Einstein’s Theory of RelativityWith the formulation of the Special Theory of Relativity, Einstein started a scientific revolution that was to change our conception of both space and time. One of the cornerstones of Einstein's theory is the assumption that nothing can travel faster than the speed of light c (roughly 300 million m/s). Once one takes into account the finite velocity with which signals such as light travel the Newtonian concept of simultaneity is destroyed. This leads naturally to a new concept, first proposed by Herman Minkowski, in which the three dimensions of space and one dimension of time are combined into a new single entity: a four dimensional continuum called spacetime.

The concept of simultaneity is destroyed in Special Relativity. Because information cannot travel faster than the speed of light, it is more natural to discuss two events as being related through their location in the four dimensional space-time.

1905 can only be described as a fabulous year for the young Albert Einstein. While working at the Bern patent office he published three ground breaking research papers. The scope of these papers, concerning the photoelectric effect, Brownian motion and the formulation of Special Relativity,

respectively, was enormous.

The simple ideas underlying Special Relativity lead to predictions of new physics:

• The Lorentz-Fitzgerald contraction - the shortening of moving rods • Time dilation - the slowing down of moving clocks• A new composition law for velocities - which means that observers and material particles can only travel at a speed less than that of light• The equivalence of mass and energy through the famous equation

As far as mathematics is concerned, the simplest way to express these results is to model the four dimensional space-time as possessing a flat metric which encodes the invariant interval which exists between events:

The fact that the metric is flat means that the stationary curves (geodesics) are straight lines. Free particles and light rays travel along certain classes of these straight lines.

ds c dt dx dy dz2 2 2 2 2 2

E mc 2

Having formulated his Special theory, Einstein wanted to generalize it to incorporate the gravitational interaction. It took him ten years to complete this task. The final version of the theory was published in 1916. It is a relativistic theory of gravitation (i.e. one consistent with Special Relativity), known as General Relativity.

The key principle on which General Relativity is based derives from Galileo's experiments in which he dropped bodies of different composition from the leaning tower of Pisa. These experiments showed that all bodies fall with the same acceleration irrespective of their mass and composition. This observation leads to the equivalence principle.

The equivalence principle is best understood in the context of Einstein's lift thought experiments where, neglecting non-local effects, a body in a linearly accelerated rocket ship behaves in the same way as one on the Earth (experiencing the pull of gravitation). On the other hand, a body in an unaccelerated rocket ship behaves in the same way as one in free fall.

In the absence of gravitation we get back to Special Relativity and a flat metric and so, in order to incorporate gravitation into the theory, Einstein proposed that the spacetime should become curved. This means that the geodesics become curved as well, which results in free bodies no longer moving in straight lines when affected by gravity. The reason that a satellite (like the Earth) orbits a central body (like the Sun) in Newtonian theory is a combination of two effects: uniform motion in a straight line (Newton's first law) and gravitational attraction between the two bodies (i.e. the satellite "falls" under the attraction of the central body). In General Relativity, the reason that a satellite orbits a central body is that the central body "curves up" space (and, in fact, time as well) in its vicinity, and the satellite travels on the "straightest path" which is available to it, namely on a curved geodesic. One major difference between the two theories is that whereas Newtonian theory describes how things move (and it does so remarkably accurately for ordinary bodies), it does not really explain what is the cause. Einstein's theory neatly provides answers for both these questions.

The meaning of the Einstein equations can be summed up in the famous words of John Archibald Wheeler:

"space tells bodies how to move and bodies tell space how to curve"

General Relativity is concerned with studying the nature of Einstein’s equations and their solutions. Since few of the known exact solutions to Einstein's equations describe physically relevant situations, these studies are often based on approximations, such as post-Newtonian expansions or perturbation techniques, or numerical simulations.

One way to think of General Relativity is to use the idea of a "rubber sheet geometry" . In the absence of gravitation, the sheet is flat, but a central massive body curves up the sheet in its vicinity so that a free body (which would otherwise have moved in a straight line) is forced to orbit the central body

The General Theory of Relativity can be stated mathematically as

These are the so-called Einstein field equations. They correspond to 10 coupled highly nonlinear partial differential equations. Their solution gives rise to a curved spacetime metric from which one can obtain the geodesics and hence investigate such things as the motion of free particles and light rays.

