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In physics, a gravitational wave is a fluctuation in the curvature of space-time which propagates as a wave. Gravitational radiation results when gravitational waves are emitted from some moving object or system of objects. Important examples of systems which emit gravitational waves are binary star systems, where the two stars in the binary are white dwarfs, neutron stars, or black holes. (Gravitational waves are sometimes called gravity waves, but this term should be reserved for a completely different kind of wave encountered in hydrodynamics.) Introduction In Einstein's theory of General relativity, the force of gravity is due to spacetime curvature. This curvature is caused by the presence of massive objects. Roughly speaking, the more massive the object, the greater the curvature it causes, and hence the more intense the gravity. The rapid motion of mass in some region will generate changes in the curvature of spacetime, which will generate ripples in the spacetime radiating outward as gravitational waves. For waves with small amplitudes, gravitational radiation is believed to travel at the speed of light—much like electromagnetic radiation. However, while electromagnetic waves are 'spin-1' particles, gravitational waves are 'spin-2'. Electromagnetic waves are associated with a massless particle called the photon. Attempts to create an analogous quantum field theory for general relativity led to an analogous concept: a massless particle called the graviton. However, quantum field theory calculations involving gravitons produce many infinite values, which cannot be readily canceled to yield a sensible finite result. (In technical terms, gravity is nonrenormalizable.) Some proposed quantum gravity theories (notably string theory) attempt to address this problem, but currently there is no known means of testing these ideas empirically. The graviton itself (if it exists) is unlikely to be easily detectable, due to the weakness of its interactions. Gravitational waves are very weak. The strongest gravitational waves we can expect to observe on Earth would be generated by very distant and ancient events in which a great deal of energy moved very violently (examples include the collision of two neutron stars, or the collision of two black holes). Such a wave should cause relative changes in distance everywhere on Earth, but these changes should be on the order of at most one part in 1021. In the case of the 4 kilometer arms of the LIGO gravitational wave detector, this is roughly one thousandth of the "diameter" of a proton. Hence, it has proven very difficult to detect even the strongest gravitational waves. The existence and indeed ubiquity of gravitational waves is an unambiguous prediction of Einstein's theory of General relativity. All competing theories of gravitation currently thought to be viable (apparently in agreement to the level of accuracy with all available evidence) feature predictions about the nature of gravitational radiation. In principle, these predictions are sometimes significantly different from those of general relativity, but unfortunately, at present it seems to be sufficiently challenging simply to directly confirm the existence of gravitational radiation, much less study its detailed properties. Although gravitational radiation has not yet been unambiguously and directly detected, there is already significant indirect evidence for its existence. Most notably, observations of the binary pulsar PSR B1913+16, which is thought to consist of two neutron stars orbiting rather tightly and rapidly around each other, have revealed a gradual inspiral at almost exactly the rate which would be predicted by general relativity. According to general relativity, this system should emit gravitational radiation which carries off energy at a specific rate, which should in turn cause the orbit to decay at a rate of roughly 7 mm per day. The simplest (and almost universally accepted) explanation for these observations is that general relativity must give an accurate account of gravitational radiation in such systems. Joseph H. Taylor Jr. and Russell A. Hulse shared the Nobel Prize in Physics in 1993 for this work. The effects of a passing gravitational wave
Sources of gravitational waves Gravitational waves are caused by certain motions of mass or energy. The type of motion required is different from electromagnetism in one very important respect however: the strongest type of electromagnetic radiation is typically dipole radiation, while the strongest type of gravitational radiation is typically quadrupole radiation. Generally, the radiation emitted is proportional to the -th time derivative of the -th multipole, which is usually largest for low values of . The strength of the wave will always die out as , where is the distance from the source to the observer. According to general relativity, the quadrupole moment (or some higher moment) of an isolated system must be time-varying in order for it to emit gravitational radiation. The simplest example is the "spinning dumbbell", tumbling end-over-end (as opposed to spinning around its long axis). The heavier the mass, and the faster its tumbling, the greater the gravitational radiation it will give off. If we imagine the two weights of the dumbbell to be massive stars like neutron stars or, black holes, orbiting each other quickly, then significant amounts of gravitational radiation would be given off. Here are some examples which illustrate when we should and should not (assuming general relativity gives accurate predictions) expect a system to emit gravitational radiation: More quantitatively, we can write the two components of gravitational radiation from a simple, "quasi-Keplerian" orbit. That is, assume that a pair of objects (planets or stars or black holes) is orbiting in the - plane in a simple circular motion. Assume that the objects are a distance away from each other, have mass and , and an angular frequency of . Assume further that the observer is a distance away, located at an angle and in the usual spherical coordinates. Then, the perturbation to the metric (roughly, the fractional change in length of anything) will be ight) , ight) . Note that systems like the Earth and the Moon would give off some gravitational radiation, but the effect would be unmeasurably small because the Earth and Moon are not very massive, and they do not move very quickly compared to the speed of light. If we use quantities from the Earth-Moon system, we have Now, the distance from the system must be much greater than both the size of the system, and the reduced wavelength of the emitted waves, for the above equations to hold. For