Black Holes

Black holes are much talked about and mysterious objects. They are in fact rather generic objects as solutions to the equations of general relativity and are essentially just a region of spacetime from which nothing that enters can escape. It is often said (although this is not always true) that once inside the black hole a person inevitably hits a "singularity" and will be crushed to death (actually you will be stretched-out until you snap). This singularity represents a break down of the known laws of physics. In addition black holes tend to be problematic in quantum theory since information can be lost forever within them and this seriously affects the wave functions outside the black hole. Originally it was thought that string theory would smooth out these singularities but it is now known that black holes are in fact of crucial importance in string theory as they are fundamental states in the theory. Indeed string theory now offers a great deal of quantitative insight into what makes up a black hole, how information is not lost and what happens at the "singularity".


Branes are higher dimensional generalisations of strings. In particular a p-brane is an object in spacetime which has p extended spatial directions and is constant in time. Thus a 0-brane is just a point particle, a 1-brane is a string and a 2-brane is a sheet. In the low energy effective supergravity theory of a string theory p-branes appear as a black hole-like solutions (where the "singularity" is not just a point but a p dimensional surface).


Ever since the 1920's physicists have pursued the idea that the universe may actually have extra dimensions. For example in string theory there are 10 dimensions. In M-theory there are 11. Clearly these extra dimensions must accounted for so it is supposed that they are curled up in to sizes so small that we could never directly detect them, i.e. they are compactified. However, the precise way in which the extra dimensions are compactified determines may important aspects of the physical world we do see. For example by compactifying one of the know string theories on a 6 dimensional space one finds many different 4 dimensional theories with various physical features.


The dimension of a space is simply the minimum number of coordinates required to uniquely specify each point on that space. The Cartesean plane has dimension 2 as you need to specify both the x-coordinate and y-coordinate of a point. The surface of the earth is also 2-dimensional as you need to specify the longitude and latitude of a point. Space itself is 3-dimensional as, in addition to longitude and latitude, one must also specify the height above the earth’s surface. 


Although the concept of duality goes back to the last century, it has only been the last few years that it string theories were seen to possess a duality structure. Simply put, two theories are said to be dual to each other if they describe the same physics using different language. Because we usually only have perturbation theory to investigate a theory with, two theories may appear completely different. However it is now believed that the non-perturbative regime (i.e. where the perturbative expansion does not provide a reliable approximation) of one string theory is actually described by the perturbative regime of another string theory. In this way it is thought that all the string theories are in fact different aspects of one underlying theory, M-theory. Duality is thus a very powerful tool in understanding the full physics of a theory.

Fine Tuning

Most models of physics require inputs which can then be used to make predictions. A predictive theory will have many more predictions than inputs. However sometimes one requires that the input parameters must be fine-tuned to particular values with a dramatically high degree of accuracy. There is nothing inconsistent or unpredictive about this but it can seem very odd and suggests that rather than having fine-tuning there is some deeper physical explanation for having such a precise value of the constants.

Gauge Theories

The standard model is built out of a type of theory known as a gauge theory. A gauge theory arises when one imposes that a particular symmetry of a theory is valid even when the parameter labelling the symmetry is allowed to vary over spacetime. In quantum theory for example it is well known that you may change the phase of the wave functions by an arbitrary amount without altering any the physical content or structure of the theory, provided that you change the all wave functions in the same way, everywhere in space. To gauge this symmetry one lets this phase change vary over spacetime. In order to maintain this as a symmetry of the theory you must introduce the electromagnetic force through the gauge potential that appears in derivatives. Thus quantum electrodynamics is an example of a gauge theory.

General Relativity

General relativity is a theory of gravity developed by Einstein between 1905 and 1916. It describes the spacetime structure of the universe as being curved. It is experimentally well tested and describes everything from planetary motion to the very large scale structure of the universe. The only known way to quantise general relativity is to embed it in a string theory or M-theory.


String theory and M-theory are unique theories. However there are so many ways to compactify them to 4 dimensions that there is a vast number of possible 4-dimensional theories that they produce. The set of all these 4-dimensional ‘vacuum’ solutions is known as the ‘landscape’ and people believe that it is astonishingly large. This presents difficulties for predictions from string theory and M-theory as it can be claimed that almost anything can happen in some string vacuum. Indeed it is even possible that the number of vacua that look very close to the standard model is so large that one could never tell if string theory were wrong or right. This has created a large amount of philosophical debate but it has also altered how we think about the various problems of fine-tuning that arise in theoretical physics.


