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Einstein’s Theory of Relativity
With 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.
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
E  mc2
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.
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:
ds2  c2 dt 2  dx 2  dy 2  dz 2
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.
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 General Theory of Relativity can be stated mathematically as
G  8T
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.
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
Was Einstein right?
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.
Black Holes
Of 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 these
revolving bodies infer the existence of the central ones.
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.
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
In May 1994 NASA announced that the Hubble Space
unable to escape and so forms a black hole. The
Telescope (left) had “seen” a black hole at the centre of M87
boundary of this region is called the event horizon.
(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 Waves
One 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 (19192000) 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.
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.
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.
Left: An aerial view of GEO 600, the laser
interferometer gravitational wave detector currently
being built near Hannover, Germany. Mirrors at the
ends 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.
Computational Relativity
The 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.
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.
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 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.
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.
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 (CauchyCharacteristic 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.
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.