The binary system of two neutron stars discovered by Hulse and Taylor gives excellent indirect evidence for gravitational radiation, but might it be possible to detect this radiation directly? Physicists and astronomers are anxious to find out. If gravitational radiation is detected, it will give us a new way to learn about violent events in the universe.
Whenever a solar mass or more is accelerated to high speeds, it is possible for large amounts of gravitational radiation to be generated. It would happen, for example, when two neutron stars collide, as will eventually happen in the Hulse-Taylor binary system. It could also happen from when the core of a massive star collapses just prior to a supernova. Events like these are largely hidden from ordinary telescopes, because neutron stars are difficult to observe directly and the core of a massive star concealed by the star's opaque outer layers. If they could measure the gravitational radiation emitted from such cataclysmic events, astronomers would be able to prone within the cataclysm itself.
Unfortunately, even the most powerful bursts of gravitational radiation are very weak and hard to detect. To appreciate how weak gravitational waves are, imagine two electrons separated by a short distance. They each possess mass and charge, so these electrons exert both gravitational and electric forces on each other. If these two electrons are made to wiggle back and forth, they will radiate both gravitational and electromagnetic waves. But since the gravitational force between the electrons is only about 10 to the power of -42 times as strong as the electric force, the resulting gravitational waves carry only 10 to the power of -42 times as much energy as the electromagnetic waves.
Undaunted by the challenge, in the 1960s, Joseph Weber at the University of Maryland constructed gravitational-wave antennas by gluing sensitive crystals onto a large aluminum cylinder. Because gravitational waves are ripples in the geometry of space, the cylinder should vibrate slightly when a gravitational wave passes through it. If this should happen, the crystals would produce an electrical signal that can be amplified and recorded. While no confirmed detections of gravitational radiation have yet been made, several research groups around the world continue to develop gravitational-wave detectors based on Weber's pioneering principal.
Even more sensitive gravitational-wave antennas are now under construction. Instead of an aluminum cylinder, these use lasers to look for tiny vibrations of mirrors. The LIGO project (short for Laser Interferometer Gravitational-wave Observatory) uses two antennas thousands of kilometers apart, one near Baton Rouge, Louisiana, and the other near Richland, Washington. Each consists of an L-shaped vacuum pipe with arms 4 km long for the laser beams. Mirrors mounted at the ends of the pipes reflect the laser beams back and forth. If a gravitational wave passes by, the mirrors should move by an infinitesimally small but measurable distance, changing the length of the light path. These changes will be detected by combining the laser beams from the two arms of the antenna. By having two antennas located thousands of kilometers apart, the LIGO scientists will be able to distinguish gravitational waves coming from space (which will be detected by both antennas) from false signals, such as seismic activity (which will be detected by only one antenna).
When LIGO becomes fully operational after the year 2000, it should be able to detect bursts of gravitational waves from colliding neutron stars or the collapsing cores of supernovae as far away as 2 x 10 to the power of 7 parsecs (about 7 x 10 to the power of 7 light years). This volume of space is so huge and contains so many galaxies that astronomers might be able to detect a burst of gravitational waves as frequently as once a year. A similar gravitational-wave observatory, called VIRGO, is under construction outside Pisa, Italy, by a French-Italian consortium.
The European Space Agency has proposed an even more ambitious gravitational-wave antenna, called LISA (Laser Interferometer Space Antenna). Six spacecraft would orbit the Sun together in a ring arrangement millions of kilometers in radius. Laser beams shone from one spacecraft to the next would continuously monitor the distances between the spacecraft. And passing gravitational wave would disturb these distances. Thanks to its great size, LISA would be much more sensitive than LIGO or VIRGO and would be unaffected by seismic disturbances that plague any Earth-based gravitational wave antenna. While LISA has not yet been funded, it may be put into orbit by 2020.
With such remarkable new instruments becoming available, the first decades of the twenty-first century will hopefully see the dawn of the new field of observational gravitational-wave astronomy.
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 allows one 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 the observed 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 completely random state. In some regions matter will be more energetic than in others and inflation could ensue, producing the observable universe.
In non-gravitational physics the approach to quantum theory that has proved most successful involves 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 a contribution 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, 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 spatial dimensions 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 four spatial 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. This spatial 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 to describe 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 for non-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 quantum gravity 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 path integral 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 in four 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 as string/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 data one 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.
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 three geometry 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 to calculate 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 within the 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
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, matter which 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 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 open universe via this bubble nucleation is known as a Coleman-De Luccia instanton.
