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If you had said we would get this stuff just 20 years ago, people would have laughed at you. It is with this realization
that I ask you to proceed.
The Holographic Principle
There is strong evidence that the area of any surface limits the information content of adjacent spacetime regions, at
10^(69) bits per square meter. We review the developments that have led to the recognition of this entropy bound, placing
special emphasis on the quantum properties of black holes. The construction of light-sheets, which associate relevant spacetime
regions to any given surface, is discussed in detail. We explain how the bound is tested and demonstrate its validity in a
wide range of examples.
A universal relation between geometry and information is thus uncovered. It has yet to be explained.
The holographic principle asserts that its origin must lie in the number of fundamental degrees of freedom involved in a unified
description of spacetime and matter. It must be manifest in an underlying quantum theory of gravity. We survey some successes
and challenges in implementing the holographic principle.
The Holographic Principle
According to the Holographic Principle, the most information you can get from this image is about 3 x 1065 bits for a normal sized computer monitor. The Holographic Principle, yet unproven, states that there is a maximum amount of information content held by regions adjacent to any surface. Therefore,
counter-intuitively, the information content inside a room depends not on the volume of the room but on the
area of the bounding walls. The principle derives from the idea that the Planck length, the length scale where quantum mechanics begins to dominate classical gravity, is one side of an area that can hold only about one bit of information. The limit was first postulated by physicist Gerard 't Hooft in 1993. It can arise from generalizations from seemingly distant speculation that the information held by a black hole is determined not by its enclosed volume but by the surface area of its event horizon.
people looking at the above image may not claim to see 3 x 1065 bits, they might claim to see a teapot.
 we owe a lot to you o hubble...happy 15th a jet traveling only 98 percent of light speed rams and mixes with interstellar material. Even higher energy jets might well
explain the structure seen around Cygnus A.
a supermassive black hole in the center of distant galaxy M84 - based on data recently recorded by Hubble's new Space Telescope Imaging Spectrograph (STIS). Very near black holes the force of gravity is so strong that even light can not escape ... but the presence of a black hole can also be revealed
by watching matter fall into it. In fact, material spiraling into a black hole would find its speed increasing at a drastic
rate. These extreme velocity increases provide a "signature" of the black hole's presence. STIS relies on the Doppler effect to measure gas velocity rapidly increasing to nearly 240 miles per second within 26 light years of the center of M84, a galaxy
in the Virgo Cluster about 50 million light years away. The STIS data show that radiation from approaching gas, shifted to blue wavelengths left of the centerline, is suddenly redshifted to the
right of center indicating a rapidly rotating disk of material near the galactic nucleus. The resulting sharp S-shape is effectively
the signature of a black hole estimated to contain at least 300 million solar masses.
Gamma-Ray Bursting
The Universe in a Nutshell by Stephen Hawking
 Stephen Hawking - science's first rock star - returns with a new journey into what makes the world tick, and this time he
is determined not to be called again "the most sold and least read" popular science writer, title earned with the publication
of A Brief History of Time.In this new book Hawking takes us to the cutting edge of theoretical physics, where truth is often
stranger than fiction, to explain in laymen’s terms the principles that control our universe. The book is a fascinating
collection of ideas, concepts, principles and connections. ranging from Quantum mechanics to M-theory, from General relativity
to11-dimensional supergravity up to 10-dimensional membranes, Superstrings, P-branes and Black holes!One of the most influential
thinkers of our time, Stephen Hawking is an intellectual icon, known not only for the adventurousness of his ideas but for
the clarity and wit with which he expresses them.
Like many in the community of theoretical physicists, Professor Hawking is seeking to uncover the grail of science —
the elusive Theory of Everything that lies at the heart of the cosmos. In his accessible and often playful style, he guides
us on his search to uncover the secrets of the universe — from supergravity to supersymmetry, from quantum theory to
M-theory, from holography to duality. He takes us to the wild frontiers of science, where superstring theory and p-branes
may hold the final clue to the puzzle. And he lets us behind the scenes of one of his most exciting intellectual adventures
as he seeks “to combine Einstein’s General Theory of Relativity and Richard Feynman’s idea of multiple histories
into one complete unified theory that will describe everything that happens in the universe.” With characteristic exuberance,
Professor Hawking invites us to be fellow travelers on this extraordinary voyage through space-time. Copious four-color illustrations
help clarify this journey into a surreal wonderland where particles, sheets, and strings move in eleven dimensions; where
black holes evaporate and disappear, taking their secret with them; and where the original cosmic seed from which our own
universe sprang was a tiny nut. Hawking admits that many of the ideas in the book are highly speculative. In fact, he revels
in his role as a provocateur, recollecting with relish the time when he and Kip Thorne pursued the "politically incorrect"
idea of time travel. On the subject of M-theory, Hawking acknowledges that extra dimensions are not required to explain any
observation. Like other theorists, he is guided by the elegance of the mathematics and by "dualities" indicating that we may
already have the fragments of a final theory that will unify gravity and quantum physics. Hawking's book is exciting and provocative,
and it poses a fundamental question: Is nature baroque or parsimonious? In the democracy of branes, all dimensions are created
equal. From the quantum state of the Big Bang, a cornucopia of universes might emerge. Hawking wields the anthropic principle
like Occam's razor to slice through these possibilities. He imagines that life can exist only in a universe with three spatial
dimensions, and that an inflated and nearly smooth universe is needed for observers to evolve. In his words, "the anthropic
principle picks out brane models from the vast zoo of universes allowed by M-theory." In epistemological circles, the anthropic
principle is suspect due to its lack of predictive power and its tendency toward tautology. In the freewheeling world of Stephen
Hawking, it is just one more device to tease and engage the reader. In a magnetically confined plasma, magnetic field lines, such as those depicted in gray,
form closed surfaces. An example is the hot-plasma surface shown in red. In this simulation, the surface is tilted a bit off
center, which indicates the onset of an instability. To see how the instability progresses and to learn about worldwide efforts
to achieve plasma simulations that integrate physics at widely separated time and space scales.
