywhere. Clearly,
this is something that one would like to believe, but it is a hypothesis. One has to test it, by
comparing the state of the universe that it would predict, with observations of what the
universe is actually like. If the observations disagreed with the predictions of the no boundary
hypothesis, we would have to conclude the hypothesis was false. There would have to be
something outside the universe, to wind up the clockwork, and set the universe going. Of
course, even if the observations do agree with the predictions, that does not prove that the no
boundary proposal is correct. But one's confidence in it would be increased, particularly
because there doesn't seem to be any other natural proposal, for the quantum state of the
universe.
The no boundary proposal, predicts that the universe would start at a single point, like the
North Pole of the Earth. But this point wouldn't be a singularity, like the Big Bang. Instead, it
would be an ordinary point of space and time, like the North Pole is an ordinary point on the
Earth, or so I'm told. I have not been there myself.
According to the no boundary proposal, the universe would have expanded in a smooth way
from a single point. As it expanded, it would have borrowed energy from the gravitational field,
to create matter. As any economist could have predicted, the result of all that borrowing, was
inflation. The universe expanded and borrowed at an ever-increasing rate. Fortunately, the
debt of gravitational energy will not have to be repaid until the end of the universe.
Eventually, the period of inflation would have ended, and the universe would have settled down
to a stage of more moderate growth or expansion. However, inflation would have left its mark
on the universe. The universe would have been almost completely smooth, but with very slight
irregularities. These irregularities are so little, only one part in a hundred thousand, that for
years people looked for them in vain. But in 1992, the Cosmic Background Explorer satellite,
COBE, found these irregularities in the microwave background radiation. It was an historic
moment. We saw back to the origin of the universe. The form of the fluctuations in the
microwave background agree closely with the predictions of the no boundary proposal. These
very slight irregularities in the universe would have caused some regions to have expanded
less fast than others. Eventually, they would have stopped expanding, and would have
collapsed in on themselves, to form stars and galaxies. Thus the no boundary proposal can
explain all the rich and varied structure, of the world we live in. What does the no boundary
proposal predict for the future of the universe? Because it requires that the universe is finite in
space, as well as in imaginary time, it implies that the universe will re-collapse eventually.
However, it will not re-collapse for a very long time, much longer than the 15 billion years it
has already been expanding. So, you will have time to sell your government bonds, before the
end of the universe is nigh. Quite what you invest in then, I don't know.
Originally, I thought that the collapse, would be the time reverse of the expansion. This would
have meant that the arrow of time would have pointed the other way in the contracting phase.
People would have gotten younger, as the universe got smaller. Eventually, they would have
disappeared back into the womb.
However, I now realise I was wrong, as these solutions show. The collapse is not the time
reverse of the expansion. The expansion will start with an inflationary phase, but the collapse
will not in general end with an anti inflationary phase. Moreover, the small departures from
uniform density will continue to grow in the contracting phase. The universe will get more and
more lumpy and irregular, as it gets smaller, and disorder will increase. This means that the
arrow of time will not reverse. People will continue to get older, even after the universe has
begun to contract. So it is no good waiting until the universe re-collapses, to return to your
youth. You would be a bit past it, anyway, by then.
The conclusion of this lecture is that the universe has not existed forever. Rather, the universe,
and time itself, had a beginning in the Big Bang, about 15 billion years ago. The beginning of
real time, would have been a singularity, at which the laws of physics would have broken
down. Nevertheless, the way the universe began would have been determined by the laws of
physics, if the universe satisfied the no boundary con***ion. This says that in the imaginary
time direction, space-time is finite in extent, but doesn't have any boundary or edge. The
predictions of the no boundary proposal seem to agree with observation. The no boundary
hypothesis also predicts that the universe will eventually collapse again. However, the
contracting phase, will not have the opposite arrow of time, to the expanding phase. So we will
keep on getting older, and we won't return to our youth. Because time is not going to go
backwards, I think I better stop now.

