|
|
| Line 1: |
Line 1: |
| {{distinguish|Gravity wave}}
| | Andrew Simcox is the title his mothers and fathers gave him and he completely enjoys this name. The favorite hobby for him and his kids is to perform lacross and he would never give it up. For a whilst I've been in Mississippi but now I'm considering other options. For many years she's been working as a travel agent.<br><br>Stop by my homepage [http://appin.co.kr/board_Zqtv22/688025 online psychic reading] |
| {{General relativity|cTopic=Phenomena}}
| |
| In [[physics]], '''gravitational waves''' are ripples in the [[curvature]] of [[spacetime]] that propagate as a [[wave]], travelling outward from the source. Predicted to exist by [[Albert Einstein]] in 1916<ref>{{cite web | last=Finley | first=Dave | title=Einstein's gravity theory passes toughest test yet: Bizarre binary star system pushes study of relativity to new limits. | url=http://phys.org/news/2013-04-einstein-gravity-theory-toughest-bizarre.html | publisher=Phys.Org}}</ref> on the basis of his theory of [[general relativity]],<ref>http://www.dpf99.library.ucla.edu/session14/barish1412.pdf The Detection of Gravitational Waves using LIGO, B. Barish</ref> gravitational waves theoretically transport energy as '''gravitational radiation'''. Sources of detectable gravitational waves could possibly include [[binary star]] systems composed of [[white dwarfs]], [[neutron stars]], or [[black holes]]. The existence of gravitational waves is a possible consequence of the [[Lorentz invariance]] of [[general relativity]] since it brings the concept of a limiting speed of propagation of the physical interactions with it. Gravitational waves cannot exist in the Newtonian theory of gravitation, in which physical interactions propagate at infinite speed.
| |
| | |
| Although gravitational radiation has not been ''directly'' detected, there is ''indirect'' evidence for its existence. For example, the 1993 [[Nobel Prize in Physics]] was awarded for measurements of the [[Hulse-Taylor binary]] system that suggests gravitational waves are more than mathematical anomalies. Various [[Gravitational-wave detector|gravitational wave detectors]] exist. However, they have not yet succeeded in detecting such phenomena.
| |
| | |
| ==Introduction==
| |
| [[File:Quadrupol Wave.gif|thumb|Linearly polarised gravitational wave]]
| |
| In Einstein's theory of [[general relativity]], [[gravity]] is treated as a phenomenon resulting from the curvature of [[spacetime]]. This curvature is caused by the presence of [[Mass in general relativity|mass]]. Generally, the more mass that is contained within a given volume of space, the greater the curvature of spacetime will be at the boundary of this volume. As objects with mass move around in spacetime, the curvature changes to reflect the changed locations of those objects. In [[Gravitational wave#Sources of gravitational waves|certain circumstances]], accelerating objects generate changes in this curvature, which propagate outwards at the [[speed of light]] in a wave-like manner. These propagating phenomena are known as gravitational waves.