G T 8

When Einstein proposed his field equations he believed they were far too complicated to allow explicit solutions to be found. Somewhat surprisingly, an exact solution was found within a year of his paper appearing in print. This solution, the Schwarzschild solution, describes a static, spherically symmetric vacuum spacetime. From this solution one can derive what are known as the four “classic tests” of the theory. These are:

• The advance of the perihelion of Mercury• Light bending in a gravitational field• The presence of a gravitational red shift• The time delay of a signal propagating in a gravitational field

These have now all been checked to an accuracy better than 1%. In fact to date, Einstein's theory has passed all experimental tests which have been proposed with flying colours.

That light is deflected as it passes by a massive object was first verified during a solar eclipse in 1919. This test of his theoretical prediction made Einstein an international celebrity. If we take the idea of gravitational light bending to the extreme we can see how black holes can arise in a curved spacetime. We can imagine an object that curves spacetime so much that it can force light rays to travel in circles, and so stop any information escaping from some enclosed region. On the right is a page from a letter from Einstein where he describes light bending by the Sun.

The gravitational deflection of light can sometimes lead to multiple images of distant quasars being observed. This is known as “gravitational lensing”. The image above shows the famous `Einstein cross', an instance where four images of the same quasar are seen surrounding the galaxy that causes the lensing.

Was Einstein right?

Black HolesOf all the conceptions of the human mind from unicorns to gargoyles to the hydrogen bomb perhaps the most fantastic is the black hole: a hole in space with a definite edge over which anything can fall and nothing can escape; a hole that curves space and warps time. (Kip Thorne 1974)

The idea of black holes can be traced back more than 200 years to the 27th of November 1783 when, in the rooms of the Royal Society of London, Henry Cavendish reported on a paper by his friend and colleague Reverend John Michell. The paper concerned potentially observable consequences of Newton's universal theory of gravitation. Michell's discussion was based on Newton's theory and the particle theory of light. Speculating that light particles ought to be attracted in the same way as all other bodies, Michell contemplated the possibility that there might exist stars much larger than the Sun, for which the escape velocity would exceed the speed of light. Michell suggested how these invisible objects may be detected:

... if any other luminous bodies should happen to revolve about them we might still perhaps from the motions of theserevolving bodies infer the existence of the central ones.

Not surprisingly, these propositions caused quite a stir among the fellows of the Royal Society. And why should they not? Not only had Michell introduced the revolutionary idea of`invisible stars, he also anticipated the way that these objects would be observed some two hundred years later.

The physics which lead to Michell's dark stars was based on Newton's law of gravity. However our current best description of gravity, Einstein's General Relativity, confirms the existence of black holes. As in the Newtonian case, if the star is sufficiently dense, it is surrounded by a region of spacetime from which light is unable to escape and so forms a black hole. The boundary of this region is called the event horizon.

In December 1974 Stephen Hawking and Kip Thorne made a famous bet on whether or not Cygnus X1 is a black hole.

The radius of the event horizon is known as the Schwarzschild radius. If we put in the appropriate numerical values we find that the Schwarszchild radius of the Sun is 3 km. In other words, if the Sun were compressed into a radius smaller than this, light could no longer escape from its surface and it would become a black hole. The Earth's Schwarzschild radius is a mere 9 mm!

A radio image of Cygnus A shows twin radio lobes stretching 160,000 light years from the centre. Consisting of beams of electrons travelling at near the speed of light, the jets slam into the intergalactic medium and spread out to form the radio lobes.The central engine fuelling this spectacular phenomenon is a gigantic black hole.

It is now generally believed that most galaxies (including our own) harbour huge black holes in their cores. The evidence for this continues to improve as astronomers gather more remarkable data on Active Galactic Nuclei.

In May 1994 NASA announced that the Hubble Space Telescope (left) had “seen” a black hole at the centre of M87 (right). The gas in the heart of M87 had been found to whirl rapidly around the very centre of the galaxy. From the observations one could deduce that the centre of M87 must hide an unseen mass of 2.4 billion solar masses.

The Search for Gravitational WavesOne of the most exciting predictions of Einstein's Theory of General Relativity is the existence of a new type of wave, known as a gravitational wave. Just as in electromagnetism, where accelerating charged particles emit electromagnetic radiation, so in General Relativity accelerating masses can emit gravitational radiation. General Relativity regards gravity as a curvature of spacetime, rather than as a force, so that these gravitational waves are sometimes described as ripples in the curvature of spacetime.

