Black Holes

Introduction to Black Holes

What is a black hole?

A black hole is a region of spacetime from which nothing can escape, even light.
To see why this happens, imagine throwing a tennis ball into the air. The harder you throw the tennis ball, the faster it is travelling when it leaves your hand and the higher the ball will go before turning back. If you throw it hard enough it will never return, the gravitational attraction will not be able to pull it back down. The velocity the ball must have to escape is known as the escape velocity and for the earth is about 7 miles a second.
As a body is crushed into a smaller and smaller volume, the gravitational attraction increases, and hence the escape velocity gets bigger. Things have to be thrown harder and harder to escape. Eventually a point is reached when even light, which travels at 186 thousand miles a second, is not travelling fast enough to escape. At this point, nothing can get out as nothing can travel faster than light. This is a black hole.

Do they really exist?

It is impossible to see a black hole directly because no light can escape from them; they are black. But there are good reasons to think they exist.
When a large star has burnt all its fuel it explodes into a supernova. The stuff that is left collapses down to an extremely dense object known as a neutron star. We know that these objects exist because several have been found using radio telescopes.
If the neutron star is too large, the gravitational forces overwhelm the pressure gradients and collapse cannot be halted. The neutron star continues to shrink until it finally becomes a black hole. This mass limit is only a couple of solar masses, that is about twice the mass of our sun, and so we should expect at least a few neutron stars to have this mass. (Our sun is not particularly large; in fact it is quite small.)
A supernova occurs in our galaxy once every 300 years, and in neighbouring galaxies about 500 neutron stars have been identified. Therefore we are quite confident that there should also be some black holes.

Forget this, I want to see some pictures

To see some observational evidence for black holes from the Hubble space telescope see the next page.We also have some numerical studies of the formation of black holes (including some movies).