 Stephen Hawking : "I want shorter, better focused, numbered questions, not a stream of
consciousness." is by his condition allowed to be curt. But if his success tells us anything, it is the folly of
reading him solely through his condition. His tone might as easily be a sign of geekiness or superiority or intolerance of
non-scientists. Hawking sits in the middle of the room attended by a nurse, one of the 10 who look after him. A PhD student
pops his head around the door and says hi - Hawking supervises a small number in his capacity as the Lucasian professor of
mathematics, a position once held by Isaac Newton - and the cosmologist either smiles or gears up to communicate. A Brief
History of Time, aroused a great deal of interest, but many found it difficult to understand. I decided to write a new version
that would be easier to follow. I took the opportunity to add material about new developments and I left out some things of
a more technical natureI would hope that people who have difficulty with A Brief History will try a Briefer History and be
pleasantly surprised." A Briefer History of Time is not exactly String Theory for Dummies. Like a lot
of specialists, Hawking has trouble imagining what it might be like not to understand what he does, or rather, where the non-scientist's
understanding will be weak and where strong. The book's range is therefore a little eccentric, lurching between explaining
what a scientific theory is ("a model of the universe") and going into quantum mechanics in the kind of vertiginous detail
that makes you open your eyes very wide as you read. It is fascinating, up to a point. I put a lot of
effort into writing A Briefer History at a time when I was critically ill with pneumonia because I think that it's important
for scientists to explain their work, particularly in cosmology. This now answers many questions once asked of religion."
There are new sections: string theory - the unproven idea that the universe is made up of lots of tiny,
vibrating strings - has apparently moved on since the first book was written, although it is still controversial. This suits
Hawking's purpose: he understands that no one, scientist or otherwise, can resist an unanswerable question. When he refers
to God it is, as he puts it, in the "impersonal sense", rather as Einstein referred to the laws of nature. It is a euphemism
and also a smart bit of marketing, anchoring the unsexy, techie bones of his subject - he once said the best hope for a theory
of everything was n=8 supergravity - with the philosophical questions everyone likes to have a stab at.Evolution has ensured
that our brains just aren't equipped to visualise 11 dimensions directly. However, from a purely mathematical point of view
it's just as easy to think in 11 dimensions, as it is to think in three or four."yet he is working on a children's book about
relativity with his daughter Lucy, because children are the best audience: "Naturally interested in space and not afraid to
ask why."It is a waste of time to be angry about my disability. One has to get on with life and I haven't done badly. People
won't have time for you if you are always angry or complaining."There are pictures of Marilyn Monroe on the wall, one of which
has been digitally manipulated to feature Hawking in the foreground. I see a card printed with the slogan: Yes,
I am the centre of the universe. At first sight, it seems remarkable that the universe is so
finely tuned. Maybe this is evidence, that the universe was specially designed to produce the human race. However, one has
to be careful about such arguments, because of what is known as the Anthropic Principle. This is based on the self-evident
truth, that if the universe had not been suitable for life, we wouldn’t be asking why it is so finely adjusted. One
can apply the Anthropic Principle, in either its Strong, or Weak, versions. For the Strong Anthropic Principle, one supposes
that there are many different universes, each with different values of the physical constants. In a small number, the values
will allow the existence of objects like carbon atoms, which can act as the building blocks of living systems. Since we must
live in one of these universes, we should not be surprised that the physical constants are finely tuned. If they weren’t,
we wouldn’t be here. The strong form of the Anthropic Principle is not very satisfactory. What operational meaning can
one give to the existence of all those other universes? And if they are separate from our own universe, how can what happens
in them, affect our universe. Instead, I shall adopt what is known as the Weak Anthropic Principle.... It
took a very long time, two and a half billion years, to go from single cells to multi-cell beings, which are a necessary precursor
to intelligence. This is a good fraction of the total time available, before the Sun blows up. So it would be consistent with
the hypothesis, that the probability for life to develop intelligence, is low. In this case, we might expect to find many
other life forms in the galaxy, but we are unlikely to find intelligent life. There used to be a project called SETI, the
search for extra-terrestrial intelligence. It involved scanning the radio frequencies, to see if we could pick up signals
from alien civilisations. I thought this project was worth supporting, though it was cancelled due to a lack of funds. But
we should have been wary of answering back, until we have develop a bit further. Meeting a more advanced civilisation, at
our present stage, might be a bit like the original inhabitants of America meeting Columbus. I don’t think they were
better off for it.'life' is based on chains of carbon atoms, with a few other atoms, such as nitrogen or phosphorous. One
can speculate that one might have life with some other chemical basis, such as silicon, but carbon seems the most favourable
case, because it has the richest chemistry. That carbon atoms should exist at all, with the properties that they have, requires
a fine adjustment of physical constants, such as the QCD scale, the electric charge, and even the dimension of space-time.
If these constants had significantly different values, either the nucleus of the carbon atom would not be stable, or the electrons
would collapse in on the nucleus. At first sight, it seems remarkable that the universe is so finely tuned. Maybe this is
evidence, that the universe was specially designed to produce the human race.
Anatomy of a Black Hole
By definition a black hole is a region where matter collapses to infinite density, and where, as a result, the curvature
of spacetime is extreme. Moreover, the intense gravitational field of the black hole prevents any light or other electromagnetic
radiation from escaping. But where lies the "point of no return" at which any matter or energy is doomed to disappear from
the visible universe?
Applying the Einstein Field Equations to collapsing stars, German astrophysicist Kurt Schwarzschild deduced the critical radius for a given mass at which matter would collapse into an infinitely dense state known as a singularity.
For a black hole whose mass equals 10 suns, this radius is about 30 kilometers or 19 miles, which translates into a critical
circumference of 189 kilometers or 118 miles.
Schwarzschild Black Hole
If you envision the simplest three-dimensional geometry for a black hole, that is a sphere (known as a Schwarzschild black
hole), the black hole's surface is known as the event horizon. Behind this horizon, the inward pull of gravity is overwhelming
and no information about the black hole's interior can escape to the outer universe.
