Relativity & Gravitation

The Relativity Group in the Department of Applied Mathematics & Theoretical Physics was started by Dr D W Sciama nearly thirty years ago, and is now led by Professor Stephen Hawking CH CBE FRS, holder of the Lucasian Chair in Mathematics until 2009.

It is internationally renowned for a number of important developments in Einstein's classical theory of gravitation, including the no hair and area theorems for black holes and the theorems indicating that singularities would occur both in gravitational collapse and at the beginning of the expansion of the Universe.

In recent years the group's main effort has been towards the inclusion of quantum effects, and the development of a theory of quantum gravity; in particular, the semiclassical quantization of black holes (leading, e.g, to the discovery of the thermal radiation produced by them) and the formulation of the Euclidean path integral approach to quantum gravity (leading to the no boundary condition for the Universe). Furthermore, the group has expertise in the areas of supergravity, string and membrane theories of gravity, cosmology, cosmic strings and other topological defects in cosmology, numerical relativity and Regge calculus.


The Standard Model, which is based on conventional quantum field theory, gives an impressive description of the strong, weak and electromagnetic forces. However, quantum field theory has profound problems when applied to the force of gravity as embodied in Einstein's general theory of relativity. The lack of a consistent formulation of gravity in the quantum regime is often considered to be the most important conceptual problem in theoretical physics. Superstring theory not only points towards a resolution of this conflict but also has the potential of describing all of the forces and particles in a unified manner.

Superstring Theory

string In conventional quantum field theories the various fundamental particles such as the quarks, the electron, the photon and others, are structureless points and are unrelated to each other. However, according to superstring theory these particles are simply different modes of vibration of a single string-like object. The average size of this string is expected to be much smaller than the smallest sizes probed experimentally, currently 10-16 centimetre, and might be as small as the Planck length, which is 10-33 centimetre. All the particles and forces of the Standard Model, including gravity, should emerge from this unifying picture at distances much larger than the string size. However, at distances of the order of the string scale all the physical laws are radically modified. The non-zero size of the string modifies physics at ultra-short distances, thereby avoiding the problems of all earlier attempts to combine quantum theory with gravity. The existence of supersymmetry is crucial for the consistency of the theory and implies that there are only five different string theories. The theory requires space-time to have ten dimensions, which is six more than we observe in the world around us. However, in any theory that describes gravity the geometry of space-time is determined in a dynamical fashion and the extra dimensions may curl up into a very small size. Clearly, in order to recover a space-time with only four large space-time dimensions the six extra space dimensions have to be smaller than 10-16 centimetre.


MTheory The string-like description of the elementary particles can only be an approximation to something deeper. It has a structure that is analogous to the wellunderstood perturbative treatment of quantum field theories embodied in Feynman diagrams, which does not take account of nonperturbative effects such as the appearance of solitons or quantum tunneling effects. In quantum field theory the Feynman diagrams are obtained as approximations to a welldefined underlying theory, such as quantum electrodynamics. Rather bizarrely, the perturbation approximation to string theory has been formulated without knowing what the underlying theory is. The search for this complete formulation of the theory, often known as 'M-theory', has been the main emphasis of research over the past decade. Some important advances have been made based on the discovery of new symmetries, known as dualities. It is now clear that what were once thought to be five distinct superstring theories in ten dimensions are actually five different approximations to M-theory. In another limit the theory reduces to the unique supersymmetric field theory, called supergravity, in eleven space-time dimensions. These different approximations to the same theory are related to one another by the duality symmetries. Members of our group were responsible for some of the most important of these developments.

Although the complete structure of M-theory is still mysterious, several of its properties are understood thanks to recent developments. Of particular significance is the presence of solitons corresponding to extended objects, or 'black p-branes'. These are generalizations of the familiar black hole solutions of Einstein's theory, which are extended in p spatial directions. A black hole would be a 0-brane while a string would be a 1-brane and a membrane a 2-brane in this terminology.


brane A particularly important class of such p-branes are known as Dirichlet branes, or simply 'D-branes'. These correspond to the extended objects where the end-points of open strings are constrained to lie.

The study of D-branes has led to important developments during the past five years. For example, by considering black holes to be composites of D-branes it has been possible to give a microscopic description of their entropy and Hawking radiation. Another development followed by considering the effect of having a large number of 3-branes. This led to the discovery that string theory in a specific kind of curved space-time, known as anti-de Sitter space, is precisely the same as a quantum field theory that lives on the boundary of that space. This is a very special example of the 'holographic principle', according to which the information inside some volume of space is encoded on the surface of that space. Although this has been conjectured to be a general property of quantum gravity, it is very difficult to demonstrate in full generality. Work in this area has led to interesting new ideas concerning the interplay of quantum gravity and gauge field theories of the kind that enter into the Standard Model of elementary particle physics.