Hayumiko

Đây nữa nè, cố đọc đi:hep-th/9409195 30 Sep 94
1. Classical Theory
S. W. Hawking
In these lectures Roger Penrose and I will put forward our related but rather dierent
viewpoints on the nature of space and time. We shall speak alternately and shall give three
lectures each, followed by a discussion on our dierent approaches. I should emphasize that
these will be technical lectures. We shall assume a basic knowledge of general relativity
and quantum theory.
There is a short article by Richard Feynman describing his experiences at a conference
on general relativity. I think it was the Warsaw conference in 1962. It commented very
unfavorably on the general competence of the people there and the relevance of what
they were doing. That general relativity soon acquired a much better reputation, and
more interest, is in a considerable measure because of Roger's work. Up to then, general
relativity had been formulated as a messy set of partial dierential equations in a single
coordinate system. People were so pleased when they found a solution that they didn't
care that it probably had no physical signicance. However, Roger brought in modern
concepts like spinors and global methods. He was the rst to show that one could discover
general properties without solving the equations exactly. It was his rst singularity theorem
that introduced me to the study of causal structure and inspired my classical work on
singularities and black holes.
I think Roger and I pretty much agree on the classical work. However, we dier in
our approach to quantum gravity and indeed to quantum theory itself. Although I'm
regarded as a dangerous radical by particle physicists for proposing that there may be loss
of quantum coherence I'm denitely a conservative compared to Roger. I take the positivist
viewpoint that a physical theory is just a mathematical model and that it is meaningless
to ask whether it corresponds to reality. All that one can ask is that its predictions should
be in agreement with observation. I think Roger is a Platonist at heart but he must answer
for himself.
Although there have been suggestions that spacetime may have a discrete structure
I see no reason to abandon the continuum theories that have been so successful. General
relativity is a beautiful theory that agrees with every observation that has been made. It
may require modications on the Planck scale but I don't think that will aect many of
the predictions that can be obtained from it. It may be only a low energy approximation
to some more fundemental theory, like string theory, but I think string theory has been
over sold. First of all, it is not clear that general relativity, when combined with various
other elds in a supergravity theory, can not give a sensible quantum theory. Reports of
1
the death of supergravity are exaggerations. One year everyone believed that supergravity
was nite. The next year the fashion changed and everyone said that supergravity was
bound to have divergences even though none had actually been found. My second reason
for not discussing string theory is that it has not made any testable predictions. By
contrast, the straight forward application of quantum theory to general relativity, which I
will be talking about, has already made two testable predictions. One of these predictions,
the development of small perturbations during in
ation, seems to be conrmed by recent
observations of
uctuations in the microwave background. The other prediction, that
black holes should radiate thermally, is testable in principle. All we have to do is nd a
primordial black hole. Unfortunately, there don't seem many around in this neck of the
woods. If there had been we would know how to quantize gravity.
Neither of these predictions will be changed even if string theory is the ultimate