Quantum Cosmology

The physical laws that govern the universe prescribe how an initial state evolves with time. In classical physics, if the initial state of a system is specified exactly then the subsequent motion will be completely predictable. In quantum physics, specifying the initial state of a system allows one to calculate the probability that it will be found in any other state at a later time. Cosmology attempts to describe the behaviour of the entire universe using these physical laws. In applying these laws to the universe one immediately encounters a problem. What is the initial state that the laws should be applied to? In practice, cosmologists tend to work backwards by using the observed properties of the universe now to understand what it was like at earlier times. This approach has proved very successful. However it has led cosmologists back to the question of the initial conditions.Inflation (a period of accelerating expansion in the very early universe) is now accepted as the standard explanation of several cosmological problems. In order for inflation to have occurred, the universe must have been formed containing some matter in a highly excited state. Inflationary theory does not address the question of why this matter was in such an excited state. Answering this demands a theory of the pre-inflationary initial conditions. There are two serious candidates for such a theory. The first, proposed by Andrei Linde of Stanford University, is called chaotic inflation. According to chaotic inflation, the universe starts off in a completely random state. In some regions matter will be more energetic than in others and inflation could ensue, producing the observable universe.
The second contender for a theory of initial conditions is quantum cosmology, the application of quantum theory to the entire universe. At first this sounds absurd because typically large systems (such as the universe) obey classical, not quantum, laws. Einstein's theory of general relativity is a classical theory that accurately describes the evolution of the universe from the first fraction of a second of its existence to now. However it is known that general relativity is inconsistent with the principles of quantum theory and is therefore not an appropriate description of physical processes that occur at very small length scales or over very short times. To describe such processes one requires a theory of quantum gravity.
In non-gravitational physics the approach to quantum theory that has proved most successful involves mathematical objects known as path integrals. Path integrals were introduced by the Nobel prizewinner Richard Feynman, of CalTech. In the path integral approach, the probability that a system in an initial state A will evolve to a final state B is given by adding up a contribution from every possible history of the system that starts in A and ends in B. For this reason a path integral is often referred to as a `sum over histories'. For large systems, contributions from similar histories cancel each other in the sum and only one history is important. This history is the history that classical physics would predict.
For mathematical reasons, path integrals are formulated in a background with four spatial dimensions rather than three spatial dimensions and one time dimension. There is a procedure known as `analytic continuation' which can be used to convert results expressed in terms of four spatial dimensions into results expressed in terms of three spatial dimensions and one time dimension. This effectively converts one of the spatial dimensions into the time dimension. This spatial dimension is sometimes referred to as `imaginary' time because it involves the use of so-called imaginary numbers, which are well defined mathematical objects used every day by electrical engineers.
The success of path integrals in describing non-gravitational physics naturally led to attempts to describe gravity using path integrals. Gravity is rather different from the other physical forces, whose classical description involves fields (e.g. electric or magnetic fields) propagating in spacetime. The classical description of gravity is given by general relativity, which says that the gravitational force is related to the curvature of spacetime itself i.e. to its geometry. Unlike for non-gravitational physics, spacetime is not just the arena in which physical processes take place but it is a dynamical field. Therefore a sum over histories of the gravitational field in quantum gravity is really a sum over possible geometries for spacetime.
The gravitational field at a fixed time can be described by the geometry of the three spatial dimensions at that time. The history of the gravitational field is described by the four dimensional spacetime that these three spatial dimensions sweep out in time. Therefore the path integral is a sum over all four dimensional spacetime geometries that interpolate between the initial and final three dimensional geometries. In other words it is a sum over all four dimensional spacetimes with two three dimensional boundaries which match the initial and final conditions. Once again, mathematical subtleties require that the path integral be formulated in four spatial dimensions rather than three spatial dimensions and one time dimension.
The path integral formulation of quantum gravity has many mathematical problems. It is also not clear how it relates to more modern attempts at constructing a theory of quantum gravity such as string/M-theory. However it can be used to correctly calculate quantities that can be calculated independently in other ways e.g. black hole temperatures and entropies.
We can now return to cosmology. At any moment, the universe is described by the geometry of the three spatial dimensions as well as by any matter fields that may be present. Given this data one can, in principle, use the path integral to calculate the probability of evolving to any other prescribed state at a later time. However this still requires a knowledge of the initial state, it does not explain it.
Quantum cosmology is a possible solution to this problem. In 1983, Stephen Hawking and James Hartle developed a theory of quantum cosmology which has become known as the `No Boundary Proposal'. Recall that the path integral involves a sum over four dimensional geometries that have boundaries matching onto the initial and final three geometries. The Hartle-Hawking proposal is to simply do away with the initial three geometry i.e. to only include four dimensional geometries that match onto the final three geometry. The path integral is interpreted as giving the probability of a universe with certain properties (i.e. those of the boundary three geometry) being created from nothing.
In practice, calculating probabilities in quantum cosmology using the full path integral is formidably difficult and an approximation has to be used. This is known as the semiclassical approximation because its validity lies somewhere between that of classical and quantum physics. In the semiclassical approximation one argues that most of the four dimensional geometries occuring in the path integral will give very small contributions to the path integral and hence these can be neglected. The path integral can be calculated by just considering a few geometries that give a particularly large contribution. These are known as instantons. Instantons don't exist for all choices of boundary three geometry; however those three geometries that do admit the existence of instantons are more probable than those that don't. Therefore attention is usually restricted to three geometries close to these.
Remember that the path integral is a sum over geometries with four spatial dimensions. Therefore an instanton has four spatial dimensions and a boundary that matches the three geometry whose probability we wish to compute. Typical instantons resemble (four dimensional) surfaces of spheres with the three geometry slicing the sphere in half. They can be used to calculate the quantum process of universe creation, which cannot be described using classical general relativity. They only usually exist for small three geometries, corresponding to the creation of a small universe. Note that the concept of time does not arise in this process. Universe creation is not something that takes place inside some bigger spacetime arena - the instanton describes the spontaneous appearance of a universe from literally nothing. Once the universe exists, quantum cosmology can be approximated by general relativity so time appears.
People have found different types of instantons that can provide the initial conditions for realistic universes. The first attempt to find an instanton that describes the creation of a universe within the context of the `no boundary' proposal was made by Stephen Hawking and Ian Moss. The Hawking-Moss instanton describes the creation of an eternally inflating universe with `closed' spatial three-geometries.
It is presently an unsolved question whether our universe contains closed, flat or open spatial three-geometries. In a flat universe, the large-scale spatial geometry looks like the ordinary three-dimensional space we experience around us. In contrast to this, the spatial sections of a realistic closed universe would look like three-dimensional (surfaces of) spheres with a very large but finite radius. An open geometry would look like an infinite hyperboloid. Only a closed universe would therefore be finite. There is, however, nowadays strong evidence from cosmological observations in favour of an infinite open universe. It is therefore an important question whether there exist instantons that describe the creation of open universes.
The idea behind the Coleman-De Luccia instanton, discovered in 1987, is that the matter in the early universe is initially in a state known as a false vacuum. A false vacuum is a classically stable excited state which is quantum mechanically unstable. In the quantum theory, matter which is in a false vacuum may `tunnel' to its true vacuum state. The quantum tunnelling of the matter in the early universe was described by Coleman and De Luccia. They showed that false vacuum decay proceeds via the nucleation of bubbles in the false vacuum. Inside each bubble the matter has tunnelled. Surprisingly, the interior of such a bubble is an infinite open universe in which inflation may occur. The cosmological instanton describing the creation of an open universe via this bubble nucleation is known as a Coleman-De Luccia instanton.