the Earth-Moon system, the wavelength is roughly one-half a light-month, and the reduced wavelength being that divided by , or about 5.8 This means that objects near the Earth-Moon system will be squeezed and stretched by just %. Thus, we see explicitly that the gravitational radiation given off by the Earth-Moon system is truly tiny. The examples above (and others) are most commonly studied using a simplified version of general relativity, sometimes called linearized general relativity, which gives indistinguishable results in the case of weak gravitational fields. (The external field of our Sun would be considered "weak" in this terminology.) Similar conclusions hold for the fully nonlinear theory, but it is much more difficult to obtain analytic results outside the domain of the linearized theory. This is one reason why so much work on phenomena such as the collision and merger of two black holes currently requires numerical analysis. Gravitational radiation carries energy away from a radiating system. Consequently, in the case of the quasi-Keplerian system discussed above, the two objects will gradually spiral in towards one another, becoming more tightly bound to compensate for this loss of energy. The predicted rate of this in-spiral can also be computed, using the linearized approximation, and the result gives excellent agreement for observed binary pulsars (this is the theoretical basis for the Nobel Prize awarded to Hulse and Taylor). In the late stages of the inspiral of two neutron stars or black holes, however, the linearized theory is no longer adequate, so one must resort to more complicated approximations, and eventually to numerical simulations. Similarly, in the case of the eccentric rotating rod, the frequency will decrease as the radiation gradually carries off energy from the system. We stress that some theories of gravitation give significantly different predictions concerning the nature and generation of gravitational radiation, while others give predictions which are almost identical to those of general relativity. All currently known theories other than general relativity are either in disagreement with observation, or in some sense more complicated than general relativity (see for example Brans-Dicke theory for an example illustrating the latter possibility). If two spinning black holes were to collide, they could emit an enormous amount of gravitational radiation and lose energy in the process. Gravitational wave detectors As noted above, the 1993 Nobel Prize in Physics was awarded for observations of a remarkable binary pulsar, the Hulse-Taylor binary. These observations appear to confirm that gravitational radiation is being given off by this binary pulsar. To directly detect gravitational waves, however, you would have to look for any motion they cause. Typically you would look for the expansion and contraction oscillations caused by the gravitational wave. A simple device to detect this motion is called a Weber bar—a large, solid piece of metal with electronics attached to detect any vibrations. This type of instrument was the first type of gravitational wave detector. Modern forms of the Weber bar are still operated. Unfortunately, Weber bars are not likely to be sensitive enough to detect anything but very powerful gravitational waves. A more sensitive version is the interferometer, with test masses placed many kilometers apart, and the distance between them measured by interferometry. Ground-based interferometers such as LIGO are now coming on line, with test masses placed several hundred meters to four kilometers apart. The motion to be detected would be very slight—a small fraction of the width of an atom. A number of teams are working on making more sensitive and selective gravitational wave detectors and analysing their results. Space-based interferometers, such as LISA, are also being developed. LISA's design calls for test masses to be placed five million kilometers apart, in separate spacecraft with lasers running between them. One reason for the lack of direct detection so far is that the gravitational waves that we expect to be produced in nature are very weak, so that the signals for gravitational waves, if they exist, are buried under noise generated from other sources. At low frequencies, for example, vibrations of the ground overpower motions due to gravitational waves. At higher frequencies, photon shot noise is the dominant noise source. Researchers are expending great effort to reduce these noises to a level such that gravitational waves are detectable. There are other prospects such as MiniGRAIL, a spherical gravitational wave antenna based at Leiden University. Some scientists even want to use the moon as a giant gravitational wave detector. The moon should be somewhat pliable to the contortions caused by gravitational waves, and the hope is that the motion of the moon caused by these waves will be detectable. Einstein@Home Bruce Allen of University of Wisconsin-Milwaukee's LIGO Scientific Collaboration (LSC) group is leading the development of the Einstein@Home project, developed to search data for signals coming from selected, extremely dense, rapidly rotating stars observed from LIGO in the US and the GEO 600 gravitational wave observatory in Germany. Such sources are believed to be either quark stars or neutron stars; a subclass of these stars are already observed by conventional means and are known as pulsars, electromagnetic wave-emitting celestial bodies. If some of these stars are not quite near-perfectly spherical, they should emit gravitational waves, which LIGO and GEO 600 may begin to detect. Einstein@Home is a small part of the LSC scientific program. It has been set up and released as a distributed computing project similar to SETI@home. That is, it relies on computer time donated by private computer users to process data generated by LIGO's and GEO 600's search for gravity waves. Prospects Scientists are eager to directly measure gravitational waves from astronomical sources, as they can probe phenomena that are difficult or impossible to study with electromagnetic radiation. For instance, although a black hole emits no visible radiation in the way that a regular star does, gravitational waves can be emitted when an object falls into a black hole, or when two black holes collide. If the inspiraling mass is significantly smaller than the central black hole, the emitted gravitational waves may, at least in some circumstances, allow physicists to directly probe the spacetime geometry around the event horizon (such observations are a primary goal of the LISA mission). Also, because gravitational waves are disturbances of spacetime itself, objects opaque to