| |
| | |
| As a gravitational wave passes a distant observer, that observer will find spacetime distorted by the effects of [[Deformation (mechanics)#Strain|strain]]. Distances between free objects increase and decrease rhythmically as the wave passes, at a frequency corresponding to that of the wave. This occurs despite such free objects never being subjected to an unbalanced [[Force (physics)|force]]. The magnitude of this effect decreases [[Inverse_proportion#Inverse_proportionality|inversely]] with distance from the source. Inspiralling [[Neutron star#Binary neutron stars|binary neutron star]]s are predicted to be a powerful source of gravitational waves as they coalesce, due to the very large acceleration of their masses as they [[orbit]] close to one another. However, due to the astronomical distances to these sources the effects when measured on Earth are predicted to be very small, having strains of less than 1 part in 10<sup>20</sup>. Scientists are attempting to demonstrate the existence of these waves with ever more sensitive detectors. The current most sensitive measurement is about one part in {{val|5|e=22}} (as of 2012) provided by the [[LIGO]] and [[VIRGO]] observatories.<ref>{{cite journal
| |
| | author1=LIGO Scientific Collaboration
| |
| | author2=Virgo Collaboration
| |
| | title=Search for Gravitational Waves from Low Mass Compact Binary Coalescence in LIGO's Sixth Science Run and Virgo's Science Runs 2 and 3
| |
| | journal=Physical Review D | volume=85 | page=082002 | year=2012
| |
| | arxiv=1111.7314
| |
| |bibcode = 2012PhRvD..85h2002A |doi = 10.1103/PhysRevD.85.082002 }}</ref> The lack of detection in these observatories provides an upper limit on the frequency of such powerful sources.<ref>{{cite journal | author1=LIGO Scientific Collaboration | author2=Virgo Collaboration | title=All-sky search for gravitational-wave bursts in the second joint LIGO-Virgo run | journal=Physical Review D | volume=85 | page=122007 | year=2012 | arxiv=1202.2788 |bibcode = 2012PhRvD..85l2007A |doi = 10.1103/PhysRevD.85.122007 }}</ref><ref>{{cite journal | author1=LIGO Scientific Collaboration | author2=Virgo Collaboration | title=Search for gravitational waves from binary black hole inspiral, merger, and ringdown in LIGO-Virgo data from 2009-2010 | journal=Physical Review D | volume=87 | page=022002 | year=2013 | arxiv=1209.6533 |bibcode = 2013PhRvD..87b2002A |doi = 10.1103/PhysRevD.87.022002 }}</ref> A space based observatory, the [[Laser Interferometer Space Antenna]], is currently under development by [[ESA]].
| |
| | |
| Gravitational waves should penetrate regions of space that electromagnetic waves cannot. It is hypothesized that they will be able to provide observers on Earth with information about black holes and other exotic objects in the distant Universe. Such systems cannot be observed with more traditional means such as [[optical telescope]]s and [[radio telescope]]s. In particular, gravitational waves could be of interest to cosmologists as they offer a possible way of observing the very early universe. This is not possible with conventional astronomy, since before [[recombination (cosmology)|recombination]] the universe was opaque to electromagnetic radiation.<ref>{{cite journal | last1 = Krauss | last2 = Dodelson | last3 = Meyer | first3 = S| year = 2010 | title = Primordial Gravitational Waves and Cosmology | url = | journal = Science | volume = 328 | issue = 5981| pages = 989–992 | doi = 10.1126/science.1179541 | pmid = 20489015 | first1 = LM | first2 = S | bibcode = 2010Sci...328..989K | arxiv = 1004.2504 }}</ref> Precise measurements of gravitational waves will also allow scientists to test the general theory of relativity more thoroughly.
| |
| | |
| In principle, gravitational waves could exist at any frequency. However, very low frequency waves would be impossible to detect and there is no credible source for detectable waves of very high frequency. [[Stephen Hawking|Stephen W. Hawking]] and [[Werner Israel]] list different frequency bands for gravitational waves that could be plausibly detected, ranging from 10<sup>−7</sup> Hz up to 10<sup>11</sup> Hz.<ref name = "HI">Hawking, S. W. and Israel, W., ''General Relativity: An Einstein Centenary Survey'', Cambridge University Press, Cambridge, 1979, 98.</ref>
| |
| | |
| ==Effects of a passing gravitational wave==
| |
| [[Image:GravitationalWave PlusPolarization.gif|thumb|150px|The effect of a plus-polarized gravitational wave on a ring of particles.]]
| |
| [[Image:GravitationalWave CrossPolarization.gif|thumb|150px|The effect of a cross-polarized gravitational wave on a ring of particles.]]
| |
| The effects of a passing gravitational wave can be visualized by imagining a perfectly flat region of spacetime with a group of motionless test particles lying in a plane (the surface of your screen). As a gravitational wave passes through the particles along a line perpendicular to the plane of the particles (i.e. following your line of vision into the screen), the particles will follow the distortion in spacetime, oscillating in a "[[cruciform]]" manner, as shown in the animations. The area enclosed by the test particles does not change and there is no motion along the direction of propagation.
