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General relativity
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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 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.)

Contents

Characteristics

In Einstein's theory of General relativity, the force of gravity is due to spacetime curvature. The rapid motion of mass in some region will generate ripples in spacetime itself which radiate outward as gravitational waves. Like electromagnetic radiation, gravitational radiation is believed to travel at the speed of light, however while electromagnetic waves are 'spin-1' (meaning their direction of oscillation is perpendicular to their direction of propagation) gravitational waves are 'spin-2'. This means that they oscillate in two directions, both perpendicular to the direction of propagation, with the oscillations being 180 degrees out of phase.

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) 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 detectable, due to the weakness of its interactions.

Just as an electromagnetic wave has electric and magnetic components, so too does a gravitational wave have gravitoelectric and gravitomagnetic components.

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 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 in-spiral at almost 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.

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]

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 and should not (assuming general relativity gives accurate predictions) expect a system to emit gravitational radiation:

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 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.

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. Imagine a perfectly flat region of spacetime, with a group of mutually motionless test particles constrained to a single plane. Then, a monochromatic linearly polarized gravitational wave arrives in the direction normal to our particles' plane. 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, after which east-west separated particles draw apart while north-south separated particles draw together, and so forth. The cross-sectional of a small box of test particles is invariant under these changes, and there is negligible motion in the direction of propagation (unless one considers gravitomagnetic effects; however, we shall assume that the relative motion of our test particles is not very rapid, and so these effects may be ignored).

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.

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:

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

Unsolved problems in physics: Is our universe filled with gravitational radiation from the big bang? From astrophysical sources, such as inspiralling neutron stars? What can this tell us about quantum gravity and general relativity?

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 of Flat Space-time

Consider that the full metric [g] is nearly the flat metric [\eta] plus some small perturbation [h].

[g_ = \eta_ + h_]
The Einstein equation in vacuum is

[R_ = \mathbf]
Where [R] is the Ricci curvature. We will expand [R] in perturbatively in powers of [h].

[R_ = \mathbf + \delta R_ + \delta^2 R_ + \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_ = \delta R_ + \mathbf]
Where [\delta R_] 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_ = \partial_ \Gamma_^\alpha - \partial_ \Gamma_^\alpha + \Gamma_^\alpha \Gamma_^\beta - \Gamma_^\alpha \Gamma_^\beta]
Where [\Gamma] are the Christoffel symbols and [\partial_] is shorthand for [\frac}]. Only first two terms which are linear in [\Gamma] will contribute to the first order correction.

[\delta R_ = \partial_ \delta \Gamma_^\alpha - \partial_ \delta \Gamma_^\alpha]
Next we need the formula for the Christoffel symbols.

[\Gamma^\alpha_ = \frac g^ \left( \partial_ g_ + \partial_ g_ - \partial_ g_ \right)]
Seeing as the flat metric is constant, the only first order terms will involve derivatives of the perturbation.

[\delta \Gamma^\alpha_ = \frac \eta^ \left( \partial_ h_ + \partial_ h_ - \partial_ h_ \right)]
The linearized Einstien equation now becomes

[\delta R_ = \frac \left( \Box^2 h_ + \partial_\alpha V_\beta + \partial_\beta V_\alpha \right)]
Where [V_\alpha] substitutes the expression [\partial_\beta h_\alpha^\beta - \frac \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 [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_ = \mathbf]
[\partial_\beta h_\alpha^\beta = \frac \partial_\alpha h_\beta^\beta]
From experience with simpler wave equations we can guess the general form of the solution.

[h_ = A_ e^]
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^ = A_\beta^\beta \partial_\alpha e^]
[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_ =\begin0 & 0 & 0 & 0\\0 & A_ & A_ & 0\\0 & A_ & -A_ & 0\\0 & 0 & 0 & 0\end e^]

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_] 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_ - \frac g_ R = \frac T_]
Our first order contributions to the curvature have previously been determined.

[\delta R_ = \frac \Box^2 h_]
We stick to the choice of Lorentz gauge, which will now be written in a very suggestive form.

[\frac \left( h_ - \frac h \eta_ \right)]
The right hand side is the divergence of the trace-reversed [h_]. The traced reversed perturbation will be abbreviated as [\overline_] from now on.

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

[\Box^2 \overline_ = - \frac T_]
This is a long solved problem from electricity and magnetism analogous to electromagnetic waves with sources. It is solved via retarded Green's functions.

[\overline^ \left(t,\vec \right) = \frac \int d^3y \frac\left(t_r,\vec\right)}-\vec \right|}]
Where [t_r = t - \frac-\vec \right|}] 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.

[\overline^ \left(t,\vec \right) \approx \frac \frac \int d^3y T^\left(t - \frac,\vec\right)]
Where [r] 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.

[\nabla \cdot \mathbf = 0]
[ \frac T^ = -\frac T^]
[ \frac T^ = - \frac T^]
[ \frac T^ = - \frac T^]
[ \frac T^ = \frac T^]
We now take this relationship and massage it into the form of our original integral and see what new information it gives us.

[ \int d^3x x^k x^l \frac T^ = \int d^3x x^k x^l \frac T^]
We wanted to multiply the right hand side with the two powers of [x] so that we can integrate by parts twice and get down to a regular volume integral.

[\frac \int d^3x x^k x^l T^ = 2 \int d^3x T^]
Assuming the stress-energy tensor takes the simple form

[T_ = \rho u_\alpha u_\beta]
Where [\rho] is the mass density and [u_\alpha] is the 4-velocity. If the source is nonrelativistic, then the energy density will be dominated by the mass density, [T^ = \rho]

[\frac \int d^3x x^k x^l \rho = 2 \int d^3x T^]
Here we see something very similar to the moment of inertia, we call it the second mass moment.

[I^(t) = \int d^3x x^k x^l \rho\left(t,\vec \right) ]
We now have our final expression that relates the gravitational waves with their source.

[\overline^ \left(t,\vec \right) \approx \frac \frac \frac I^ \left(t,\vec \right)]

Perturbative versus Exact

Gravitational waves differ markedly from electromagnetic waves in that electromagnetic waves can be derived exactly from Maxwell's equations whereas gravitational waves, interpreted as linear spin-2 waves, 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 provides one argument against the existence of gravitons.

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.[[Citing sources citation needed]]

See also

External links

References

 

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