The Coleman-De Luccia InstantonRemember that this scenario requires the existence of a false vacuum for the matter in the early universe. Moreover, the condition for inflation to occur once the universe has been created strongly constrains the way the matter decays to its true vacuum. Therefore the creation of open inflating universes appears to be rather contrived in the absence of any explanation of these specific pre-inflationary initial conditions.
Recently, Stephen Hawking and Neil Turok have proposed a bold solution to this problem. They constructed a class of instantons that give rise to open universes in a similar way to the instantons of Coleman and De Luccia. However, they did not require the existence of a false vacuum or other very specific properties of the excited matter state. The price they pay for this is that their instantons have singularities: places where the curvature becomes infinite. Since singularities are usually regarded as places where the theory breaks down and must be replaced by a more fundamental theory, this is a quite controversial feature of their work.

The Hawking-Turok InstantonThe question of course arises which of these instantons describes correctly the creation of our own universe. The way one might hope to distinguish between different theories of quantum cosmology is by considering quantum fluctuations about these instantons. The Heisenberg uncertainty principle in quantum mechanics implies that vacuum fluctuations are present in every quantum theory. In the full quantum picture therefore, an instanton provides us just with a background geometry in the path integral with respect to which quantum fluctuations need to be considered.
During inflation, these quantum mechanical vacuum fluctuations are amplified and due to the accelerating expansion of the universe they are stretched to macroscopic length scales. Later on, when the universe has cooled, they seed the growth of large scale structures (e.g. galaxies) like those we see today. One sees the imprint of these primordial fluctuations as small temperature perturbations in the cosmic microwave background radiation.
Since different types of instantons predict slightly different fluctuation spectra, the temperature perturbations in the cosmic microwave background radiation will depend on the instanton from which the universe was created. In the next decade the satellites MAP and PLANCK will be launched to measure the temperature of the microwave background radiation in different directions on the sky to a very high accuracy. The observations will not only provide us with a very important test of inflation itself but may also be the first possibility to observationally distinguish between different theories for quantum cosmology.
The observations made by MAP and PLANCK will therefore turn the `no boundary' proposal and instanton cosmology into real testable science

M-theory, the theory formerly known as Strings

The Standard Model

In the standard model of particle physics, particles are considered to be points moving through space, tracing out a line called the World Line. To take into account the different interactions observed in Nature one has to provide particles with more degrees of freedom than only their position and velocity, such as mass, electric charge, color (which is the "charge" associated with the strong interaction) or spin.The standard model was designed within a framework known as Quantum Field Theory (QFT), which gives us the tools to build theories consistent both with quantum mechanics and the special theory of relativity. With these tools, theories were built which describe with great success three of the four known interactions in Nature: Electromagnetism, and the Strong and Weak nuclear forces. Furthermore, a very successful unification between Electromagnetism and the Weak force was achieved (Electroweak Theory), and promising ideas put forward to try to include the Strong force. But unfortunately the fourth interaction, gravity, beautifully described by Einstein's General Relativity (GR), does not seem to fit into this scheme. Whenever one tries to apply the rules of QFT to GR one gets results which make no sense. For instance, the force between two gravitons (the particles that mediate gravitational interactions), becomes infinite and we do not know how to get rid of these infinities to get physically sensible results.

String Theory

In String Theory, the myriad of particle types is replaced by a single fundamental building block, a `string'. These strings can be closed, like loops, or open, like a hair. As the string moves through time it traces out a tube or a sheet, according to whether it is closed or open. Furthermore, the string is free to vibrate, and different vibrational modes of the string represent the different particle types, since different modes are seen as different masses or spins.One mode of vibration, or `note', makes the string appear as an electron, another as a photon. There is even a mode describing the graviton, the particle carrying the force of gravity, which is an important reason why String Theory has received so much attention. The point is that we can make sense of the interaction of two gravitons in String theory in a way we could not in QFT. There are no infinities! And gravity is not something we put in by hand. It has to be there in a theory of strings. So, the first great achievement of String Theory was to give a consistent theory of quantum gravity, which resembles GR at macroscopic distances. Moreover String Theory also possesses the necessary degrees of freedom to describe the other interactions! At this point a great hope was created that String Theory would be able to unify all the known forces and particles together into a single `Theory of Everything'.