Apparent versus Event HorizonAs a doomed star reaches its critical circumference, an "apparent" event horizon forms
suddenly. Why "apparent?" Because it separates light rays that are trapped inside a black hole from those that can move away
from it. However, some light rays that are moving away at a given instant of time may find themselves trapped later if more
matter or energy falls into the black hole, increasing its gravitational pull. The event horizon is traced out by "critical"
light rays that will never escape or fall in.
Apparent versus Event Horizon
Caption
Even before the star meets its final doom, the event horizon forms at the center, balloons out and breaks through the star's
surface at the very moment it shrinks through the critical circumference. At this point in time, the apparent and event horizons
merge as one: the horizon. For more details, see the caption for the above diagram.
The distinction between apparent horizon and event horizon may seem subtle, even obscure. Nevertheless the difference becomes
important in computer simulations of how black holes form and evolve.
Beyond the event horizon, nothing, not even light, can escape. So the event horizon acts as a kind of "surface" or "skin"
beyond which we can venture but cannot see. Imagine what happens as you approach the horizon, then cross the threshold.
Care to take a one-way trip into a black hole?
At the center of a black hole lies the singularity, where matter is crushed to infinite density, the pull of gravity
is infinitely strong, and spacetime has infinite curvature. Here it's no longer meaningful to speak of space and time, much
less spacetime. Jumbled up at the singularity, space and time cease to exist as we know them.
The Limits of Physical LawNewton and Einstein may have looked at the universe very differently, but they would have
agreed on one thing: all physical laws are inherently bound up with a coherent fabric of space and time.
At the singularity, though, the laws of physics, including General Relativity, break down. Enter the strange world of quantum
gravity. In this bizzare realm in which space and time are broken apart, cause and effect cannot be unraveled. Even today,
there is no satisfactory theory for what happens at and beyond the singularity.
Cosmic CensorshipIt's no surprise that throughout his life Einstein rejected the possibility of singularities. So
disturbing were the implications that, by the late 1960s, physicists conjectured that the universe forbade "naked singularities."
After all, if a singularity were "naked," it could alter the whole universe unpredictably. All singularities within the universe
must therefore be "clothed."
But inside what? The event horizon, of course! Cosmic censorship is thus enforced. Not so, however, for that ultimate cosmic singularity that gave rise to the Big Bang.
Science versus SpeculationWe can't see beyond the event horizon. At the singularity, randomness reigns supreme. What,
then, can we really "know" about black holes? How can we probe their secrets? The answer in part lies in understanding their
evolution right after they form
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Rotation, Nut charge and Anti de Sitter space
This lecture
is the intellectual property of Professor S.W. Hawking. You may not reproduce, edit or distribute this document in anyway
for monetary advantage. The work I'm going to talk about has been carried out with Chris Hunter and Marika Taylor Robinson
at Cambridge, and Don Page at Alberta.References It is described in the first three papers shown on the screen. Related work
has been carried out by Chamblin, Emparan, Johnson, and Myers. However, they seemed a bit uncertain what reference background
to use. I have also shown a reference to Dowker which is relevant.
It has been known for quite a time, that black holes
behave like they have entropy. The entropy is the area of the horizon, divided by 4 G, where G is Newton's constant. Black
Hole Entropy
The idea is that the Euclidean sections of black hole metrics are periodic in the imaginary time coordinate.
Thus they represent black holes in equilibrium with thermal radiation. However there are problems with this interpretation.Problems
with thermodynamic interpretation
(first problem appear)
First, one can not have thermal radiation in asymptotically
flat space, all the way to infinity, because the energy density would curve the space, and make it an expanding or collapsing
Friedmann universe. Thus, if you want a static situation, you have to resort to the dubious Gedanken experiment, of putting
the black hole in a box. But you don't find black hole proof boxes, advertised on the Internet.
(second problem appear)
The
second difficulty with black holes in equilibrium with thermal radiation is that black holes have negative specific heat.
In many cases, when they absorb energy, they get larger and colder. This reduces the radiation they give off, and so they
absorb faster than they radiate, and the equilibrium is unstable. This is closely related to the fact that the Euclidean metric
has a negative mode. Thus it seems that asymptotically flat Euclidean black holes, describe the decay of hot flat space, rather
than a black hole in equilibrium with thermal radiation.
(third problem appear)
The third difficulty with the idea of
equilibrium is that if the black hole is rotating, the thermal radiation should be co-rotating with it. But far away from
the black hole, the radiation would be co-rotating faster than light, which is impossible. Thus, again one has to use the
artificial expedient, of a box of finite size.
Way back in pre-history, Don page and I, realized one could avoid the first
two difficulties, if one considered black holes in anti de Sitter space, rather than asymptotically-flat space. In anti de
Sitter space, the gravitational potential increases as one goes to infinity. This red shifts the thermal radiation, and means
that it has finite energy. Thus anti de Sitter space can exist at finite temperature, without collapsing. In a sense, the
gravitational potential in anti de Sitter space, acts like a confining box.
Anti de Sitter space can also help with the
second problem, that the equilibrium between black holes and thermal radiation, will be unstable. Small black holes in anti
de Sitter space, have negative specific heat, like in asymptotically flat space, and are unstable. But black holes larger
than the curvature radius of anti de Sitter space, have positive specific heat, and are presumably stable. At the time, Don
page and I, did not think about rotating black holes. But I recently came back to the problem, along with Chris Hunter, and
Marika Taylor Robinson. We realized that thermal radiation in anti de Sitter space could co-rotate with up to some limiting
angular velocity, without having to travel faster than light. Thus anti de Sitter boundary conditions, can solve all three
problems, in the interpretation of Euclidean black holes, as equilibria of black holes, with thermal radiation. Anti de Sitter
black holes may not seem of much interest, because we can be fairly sure, that the universe is not asymptotically anti de
Sitter. However, they seem worth studying, both for the reasons I have just given, and because of the Maldacena conjecture,
relating asymptotically anti de Sitter spaces, to conformal field theories on their boundary. I shall report on two pieces
of work in relation to this conjecture. One is a study of rotating black holes in anti de Sitter space. We have found Kerr
anti de Sitter metrics in four and five dimensions. As they approach the critical angular velocity in anti de Sitter space,
their entropy, as measured by the horizon area, diverges. We compare this entropy, with that of a conformal field theory on
the boundary of anti de Sitter space. This also diverges at the critical angular velocity, when the rotational velocity, approaches
the speed of light. We show that the two divergences are similar.