theory of nature. But string theory, at least at its current state of development, is quite
incapable of making these predictions except by appealing to general relativity as the low
energy eective theory. I suspect this may always be the case and that there may not be
any observable predictions of string theory that can not also be predicted from general
relativity or supergravity. If this is true it raises the question of whether string theory is a
genuine scientic theory. Is mathematical beauty and completeness enough in the absence
of distinctive observationally tested predictions. Not that string theory in its present form
is either beautiful or complete.
For these reasons, I shall talk about general relativity in these lectures. I shall concentrate
on two areas where gravity seems to lead to features that are completely dierent
from other eld theories. The rst is the idea that gravity should cause spacetime to have
a begining and maybe an end. The second is the discovery that there seems to be intrinsic
gravitational entropy that is not the result of coarse graining. Some people have claimed
that these predictions are just artifacts of the semi classical approximation. They say that
string theory, the true quantum theory of gravity, will smear out the singularities and will
introduce correlations in the radiation from black holes so that it is only approximately
thermal in the coarse grained sense. It would be rather boring if this were the case. Gravity
would be just like any other eld. But I believe it is distinctively dierent, because
it shapes the arena in which it acts, unlike other elds which act in a xed spacetime
background. It is this that leads to the possibility of time having a begining. It also leads
to regions of the universe which one can't observe, which in turn gives rise to the concept
of gravitational entropy as a measure of what we can't know.
In this lecture I shall review the work in classical general relativity that leads to these
ideas. In the second and third lectures I shall show how they are changed and extended
2
when one goes to quantum theory. Lecture two will be about black holes and lecture three
will be on quantum cosmology.
The crucial technique for investigating singularities and black holes that was introduced
by Roger, and which I helped develop, was the study of the global causal structure
of spacetime.
Time
Space
Null geodesics through p
generating part of
Null geodesic in (p) which
does not go back to p and has
no past end point
Point removed
from spacetime
Chronological
future
p +
.
I (p)
+ I (p)
+
.
I
Dene I+(p) to be the set of all points of the spacetime M that can be reached from p by
future directed time like curves. One can think of I+(p) as the set of all events that can
be in
uenced by what happens at p. There are similar denitions in which plus is replaced
by minus and future by past. I shall regard such denitions as self evident.
q
p
+
.
I (S)
.
timelike curve
+ I (S)
+ I (S) can't be timelike
q
+
.
I (S)
.
+ I (S)
+ I (S) can't be spacelike
All timelike curves from q leave + I (S)
One now considers the boundary _ I+(S) of the future of a set S. It is fairly easy to
see that this boundary can not be time like. For in that case, a point q just outside the
boundary would be to the future of a point p just inside. Nor can the boundary of the
3
future be space like, except at the set S itself. For in that case every past directed curve
from a point q, just to the future of the boundary, would cross the boundary and leave the
future of S. That would be a contradiction with the fact that q is in the future of S.
q
+
.
I (S) null geodesic segment in
+ I (S)
q
+ I (S)
+
.
I (S) null geodesic segment in
+
.
I (S) future end point of generators of