This is seen most concretely in the superstring theory realization of the so-called 'brane world scenario'. This suggests that our universe is in reality a 3-brane, or 3-dimensional surface, embedded in higher dimensions. The theory predicts that the nongravitational forces the electroweak force and the strong force are trapped on the 3-brane and do not sense the extra dimensions while the gravitational force probes all the dimensions. Not only does this picture have obvious conceptual implications but it also allows for the possibility of having the extra dimensions as large as a fraction of one millimetre without contradicting any experiment. It is possible that the string scale may be much larger than the Planck length. This would have interesting phenomenological and cosmological implications which are currently under study.

Relativity & Quantum Gravity

The work in this area covers classical general relativity and quantum gravity. However, there is considerable overlap, both in people and subject, with cosmology and Superstrings/M-Theory.

General Relativity

blackhole Group members continue to advance the understanding of Einstein's theory of gravity (general relativity), notably by studying the formation of singularities and black holes during gravitational collapse. Numerical studies have shown that in classical general relativity, spherically symmetric collapse of a massless scalar field produces critical phenomena and a violation of the Cosmic Censorship Conjecture; this states that singularities are always hidden behind the event horizon of a black hole. Sophisticated adaptive mesh algorithms are being developed to resolve the fine structure in more general collapse situations and near cosmological singularities.

The non-linearity of the equations used to describe general relativity make it highly likely that typical gravitational systems will exhibit chaos or turbulence. Evidence for this behaviour has been found in theoretical models describing the Big Bang and orbits around black holes. More recently the non-linear evolution of density fluctuations in the early universe has been studied numerically. Strong fluctuations can collapse to form primordial black holes. The hydrodynamics is complicated by the occurrence of strong ultrarelativistic shocks. Before these shocks appear, the assumptions made in earlier analytical approximations to primordial black hole formation appear to be surprisingly accurate. There is a minimum mass for such black holes, about one ten thousandth of the mass within the cosmological horizon, and this mass is the maximum that subsequently accretes onto the black hole.

Attention is now switching to the evolution of axisymmetric rotating systems. In order to avoid the instabilities which plague other calculations, more sophisticated analytic and numerical techniques are under development. Besides gravitational collapse calculations this study is relevant also for the evolution of neutron stars.

Gravitational Waves

The quest for gravitational waves, propagating perturbations in spacetime, is nearing fruition with major detector experiments coming online very shortly, such as LIGO and VIRGO. The discovery of gravitational waves from black hole collisions could be used for precision tests of many of the developments in black hole theory made by members of the Relativity group, including uniqueness, area addition theorems and cosmic censorship.

Ongoing efforts in the study of gravitational waves relate to the distinct observable signatures made by other possible sources such as superheavy cosmic strings.

Gravity Theories

Gravity Theories in higher dimensions have long played a role in research in M-Theory and String Theory. Recently, with the suggestion that the scale of the extra dimensions may be around a TeV, the detailed study of black hole properties in higher dimensions has become important. Members of the group are currently active in this area, which may be relevant to the possible creation of black holes in accelerator experiments. It also throws some light on long-standing and hitherto intractable problems in the theory of black holes in three dimensions.

Quantum Gravity

The Euclidean formulation of quantum gravity allows the possibility that on short length scales spacetime can have a foam-like structure with highly non-trivial topology. One can interpret the bubbles as pairs of virtual black holes that appear and disappear. Particles can fall into these holes and re-emerge as different particles. An important unsolved problem is whether information and quantum coherence are lost in this process and in the formation and evaporation of macroscopic black holes. If they are lost, it would mean that quantum gravity would add a new level of uncertainty to physics over and above that normally associated with quantum theory.

Regge Calculus

Another approach to gravity theories being developed is Regge calculus. This is a discrete approximation in which spacetime is represented by a collection of flat blocks with curvature restricted to certain subspaces. This technique has applications in classical and quantum field theories. Members of the group have also worked recently on the related field of spin foam models, which are attempts to obtain a backgroundindependent and finite formulation of discrete quantum gravity. This has been achieved in three dimensions with the state sum of Turaev and Viro, an invariant of 3-manifolds.


blackhole A major goal of cosmology is to explain the present state of the universe in terms of processes that happened early on in its history. The Big Bang model provides an empirically tested explanation for the evolution of the universe from about one hundredth of a second after its creation through to the present day, some fifteen billion years later. However, despite its successes, many unanswered questions remain including: What was the origin of the Big Bang and the expansion of the universe? How did the density inhomogeneities arise which are needed to seed the formation of galaxies? Why does the vacuum energy (or cosmological constant) seem to be non-zero, causing the universe to accelerate today? Recent advances in fundamental theory may provide explanations for many of these outstanding problems, but these ideas are being confronted with a flood of new observational information. Cosmology is at an historic juncture because of this fruitful tension, with the new data providing important clues about the very early universe.