Although there has been no direct detection of gravitational waves, there does exist powerful indirect evidence to support their existence. General Relativity predicts that the gravitational waves emitted by two neutron stars orbiting each other should carry away energy and angular momentum from the system. Thus the two stars should slowly spiral in toward one another. Such a spiraling motion was measured by Russell Hulse and Joseph Taylor for the “binary pulsar” 1913+16. The rate of inspiral has been found to match the value predicted by General Relativity to better than 1%.. This was a spectacular success for General Relativity, and earned Hulse and Taylor the 1993 Nobel prize.

In principle, almost all motions of bodies will produce gravitational waves. However, it turns out that the strongest waves are produced not here on Earth, but instead by violent astrophysical events such as the explosion of a star or the collision of two black holes. Since the 1960s, experimenters have been developing gravitational wave detectors to observe these waves. A new generation of detectors - the first to have a realistic chance of detecting gravitational waves - is due to start collecting data in the next few years. By analyzing the exact form of the waves, we hope to improve our understanding of the distant astrophysical objects that produce them.

In order to understand how to detect gravitational waves, it is necessary to understand their effect on physical objects. This effect is best illustrated by considering a ring of test particles hit by a gravitational wave, with the wave travelling in a direction perpendicular to the plane of the ring. As the wave passes through the particle ring, the initially circular configuration will start oscillating. It is these periodic displacements that gravitational wave detectors aim to measure. The change of separation of the particles is minute, which makes its detection a highly non-trivial task.

Even though the prediction of gravitational waves dates back as far as the early twentieth century and the first papers on General Relativity, it was not until the sixties that the first serious attempts were made to detect them. In his pioneering work Joe Weber (1919-2000) initiated work on bar detectors - large metallic cylinders isolated from terrestrial seismic noise. A passing gravitational wave would tend to make the bar oscillate and these oscillations might then be recorded by sensitive electronic instruments.

A new generation of laser interferometer gravitational wave detectors is currently being developed. Each L-shaped detector consists of two arms, typically a few kilometres long, along which laser beams are reflected back and forth. By analysing the phase of the reflected light, tiny displacements of the test masses at the end of each interferometer arm can be detected. Even for these large instruments, the change in length will be of the order of one ten-thousandth of the diameter of an atomic nucleus. It is only the recent advance in laser technology that enables us to measure displacements of this microscopic size. Currently five such gravitational detectors are under construction, the UK -German GEO 600, the Japanese TAMA, the Franco-Italian VIRGO project, and two American LIGO detectors. The Japanese TAMA detector is already gathering data and the other interferometers are likely to go online within the next year.

Left: An aerial view of GEO 600, the laser interferometer gravitational wave detector currently being built near Hannover, Germany. Mirrors at theends of the two 600 m long arms of the L-shape monitor the change in arm-length as a gravitational wave passes through the instrument. Right: A further project currently under consideration by ESA and NASA is the Laser Interferometer Space Antenna or LISA, which is planned to consist of three spacecraft forming an equilateral triangle, with the whole configuration orbiting the Sun. LISA will complement the ground based detectors by operating with a much larger baseline of 5 million km, thus making accessible a different frequency range in the spectrum of gravitational waves.

The gravitational-wave sky: The sources that might be detected by the space based LISA mission, and a ground based detector (LIGO). Wave frequency is plotted on the horizontal axis and wave strength on the vertical. Sources that lie above the red curve could be detected by LISA and sources above the yellow could be detected by LIGO. Because of its larger size, LISA is sensitive to much lower frequencies than the ground based detectors.

Detection of gravitational waves from known possible sources would not only serve as confirmation of General Relativity, it could also provide a whole new window to the universe. As well as finding out new information about known astrophysical objects, gravitational wave astronomy could lead to discoveries of previously unknown objects far beyond our imagination.

Computational RelativityThe largest field of enquiry historically has been the field of exact solutions of Einstein's field equations. The Einstein field equations constitute an extremely complicated set of non-linear partial differential equations. It came as something of a surprise when Schwarzschild found an exact solution within a year of the field equations being published. In the ensuing years very few exact solutions were found until new invariant techniques were introduced in the mid 1960s. This led to an explosion of exact solutions being discovered.

Many of the calculations associated with exact solutions are straightforward but extremely long and complicated to perform which can easily lead to errors. Around the mid 1960s the field of Algebraic Computing in General Relativity came into existence and it soon became possible to undertake calculations with computers that would take more than a lifetime to complete by hand. An example (and one which has been used as a standard for comparing algebraic computing systems) is the calculation of the Ricci tensor for Bondi's radiating metric and was first undertaken by Ray d’Inverno using his algebraic computing system LAM (which eventually developed into the system SHEEP). The original hand calculation was undertaken over a period of some six months and now takes less than a second on a reasonable spec PC.