Hayumiko

Continue: (Ko post được công thức và hình, thông cảm)
One therefore concludes that the boundary of the future is null apart from at S itself.
More precisely, if q is in the boundary of the future but is not in the closure of S there
is a past directed null geodesic segment through q lying in the boundary. There may be
more than one null geodesic segment through q lying in the boundary, but in that case q
will be a future end point of the segments. In other words, the boundary of the future of
S is generated by null geodesics that have a future end point in the boundary and pass
into the interior of the future if they intersect another generator. On the other hand, the
null geodesic generators can have past end points only on S. It is possible, however, to
have spacetimes in which there are generators of the boundary of the future of a set S that
never intersect S. Such generators can have no past end point.
A simple example of this is Minkowski space with a horizontal line segment removed.
If the set S lies to the past of the horizontal line, the line will cast a shadow and there
will be points just to the future of the line that are not in the future of S. There will be
a generator of the boundary of the future of S that goes back to the end of the horizontal
4
+
.
I
+ I (S)
S
line removed from
Minkowski space
generator of (S)
with no end point on S
+
.
I generators of (S)
with past end point on S
line. However, as the end point of the horizontal line has been removed from spacetime,
this generator of the boundary will have no past end point. This spacetime is incomplete,
but one can cure this by multiplying the metric by a suitable conformal factor near the
end of the horizontal line. Although spaces like this are very articial they are important
in showing how careful you have to be in the study of causal structure. In fact Roger
Penrose, who was one of my PhD examiners, pointed out that a space like that I have just
described was a counter example to some of the claims I made in my thesis.
To show that each generator of the boundary of the future has a past end point on
the set one has to impose some global con***ion on the causal structure. The strongest
and physically most important con***ion is that of global hyperbolicity.
q
p
? + I (p) _
I (q)
An open set U is said to be globally hyperbolic if:
1) for every pair of points p and q in U the intersection of the future of p and the past
of q has compact closure. In other words, it is a bounded diamond shaped region.
2) strong causality holds on U. That is there are no closed or almost closed time like
curves contained in U.
5
p
every timelike curve
intersects (t)
(t)
S
S
The physical signicance of global hyperbolicity comes from the fact that it implies
that there is a family of Cauchy surfaces (t) for U. A Cauchy surface for U is a space
like or null surface that intersects every time like curve in U once and once only. One can
predict what will happen in U from data on the Cauchy surface, and one can formulate a
well behaved quantum eld theory on a globally hyperbolic background. Whether one can
formulate a sensible quantum eld theory on a non globally hyperbolic background is less
clear. So global hyperbolicity may be a physical necessity. But my view point is that one
shouldn't assume it because that may be ruling out something that gravity is trying to
tell us. Rather one should deduce that certain regions of spacetime are globally hyperbolic
from other physically reasonable assumptions.
The signicance of global hyperbolicity for singularity theorems stems from the following.
q
p
geodesic of
maximum length
6
Let U be globally hyperbolic and let p and q be points of U that can be joined by a
time like or null curve. Then there is a time like or null geodesic between p and q which
maximizes the length of time like or null curves from p to q. The method of proof is to
show the space of all time like or null curves from p to q is compact in a certain topology.
One then shows that the length of the curve is an upper semi continuous function on this
space. It must therefore attain its maximum and the curve of maximum length will be a
geodesic because otherwise a small variation will give a longer curve.
q
r
p
geodesic
point conjugate
to p along
neighbouring
geodesic
g
g
p
q
r
non-minimal
geodesic
minimal geodesic
without conjugate points
point conjugate to p
One can now consider the second variation of the length of a geodesic
. One can show
that
can be varied to a longer curve if there is an innitesimally neighbouring geodesic
from p which intersects
again at a point r between p and q. The point r is said to be
conjugate to p. One can illustrate this by considering two points p and q on the surface of
the Earth. Without loss of generality one can take p to be at the north pole. Because the
Earth has a positive denite metric rather than a Lorentzian one, there is a geodesic of
minimal length, rather than a geodesic of maximum length. This minimal geodesic will be
a line of longtitude running from the north pole to the point q. But there will be another
geodesic from p to q which runs down the back from the north pole to the south pole and
then up to q. This geodesic contains a point conjugate to p at the south pole where all the
geodesics from p intersect. Both geodesics from p to q are stationary points of the length
under a small variation. But now in a positive denite metric the second variation of a
geodesic containing a conjugate point can give a shorter curve from p to q. Thus, in the
example of the Earth, we can deduce that the geodesic that goes down to the south pole
and then comes up is not the shortest curve from p to q. This example is very obvious.
However, in the case of spacetime one can show that under certain assumptions there
7
ought to be a globally hyperbolic region in which there ought to be conjugate points on
every geodesic between two points. This establishes a contradiction which shows that the
assumption of geodesic completeness, which can be taken as a denition of a non singular
spacetime, is false.
The reason one gets conjugate points in spacetime is that gravity is an attractive force.
It therefore curves spacetime in such a way that neighbouring geodesics are bent towards
each other rather than away. One can see this from the Raychaudhuri or Newman-Penrose
equation, which I will write in a unied form.
Raychaudhuri - Newman - Penrose equation
d
dv
= 2 + ijij +
1
n
Rablalb
where n = 2 for null geodesics
n = 3 for timelike geodesics
Here v is an ane parameter along a congruence of geodesics, with tangent vector la
which are hypersurface orthogonal. The quantity  is the average rate of convergence of
the geodesics, while  measures the shear. The term Rablalb gives the direct gravitational
eect of the matter on the convergence of the geodesics.
Einstein equation
Rab

Hayumiko

Text
x2+x=0
H2
Được kankuli sửa chữa / chuyển vào 10:19 ngày 26/02/2003

Em chịu không đọc được ! bác nào siêu dịch hộ anh em cái !
***************
Với thế giới bạn chỉ là một người , nhưng có thể với một người bạn là cả thế giới.