Remember that this scenario requires the existence of a false vacuum for the matter in the early universe. 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 these specific 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 is that 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.
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 every quantum 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 temperature perturbations 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 be launched 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 observationally distinguish between different theories for quantum cosmology.
Lorentz transformation for time
T = time interval
measured by an observer moving relative to the phenomenon
Lorentz transformation for length
L = length of a moving object along the direction of motion
The Schwarzchild Radius
= Schwarzchild radius of a black hole
When using this formula,
be careful to express the mass in kilograms, not solar masses. That is
because of the units in which G and c are commonly expressed:
EXAMPLE: Consider a black hole of mass . One solar mass is 1.99 X 10kg, so in this case M = 10 X 1.99 X 10kg = 1.99 X 10kg. The Schwarzschild radius of this black hole is then
One kilometre is 10 metres, so the Schwarzschild Radius of this black hole is 30km.
General relativity is so rich, complex, and fascinating - and so may be the geometry of black holes. For example, in the 1930s, Einstein and his colleague, Nathan Rosen, discovered that a black hole could possibly connect our universe with a second domain of space and time that is separate from ours. The first diagram shows this connection, called an Einstein-Rosen bridge. You could think of an upper surface as our universe and the lower surface as a "parallel universe". Alternatively, the upper and lower surfaces could be different regions of our own universe. In that case, and Einstein-Rosen bridge would connect our universe with itself, forming a wormhole, shown in the diagram above.
Scientists and science-fiction authors have speculated that a wormhole could be a shortcut to distant places in our universe, but detailed calculations reveal a major obstacle. The powerful gravity of a black hole causes the wormhole to collapse almost as soon as it forms. As a result, to get from one side of a wormhole to the other, you would have to travel faster than the speed of light, which is not possible.
Caltech physicist Kip Thorne, Michael Morris, and Ulvi Yurtsever have proposed a scheme that might get around the difficulty of a collapsing wormhole. According to general relativity, they point out, pressure as well as mass can be a source of gravity. Normally we do not see the gravitational effects of pressure because they are too small. Thorne and his colleagues speculate that a technologically advanced civilization might someday be able to use pressure to produce antigravity strong enough to keep the wormhole open.
If a wormhole could be held open, it could also be a "time machine." To see how, imagine you take one end of a wormhole and move it around for a while at speeds very near the speed of light. As discussed in Box 24-1, such motion causes clocks to slow down. Thus, when you stop moving that end of the wormhole, you find that it has not aged as much as the stationary end. In other words, one side of the wormhole has a different time than the other. As a result, you could go into one end of a wormhole at a late time and come out at an early time. For example, you might go in 10 A.M. and come out at 9A.M.
Time machines are
illogical. If you could get back from a trip an hour before you left,
you could meet yourself and tell yourself what a nice journey you had.
Then both of you could take the trip. If you and your twin just return
before you both left, there would be four of you. And all four of you
could take the trip again. And then all eight of you. Then all sixteen
We have never seen a violation of causality. To the contrary, the universe seems remarkably rational. Scientists would therefore like to show that time machines couldn't exist. British astrophysicist Stephen Hawking points to one strong bit of observational evidence against time machines: We are not visited by hordes of tourists from the future. If we could discover why nature precludes time machines, we would have a much deeper understanding of the nature of space and time.
The axiom of fusion allows us to imagine a unit interval as a set of
infinitesimals, with each infinitesimal containing N1 figments (elements
which cannot be accessed by the axiom of choice) in it. We consider these
infinitesimals as integral unites which cannot be broken up any further.
To facilitate the discussion, in addition to Dedekind cuts, we will use
also the concept of a Dedekind knife, and assume that the knife can cut
any interval given to it, exactly in the middle. From this it follows
that every infinite recursive subset of positive integers, or equivalently,
a binary number in the unit interval, represents the use of Dedekind knife
an infinite number of times. The result we get when we use the knife N0
times, according to an infinite binary sequence, is what we call an infinitesimal
and the location of that infinitesimal is what we call a number.
Having stated this, we continue our second infinite cutting, this time,
without bothering to restrict ourselves to recursive subsets of positive
integers as in the original case. The justification for this is that our
operation is in the realm of the imaginable and not physical. The result
of the cutting is N1 figments and they are to be considered only as a
figment of our imagination. These arguments, of course, do not prove the
consistency of the axiom of fusion, but hopefully makes it plausible.