introduction to black holes

8/8/2019 Introduction to Black Holes

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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 thetennis 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, isnot 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 areblack. 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 existbecause 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 besome black holes.

Forget this, I want to see some pictures
http://www.damtp.cam.ac.uk/research/gr/public/bh_home.html


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To see some observational evidence for black holes from the Hubble space telescope see the next

page.

We also have some numerical studiesof the formation of black holes (including some movies).

[Back][Cosmology][Black holes][Cosmic strings][Inflation][Quantum gravity][Home] [Next]
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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 gobefore 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 theescape 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 fasterthan 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 andcollapse 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 notparticularly 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.


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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 studiesof the formation of black holes (including some movies).

http://www.damtp.cam.ac.uk/research/gr/public/bh_obsv.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/bh_critical.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/bh_critical.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/bh_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/cos_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/bh_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/cs_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/inf_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/qg_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/index.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/bh_obsv.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/bh_obsv.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/index.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/qg_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/inf_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/cs_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/bh_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/cos_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/bh_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/bh_obsv.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/bh_critical.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/bh_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/cos_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/bh_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/cs_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/inf_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/qg_home.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/index.htmlhttp://www.damtp.cam.ac.uk/research/gr/public/bh_obsv.html


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Further Explanation

How do astronomers weigh stars & galaxies?

If a planet moves around the sun in a circular orbit of radius

r at speed v, then the mass M of the sun is given by

where G is a gravitational constant. If we can measure speedand size of orbit, we can estimate the mass of the central

object. For our sun it is about

2,000,000,000,000,000,000,000,000,000,000 kg.

Most solar systems are elliptical rather than circular. The principle's the same, but the arithmeticis harder!

How do astronomers measure distances & sizes?

The universe is expanding, and so most objects are receding from us. By measuring the redshift

we can find out how fast they are moving away.

Cosmological models predict a definite relation between speed and distance; for nearby objects,distance is proportional to speed. This makes it straightforward to decide the distance of say a

galaxy.

For objects which are not too far away (and not too small) the difference in angular direction

between one boundary and another is directly proportional to its size:

size = angular separation x distance.
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Light Years

Particles of light, called photons, move faster than any other object with a speed of three hundredmillion metres per second. In one year they move a distance

9,500,000,000,000 kilometres,

which is called a light year.

If we see light coming from a distant source one million light years away, we are seeing the

object not as it is now, but how it was one million years ago.

How do astronomers measure speeds?

Express trains usually sound their horns as they pass through stations at which they do not stop.Passengers on the platform hear a higher note as the train approaches and a lower one as it

recedes. The difference in frequency is proportional to the speed of the train. Physicists call this

theDoppler effect.

The same effect happens with light. If a star is approaching us it appears bluer, and if it isreceding it appears redder. The shift in frequency, usually called the redshift, is proportional to

the speed at which it is moving away from us.

NGC 4261NGC 4261 is galaxy number 4261 in the New Galactic Catalogue. It is one hundred million lightyearsaway in the constellation (group of galaxies) called Virgo.

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Black Holes and Critical Phenomena

A scalar field is a simple form of matter which feels two forces only, gravity produced by itself,

and its own pressure. If you try to build a ball in which the field is initially weak, pressure wins

and the ball expands away leaving nothing behind. If the field is initially strong gravity wins andthe ball collapses to form ablack hole

There is a special critical initial strength such that the field cannot decide whether to evaporate

away or collapse to form a black hole. Instead it oscillates increasingly rapidly, performing aninfinite number of oscillations in a finite time. The two animations show first an evolution whichdoesn't form a black hole, and second an evolution which does. The initial strengths differ only

by one in the fifteenth decimal place. Each frame shows the scalar field as a function of position.

The gravitational field changes the frequency of the field and so the colour indicates what adistant observer would see.

Movies

This evolution doesn't form a black hole.

This collapse does form a black hole.

Because the animation moves so quickly it is hard to see what is happening. The next picturesummarises the animations.

Time is increasing from left to right (downwards).

Radius is increasing from left to right (upwards).

The colour indicates the curvature of spacetime. Far away (back of picture) the spacetime is

almost flat (blue) while on the axis the field is becoming stronger and stronger (red turningblack). Although there are an infinite number of oscillations in a finite time, all bunched up on

each other, we have stretched out the time coordinate to separate them.
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The next group of figures show the paths of photons emitted from the axis. Each blue line in the

left picture represents a photon path. Radius is plotted horizontally and time vertically. Thus

photons emitted at late times have higher lines in the figure. In the first picture there is some finedetail near the axis. the second increases the scale by one thousand, and the third by a further one

thousand. Here you can see that all the photons escape.


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The next set are the same photon paths but for the case where a black hole forms. At the finestresolution you can see that photons emitted at late times stop moving outwards and return back

towards the axis. This is the moment at which the black hole forms.