Các định luật khoa học không biệt quá khứ với tương lai .Nói chính xác hơn,các định luật khoa học không thay đổi dưới tổ hợp các toán tử (hay là các phép đối xứng).Như ví dụ của dz_slums cái cốc rơi từ từ bàn xuống và vỡ tan dưới sàn ,nếu ta nhìn phim ghi lại hiện tượng đó ,ta có thể dễ dàng nói rằng đang quay tới hay quay lui.Nếu phim bị quay lui thì ta có thể thấy được chiếc cốc bỗng dưng được tập kết lại và trở lại bản và trở thành chiếc cốc nguyên vẹn.Sở dĩ ta có thể nói được phim đang quay lui là vì một tiến trình như vậy không thể quan sát được trong cuộc sống hàng ngày.
Song để phân biệt quá khứ và tương lại người ta dựa vào ba mũi tên thới gian đó là :mũi tên nhiệt động học,mũi tên tâm lí học ,mũi tên vũ trụ học.
1.Mũi tên nhiệt động học chỉ hướng theo thời gian theo đó trạng thái vô trật tự có nhiều hơn trạng thái trật tự.(định luật II của nhiệt đọng học).
Ví dụ:hãy xết những miếng lắp hình trong một trò chơi ,có một và chỉ một cách xếp những miếng lắp hình này thành một hình cho trước.Mặt khác có vô số cách xếp trong đó có những miếng lắp hình vô trật tự không tạo ra hình nào cả
Càng về sau trạng thái vô trật tự càng nhiều hơn trạng thái vô trật tự trước.Như vậy vô trật tự có chiều hướng tăng lên với thời gian nếu hệ lúc đầu có một trật tự cao .
Giả sử rằng vũ trụ sẽ kết thúc bằng vô trật tự cao bất kể trạng thái ban đầu là như thế nào ?Như vậy ở trạng thái ban đầu của vũ trụ thì có nhiều xác suất ở trạng thái vô trật tự .Điều đó có nghĩa là vô trật tự sẽ giảm theo thời gian và ta sẽ thấy cốc vở sẽ tâpk kết lại thành cốc lành và nhảy lên bàn .Những còn người quan sát được việc đó phải sống trong vũ trụ ở đây vũ trụ giảm theo thời gian ,như vậycon người như thế sẽ có mũi tên tâm lí học của thời gian sẽ hướng về phía sau.Nghĩa là họ sẽ nhớ các sự kiện trong tương lai chứ không nhớ các sự kiện trong quá khứ .Khi cốc vỡ ,họ chỉ nhớ chiếc cố trên bàn chứ không nhớ chiêc cốc ở dưới sàn.Điều này trái với thực tế ,vậy vô trật tự sẽ tăng lên theo thời gian.
1.Mũi tên tâm lí học :thử so sánh rằng khi mũi tên tâm lí học của thời gian của máy tình trùng với mũi tên tâm lí học của con người thì ta sẽ giàu to ,ta chỉ cần dùng một máy tính điện tử nhớ giá cả chứng khoán chẳng hạn của ngay mai .
Bộ nhớ của máy tính lại được xếp một cách vô trật tự trước khi ta ghi lại chúng từ bàn phím ,sau khi bộ nhớ tương tác với hệ cần nhớ thì bộ nhớ sẽ ở trạng thái nhất định.Như vậy bộ nhớ chuyển từ một trạng thái vô trật tự sang một trạng thái trật tư.Song để thực hiện trạng thái trật tự đó thì cần phải tiếu tốn một năng lượng (để cung cấp điện năng cho máy tính ) và năng lượng này sẽ khuếch tán nhiệt năng và làm tăng vô trật tự của vũ trụ.Như thế ,ta có thể nói rằng mũi tên tâm lí học tăng lên theo thời gian và phụ thuộc vào mũi tên nhiêt động học (coi như là một vì cùng hướng )như vậy ta chỉ nhớ quá khứ chứ không nhớ tương lai.
3.Mũi tên vũ trụ học chỉ hướng thời gian theo đó vũ trụ sẽ giãn nở trứ không co lai(các bac có thể đọc vấn đề này ở mục vũ trụ sẽ nở ra mãi mãi"vật trất tối và số phạn bát định cảu vũ trụ".Tóm lại pha co lại không phù hợp với sự sống có trí tuệ vì trong pha này không tồn tại mũi tên nhiệt dộng học rõ ràng .
Như vậy theo em ta có thể quay lại qua khứ nhưng ta không phân biệt đâu là quá khứ đâu là tương lai.
VXH
Iam the wind.You are the sun.And one day we'll all be one.