From Strings to Superstrings

The particles known in nature are classified according to their spin into bosons (integer spin) or fermions (odd half integer spin). The former are the ones that carry forces, for example, the photon, which carries electromagnetic force, the gluon, which carries the strong nuclear force, and the graviton, which carries gravitational force. The latter make up the matter we are made of, like the electron or the quark. The original String Theory only described particles that were bosons, hence Bosonic String Theory. It did not describe Fermions. So quarks and electrons, for instance, were not included in Bosonic String Theory.By introducing Supersymmetry to Bosonic String Theory, we can obtain a new theory that describes both the forces and the matter which make up the Universe. This is the theory of superstrings. There are three different superstring theories which make sense, i.e. display no mathematical inconsistencies. In two of them the fundamental object is a closed string, while in the third, open strings are the building blocks. Furthermore, mixing the best features of the bosonic string and the superstring, we can create two other consistent theories of strings, Heterotic String Theories.
However, this abundance of theories of strings was a puzzle: If we are searching for the theory of everything, to have five of them is an embarrassment of riches! Fortunately, M-theory came to save us.

Extra dimensions...

One of the most remarkable predictions of String Theory is that space-time has ten dimensions! At first sight, this may be seen as a reason to dismiss the theory altogether, as we obviously have only three dimensions of space and one of time. However, if we assume that six of these dimensions are curled up very tightly, then we may never be aware of their existence. Furthermore, having these so-called compact dimensions is very beneficial if String Theory is to describe a Theory of Everything. The idea is that degrees of freedom like the electric charge of an electron will then arise simply as motion in the extra compact directions! The principle that compact dimensions may lead to unifying theories is not new, but dates from the 1920's, since the theory of Kaluza and Klein. In a sense, String Theory is the ultimate Kaluza-Klein theory.For simplicity, it is usually assumed that the extra dimensions are wrapped up on six circles. For realistic results they are treated as being wrapped up on mathematical elaborations known as Calabi-Yau Manifolds and Orbifolds.

M-theory

Apart from the fact that instead of one there are five different, healthy theories of strings (three superstrings and two heterotic strings) there was another difficulty in studying these theories: we did not have tools to explore the theory over all possible values of the parameters in the theory. Each theory was like a large planet of which we only knew a small island somewhere on the planet. But over the last four years, techniques were developed to explore the theories more thoroughly, in other words, to travel around the seas in each of those planets and find new islands. And only then it was realized that those five string theories are actually islands on the same planet, not different ones! Thus there is an underlying theory of which all string theories are only different aspects. This was called M-theory. The M might stand for Mother of all theories or Mystery, because the planet we call M-theory is still largely unexplored.
There is still a third possibility for the M in M-theory. One of the islands that was found on the M-theory planet corresponds to a theory that lives not in 10 but in 11 dimensions. This seems to be telling us that M-theory should be viewed as an 11 dimensional theory that looks 10 dimensional at some points in its space of parameters. Such a theory could have as a fundamental object a Membrane, as opposed to a string. Like a drinking straw seen at a distance, the membranes would look like strings when we curl the 11th dimension into a small circle.

Black Holes in M-theory

Black Holes have been studied for many years as configurations of spacetime in General Relativity, corresponding to very strong gravitational fields. But since we cannot build a consistent quantum theory from GR, several puzzles were raised concerning the microscopic physics of black holes. One of the most intriguing was related to the entropy of Black Holes. In thermodynamics, entropy is the quantity that measures the number of states of a system that look the same. A very untidy room has a large entropy, since one can move something on the floor from one side of the room to the other and no one will notice because of the mess - they are equivalent states. In a very tidy room, if you change anything it will be noticeable, since everything has its own place. So we associate entropy to disorder. Black Holes have a huge disorder. However, no one knew what the states associated to the entropy of the black hole were. The last four years brought great excitement in this area. Similar techniques to the ones used to find the islands of M-theory, allowed us to explain exactly what states correspond to the disorder of some black holes, and to explain using fundamental theory the thermodynamic properties that had been deduced previously using less direct arguments.Many other problems are still open, but the application of string theory to the study of Black Holes promises to be one of the most interesting topics for the next few years.

String Theory

Physics: String Theory
The Wave Structure of Matter (WSM) explains why String Theory is a Mathematical Theory Only (Founded on Wave Theory & Resonance)

In String Theory, the myriad of particle types is replaced by a single fundamental building block, a `string'. These strings can be closed, like loops, or open, like a hair. As the string moves through time it traces out a tube or a sheet, according to whether it is closed or open. Furthermore, the string is free to vibrate, and different vibrational modes of the string represent the different particle types, since different modes are seen as different masses or spins. One mode of vibration, or `note', makes the string appear as an electron, another as a photon.