From the point of view of supersymmetry 11 is the maximum number of dimensions spacetime can have and indeed there is a unique supergravity theory in 11 dimensions. However when the string theories where discovered and found to only exist in 10 dimensions, this 11-dimensional theory was largely ignored (prior to string theory a 4-dimensional form of it was once heralded by Stephen Hawking as the fundamental theory of nature). However, it has since been realised that the 10 dimensional nature of string theory is in fact an artifact of the perturbation theory used. There is now a great deal of evidence that all of the string theories are related by dualities and can be viewed as approximations to a single, as of yet unknown, 11-dimensional theory. This theory is called M-theory (it is not even known what the M stands for!) and has 11-dimensional supergravity as its low energy effective theory. Despite the fact that the fundamental dynamics of M-theory are unknown, its mere existence has lead to a great deal of insight into non-perturbative physics in string theories and perhaps most remarkably 4 dimensional gauge theories.

Perturbation Theory

A physical theory which applies to nature is necessarily a rather complicated object, even if the underlying principle behind it is simple and beautiful. Therefore to examine the detailed physics which the theory predicts requires some kind of approximation. Perturbation theory is a type of approximation whereby one expands the equations in powers of some small parameter, much like a decimal expansion is an expansion in powers of 1/10th. If there is a suitable small parameter in the theory then perturbation techniques can be very successful. However, if there is no such parameter then perturbation theory will fail and in general there is no known alternative. Even if a perturbative parameter is available it is inevitable that some physics will be obscured by the perturbative expansion. The confinement of quarks into protons and neutrons and high temperature super-conductivity are examples of phenomena for which perturbative techniques have failed. Thus the understanding of non-perturbative effects is an important if not crucial goal for physics.

Quantum Field Theory

Quantum field theories are physical theories which encapsulate both special relativity and quantum mechanics. The fundamental objects in a quantum field theory (QFT) are point like particles such as electrons. The quintessential example of a QFT is quantum electrodynamics or QED for short. This theory was developed in the 1940's and 50's and describes the interaction of light (electromagnetic radiation) with matter. The standard model of particle physics is also a QFT.

The Standard Model

The standard model of particle physics is a theory of interactions of the most fundamental particles of nature known (e.g. electrons, quarks, etc.). It was developed in the late 1960's and early 1970's and has stood up to various experimental tests ever since. However, it is strongly believed that this theory cannot be a true fundamental theory of nature since it does not contain gravity and only makes sense if viewed as a low energy approximation.

String Theory

Sting theories (there are five consistent string theories known) are quantum theories where the underlying object is a one-dimensional string in spacetime (as opposed to particles in quantum field theories). It is known that the low energy effective theories of strings are supergravities (i.e. quantum field theories coupled to gravity) but they offer a consistent and finite quantum description of the gauge forces and gravity. Furthermore these forces appear in a unified manner and the dimension of spacetime is restricted to be ten. Although string theories have been studied since the late 1960's it wasn't until 1984 that their application to quantum gravity and unification was fully realised. Most recently it has become clear that string theory is not just a theory of strings but that they contain branes as well. It has also become clear recently that the five known string theories may actually be viewed as parts of a single 11-dimensional theory called M-theory.


Supersymmetry is an extension of the Lorentz symmetry of spacetime. Therefore it is natural to enlarge the theory of general relativity (which may be thought of as gauge theory of the Lorentz group) to include supersymmetry. This yields a so-called supergravity theory and it turns out that there are many examples of supergravities in various dimensions up to 11. Although they were discovered prior to the advent of string theories supergravities are now primarily viewed as the low energy effective theories for strings.


Supersymmetry is a special type of symmetry which related bosons (force carriers) to fermions (matter) which has been studied since the 1970's as an extension of the Lorentz symmetry of spacetime. In a supersymmetric theory there are equal number of bosons and fermions and their dynamics takes on a very special form. Although supersymmetry has not been seen in nature it is believed by some that we will see it, possibly relatively soon at the LHC in CERN. Supersymmetry is actually predicted by the string theories and it could be used to explain why we see any light particles and low energy physics at all in a universe where extremely high energies once existed. Another important feature of supersymmetry is that supersymmetric theories may be studied much more readily and their properties deduced even at the non-perturbative level.


As the word suggests electromagnetism in fact describes what were once seen as two different forces, electricity and magnetism. However with the advent of Maxwell's equations and special relativity they could be realised as two manifestation of the same force. The standard model is a theory in which the electromagnetic and weak forces are in fact two aspects of the same force. The symmetry between the two is restored only a high energies. A general aim of particle physics has been to formulate all the known forces of nature in as a single unified force at a very high energy. Although there is no conclusive evidence for unification there are several strong indications that electromagnetism, the weak and the strong nuclear forces are unifed at an energy-scale that is about 1/1000 the Planck scale.




1 February 2008

Neil Lambert
Dept. Mathematics
King's College, London
The Strand