trời,bác hayumiko viết toàn bằng tiếng anh,em ngại đọc quá,em chẳng giỏi anh gì cho cam,chịu.em định post về việc đi tới tương lai thì bác slums lại post trước em rồi,rất cám ơn bác,em nói cho dễ hiểu hơn 1 tí nhé.ví dụ như 1 nhà du hành vũ trụ rời trái đất với vận tốc nhỏ hơn 1/20000 tốc độ của ánh sáng ,nếu anh ta đi 1 năm rồi lại quay về trái đất.đối với đồng hồ của anh ta thì thời gian chỉ là 2 năm nhưng khi về đến trái đất thì thực chất thời gian đã được 200 năm rồi.như vậy không phải anh ta đã tới tương lai đấy thôi.điều này được kiểm chứng đoàng hoàng chứ không phải vô căn cứ đâu,chúng đã được kiểm chứng bằng thí nghiệm thực hành.những chếc đồng hồ trên phi thuyền chạy lệch đi so với những chiếc đồng hồ ở dưới mặt đất nguyên nhân là do trường hấp dẫn không đồng nhất ở 2 nơi.

con chó là con chó con
có đôi là đôi mắt tròn

Mình tạm dịch một phần nhỏ của chương đầu tiên của bác Hayumiko ,phần này mình chỉ tạm thờI dịch được một phần nhỏ ,định dịch tiêp nhưng khó quá lại dài thôi các bac tự dịch nhé.
Sự bắt đầu của thờI gian(the beginning of time)
Trong bài giảng này ,tôi thích bàn luận về liều thờI gian có điểm bắt đầu và liệu nó sẽ có một sự kết thúc .Tất cả bằng chứng hình như chỉ báo trước cho ta một điều ,vũ trụ không tồn tạI mãi mãi,trừ khi nó có sự bắt đầu khoảng 15 tỉ năm trước đây . Đây có lẽ là khám phá đáng chú ý nhất của vũ trụ hiên đại.Vậy điều đó được xem là điều tất nhiên.Chúng ta chưa nhất định rằng vũ trụ liệu có một sự kết thúc .Khi tôi đưa ra giang dạy ở nhật bản ,tôi được bảo rằng không nên đề cập đến vấn đè này có thể sự sụp đổ của vũ trụ sẽ ảnh hưởng đén thì trường chứng khoán.(chiu khong biết có phảI là thì trường chứng khoán không vì mình không nhơ rõ tư này nắm các bác thôm cảm hen)
Tuy nhiên ,tôi có thể khẳng định-đảm bảo bất cứ ai có thần kinh kém thì những cái cổ phiếu của họ đang đàu tư mà bán đi thì còn hơi sớmù vũ trụ đi đén một kết thúc ,nó sẽ còn tồn tạI được ít nhất 20 tỉ năm nữa.
Trước thờI gian ấy ,có thể GATT(KHÔNG BIẾT ĐÂY CÓ PHẢI LÀ CHỈ SỐ CHỨNG KHOAN KHÔNG)sẽ đi vào hiệu ứng (vòng xoáy của cổ phiếu ).
ThờI gian quy mô của vũ trụ còn quá dài so vớI thực tếcuộc sống của con người.BởI vậy không phảI ngạc nhiên giả thiết đó cho đén mơi đây ,vũ trụ được coi là tĩnh và không thay đổI .Mặt khác ,nó chắc rằng điều đó đã là hiển nhiên,xã hộI đang tiến triển trong nền văn hoá của cônng nghệ. Điều đó chỉ bảo rằng cái hiện hữa của lịch sử con người đã tiến hơn một ít ngàn năm .Cách khác ,chúng ta càng ngày càng hiểu biết hơn chúng ta bây giờ . Đó là tự nhiên bởI vậy để tin tưởng vào loài người,và có thể là toàn bộ vũ trụ có một sự bắt đầu vào thờI điểm khá gần đây.Tuy nhiên,nhiều ngườI không đồng ý rằng vũ trụ có một sự bắt đầu ,bởI vì họ nghĩ rằng vũ trụ tồn tạI là một sự hiện thân của siêu nhiên mà tạo ra vũ trụHọ thích tin tưởng vũ trụ đó,và loài ngườI , đã tồn tạI mãi mãi.Cách giảI thích của họ cho sự tiến bộ của loài người là đã có sự tuần hoàn nhưng nạn lụt hoặc những tai hoạ tự nhiên khác ,mà nhiều lần đã kéo con chở lạI sự nguyên thuỷ ban đầu.