Particular attention is being focused by group members on competing theories for large-scale structure formation. One hypothesis, first proposed here, was that these were born as microscopic quantum fluctuations during an inflationary epoch when the universe underwent a period of exponential expansion. An alternative idea, also with pioneers in the group, is that we live in a brane world and that the Big Bang was a collision between branes. Quantitative predictions for these two classes of models are being actively studied and compared with improving observations.

Cosmic microwave sky

blackhole Rapid progress in accurately mapping the cosmic microwave background (CMB), relic radiation left over from the Big Bang, is the most exciting current development in cosmology; it offers a unique window on the early universe with which to test models for the origin of the universe and its structure. As well as refining theoretical predictions of CMB anisotropies (how the temperature of the sky varies in different directions), group members are actively involved in the world's most ambitious CMB experiment, the ESA Planck Surveyor satellite (right) due for launch in 2007; they are part of the Cambridge Planck Analysis Centre (CPAC), an interdisciplinary team responsible for the highest level data analysis in the experiment.

Inflation and the cosmological constant

Group members continue to tackle open questions about the origin and role of inflation in fundamental theory, such as Superstring/M-theory. A closely related issue is the apparent existence today of a cosmological constant, as inferred from recent supernovae evidence that the universe is accelerating. Inflationary scenarios come in many guises and group members are actively investigating means by which to distinguish between them using observational data, whether through the power spectrum of CMB anisotropies or other distinct (non-Gaussian) signatures.


blackhole Inspired by the realization that M-theory entails the existence of 'branes', surfaces of various dimensions to which matter is bound, a new area that may be described as 'braneworld cosmology' has opened up. From this perspective, the visible universe is a three dimensional object (called a brane) embedded in a higher dimensional spacetime (called the bulk). Group members have devoted much effort to understanding the detailed predictions of, and experimental constraints on, such 'brane world' scenarios, including the consequences for large-scale structure formation, late-time acceleration due to brane/bulk couplings, and the implications of brane collisions. As shown in the Ekpyrotic model developed here, collisions between brane worlds result in dramatic phenomena which may explain many cosmic conundra.The sudden release of thermal energy could account for the origin of the hot Big Bang and, as the branes draw together, quantum effects create a spectrum of ripples which could seed galaxy formation. A cyclic version of the model entails repeated collisions each followed by a late-time period of cosmic acceleration.

The observational implications of this and other braneworld scenarios, such as the colliding bubble model, are being vigorously pursued.

Other topics

Other research interests in cosmology being actively pursued are broad and varied, including:

  • The cosmological evolution and potential implications of cosmic strings and other topological defect networks.
  • The creation of the matter/antimatter asymmetry of the universe and the origin of dark matter, primordial magnetic fields and ultra-high energy cosmic rays.
  • Computer N-body simulations of galaxy formation, notably studying the influence of different types of dark matter on structure formation.
  • Constraints on models with varying fundamental constants using astrophysical data; recent work with quasar absorption lines indicates that there may have been small changes in the fine structure constant alpha over the age of the universe.

National Cosmology Supercomputer

blackhole COSMOS, the UK's first national cosmology supercomputer, is housed and operated by the Cambridge Relativity group within DAMTP, but it is owned by a consortium made up of researchers from around the UK. Since its inception in 1997, the fifth upgrade in 2003 marks more than a twenty fold increase in computational power, a reflection of its success in advancing the confrontation between accurate theoretical predictions and improving observations. As well as simulations of early universe physics and late-time galaxy formation, particular effort is being focused on the cosmic microwave sky, COSMOS being the CPAC development platform for the Planck Surveyor satellite. A combination of shared-memory ease-of-use and advanced visualization, places the group at the international forefront computationally.

Black Holes

Black holes are probably the most widely celebrated theoretical prediction of General Relativity. In fact the idea that such objects might exist predates Einstein by over 100 years. Only in this century, however, did the strange behaviour of such objects become fully appreciated. Today they have outgrown mere speculation and become a phenomenon we definitely expect to find in reality, perhaps lurking at the centres of galaxies.

It was originally thought that nothing, not even light, could escape from the gravity of a black hole (hence the word black). Here in Cambridge the physics of black holes was revolutionized when Stephen Hawking, applying quantum mechanics, realized that in fact a black hole must glow gently, slowly radiating its substance away into space at a rate which increases as the black hole shrinks. Hence black holes slowly evaporate. The quantum physics of black holes may also have cosmological implications.

Topological Defects

In understanding cosmology it is necessary to examine the consequences of certain proposed models of particle physics. Some of these contain the possibility of exotic objects, so-called topological defects, such as cosmic strings, which may have dramatic cosmological effects. Study of these may help us to narrow down the candidates for a correct model of particle behaviour.