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In physics, in terms of a metric theory of gravitation, 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 object or system of gravitating objects.

(Gravitational waves are sometimes called gravity waves, but this term should be reserved for a completely different kind of wave encountered in hydrodynamics.)

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 super massive 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 arms of the LIGO gravitational wave detector, this is less than one thousandth of the "diameter" of a proton. This should give some indication of why 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 gravitation theories 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 PSR1913+16, which is thought to consist of two neutron stars orbiting rather tightly and rapidly around each other, have revealed a gradual in spiral at exactly the rate which would be predicted by general relativity. 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.

## Overview

In Einstein's theory of General Relativity, gravitation is, essentially, identified with spacetime curvature. In the famous slogan promulgated by John Archibald Wheeler, matter tells spacetime how to curve, and spacetime tells matter how to move. For example, humans feel the ground pressing against their feet (or behind, according to stance). From the viewpoint of general relativity, this means that contact with the ground is preventing them from falling freely, thereby accelerating them. Since acceleration is identified with bending of world lines, this means that the world line of a human who is not in freefall is not a geodesic. On the other hand, far from any mass-energy, spacetime is almost perfectly flat, so geodesics behave much like straight lines in familiar solid geometry, so small objects can exhibit rectilinear inertial motion.

General relativity (and similar theories of gravitation) are expressed by writing down a field equation (and possibly an equation of motion, if this is not implicit from the field equation, as it is, remarkably, in the case of general relativity). That is, these are classical relativistic field theories in which the gravitational field is at least partially identified with spacetime curvature. As a more or less inevitable consequence, in these theories, the rapid motion of mass-energy in some region will generate ripples in spacetime itself which radiate outward as gravitational waves. Indeed, this is in a sense how "field updating information" is typically conveyed from place to place.

Like electromagnetic radiation, in general relativity (and many other theories), gravitational radiation travels at the speed of light and is transverse (meaning that the major effects of a gravitational wave on the motion of test particles occurs in a plane orthogonal to the direction of propagation). However (roughly speaking):

• gravitational waves represent perturbations in a second rank tensor field (and, borrowing a term from quantum field theory, are said to spin-two),
• electromagnetic waves result from perturbations in a vector field (and are said to be spin-one).
• various other waves treated in physics result from perturbations in a scalar field (and are said to be spin-zero).

In electromagnetism, certain motions of charged particles, like electrons, will radiate electromagnetic waves. Analogously in gravity, certain motions of mass or energy will radiate gravitational waves. In the quantum field theory which arises from classical electromagnetism, called quantum electrodynamics, there is a massless particle associated with electromagnetic radiation, called the photon. Attempts to create an analogous quantum field theory for classical gravity (general relativity) led to an analogous concept, a massless particle called the graviton. However, it turns out that this route to quantizing general relativity ultimately fails, which renders the possible role of the graviton somewhat problematic in gravitational physics. Probably it is best to think of this notion as one which arises in an approximation and which has some virtues, but which is probably not as the notion of the photon.

## The Nature of Gravitational Waves

Gravitational waves represent fluctatations in the metric of space-time. That is, they alter the relative distance between test particles. It follows that to directly detect a gravitational wave, you should in essence look for tiny relative motions between two objects. In the case of the LIGO detectors, this is essentially relative motion between two suspended mirrors, and as we saw above the motion to be detected is far smaller than the size of an atom, in fact smaller than the "size" of an atomic nucleus. Since thermal motion in each mirror is far larger than this, understanding why anyone would expect LIGO to work takes some explaining! (See LIGO.)

Imagine a perfect flat region of spacetime, with a bunch of mutually motionless test particles. Along comes a monochromatic linearly polarized gravitational wave. What happens to the test particles? Roughly speaking, they will oscillate in a cruciform manner, orthogonal to the direction of motion:

• first, East/West separated particles draw together while North/South separated particles draw apart,
• next, East/West separated particles draw apart while North/South separated particles draw together,

and so forth. (Diagonally separated particles exhibit a relative motion which is more difficult to describe verbally, but which is more or less implied by this description.) The cross-sectional of a small box of test particles is invariant under these changes, and there is neglible motion in the direction of propagation (at least, neglecting gravitomagnetic effects; that is, we are tacitly assuming that the relative motion of our test particles is not very rapid).

A monochromatic circularly polarized induces similar cruciform oscillation, except that the crucifix rotates with the same frequency as the frequency of cruciform oscillation.

Interestingly enough, after the wave has passed, there may be some residual "secular" relative motion of the test particles. There are also some interesting optical effects. If, before the wave arrives, we look through the oncoming wavefronts at objects behind these wavefronts, we can see no optical distortion (if we could, of course, we would have advance notice of its impending arrival, in violation of the principle of causality). But if, after the wave has passed by, we turn and look through the departing wavefronts at objects which the wave has not yet reached, we will see optical distortions in the images of small shapes such as galaxies. Unfortunately, this is an utterly impractical method of detecting the very weak waves we can expect to occur in the vicinity of the solar system.