Just like an ordinary guitar string, a fundamental string can vibrate in different modes. And it is these different modes of vibration of the string that are understood in string theory as being the different elementary particles. (Smolin, 1997)

Introduction to String Theory
Superstring Theory, M Theory and Brane Theory all evolved from the original String Theory back in the mid 1980s. As I see it, String Theory (and its derivatives) are an example of theorists being taught mathematical Physics while completely neglecting Philosophy and Metaphysics. Thus there is no awareness that a complete description of Reality must be founded on ONE thing (not on many discrete things / strings!). The reason for the partial success of String Theory is that it utilises Wave Theory and resonant frequencies. The strings can only vibrate at discrete frequencies which correspond to different 'particles' and their energy states.
I must admit that I look forward to the future when the Wave Structure of Matter is equally famous, for then they will see how truly absurd string theory is (founded on 9 dimensions of Space, but six dimensions 'magically' curl up so we don't see them!).

Below you will find several summaries of String Theory from the main web sites that promote this theory. I think that after reading on the Wave Structure of Matter you will understand why String Theory evolved as a mathematical theory (founded on waves / resonance), and also why its literal interpretation is nonsense.

M Theory, the theory formerly known as String Theory
The Standard Model
In the standard model of particle physics, particles are considered to be points moving through space, tracing out a line called the World Line. To take into account the different interactions observed in Nature one has to provide particles with more degrees of freedom than only their position and velocity, such as mass, electric charge, color (which is the "charge" associated with the strong interaction) or spin.

The standard model was designed within a framework known as Quantum Field Theory (QFT), which gives us the tools to build theories consistent both with quantum mechanics and the special theory of relativity. With these tools, theories were built which describe with great success three of the four known interactions in Nature: Electromagnetism, and the Strong and Weak nuclear forces. Furthermore, a very successful unification between Electromagnetism and the Weak force was achieved (Electroweak Theory), and promising ideas put forward to try to include the Strong force. But unfortunately the fourth interaction, gravity, beautifully described by Einstein's General Relativity (GR), does not seem to fit into this scheme. Whenever one tries to apply the rules of QFT to GR one gets results which make no sense. For instance, the force between two gravitons (the particles that mediate gravitational interactions), becomes infinite and we do not know how to get rid of these infinities to get physically sensible results.

String Theory
In String Theory, the myriad of particle types is replaced by a single fundamental building block, a `string'. These strings can be closed, like loops, or open, like a hair. As the string moves through time it traces out a tube or a sheet, according to whether it is closed or open. Furthermore, the string is free to vibrate, and different vibrational modes of the string represent the different particle types, since different modes are seen as different masses or spins. One mode of vibration, or `note', makes the string appear as an electron, another as a photon. There is even a mode describing the graviton, the particle carrying the force of gravity, which is an important reason why String Theory has received so much attention. The point is that we can make sense of the interaction of two gravitons in String theory in a way we could not in QFT. There are no infinities! And gravity is not something we put in by hand. It has to be there in a theory of strings. So, the first great achievement of String Theory was to give a consistent theory of quantum gravity, which resembles GR at macroscopic distances. Moreover String Theory also possesses the necessary degrees of freedom to describe the other interactions! At this point a great hope was created that String Theory would be able to unify all the known forces and particles together into a single `Theory of Everything'.

From Strings to Superstrings
The particles known in nature are classified according to their spin into bosons (integer spin) or fermions (odd half integer spin). The former are the ones that carry forces, for example, the photon, which carries electromagnetic force, the gluon, which carries the strong nuclear force, and the graviton, which carries gravitational force. The latter make up the matter we are made of, like the electron or the quark. The original String Theory only described particles that were bosons, hence Bosonic String Theory. It did not describe Fermions. So quarks and electrons, for instance, were not included in Bosonic String Theory.
By introducing Supersymmetry to Bosonic String Theory, we can obtain a new theory that describes both the forces and the matter which make up the Universe. This is the theory of superstrings. There are three different superstring theories which make sense, i.e. display no mathematical inconsistencies. In two of them the fundamental object is a closed string, while in the third, open strings are the building blocks. Furthermore, mixing the best features of the bosonic string and the superstring, we can create two other consistent theories of strings, Heterotic String Theories.
However, this abundance of theories of strings was a puzzle: If we are searching for the theory of everything, to have five of them is an embarrassment of riches! Fortunately, M-theory came to save us.