The other piece of work, is a study of gravitational
entropy, in a more general setting. The quarter area law, holds for black holes or black branes in any dimension, d, that
have a horizon, which is a d minus 2 dimensional fixed point set, of a U1 isometry group. However Chris Hunter and I, have
recently shown that entropy can be associated with a more general class of space-times. In these, the U1 isometry group can
have fixed points on surfaces of any even co-dimension, and the space-time need not be asymptotically flat, or asymptotically
anti de Sitter. In these more general class, the entropy is not just a quarter the area, of the d minus two fixed point set.
Among the more general class of space-times for which entropy can be defined, an interesting case is those with Nut charge.
Nut charge can be defined in four dimensions, and can be regarded as a magnetic type of mass. Solutions with nut charge are
not asymptotically flat in the usual sense. Instead, they are said to be asymptotically locally flat, or ALF. In the Euclidean
regime, in which I shall be working, the difference can be described as follows. An asymptotically flat metric, like Euclidean
Schwarzschild, has a boundary at infinity, that is a two-sphere of radius r, cross a circle, whose radius is asymptotically
constant.
To get finite values for the action and Hamiltonian, one subtracts the values for flat space, periodically identified.
In asymptotically locally flat metrics, on the other hand, the boundary at infinity, is an S1 bundle over S2. These bundles
are labeled by their first Chern number, which is proportional to the Nut charge. If the first Chern number is zero, the boundary
is the product, S2 cross S1, and the metric is asymptotically flat. However, if the first Chern number is k, the boundary
is a squashed three sphere, with mod k points identified around the S1 fibers. Such asymptotically locally flat metrics, can
not be matched to flat space at infinity, to give a finite action and Hamiltonian, despite a number of papers that claim it
can be done. The best that one can do, is match to the self-dual multi Taub nut solutions. These can be regarded as defining
the vacuums for ALF metrics.
In the self-dual Taub Nut solution, the U1 isometry group, has a zero dimensional fixed point
set at the center, called a nut. However, the same ALF boundary conditions, admit another Euclidean solution, called the Taub
bolt metric, in which the nut is replaced by a two dimensional bolt. The interesting feature, is that according to the new
definition of entropy, the entropy of Taub bolt, is not equal to a quarter the area of the bolt, in Planck units. The reason
is that there is a contribution to the entropy from the Misner string, the gravitational counterpart to a Dirac string for
a gauge field. The fact that black hole entropy is proportional to the area of the horizon has led people to try and identify
the microstates, with states on the horizon. After years of failure, success seemed to come in 1996, with the paper of Strominger
and Vafa, which connected the entropy of certain black holes, with a system of D-branes. With hindsight, this can now be seen
as an example of a duality, between a gravitational theory in asymptotically anti de Sitter space, and a conformal field theory
on its boundary.
It would be interesting if similar dualities could be found for solutions with Nut charge, so that one
could verify that the contribution of the Misner string was reflected in the entropy of a conformal field theory. This would
be particularly significant for solutions like Taub bolt, which don't have a spin structure. It would show whether the duality
between anti de Sitter space, and conformal field theories on its boundary, depends on super symmetry. In fact, I will present
evidence, that the duality requires super symmetry.
To investigate the effect of Nut charge, we have found a family of
Taub bolt anti de Sitter solutions. These Euclidean metrics are characterized by an integer, k, and a positive parameter,
s. The boundary at large distances is an S1 bundle over S2, with first Chern number, k. If k=0, the boundary is a product,
S1 cross S2, and the space is asymptotically anti de Sitter, in the usual sense. But if k is not zero, they are what may be
called, asymptotically locally anti de Sitter, or ALADS. The boundary is a squashed three sphere, with k points identified
around the U1 direction. This is just like asymptotically locally flat, or ALF metrics. But unlike the ALF case, the squashing
of the three-sphere, tends to a finite limit, as one approaches infinity. This means that the boundary has a well-defined
conformal structure. One can then ask whether the partition function and entropy, of a conformal field theory on the boundary,
is related to the action and entropy, of these asymptotically locally anti de Sitter solutions.
To make the ADS, CFT correspondence
well posed, we have to specify the reference backgrounds, with respect to which the actions and Hamiltonians are defined.
For Kerr anti de Sitter, the reference background is just identified anti de Sitter space. However, as in the asymptotically
locally flat case, a squashed three sphere, can not be imbedded in Euclidean anti de Sitter. One therefore can not use it
as a reference background, to make the action and Hamiltonian finite. Instead, one has to use Taub Nut anti de Sitter, which
is a limiting case of our family. If mod k is greater than one, there is an orbifold singularity in the reference backgrounds,
but not in the Taub bolt anti de Sitter solutions. These orbifold singularities in the backgrounds could be resolved, by replacing
a small neighbourhood of the nut, by an ALE metric. We shall take it, that the orbifold singularities are harmless.
Another
issue that has to be resolved, is what conformal field theory to use, on the boundary of the anti de Sitter space. For five
dimensional Kerr anti de Sitter space, there are good reasons to believe the boundary theory is large N Yang Mills. But for
four-dimensional Kerr anti de Sitter, or Taub bolt anti de Sitter, we are on shakier ground. On the three dimensional boundaries
of four dimensional anti de Sitter spaces, Yang Mills theory is not conformally invariant. The folklore is that one takes
the infrared fixed point, of three-dimensional Yang Mills, but no one knows what this is. The best we can do, is calculate
the determinants of free fields on the squashed three sphere, and see if they have the same dependence on the squashing, as
the action. Note that as the boundary is odd dimensional, there is no conformal anomaly. The determinant of a conformally
invariant operator, will just be a function of the squashing. We can then interpret the squashing, as the inverse temperature,
and get the number of degrees of freedom, from a comparison with the entropy of ordinary black holes, in four dimensional
anti de Sitter. I now turn to the question, of how one can define the entropy, of a space-time. A thermodynamic ensemble,
is a collection of systems, whose charges are constrained by La-grange multipliers.
