Gravitational Wave

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 Wave s, 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 . 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 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 PSR B1913+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 Physic s 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. 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) is 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 be 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.

Just as an electromagnetic wave has electric and magnetic components, so too does a gravitational wave have gravitoelectric and gravitomagnetic components. A "+" wave has a "X" gravitomagnetic component, and vice versa.

THE NATURE OF GRAVITATIONAL WAVES

Gravitational waves represent fluctuations in the Metric of space-time. That is, they alter the relative distance between 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 perfectly flat region of spacetime, with a group of mutually motionless test particles constrained to a single plane. 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,

A monochromatic circularly polarized gravitational wave induces a similar cruciform oscillation, except the crucifix rotates with the same frequency as the above described 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 in one very important respect however: the strongest type of electromagnetic radiation is ''dipole'' radiation, while the strongest type of gravitational radiation is ''quadrupole'' radiation. {Link without Title}

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 with respect to 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 $\omega$ , 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 $\omega$ . 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 $\omega$ 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 $2 \omega$ . Observers far from the system and near its axis of symmetry will observe ''circularly polarized'' radiation with frequency $\omega$ .

Two objects orbiting each other with angular frequency $\omega$ 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 $2 \omega$ . Observers far from the system and lying on its axis of symmetry will observe ''circularly polarized'' radiation with frequency $\omega$ .

The last three examples illustrate a general rule-of-thumb: far from a radiating 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).

If two spinning black holes were to collide, they could emit an enormous amount of gravitational radiation and lose energy in the process.

DETECTION

Russell Alan Hulse and Joseph Hooton Taylor Jr. were awarded the Nobel Prize In Physics in 1993 for their observations of a remarkable binary pulsar, PSR B1913+16 . 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. This prediction agrees with the observations of Hulse and Taylor.

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. 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.

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 .

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:

Laser Interferometer s, which use long light paths, such as GEO , LIGO , TAMA , VIRGO , ACIGA and the space-based LISA ;

Resonant Mass Gravitational Wave Detector s which use large masses at very low temperatures such as AURIGA , ALLEGRO , EXPLORER and NAUTILUS .

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.

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 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:

Supernova s or Gamma Ray Burst s;

" Chirp s" from inspiraling coalescing Binary Star s;

periodic signals from spherically asymmetric Neutron Star s or Quark Star s;

Stochastic Gravitational Wave Background sources.

DERIVATION

Perturbation of Flat Space-time

Consider that the full metric

g

is nearly the flat metric

\eta

plus some small perturbation

h

.

:

g_{\mu 
u} = \eta_{\mu 
u} + h_{\mu 
u}

The Einstein equation in vacuum is

:

R_{\mu 
u} = \mathbf{0}

Where

R

is the Ricci Curvature . We will expand

R

in perturbatively in powers of

h

.

:

R_{\mu 
u} = \mathbf{0} + \delta R_{\mu 
u} + \delta^2 R_{\mu 
u} + \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 
u} = \delta R_{\mu 
u} + \mathbf{0}

Where

\delta R_{\mu 
u}

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 
u} = \partial_{\alpha} \Gamma_{\mu 
u}^\alpha - \partial_{
u} \Gamma_{\mu \alpha}^\alpha + \Gamma_{\mu 
u}^\alpha \Gamma_{\alpha \beta}^\beta - \Gamma_{\mu \beta}^\alpha \Gamma_{
u \alpha}^\beta

Where

\Gamma

are the Christoffel Symbols and

\partial_{\alpha}

is shorthand for

rac{\partial}{\partial x^{\alpha}}

. Only first two terms which are linear in

\Gamma

will contribute to the first order correction.

:

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

Next we need the formula for the Christoffel symbols.

:

\Gamma^\alpha_{\mu 
u} = rac{1}{2} g^{\alpha \gamma} \left( \partial_{
u} g_{\gamma \mu} + \partial_{\mu} g_{\gamma 
u} - \partial_{\gamma} g_{\mu 
u} 
ight)

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

:

\delta \Gamma^\alpha_{\mu 
u} = rac{1}{2} \eta^{\alpha \gamma} \left( \partial_{
u} h_{\gamma \mu} + \partial_{\mu} h_{\gamma 
u} - \partial_{\gamma} h_{\mu 
u} 
ight)

The linearized Einstien equation now becomes

:

\delta R_{\mu 
u} = rac{1}{2} \left( \Box^2 h_{\mu 
u} + \partial_\alpha V_\beta + \partial_\beta V_\alpha 
ight)

Where

V_\alpha

substitutes the expression

\partial_\beta h_\alpha^\beta - rac{1}{2} \partial_\alpha h_\beta^\beta

and

\Box^2 = \partial_t^2 - 
abla^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

V_\alpha

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 
u} = \mathbf{0}

\partial_\beta h_\alpha^\beta = rac{1}{2} \partial_\alpha h_\beta^\beta

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

:

h_{\mu 
u} = A_{\mu 
u} 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 
u} =
\begin{bmatrix}
0 & 0 & 0 & 0\
0 & A_{+} & A_{	imes} & 0\
0 & A_{	imes} & -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. Light only has one!

A_{+}

is called the plus polarization and

A_{	imes}

is called the cross polarization.

Perturbation with Sources

The Einstein equation is the relationship between space-time curvature and the matter or source of that curvature.

:

R_{\alpha \beta} - rac{1}{2} g_{\alpha \beta} R = rac{8 \pi G}{c^4} T_{\alpha \beta}

Our first order contributions to the curvature have previously been determined.

:

\delta R_{\mu 
u} = rac{1}{2} \Box^2 h_{\mu 
u}

We stick to the choice of Lorentz gauge, which will now be written in a very suggestive form.

:

rac{\partial}{\partial^\beta} \left( h_{\alpha \beta} - rac{1}{2} h \eta_{\alpha \beta} 
ight)

The right hand side is the divergence of the trace-reversed

h_{\alpha \beta}

. The traced reversed perturbation will be abbreviated as

\overline{h}_{\alpha \beta}

from now on.

We can now combine these equations into the linearized Einstein equation.

:

\Box^2 \overline{h}_{\alpha \beta} = - rac{16 \pi G}{c^4} T_{\alpha \beta}

This is a long solved problem from electricity and magnetism analogous to electromagnetic waves with sources. It is solved via retarded Green's functions.

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Gravitational Wave