Extra Dimensions
One of the most remarkable predictions of String Theory is that space-time has ten dimensions! At first sight, this may be seen as a reason to dismiss the theory altogether, as we obviously have only three dimensions of space and one of time. However, if we assume that six of these dimensions are curled up very tightly, then we may never be aware of their existence. Furthermore, having these so-called compact dimensions is very beneficial if String Theory is to describe a Theory of Everything. The idea is that degrees of freedom like the electric charge of an electron will then arise simply as motion in the extra compact directions! The principle that compact dimensions may lead to unifying theories is not new, but dates from the 1920's, since the theory of Kaluza and Klein. In a sense, String Theory is the ultimate Kaluza-Klein theory.
For simplicity, it is usually assumed that the extra dimensions are wrapped up on six circles. For realistic results they are treated as being wrapped up on mathematical elaborations known as Calabi-Yau Manifolds and Orbifolds.

M-Theory
Apart from the fact that instead of one there are five different, healthy theories of strings (three superstrings and two heterotic strings) there was another difficulty in studying these theories: we did not have tools to explore the theory over all possible values of the parameters in the theory. Each theory was like a large planet of which we only knew a small island somewhere on the planet. But over the last four years, techniques were developed to explore the theories more thoroughly, in other words, to travel around the seas in each of those planets and find new islands. And only then it was realized that those five string theories are actually islands on the same planet, not different ones! Thus there is an underlying theory of which all string theories are only different aspects. This was called M-theory. The M might stand for Mother of all theories or Mystery, because the planet we call M-theory is still largely unexplored.
There is still a third possibility for the M in M-theory. One of the islands that was found on the M-theory planet corresponds to a theory that lives not in 10 but in 11 dimensions. This seems to be telling us that M-theory should be viewed as an 11 dimensional theory that looks 10 dimensional at some points in its space of parameters. Such a theory could have as a fundamental object a Membrane, as opposed to a string. Like a drinking straw seen at a distance, the membranes would look like strings when we curl the 11th dimension into a small circle.

Black Holes in M-Theory
Black Holes have been studied for many years as configurations of spacetime in General Relativity, corresponding to very strong gravitational fields. But since we cannot build a consistent quantum theory from GR, several puzzles were raised concerning the microscopic physics of black holes. One of the most intriguing was related to the entropy of Black Holes. In thermodynamics, entropy is the quantity that measures the number of states of a system that look the same. A very untidy room has a large entropy, since one can move something on the floor from one side of the room to the other and no one will notice because of the mess - they are equivalent states. In a very tidy room, if you change anything it will be noticeable, since everything has its own place. So we associate entropy to disorder. Black Holes have a huge disorder. However, no one knew what the states associated to the entropy of the black hole were. The last four years brought great excitement in this area. Similar techniques to the ones used to find the islands of M-theory, allowed us to explain exactly what states correspond to the disorder of some black holes, and to explain using fundamental theory the thermodynamic properties that had been deduced previously using less direct arguments.
Many other problems are still open, but the application of string theory to the study of Black Holes promises to be one of the most interesting topics for the next few years.


Basic Ideas of Superstring Theory

In 1984-85 there was a series of discoveries that convinced many theorists that superstring theory is a very promising approach to unification. Almost overnight, the subject was transformed from an intellectual backwater to one of the most active areas of theoretical physics, which it has remained ever since.
By the time the dust settled in 1985, it seemed clear that there are five different superstring theories, each requiring ten dimensions (nine space and one time). To have a chance of being realistic, the six extra space dimensions must curl up into a tiny geometrical space, whose size should be comparable to the string length Lst..
A string's space-time history is described by functions Xm(s,t) which describe how the string's two-dimensional 'world sheet,' represented by coordinates (s,t), is mapped into space-time Xm. There are also functions defined on the two-dimensional world-sheet that describe other degrees of freedom, such as those associated with supersymmetry and gauge symmetries.
Surprisingly, classical string theory dynamics is described by a conformally invariant 2D quantum field theory. (Roughly, conformal invariance is symmetry under a change of length scale.) What distinguishes one-dimensional strings from higher dimensional analogs is the fact that this 2D theory is renormalizable (no bad short-distance infinities). By contrast, objects with p dimensions, called 'p-branes,' have a (p+1)-dimensional world volume theory. For p > 1, those theories are non-renormalizable. This is the feature that gives strings a special status, even though, as we will discuss later, higher-dimensional p-branes do occur in superstring theory.

In the IIA case the 11th dimension is a circle, whereas in the HE case it is a line interval (so that the eleven-dimensional space-time has two ten-dimensional boundaries). The strong coupling limit of either of these theories gives an 11-dimensional space-time. The eleven-dimensional description of the underlying theory is called "M theory." As yet, it is less well understood than the five 10-dimensional string theories.