Partition functionOne such charge,
is the energy or mass, with the Lagrange multiplier being the inverse temperature, beta. But one can also constrain the angular
momentum, and gauge charges. The partition function for the ensemble, is the sum over all states, of e to the minus, La-grange
multipliers, times associated charges.
Thus it can be written as, trace of e to the minus Q. Here Q is the operator that
generates a Euclidean time translation, beta, a rotation, delta phi, and a gauge transformation, alpha. In other words, Q
is the Hamiltonian operator, for a lapse that is beta at infinity, and a shift that is a rotation through delta phi. This
means that the partition function can be represented by a Euclidean path integral.
The path integral is over all metrics
which at infinity, are periodic under the combination of a Euclidean time translation, beta, a rotation through delta phi,
and a gauge rotation, alpha. The lowest order contributions to the path integral for the partition function will come from
Euclidean solutions with a U1 isometry, that agree with the periodic boundary conditions at infinity. The Hamiltonian in general
relativity or supergravity, can be written as a volume integral over a surface of constant tau, plus surface integrals over
its boundaries. Gravitational Hamiltonian The volume integral vanishes by the constraint equations. Thus the numerical value
of the Hamiltonian, comes entirely from the surface terms. The action can be related to the Hamiltonian in the usual way.
Because the metric has a time translation isometry, all dotted quantities vanish. Thus the action is just beta times the Hamiltonian.If
the solution can be foliated by a family of surfaces, that agree with Euclidean time at infinity, the only surface terms will
be at infinity.
Family of time surfaces In this case, a solution can be identified under any time translation, rotation,
or gauge transformation at infinity. This means that the action will be linear in beta, delta phi, and alpha. If one takes
such a linear action, for the partition function, and applies the standard thermodynamic relations, one finds the entropy
is zero.The situation is very different however, if the solution can't be foliated by surfaces of constant tau, where tau
is the parameter of the U1 isometry group, which agrees with the periodic identification at infinity. Breakdown of foliation
The break down of foliation can occur in two ways. The first is at fixed points of the U1 isometry group. These occur on surfaces
of even co-dimension. Fixed-point sets of co-dimension-two play a special role. I shall refer to them as bolts. Examples include
the horizons of non-extreme black holes and p-branes, but there can be more complicated cases, like Taub bolt.
The other
way the foliation by surfaces of constant tau, can break down, is if there are what are called, Misner strings. Kaluza Klein
metricTo explain what they are, write the metric in the Kaluza Klein form, with respect to the U1 isometry group. The one
form, omega, the scalar, V, and the metric, gamma, can be regarded as fields on B, the space of orbits of the isometry group.
If
B has homology in dimension two, the Kaluza Klein field strength, F, can have non-zero integrals over two cycles. This means
that the one form, omega, will have Dirac strings in B. In turn, this will mean that the foliation of the spacetime, M, by
surfaces of constant tau, will break down on surfaces of co-dimension two, called Misner strings. In order to do a Hamiltonian
treatment using surfaces of constant tau, one has to cut out small neighbourhoods of the fixed point sets, and the Misner
strings. This modifies the treatment, in two ways. First, the surfaces of constant tau now have boundaries at the fixed-point
sets, and Misner strings, as well as the boundary at infinity. This means there can be additional surface terms in the Hamiltonian.
In fact, the surface terms at the fixed-point sets are zero, because the shift and lapse vanish there. On the other hand,
at a Misner string, the lapse vanishes, but the shift is non zero. The Hamiltonian can therefore have a surface term on the
Misner string, which is the shift, times a component of the second fundamental form, of the constant tau surfaces. The total
Hamiltonian, will be the sum of this Misner string Hamiltonian, and the Hamiltonian surface term at infinity. Consequences
of non-foliation
As before, the action will be beta times the Hamiltonian. However, this will be the action of the space-time,
with the neighborhoods of the fixed-point sets and Misner strings removed. To get the action of the full space-time, one has
to put back the neighborhoods. When one does so, the surface term associated with the Einstein Hilbert action, will give a
contribution to the action, of minus area over 4G, for the bolts and Misner strings. Here G is Newton's constant in the dimension
one is considering. The surface terms around lower dimensional fixed-point sets make no contribution to the action. The action
of the space-time, will be the lowest order contribution to minus log Z, where Z is the partition function. But log Z is equal
to the entropy, minus beta times the Hamiltonian at infinity. So the entropy is a quarter the area of the bolts and Misner
strings, minus beta times the Hamiltonian on the Misner strings. In other words, the entropy is the amount by which the action
is less than the value, beta times the Hamiltonian at infinity, that it would have if the surfaces of constant tau, foliated
the space-time.
This formula for the entropy applies in any dimension and for any class of boundary condition at infinity.
Thus one can use it for rotating black holes, in anti de Sitter space. In this case, the reference background is just Euclidean
anti de Sitter space, identified with imaginary time period, beta, and appropriate rotation.
Four-dimensional Kerr-AdS
The
four-dimensional Kerr anti de Sitter solution, was found by Carter, and is shown on the slide. The parameter, a, determines
the rate of rotation. When a-l approaches 1, the co-rotation velocity approaches the speed of light at infinity. It is therefore
interesting to examine the behavior of the black hole action, and the conformal field theory partition function, in this limit.
To
calculate the action of the black hole is quite delicate, because one has to match it to rotating anti de Sitter space, and
subtract one infinite quantity, from another.
Euclidean action.
Nevertheless, this can be done in a well-defined
way, and the result is shown on the slide. As you might expect, it diverges at the critical angular velocity, at which the
co-rotating velocity, approaches the speed of light.The boundary of rotating anti de Sitter, is a rotating Einstein universe,
of one dimension lower. Thus it is straightforward in principle, to calculate the partition function for a free conformal
field on the boundary. Someone like Dowker might have calculated the result exactly. However, as we are only human, we looked
only at the divergence in the partition function, as one approaches the critical angular velocity.