## Sources of Gravitational Waves

Gravitational waves are caused by certain motions of mass or energy. The type of motion required is different from electromagnetism where any accelerating charge will radiate electromagnetic radiation. [1]

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. Here are some examples which illustrate when we should (assuming general relativity gives accurate predictions) expect a system to emit gravitational radiation:

• An isolated object in approximately "rectilinear" motion will not radiate. (Needless to say, this motion is wrt some observer and can be only approximately rectilinear. Technically, this entails defining a weakly gravitating system possessing a time varying dipole moment but stationary quadrupole moment, with all moments being taken with respect to the origin.) This can be regarded as a consequence of the principle of conservation of linear momentum. (Caveat: this example is trickier than it looks, and in the case of a small object falling toward a large one, say, it leads to one of the most vexed questions in general relativity, the problem of treating radiation reaction).
• A spherically pulsating spherical star (nonzero and non-stationary monopole moment or mass, but vanishing and hence stationary quadrupole moment) will not radiate, in agreement with Birkhoff's theorem.
• A spinning disk (nonzero but stationary monopole and quadrupole moments) will not radiate. This can be regarded as a consequence of the principle of conservation of angular momentum. (Caveat: in general relativity, unlike Newtonian gravitation, a spinning disk will not generate an external field identical to the field of an equivalent but non-spinning disk, due to gravitomagnetic effects, but this does not contradict the absence of radiation. Roughly speaking, the field is generated as we concentrate matter, and if that matter has some angular momentum, but we end up with a stationary external gravitational field, that field will exhibit gravitomagnetism but not radiation.)
• Two objects mounted on the endpoints of an isolated extensible curtain rod, which is provided with some kind of engine and which oscillates long/short/long with frequency ω, gives a system with time-varying quadrupole moment, so this system will radiate. Observers far from the rod and in the equatorial plane of the rod will observe linearly polarized radiation (aligned with the rod) with frequency ω. Observers lying on the axis of symmetry of the rod will observe no radiation, however.
• A spinning non-axisymmetric planetoid (say with a large bump or dimple on the equator) will define a system with a time-varying quadrupole moment, so this system will radiate. As an idealization, one can study an isolated uniform mass curtain rod which is spinning with angular frequency ω about a rotation axis orthogonal to the rod, but passing through some point other than the centroid of the rod. This gives a system with time varying quadrupole moment, so the system will radiate. Observers far from the system and lying in the plane of rotation will observe linearly polarized radiation with frequency . Observers far from the system and near its axis of symmetry will observe circularly polarized radiation with frequency ω.
• Two objects orbiting each other with angular frequency ω in a quasi-Keplerian planar orbit, gives a system with time-varying quadrupole moment, so this system will radiate. Observers far from the system and in its equatorial plane will observe linearly polarized radiation (aligned with the rod) with frequency . Observers far from the system and lying on its axis of symmetry will observe circularly polarized radiation with frequency ω.

The last three examples illustrate a general rule-of-thumb: far from a radiatiating system, projection of the system on the "viewing plane" affords a rough and ready indication of what kind of radiation will be observed.

These examples (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 inspiral 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 result 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).

### Examples

• Our solar system can not radiate very much gravitational radiation because the planets are mostly on the same plane of orbit with spins aligned to that plane. And even if this were not the case, the gravitational forces in our solar system are very weak with the exception of Mercury and the Sun.
• If two spinning black holes were to collide, they could emit an enormous amount of gravitational radiation and lose energy in the process.

## Detection

As mentioned above, observations of the binary pulsar PSR J0737-3039 appeared to confirm predictions of general relativity with respect to energy emitted by gravitational waves, with the system's orbit observed to shrink 7 mm per day. This further confirms predictions made by Russell Alan Hulse and Joseph Hooton Taylor Jr. observing binary pulsar PSR B1913+16, for the discovery and analysis of which they were awarded the Nobel Prize in Physics in 1993.

But to directly detect gravitational waves 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 version of this setup is called a Weber bar -- a large, solid piece of metal with electronics attached to detect any vibrations. 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 as many as four kilometers apart. Ground-based interferometers such as LIGO are now coming on line. The motion to be detected would be very slight -- a small fraction of the width of an atom, over a distance of four kilometers. Space-based interferometers, such as LISA are also being developed.

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. Reportedly, ordinary terrestrial sources would be undetectable, despite their closeness, because of the great relative weakness of the gravitational force.

Gravitational radiation has not been directly observed, although there are a number of existing and proposed experiments such as LIGO that intend to do so. A number of teams are working on making more sensitive and selective gravitational wave detectors and analysing their results. A commonly used technique to reduce the effects of noise is to use coincidence detection to filter out events that do not register on both detectors. There are two common types of detectors used in these experiments:

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.

In November 2002, a team of Italian researchers at the Instituto Nazionale di Fisica Nucleare and the University of Rome La Sapienza produced an analysis of their experimental results that may be further indirect evidence of the existence of gravitational waves. Their paper, entitled "Study of the coincidences between the gravitational wave detectors EXPLORER and NAUTILUS in 2001", is based on a statistical analysis of the results from their detectors which shows that the number of coincident detections is greatest when both of their detectors are pointing into the center of our Milky Way galaxy.