Another source of insight into non-perturbative properties of superstring theory has arisen from the study of a special class of p-branes called Dirichlet p-branes (or D-branes for short). The name derives from the boundary conditions assigned to the ends of open strings. The usual open strings of the type I theory satisfy a condition (Neumann boundary condition) that ensures that no momentum flows on or of the end of a string. However, T duality implies the existence of dual open strings with specified positions (Dirichlet boundary conditions) in the dimensions that are T-transformed. More generally, in type II theories, one can consider open strings with specified positions for the end-points in some of the dimensions, which implies that they are forced to end on a preferred surface. At first sight this appears to break the relativistic invariance of the theory, which is paradoxical. The resolution of the paradox is that strings end on a p-dimensional dynamical object -- a D-brane. D-branes had been studied for a number of years, but their significance was explained by Polchinski only recently.
The importance of D-branes stems from the fact that they make it possible to study the excitations of the brane using the renormalizable 2D quantum field theory of the open string instead of the non-renormalizable world-volume theory of the D-brane itself. In this way it becomes possible to compute non-perturbative phenomena using perturbative methods. Many (but not all) of the previously identified p-branes are D-branes. Others are related to D-branes by duality symmetries, so that they can also be brought under mathematical control.
D-branes have found many interesting applications, but the most remarkable of these concerns the study of black holes. Strominger and Vafa[8] (and subsequently many others) have shown that D-brane techniques can be used to count the quantum microstates associated to classical black hole configurations. The simplest case, which was studied first, is static extremal charged black holes in five dimensions. Strominger and Vafa showed that for large values of the charges the entropy (defined by S = log N, where N is the number of quantum states that system can be in) agrees with the Bekenstein-Hawking prediction (1/4 the area of the event horizon).
This result has been generalized to black holes in 4D as well as to ones that are near extremal (and radiate correctly) or rotating. In my opinion, this is a truly dramatic advance. It has not yet been proved that there is no breakdown of quantum mechanics due to black holes, but I expect that result to follow in due course.

In the euphoria following the first superstring revolution in 1985, some of the less experienced participants in the enterprise thought that we were on the verge of constructing a complete fundamental theory of the physical world. To put it mildly, I found this naive. In this setting, the phrase "Theory of Everything" was introduced and propagated by the public media. This was very unfortunate for several reasons.
The TOE phrase is very misleading on several counts. First of all, the theory is not yet fully formulated, and when it is (which might still take decades) it is not entirely clear that it will be the last word in fundamental physics.
Furthermore, even if the theory is a complete description of quantum dynamics, it seems unlikely that it will also provide a theory of initial conditions, which is another key ingredient required to explain why we observe the particular universe that we do.
But even if a theory of initial conditions is also obtained, there will still be much about this universe that cannot be explained. Many things, such as our very existence, are a consequence of the inherent quantum indeterminacy of nature. I believe that cannot be overcome. Maybe that is just as well, because if we had old-fashioned classical determinism, the future would be fully determined, which would undermine our humanity.
There is also a more mundane sort of unpredictability that is also to be expected. Many of the things that the theory predicts unambiguously in principle could require intractable calculations. Part of the art of physics is to identify those things that can be calculated.
The other reason the TOE phrase upset me is that it alienated many of our physics colleagues, some of whom had serious doubts about the subject anyway.
Quite understandably, it gave them the impression that people who work in this field are a very arrogant bunch. Actually, we are all very charming and delightful.

Physics Slide

MIT OpenCourseWare | Physics | 8.01L Physics I: Classical Mechanics, Fall 2005 | Home

MIT OpenCourseWare | Physics | 8.01L Physics I: Classical Mechanics, Fall 2005 | Home

Oscillations

You may have heard the one about the opera singer breaking a wineglass with their voice...(in other versions it's a greenhouse(?!), or even the specs worn by members of the audience). Well, if the singer could pull it off, they would be exploiting oscillations. Later in this section we'll show you how to break a wineglass with a loudspeaker...but first we'll try to work out some rules for things which vibrate, or oscillate. There are many forms of oscillation in the real world - oscillations determine the sound of a musical instrument, the colour of a rainbow, the ticking of a clock and even the temperature of a cup of tea. A mechanical oscillation is a repeating movement - an electrical oscillation is a repeating change in voltage and current.
Mechanical oscillations are probably easiest to think about, so we'll start there. Let's think about the movement of an object back and forth over a fixed range of positions, such as the movement of a swing in a playground, or the bouncing up and down of a weight on the end of a spring (this last one is one of those examples which teachers / scientists / engineers love, but which doesn't seem to have anything to do with the real world. As we'll see, it does!!).