This divergence arises
because in the mode sum for the partition function, one has Bose-Einstein factors with a correction because of the rotation.
As one approaches the critical angular velocity, this causes a Bose-Einstein condensation in modes with the maximum axial
quantum number, m.
Conformal field theory The divergence in the conformal field theory partition function has the same
divergence as the black hole action, at the critical angular velocity. I haven't compared the residues. This is difficult,
because it is not clear what three-dimensional conformal field theory one should use on the boundary of four dimensional anti
de Sitter. Five-dimensional Kerr-AdS The case of rotating black holes in anti de Sitter five, is broadly similar, but with
some differences. One of these is that, because the spatial rotation group, O4, is of rank 2, there are two rotation parameters,
a & b. Each of these must have absolute value less than l to the minus one, for the co-rotation velocity to be less than
the speed of light, all the way out to infinity. If just one of a & b, approaches the limiting value, the action of the
black hole, and the partition function of the conformal field theory, both diverge in a manner similar to the four dimensional
case. Action of five-dimensional Kerr-AdS But if a = b, and they approach the limit together, the action and the partition
function, both have the same stronger divergence. Again, I haven't compared residues, but this might be worth doing. It may
be that in the critical angular velocity limit, the interactions between the particles of super Yang Mills theory, become
unimportant. If this is the case, one would expect the action and partition function to agree, rather than differ by a factor
of four thirds, as in the non rotating case. Asymptotically locally flat I now turn the case of Nut charge. For asymptotically
locally flat metrics in four dimensions, the reference background is the self-dual Taub Nut solution. The Taub bolt solution,
has the same asymptotic behavior, but with the zero-dimensional nut fixed point, replaced by a two-dimensional bolt. The area
of the bolt is 12 pi N squared, where N is the Nut charge. The area of the Misner string is minus 6 pi N squared. That is
to say, the area of the Misner string in Taub bolt, is infinite, but it is less than the area of the Misner string in Taub
nut, in a well-defined sense. The Hamiltonian on the Misner string, is N over 8. Again the Misner string Hamiltonian is infinite,
but the difference from Taub nut, is finite. And the period, beta, is 8pi N. Thus the entropy, is pi N squared. Note that
this is less than a quarter the area of the bolt, which would give 3 pi N squared. It is the effect of the Misner string that
reduces the entropy. Taub Nut Anti de Sitter We would like to confirm the effect of Misner strings on entropy, by seeing what
effect they have on conformal field theories, on the boundary of anti de Sitter space. For this purpose we constructed versions
of Taub nut and Taub bolt, with a negative cosmological constant. The Taub nut anti de Sitter metric is shown on the transparancy.
The parameter E, is the squashing of the three-sphere at infinity. If E=1, the three spheres are round, and the metric is
Euclidean anti de Sitter space. However, if E is not equal to one, the metric can not be matched to anti de Sitter space at
large distance. Each value of E, therefore, defines a different sector of ALADS metrics. This is an important point, which
did not seem to have been realized by Chamblin et al. Taub Bolt Anti de Sitter One can also find a family of Taub bolt anti
de Sitter metrics, with the same asymptotic behavior. These are characterized by an integer, k, and a positive quantity, s.
These determine the asymptotic squashing parameter, E, and the area of the bolt, A-. K is the self-intersection number of
the bolt. Thus the spaces do not have spin structure if k is odd. At infinity, the squashed three sphere has k points identified
around the U1 fiber. This means that the reference background, is Taub nut anti de Sitter, with k points identified. If k
is greater than one, the reference background will have an orbifold singularity at the nut. However, as I said earlier, I
shall take it that such singularities are harmless.
Action To calculate the action, one matches the Taub bolt solution
on a squashed three sphere, to a Taub nut solution. To do this, one has to re-scale the squashing parameter, E, as a function
of radius. The surface term in the action, is the same asymptotically for Taub nut and Taub bolt. Thus the action comes entirely
from the difference in volumes. Action for k = 1 For k greater than one, the action is always negative, while for k=1, it
is positive for small areas of the bolt, and negative for large areas. This behavior is similar to that for Schwarzschild
anti de Sitter space, and might indicate a phase transition in the corresponding conformal field theory. However, as I will
argue later, there are problems with the ADS, CFT duality, if k=1. On the other hand, our results seem to indicate that there
will be no phase transition, if more than one point is identified around the fiber. It will be interesting to see if this
is indeed the case, for a conformal field theory on an identified squashed three-sphere. In these Taub bolt anti de Sitter
metrics, one can calculate the area of the Misner string, and the Hamiltonian surface term. Both will be infinite, but if
one matches to Taub nut anti de Sitter on a large squashed three-sphere, the differences will tend to finite limits. As in
the asymptotically locally flat case, the entropy is less than a quarter the area of the bolt, because of the effect of the
Misner string. Entropy One can also calculate the entropy from the partition function, by the usual thermodynamic relations.
The mass will be given by taking the derivative of the action with respect to beta. This is equal to the Hamiltonian surface
term at infinity. The mass or energy, is the only charge that is constrained in the ensemble. The nut charge is fixed by the
boundary conditions, and so doesn't need a La-grange multiplier. Thus the entropy is beta M, minus I. This agrees with the
entropy calculated from the bolts and Misner strings, showing our definition, is consistent.Formally at least, one can regard
Euclidean conformal field theory on the squashed three sphere, as a twisted 2+1 theory, at a temperature, beta to the minus
one. Thus one would expect the entropy to be proportional to beta to the minus two, at least for small beta. This has been
confirmed by calculations by Dowker, of the determinants of scalar and fermion operators on the squashed three sphere, for
k=1. Dowker has not so far calculated the higher k cases, but one would expect that these would have similar leading terms,
but with beta replaced by beta over k. The next leading order terms in the determinant, are beta to the minus one, log beta.
No terms like this appear in the bulk theory, so if there really is an ADS, CFT duality in this situation, the log beta terms
have to cancel between the different spins.