### Einstein@Home

Bruce Allen of UWM'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 so weak (and thus difficult to detect), 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:

## Derivation

### Perturbation off Flat Space-time

Consider that the full metric g is nearly the flat metric η plus some small perturbation h.

gμν = ημν + hμν

The Einstein equation in vacuum is

$R_{\mu \nu} = \mathbf{0}$

Where R is the Ricci curvature. We will expand R in perturbatively in powers of h.

$R_{\mu \nu} = \mathbf{0} + \delta R_{\mu \nu} + \delta^2 R_{\mu \nu} + \cdots$

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.

$R_{\mu \nu} = \delta R_{\mu \nu} + \mathbf{0}$

Where δRμν is the deviation from the flat (and thus zero) Ricci curvature that depends linearly on the perturbation h.

Now we need the formula for the Ricci curvature.

$R_{\mu \nu} = \partial_{\alpha} \Gamma_{\mu \nu}^\alpha - \partial_{\nu} \Gamma_{\mu \alpha}^\alpha + \Gamma_{\mu \nu}^\alpha \Gamma_{\alpha \beta}^\beta - \Gamma_{\mu \beta}^\alpha \Gamma_{\nu \alpha}^\beta$

Where Γ are the Christoffel symbols and $\partial_{\alpha}$ is shorthand for $\frac{\partial}{\partial x^{\alpha}}$. Only first two terms which are linear in Γ will contribute to the first order correction.

$\delta R_{\mu \nu} = \partial_{\alpha} \delta \Gamma_{\mu \nu}^\alpha - \partial_{\nu} \delta \Gamma_{\mu \alpha}^\alpha$

Next we need the formula for the Christoffel symbols.

$\Gamma^\alpha_{\mu \nu} = \frac{1}{2} g^{\alpha \gamma} \left( \partial_{\nu} g_{\gamma \mu} + \partial_{\mu} g_{\gamma \nu} - \partial_{\gamma} g_{\mu \nu} \right)$

Seeing as the flat metric is constant, the only first order terms will involve derivatives of the perturbation.

$\delta \Gamma^\alpha_{\mu \nu} = \frac{1}{2} \eta^{\alpha \gamma} \left( \partial_{\nu} h_{\gamma \mu} + \partial_{\mu} h_{\gamma \nu} - \partial_{\gamma} h_{\mu \nu} \right)$

The linearized Einstien equation now becomes

$\delta R_{\mu \nu} = \frac{1}{2} \left( \Box^2 h_{\mu \nu} + \partial_\alpha V_\beta + \partial_\beta V_\alpha \right)$

Where Vα substitutes the expression $\partial_\beta h_\alpha^\beta - \frac{1}{2} \partial_\alpha h_\beta^\beta$ and $\Box^2 = \partial_t^2 - \nabla^2$ 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 Vα 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.

$\Box^2 h_{\mu \nu} = \mathbf{0}$
$\partial_\beta h_\alpha^\beta = \frac{1}{2} \partial_\alpha h_\beta^\beta$

From experience with simpler wave equations we can guess the general form of the solution.

$h_{\mu \nu} = A_{\mu \nu} e^{\imath k \cdot x}$

Where $k \cdot k = 0$ is a null vector. The wave equation is now satisfied, but what choices of A will satisfy the gauge condition we used.

$A_\alpha^\beta \partial_\beta e^{\imath k \cdot x} = A_\beta^\beta \partial_\alpha e^{\imath k \cdot x}$
$A_\alpha^\beta k_\beta = A_\beta^\beta k_\alpha$

If we don't want transformations to disturb our choice of gauge, then we better make the wave traceless, $A_\beta^\beta = 0$, and transverse, $A_\alpha^\beta k_\beta = 0$.

For a wave traveling in the z direction, k = (1,0,0,1), the perturbation will take the following form.

$h_{\mu \nu} = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & a & b & 0\\ 0 & b & -a & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} e^{\imath k \cdot x}$

Thus the oscillations are transverse spacial distortions. The wave is called spin-2 because there are 2 different polarizations a,b. Light only has one! a is called the plus polarization and b is called the cross polarization.

## Perturbative versus Exact

Gravitational waves differ markedly from electromagnetic waves in that electromagnetic waves can be derived exactly from Maxwell's equations. However gravitational waves, as a linear, spin-2 wave, as they are often thought of are only perturbations to certain space-time geometries. In other words, classically there are always linear, spin-1 E&M waves, but there are never linear, spin-2 gravitational waves. There are still wave-like fluctuations, but in general things are nonlinear, as is always the case in General Relativity. This is one of the reasons there may be no graviton.

The thing that is analogous to electromagnetic radiation is the Weyl curvature, not the linear, spin-2 wave.

## Gravitational waves transmit energy

Within parts of the scientific community there was initially some confusion as to if gravitational waves could transmit energy like 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 field was thought to not contain energy, like a gravitational potential would. But the field can most certainly carry energy as it can do mechanical work at a distance. And this has been proven using stress-energy pseudo tensors that transport energy as well as seeing how radiation can carry energy out to infinity.