Part of the SHEEP output for the famous Bondi radiating metric. The user asks SHEEP to make and write (wmake) the line element which is called ds2 and then to make the Ricci tensor and the first 23 terms in the output are displayed above.

The advent of Algebraic Computing led to a new attack on the famous equivalence problem of General Relativity, namely: given two metrics, does there exist a local transformation which transforms one into the other? Cartan showed in some classic work that the problem can be solved but depends on computing the 10th covariant derivative of the Riemann tensor of each metric. Even with modern computer algebra systems this is out of the question. The work of Karlhede significantly improved the situation. Karlhede's approach provides an invariant classification of a metric. Thus, if two metrics have different classifications then they are necessarily inequivalent, whereas if they have the same classification then they are candidates for equivalence. The problem then reduces to solving four algebraic equations. Karlhede's algorithm reduces the derivative bounds significantly below the original 10 of Cartan. There is particular interest in the Southampton group in reducing the bounds which might occur in various cases, for if they can be reduced sufficiently then it becomes possible to use algebraic computing systems to classify exact solutions.

The advent of the Karlhede classification algorithm has lead to the setting up of a computer database of exact solutions through collaborative work between UERJ, Rio de Janeiro, Brazil and Southampton. At present some 200 metrics exist in the computer database of exact solutions. The ultimate hope is that it will contain all known solutions, fully documented and classified. Then any “newly” discovered solution can be compared with the contents of the database and, if indeed it is new, then the database can be updated accordingly. Were this to be fully realised then it would provide a valuable resource for the international community of relativists.

While several hundred exact solutions to the field equations are known to exist, few of these solutions describe physically relevant situations. If we want to solve the equations in scenarios of astrophysical interest, such as the birth of a neutron star in a supernova or the collision of two black holes in a binary system, we need to perform large scale numerical simulations. In the last ten years or so, there has been an international effort to develop Numerical Relativity (NR) to a level where simulations can provide reliable information about the most violent events in the Universe. It is only recently that computers have become powerful enough that one can hope to achieve the desired computational precision (in a reasonable computing time). After all, Einstein's equations are extremely complicated (they involve over 100,000 terms in the general case) and solving them numerically makes enormous demands on the processing power and memory of a computer.

One of the major problems in NR is that there is no known local expression for gravitational waves - they can only be properly described asymptotically (that is, out at infinity). In most previous work, numerical simulations are carried out on a central finite grid extending into the vacuum region surrounding the sources present and ad hoc conditions are imposed at the edge of the grid to prevent incoming waves (which would be unphysical). Unfortunately, these ad hoc conditions themselves generate spurious reflected numerical waves.

The basic formalism that the numerical codes employ is the Arnowitt-Deser-Misner 3+1 formalism which decomposes four dimensional spacetime into a family of constant time three dimensional spatial slices. Although this formalism is well adapted to central regions it does not work well in asymptotic regions. Instead, the d’Inverno-Stachel-Smallwood 2+2 formalism decomposes space-time into two families of two dimensional spacelike surfaces. The formalism encompasses 6 different cases, one of which is called the null-timelike case in which one of the families can be taken to form null 3-surfaces and the other to form timelike 3-surfaces. Null surfaces are especially important in General Relativity because they are ruled by null geodesics and these are the curves along which gravitational information is propagated. The idea behind the Southampton CCM (Cauchy-Characteristic Matching) project is to combine a central 3+1 numerical code with an exterior null-timelike 2+2 code connected across a timelike interface residing in the vacuum. In addition, the exterior region is compactified so as to incorporate null infinity where gravitational radiation can be unambiguously defined. The significance of this work is that it leads to wave forms asymptotically, and it is these exact templates which are needed in the search for gravitational waves.

A Teukolsky gravitational wave on a finite 3+1 grid (above) instead of dissipating off of the grid leads to a small spurious reflection (below). The Southampton CCM code provides a global code in which the wave reaches infinity unaltered and where it can be unambiguously characterised.

Results from the US Binary Black Hole Grand Challenge supercomputer simulations of the coalescence of two black holes. The merger of the two horizons lead to the famous “pair of pants” picture.

NASA supported Neutron Star Grand Challenge: the simulation of the merger of two Neutron stars emitting gravitational waves.