In fact, the scalar and fermion log beta terms will cancel each other, if
there are twice as many scalars as fermions. This would be implied by super symmetry, suggesting that super symmetry is indeed
necessary for the ADS, CFT duality.The Misner string contributions to the entropy are of order beta squared. Thus Dowker's
calculations will have to be extended to this order, to k greater than one, to fermion fields with anti periodic boundary
conditions, and to spin one fields. All this is quite possible, but it will probably require Dowker to do it. One might ask,
how can a conformal field theory on the Euclidean squashed three sphere, correspond to a theory in a spacetime of Lorentzian
signature. The answer is that, unlike the Schwarzschild anti de Sitter case, one has to continue the period, beta, to imaginary
values. This makes the spacetime periodic in real time, rather than imaginary time. One gets a 2+1 rotating spacetime, rather
like the Goedel universe, with closed time like curves. Although field theory in such a spacetime, may seem pathological,
it can be obtained by analytical continuation, and is well defined despite the lack of causality. It is interesting that the
analytically continued entropy, is negative, suggesting that causality violating spacetimes, are quantum suppressed. However,
it is probably a mistake, to attach physical significance, to the Lorentzian conformal field theory. To sum up, I discussed
the ADS, CFT duality in two new contexts. That of rotating black holes and that of solutions with nut charge. I showed how
gravitational entropy can be defined in general. The partition function for a thermodynamic ensemble can be defined by a path
integral over periodic metrics. The lowest order contributions to the partition function will come from metrics with a U1
isometry, and given behavior at infinity. The entropy of such metrics will receive contributions from horizons or bolts, and
from Misner strings, which are the Dirac strings of the U1 isometry, under Kaluza Klein reduction. One would like to relate
this gravitational entropy, to the entropy of a conformal field theory on the boundary. For this reason, we considered a new
class of asymptotically locally anti de Sitter spaces. Other people have investigated the Maldacena conjecture, by deforming
the compact part of the metric, but this is the first time deformed anti de Sitter boundary conditions, have been considered.
We studied Taub bolt anti de Sitter solutions, with Taub nut anti de Sitter, as the reference background. The entropy we obtained
obeyed the right thermodynamic relations, and had the right temperature dependence, to be the entropy of a conformal field
theory, on the squashed three sphere. Because the Taub bolt solutions for odd k, do not have spin structures, this may indicate
that the anti de Sitter, conformal field theory correspondence, does not depend on super symmetry.
I will end by saying
that gravitational entropy, is alive and well, 34 years on. But there's more to entropy, than just horizon area. We need to
look at the nuts and bolts. |
What happens to a black hole after it forms? Does it vibrate? Radiate? Lose mass? Grow? Shrink?
Partial solutions of the Einstein equations point to two possible outcomes:
- A non-rotating, spherically symmetric black hole, first postulated by Schwarzschild.
- A rotating, spherical black hole, predicted in 1964 by the New Zealand mathematician Roy Kerr.
These two types of black holes have become known as Schwarzschild and Kerr black holes, respectively. Both types of black
holes are "stationary" in that they do not change in time, unless they are disturbed in some way. As such, they are among
the simple st objects known in General Relativity. They can be completely described in terms of just 2 numbers: their mass
M and their angular momentum J.
Theoretically, black holes may also possess electric charge, Q, but it would quickly attract enough charge of the opposite
sign. The net result is that any "realistic" or astrophysical black hole would tend to exhibit zero charge. This simplicity
of black holes is summed up in the saying "black holes have no hair," meaning that, apart from its mass and momentum, there
is no other characteristic (or "hair") that a black hole can exhibit.
(But things may not be quite so simple. Yes -- you've guessed it -- there's more to this story. To explore it, though,
is beyond the scope of this exhibit.)
However, both the Schwarzschild and Kerr black holes represent end states. Their formation may result from various
processes, all of them quite complicated. When a "real" black hole forms from, say, the collapse of a very mass ive star, or when a black hole is disturbed by, say, another black hole spiralling
into it, the resulting dynamics cause disturbances in spacetime that should lead to the generation of gravitational waves.
By numerically solving the Einstein equations on powerful computers, scientists have been able to simulate the gravitational
waves emitted by perturbed or interacting black holes. When visualized in movies generated by advan ced computer graphics, the unfolding wave patterns are not only intriguing but strikingly beautiful.
By emitting gravitational waves, non-stationary black holes lose energy, eventually become stationary and cease to radiate
in this manner. In other words, they "decay" into stationary black holes, namely holes that are perfectly spherical or whose
rotatio n is perfectly uniform. According to Einstein's Theory of General Relativity, such objects cannot emit gravitational waves.
Eventually, it seems the black hole will get down to zero mass, and will disappear altogether. What then will happen
to all the objects that fell into the hole, and all the people that either jumped in, or were pushed? They can't come out
again, because there isn't enough mass or energy left in the black hole, to send them out again. They may pass into another
universe, but that is not something that will make any difference, to those of us prudent enough not to jump into a black
hole.
What are Gravitational Waves?Predicted in Einstein's General Theory of Relativity, gravitational waves are disturbances in the curvature of spacetime caused by the motions of matter. Propagating at (or near)
the speed of light, gravitational waves do not travel "through" spacetime as such -- the fabric of spacetime itself is oscillating.
Though gravitational waves pass straight t hrough matter, their strength weakens proportionally to the distance traveled from
the source. A gravitational wave arriving on Earth will alternately stretch and shrink distances, though on an incredibly
small scale -- by a factor of  for very strong sources. That's roughly equivalent to measuring a change the size of an atom in the distance from the Sun
to Earth!
No wonder these waves are so hard to detect.
Are Gravitational Waves Real?The first test of Einstein's General Theory of Relativity ( the bending of light by the gravity of a large mass, seen in a solar eclipse) was made by a team led by Sir Arthur Eddington, who became one of
the strongest supporters of the new theory. But when it came to gravity wave s, Eddington was skeptical and reportedly commented,
"Gravitational waves propagate at the speed of thought."