The pendulum and the mass-spring system are both oscillating. How can we describe their oscillation?

Displacement

The animation shows how displacement describes the distance (and also direction...displacement can be negative as well as positive) of the object from its equilibrium position.
The displacement amplitude tells us how 'big' the oscillations are - we can use the peak value (the maximum positive displacement from the equilibrium position) or the peak-to-peak value (the distance between negative-maximum to positive-maximum).
The time period of the oscillation is the time taken for the object to travel through one complete cycle. We can measure from the equilibrium position, as in the animation, or from any other point on the cycle, as long as we measure the time taken to return to the same point with the same direction of travel.
Can you see why the first time the mass returns to the equilibrium position, we've only reached half a cycle?
Displacement on its own will sometimes do...but often it is also useful to understand the oscillation in terms of velocity oracceleration. These three things are very similar...read on.

Velocity and Acceleration

When our oscillating object reaches maximum displacement (when it is as far from equilibrium as it can get) it changes direction. This must mean that there is an instant in time when it is not moving...and so at the point of maximum displacement, its speed is zero. Then it speeds up in the opposite direction, and travels fast through the equilibrium position before starting to slow again in preparation for the next change in direction.
Velocity is just another word for speed, with the extra feature that it has direction and therefore can be negative or positive. (If you walk backwards at 4m.p.h. - difficult without falling over - then your velocity = -4m.p.h.)

So - when velocity is maximum, displacement is zero, and when displacement is maximum, velocity is zero. The period of oscillation is the same, but the plots for displacement and velocity do not 'line-up' in time...one is shifted along compared to the other. This shift along in time is called a phase angle, which is an idea we'll come back to later.
At maximum displacement the object has to come to rest, and then start moving in the opposite direction. Imagine braking in a car (moving forwards) before immediately moving off fast in reverse - you'd feel rapid deceleration during braking, followed acceleration in reverse gear. Like displacement and velocity, acceleration can be negative or positive depending on direction - in this example, braking would cause negative acceleration, as would moving off in reverse. It sounds wierd, but the point in time where the acceleration would be (negative) maximum is the moment when the car stops moving - just as it changes direction.
Thinking about our animated mass-spring system, we can see this effect. Conversely as the object moves through its equilibrium position, the velocity is only changing very slowly and the acceleration actually becomes zero. This is shown below:

Note again - the period is the same, but things don't line up in time...there's some phase in there too.
All these masses and springs are OK - but you can experience this by playing on a swing. Look at these videos, and try to work out when displacement, velocity and acceleration are 1) a maximum, and 2) zero, for someone swinging...

In our animations, the mass on a spring bounces up and down with a regular 'pattern' or waveform. Why does this happen?

Equilibrium and Restoring Forces

Let's think some more about playground swings. What's the first thing we do to get the swing moving?

The swing is first pulled back, and then pushed forward. It then oscillates back and forth 'on its own' until it slowly comes to rest at its equilibrium position - the middle position between the two extremes of displacement. This is usually the place where an object will naturally rest if no external forces are applied to it.
Once the swing has been pulled away from its equilibrium position, the force of gravity will act to bring it back. This force will always act in a direction towards the equilibrium position, and is known as a restoring force. If the swing is pulled higher into the air (larger displacement amplitude) then there will be a larger restoring force acting on it, and when the pusher lets go it will travel further. This shows that the restoring force is proportional to the distance of the swing from the equilibrium position.
The animation below shows these forces in action:

So why doesn't the swing just return directly to the equilibrium position, and stay there?
The restoring forces are large enough to make the swing over-shoot and travel 'up the other side'; this is when a restoring force in the opposite direction takes over and pushes the swing back down. Again, the force is large enough for the swing to over-shoot equilibrium and travel 'up the other side'. The process keeps repeating and an oscillation is formed. If no other forces are applied to the system, the swing will keep swinging back and forth forever, with a constant period and constant amplitude.
Does this happen in real life? (Silly question - of course not). Then why not?
In all real-life oscillating systems, damping comes into play. Damping is all about the loss of energy, often due toresistance or friction. The air displaced by the swinger adds some resistance, and even if the swing were in a vacuum (and the swinger were, therefore, dead) the pivots at the end of the chain generate some friction with their supports and slowly, almost immeasurably, heat up. This uses enery, which has to come from somewhere...and so the amplitude of oscillation decreases.
If you have no pusher, and have to propel yourself on the swing, you have to put some effort into it to keep the swing moving. This effort is overcoming damping. If there were no damping (and also if you could switch off your heart, brain and other energy-consumptive body functions) you could swing forever without needing to eat!