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This is an enlargement of the region where the black hole forms.


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In the next picture we superpose the evolutions with (red) and without (green) black holeformation. At early times (bottom of the picture) the spacetimes differ only in the fifteenth

decimal place. Once the spacetime decides what to do, the solutions then bifurcate.


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Black Holes and Quantum Gravity

Why bother?

Black holes arise in general relativity, a classical theory of gravity. However, we need to include

quantum effects to understand black holes properly.

Roger Penrose and Stephen Hawking showed thirty years ago that, according to generalrelativity, any object that collapses to form a black hole will go on to collapse to a singularity

inside the black hole. This means that there are strong gravitational effects on arbitrarily short

distance scales inside a black hole. On short disctance scales, we certainly need to use aquantum theory to describe the collapsing matter. The presence of a singularity in the classical

thoery also means that once we go sufficiently far into the black hole, we can no longer predict

what will happen. It is hoped that this failure of the classical theory can be cured by quantisinggravity as well.

Black holes radiate

We can try to describe the interaction of some quantum matter with gravity by quantising thematter on a fixed, classical gravitational background. That is, we can try quantising the matter,but not the gravity. This will work only if the gravity is weak. It should work outside a large

black hole, but not near the singularity.

Using this approach, Hawking has shown that a black hole will radiate thermally. That is, if we

study quantum matter fields on a classical black hole background, we find that, when the matterfields are initially in the vacuum (that is, there is no matter falling into the black hole), there is a

steady stream of outgoing radiation, which has a temperature determined by its mass and charge.

This is an extremely startling discovery; classically, no radiation can escape from a black hole,

but if we quantise the matter fields, we find there is steady flux of radiation coming out of theblack hole! This outgoing radiation decreases the mass of the black holes, so eventually the black

hole will disappear. The temperature goes up as the black hole gets smaller (unlike most things,

which cool off as they lose energy), so the black hole will disappear abruptly, in a final flash ofradiation.

Black hole thermodynamics


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There is an analogy between the classical laws governing black holes, and the laws ofthermodynamics. But thermodynamics is just an approximate description of the behaviour of

large groups of particles, which works because the particles obey statistical mechanics (a branchof quantum theory). Since black holes have a non-zero temperature, the classical laws of black

holes are the laws of thermodynamics applied to black holes, so there must be some more

fundamental description of the classical laws governing black holes in terms of statisticalmechanics.

Quantising gravity

Quantising matter fields on a black hole background teaches us a lot about black holes. However,

we need a quantum theory of gravity to understand the fundamental principles underlying blackhole thermodynamics. We also need a quantum theory to tell us what happens near the

singularity. However, quantising gravity is extremely difficult. One theory which offers some

hope, particularly for understanding black holes, is string theory.

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QUANTUM GRAVITY

Quantum cosmology

M-theory, the theory formerly known as Strings

The Holographic Principle and M-theory

[Back][Cosmology][Black holes] [Cosmic strings][Inflation][Home][Next]
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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 allowsone 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 theobserved 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 completelyrandom 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 withthe 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 successfulinvolves 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 acontribution 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,
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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 spatialdimensions 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 fourspatial 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. Thisspatial 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 todescribe 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 fornon-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 quantumgravity 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 pathintegral 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 infour 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 asstring/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 dataone 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 JamesHartle 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-Hawkingproposal is to simply do away with the initial three geometry i.e. to only include four
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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 semiclassicalapproximation because its validity lies somewhere between that of classical and quantum

physics. In the semiclassical approximation one argues that most of the four dimensionalgeometries 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. Instantonsdon'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 threegeometry 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 tocalculate 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 withinthe 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 arealistic 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 importantquestion 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, matterwhich 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


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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 openuniverse via this bubble nucleation is known as a Coleman-De Luccia instanton.

The Coleman-De Luccia Instanton

Remember that this scenario requires the existence of a false vacuum for the matter in the earlyuniverse. 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 thesespecific 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 isthat 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.


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The Hawking-Turok Instanton

The 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 everyquantum 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 temperatureperturbations 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 belaunched 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 observationallydistinguish between different theories for quantum cosmology.
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The observations made by MAP and PLANCK will therefore turn the `no boundary' proposal

and instanton cosmology into real testable science!

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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 theirposition 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 thespecial 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 toinclude 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 triesto apply the rules of QFT to GR one gets results which make no sense. For instance, the forcebetween 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 representthe 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 canmake 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
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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 agreat 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) orfermions (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, henceBosonic 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 thatdescribes 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 inthe 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 ofeverything, to have five of them is an embarrassment of riches! Fortunately, M-theorycame 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, sincethe 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 asCalabi-Yau Manifolds and Orbifolds.

M-theory
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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 newislands. 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 alltheories 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 tobe 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


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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 GeneralRelativity, 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. Inthermodynamics, 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 areequivalent 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 disorderof 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.

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introduction to black holes

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