Ed Seidel, NCSA/Univ. of Illinois, on-camera
QuickTime Movie (1.0 MB); Sound File (615K); Text
Eddington was not the only skeptic. Many physicists thought the waves predicted by the theory were simply a mathematical artifact. Yet others continued to further
develop and test the concept. By the 1960s, theorists had showed that if an object emits gravitational waves, its mass should
decrease. Then, in the mid 1970s, American researchers observed a binary pulsar system (named PSR1913+16) that was thought to consist of two neutron stars orbiting each other closely and rapidly. Radio pulses from one of the stars showed that its orbital period decreases by 75
microseconds per year. In other words, the stars are spiralling in towards each other -- and by just the amount predicted
if the system were losing energy by radiating gravity waves.
Why Should We Care About Gravity Waves? Gravitational wave astronomy could expand our knowledge of the cosmos dramatically.
For starters, gravitational waves, though weakening with distance, are thought to be unchanged by any material they pass through
and, therefore, should carry signals unalt ered across the vast reaches of space. By comparison, electromagnetic radiation
tends to be modified by intervening matter.
Aside from demonstrating the existence of black holes and revealing a wealth of data on supernovae and neutron stars, gravitational
wave observations could also provide an independent means of estimating cosmological distances and help further our understanding
of how the universe came to be the way it looks today and of its ultimate fate. Gravitational waves might unveil phenomena never considered before.
Nature is smarter than any theorist trying to imagine or calculate what might be out there!
From supercomputer simulations performed at NCSA and other advanced computational facilities, relativity researchers expect different types of cosmic events
to possess characteristic gravitational wave signatures.
Consider the waves emitted by a single, distorted black hole, for example.
Distorted Black Hole
The remarkable thing about a black hole when simulated on a computer is that no matter how it forms or is perturbed,
whether by infalling matter, by gravitional waves, or via a collision with another object (including a second black hole),
it will "ring" with a unique frequency known as its natural mode of vibration. It's this unique wave signature that
will allow scientists to know if they've really detected a black hole. But that's not all. The signal will tell them how big
the black hole is and how fast it's spinning.
as I have written before and I would be thrilled if you post it is in the holgraphic principle there remains a higher
dimension that explains why we can calculate facts about black holes. My theory suggests that this greater dimension may be
interferring causing short grbs, also I have suggested possible brane interference grb.sonoma.edu GRBs appear to be divided into two classes: those shorter than 2 seconds, and those longer than 2 seconds. If this classification
scheme is correct, then it would follow that there are two different causes for the bursts. However, there are other properties
of the bursts which differ and could lead to alternative explanations. The jury is still out as to the classification scheme
and the associated mechanisms, especially for the shorter class of bursts SGR 1806-20 Blast I noticed that seconds after the initial blast of light, a doppler-like wave emanated from
that point and seemed to fly past the SWIFT camera. This "wave" appeared to be traveling at close to light speed, what was
it's composition? Referring Page: http://swift.sonoma.edu/resources/multimedia/images/
Answer provided by Lynn Cominsky (SSU E/PO Director):
The wave in question represented the light coming from the SGR
in the initial blast. The animation showed it as coming out afterwards, this was incorrect.
A common event often mistaken by people in the sky to be a Gamma-Ray burst is an Iridium Flare. If you visit this link
http://www.heavens-above.com, then put in your latitude and longitude, you will be able to see if there were any of these flares reported during this
time.
Fomalhaut, a 200-million-year-old
star, is a mere infant compared to our own 4.5-billion-year-old Sun. It resides 25 light-years away from the Sun. Located
in the constellation Piscis Austrinus (the Southern Fish), the Fomalhaut ring is ten times as old as debris disks seen previously
around the stars AU Microscopii and Beta Pictoris, where planets may still be forming. If our solar system is any example,
planets should have formed around Fomalhaut within tens of millions of years after the birth of the star. The Hubble images also provide a glimpse of the outer planetary region
surrounding a star other than our Sun. Many of the more than 100 planets detected beyond our solar system are orbiting close
to their stars. Most of the current planet-detecting techniques favor finding planets that are close to their stars. "The size of Fomalhaut's dust ring suggests that not all planetary
systems form and evolve in the same way — planetary architectures can be quite different from star to star," Kalas explained.
"While Fomalhaut's ring is analogous to the Kuiper Belt, its diameter is four times greater than that of the Kuiper Belt."The astronomers used the Advanced Camera for Surveys' (ACS) coronagraph
aboard Hubble to block out the light from the bright star so they could see details in the faint ring.
A huge, billowing pair of gas and dust clouds is the super-massive
star Eta Carinae.
Even though Eta Carinae is more than 8,000 light-years away,
features 10 billion miles across (about the diameter of our solar system) can be distinguished. Eta Carinae suffered giant
outburst about 150 years ago, when it became one of the brightest stars in the southern sky. Though the star released as much
visible light as a supernova explosion, it survived the outburst. Somehow, the explosion produced two lobes and a large, thin
equatorial disk, all moving outward at about 1.5 million miles per hour. Estimated to be 100 times heftier than our Sun, Eta
Carinae may be one of the most massive stars in our galaxy.
A Martian Dust Devil Passes
we are so fortunate that the general science community is so benevolent with their freedom to access all sorts of
wonderful information. This awesome NASA pic is no exception. Moving pictures from the surface of Mars...?
Our Business Philosophy
Our mission statement..... A Hitchhiker's Guide to the Moon  Imagine trekking in a lunar rover across miles of the Moon's rough surface. Your mission: to explore a crater with suspected
deposits of ice. Image left: When traveling on the Moon, don't forget your map!
Ion Thrusters Propel NASA into Future
|
05.27.05
|
We are a curious species with amazing capacities to imagine and dream. We wonder
about what we cannot see, are fascinated by what we do not know and are driven to explore.
In keeping with our continuous
quest for knowledge, President George W. Bush announced a new plan for NASA in Jan. 2004. A renewed focus on space exploration,
he explained, would strengthen our leadership in the world, improve our economy and enhance the quality of our lives.
 The Vision for Space Exploration calls for human and robotic missions to the Moon, Mars and beyond. To realize these ambitious
goals, we will need more powerful and efficient propulsion and power-generation systems -- systems that can thrust a spacecraft
out of Earth's orbit to the far reaches of the Universe. |
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black hole