Lý lẽ này (không phảI vũ trụ có một sự bắt đầu ) dù sao cúng đã được thừa nhận ở trong thế kỷ 19 và những năm của thế kỷ 20 .Nó được dựa trên quan điểm chủ yếu của thần học và triết học,vớI sự xem xét nhỏ bé của bằng chứng quan sát .Tháng năm này đã hợp lý, đưọc đưa từ đặc tính không đáng tin cậy của Việcquan sát.Cho đến khi nhà vu trụ học, ông Arthur Eddington ,một lần đã phát biểu rằng?Hãy không nên lo lắn lý thuyết của bạn nếu bạn không đồng ý vớI sự quan sát ,bởI vì chúng có thể sai?Nhưng nếu lý thuyết của bạn đồng nhất vớI định luật thứ hai của nhiệt động học ,nó gần như rắc rối.Thật ra ,lý thuyết vũ trụ tồn tạI mãi mãi không khó khăn gì nếu được giả thích bằng định luật thứ hai của nhiệt động học. Định luật thứ hai phát biểu rằng?sự mất trật tự luôn luôn tăng thêm vớI thời gian ?o(bạn có thể đọc được điều này ở chuyên mục trước mình vừa post ).Như lý lẽ rằng con ngườI luôn tiến triển ,nó chỉ cho ta rằng ở đó có sự bắt đầu .Nói cách khác,vũ trụ trước kia tồn tại mọt cách lộn xộn gần như bây giờ và mộI thứ cùng ơ một nhiệt độ như bây giờ .Trong một vũ trụ vĩnh cửu và vô hạn ,mỗI dòng tầm nhìn chấm dứt trên một bề mặt của một ngôi sao .Cái này có nghĩa là bầu trờI đêm đã sáng sủa như bề mặt của mặt trờI .Cách tránh vấn đề này chỉ có thể giảI thích những ngôi sao toả sang trước một nhất định thờI gian .
Trong một vũ trụ mã tĩnh thực chất ở đó không có bất kỳ khả năng chuyển độnh nào ,tạI sao những ngôi sao cần phảI quay ,vào thờI gian nào?Bất cứ ?oánh sáng nào xuất hiện ở thờI gian này?phảI chịu tác động từ bên ngoài vũ trụ .Tình trạng thì khác nhau ,tuy nhiên khi nó đuợc nhận thức rằng vũ trũ không phảI là tĩnh ,và nó đang được mở rộng ra.Những thiên hà là chuyển động đều đều ngoài lẫn nhau.Có nghĩa là chúng (thiên hà)là một truớc đây.
Ta có thể phác hoạ sự tách ra của hai thiên hà ,như một hàm của thờI gian .Nếu không có gia tốc của vì sức nặng ,thì đồ thị là một hang thẳng .Nó đí xuống và tách ra zero,khoảng hai mươi tỉ năm trước đây .Có một sức nặng nào đó gây ra những thiên hà tăng tốc về phía lẫn nhau . Điều này có nghĩa là rằng đồ thị sẽ tách ra ,hai thiên hà sẽ cong xuống phía dướI , ở dướI hàng thẳng.Như vậy thờI gian tách ra từ zero(0) đã ít hơn Hơn 20 tỉ năm trước đây.
VXH
Iam the wind.You are the sun.And one day we'll all be one.
Được vu xuan ha sửa chữa / chuyển vào 15:51 ngày 27/02/2003

bác Hayumiko cho em hỏi cái ........... mấy bài viết tiếng A của bác , bác lấy từ trang web nào vậy , hay là sách nào vậy............. bác chỉ em mấy ............ em ko tìm nổi sách của Stephen Hawking

THE GODMOTHER
Sống là để khỏi bị chết chứ ko fải để trở thành anh hùng