light are often transparent to gravitational radiation. In particular, gravitational waves could propagate while the universe was still opaque to light (i.e., at times before recombination). In this way, gravitational waves could help reveal information about the very structure of the universe. In contrast to electromagnetic radiation, it is not fully understood what difference the presence of gravitational radiation would make for the workings of the universe. A sufficiently strong sea of primordial gravitational radiation, with an energy density exceeding that of the big bang electromagnetic radiation by a few orders of magnitude, would shorten the life of the universe, violating existing data that show it is at least 13 billion years old. More promising is the hope to detect waves emitted by sources on astronomic size scales, such as: Perturbation of Flat Space-time Consider that the full metric is nearly the flat metric plus some small perturbation . The Einstein equation in vacuum is where is the Ricci curvature. We will expand in perturbatively in powers of . The zeroth order term can only be a function of the flat metric and therefore is identically zero. As the perturbation is to be small, we will solve only for the first order term and ignore all higher orders. where is the deviation from the flat (and thus zero) Ricci curvature that depends linearly on the perturbation . Now we need the formula for the Ricci curvature. where are the Christoffel symbols and is shorthand for . Only first two terms which are linear in will contribute to the first order correction. Next we need the formula for the Christoffel symbols. ight) Seeing as the flat metric is constant, the only first order terms will involve derivatives of the perturbation. ight) The linearized Einstein equation now becomes ight) where substitutes the expression and is the d'Alembertian or 4-Laplacian. Raising and lowering indices can be tricky. To first order you only use the flat metric. Also note the inverse metric has a negative perturbation plus higher order terms. Next we choose a particular coordinate system where is identically zero. Some proof is necessary to make sure this is possible, but it is. We are left with a wave equation and our gauge condition. From experience with simpler wave equations we can guess the general form of the solution. Where is a null vector. The wave equation is now satisfied, but what choices of will satisfy the gauge condition we used. If we don't want transformations to disturb our choice of gauge, then we better make the wave traceless, , and transverse, . For a wave traveling in the direction, , the perturbation will take the following form. Thus the oscillations are transverse spacial distortions. The two polarizations are known as the "plus" and "cross" polarizations from the distortion patterns they produce, and their amplitudes are and respectively. Perturbation with Sources The Einstein equation is the relationship between space-time curvature and the matter or source of that curvature. Our first order contributions to the curvature have previously been determined. We stick to the choice of Lorentz gauge, which will now be written in a very suggestive form. ight) The right hand side is the divergence of the trace-reversed . The traced reversed perturbation will be abbreviated as from now on. We can now combine these equations into the linearized Einstein equation. This is a long solved problem from electricity and magnetism analogous to electromagnetic waves with sources. It is solved via retarded Green's functions. ight) = rac int d^3y rac Where is the retarded time. Far from Source Approximation If we want to study metric perturbations far from the source then we can envoke a very useful approximation. ight) approx rac rac int d^3y T^left(t - rac,vec ight) Where is the approximate distance to the source. We now invoke the local conservation of energy-momentum (to first order) to find useful interrelationships in the stress-energy tensor. abla cdot mathbf = 0 We now take this relationship and massage it into the form of our original integral and see what new information it gives us. We wanted to multiply the right hand side with the two powers of so that we can integrate by parts twice and get down to a regular volume integral. Assuming the stress-energy tensor takes the simple form ho u_alpha u_eta Where is the mass density and is the 4-velocity. If the source is nonrelativistic, then the energy density will be dominated by the mass density, ho = 2 int d^3x T^ Here we see something very similar to the moment of inertia, we call it the second mass moment. holeft(t,vec ight) We now have our final expression that relates the gravitational waves with their source. ight) approx rac rac rac I^ left(t,vec ight) Gravitational waves transmit energy Within parts of the scientific community there was initially some confusion as to whether gravitational waves could transmit energy as electromagnetic waves can. This confusion came from the fact that gravitational waves have no local energy density - no contribution to the stress-energy tensor. Unlike Newtonian gravity, Einstein gravity is not a force theory. Gravity is not a force in general relativity; it is geometry. Therefore the gravitational field was thought not to contain energy, as would a gravitational potential. But the field can most certainly carry energy as it can do mechanical work at a distance. For a number of years this issue was addressed by using gravitational stress-energy pseudotensors that transport energy. Among a number of candidates a frequently used one is the symmetric Landau-Lifshitz pseudotensor. The analysis based on pseudotensors is subject to the general criticism that they are not proper tensors and thus physical conclusions based on them may not be coordinate independent. Since the components of a pseudotensor can vanish in a coordinate system but not in others, the gravitational energy density is not localizable. In physical terms this is simply a reflection of the fact that gravitational field is locally transformed away for a freely falling observer because of the Equivalence Principle. A major conceptual advance came when in 1962 Hermann Bondi and his coworkers analyzed gravitational radiation in Einstein gravity using a specially devised coordinate system which showed explicitly how radiation can carry energy out to infinity from an isolated source causing it to lose mass. The work by Bondi et al., Sachs and by Newman and Penrose form the basis of much of the current theoretical understanding of the structure of gravitational radiation field far from the sources. See also | |||||||||
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