| |
| | |
| The oscillations depicted here in the animation are exaggerated for the purpose of discussion—in reality a gravitational wave has a very small amplitude (as formulated in [[linearized gravity]]). However they enable us to visualize the kind of oscillations associated with gravitational waves as produced for example by a pair of masses in a [[circular orbit]]. In this case the amplitude of the gravitational wave is a constant, but its plane of [[Polarization (waves)|polarization]] changes or rotates at twice the orbital rate and so the time-varying gravitational wave size (or 'periodic spacetime strain') exhibits a variation as shown in the animation.<ref name = "LL75">Landau, L. D. and Lifshitz, E. M., ''The Classical Theory of Fields''. Fourth Revised English Edition, Pergamon Press., 1975, 356–357.</ref> If the orbit is elliptical then the gravitational wave's amplitude also varies with time according Einstein's [[quadrupole formula]].<ref>Einstein, A., "Über Gravitationswellen." ''Sitzungsberichte, Preussische Akademie der Wissenschaften'', 154, (1918).</ref>
| |
| | |
| Like other [[wave]]s, there are a few useful characteristics describing a gravitational wave:
| |
| * '''Amplitude''': Usually denoted <math>h</math>, this is the size of the wave — the fraction of stretching or squeezing in the animation. The amplitude shown here is roughly <math>h=0.5</math> (or 50%). Gravitational waves passing through the Earth are many billions times weaker than this — <math>h \approx 10^{-20}</math>. Note that this is not the quantity that would be analogous to what is usually called the amplitude of an electromagnetic wave, which would be <math>\frac{\mathrm{d}h}{\mathrm{d}t}</math>.
| |
| * '''[[Frequency]]''': Usually denoted ''f'', this is the frequency with which the wave oscillates (1 divided by the amount of time between two successive maximum stretches or squeezes)
| |
| * '''[[Wavelength]]''': Usually denoted <math>\lambda</math>, this is the distance along the wave between points of maximum stretch or squeeze.
| |
| * '''[[Speed]]''': This is the speed at which a point on the wave (for example, a point of maximum stretch or squeeze) travels. For gravitational waves with small amplitudes, this is equal to the [[speed of light]], <math>c</math>.
| |
| | |
| The speed, wavelength, and frequency of a gravitational wave are related by the equation ''c = λ f'', just like the equation for a [[Electromagnetic radiation#Wave model|light wave]]. For example, the animations shown here oscillate roughly once every two seconds. This would correspond to a frequency of 0.5 Hz, and a wavelength of about 600,000 km, or 47 times the diameter of the Earth.
| |
| | |
| In the example just discussed, we actually assume something special about the wave. We have assumed that the wave is [[linear polarization|linearly polarized]], with a "plus" polarization, written <math>h_{\,+}</math>. Polarization of a gravitational wave is just like polarization of a light wave except that the polarizations of a gravitational wave are at 45 degrees, as opposed to 90 degrees. In particular, if we had a "cross"-polarized gravitational wave, <math>h_{\,\times}</math>, the effect on the test particles would be basically the same, but rotated by 45 degrees, as shown in the second animation. Just as with light polarization, the polarizations of gravitational waves may also be expressed in terms of [[circular polarization|circularly polarized]] waves. Gravitational waves are polarized because of the nature of their sources. The polarization of a wave depends on the angle from the source, as we will see in the next section.
| |
| | |
| ==Sources of gravitational waves==
| |
| In general terms, gravitational waves are radiated by objects whose motion involves acceleration, provided that the motion is not perfectly spherically [[symmetric]] (like an expanding or contracting sphere) or cylindrically symmetric (like a spinning disk or sphere). A simple example of this principle is provided by the spinning [[dumbbell]]. If the dumbbell spins like wheels on an axle, it will not radiate gravitational waves; if it tumbles end over end like two planets orbiting each other, it will radiate gravitational waves. The heavier the dumbbell, and the faster it tumbles, the greater is the gravitational radiation it will give off. If we imagine an extreme case in which the two weights of the dumbbell are massive stars like neutron stars or black holes, orbiting each other quickly, then significant amounts of gravitational radiation would be given off.
| |
| | |
| Some more detailed examples:
| |
| *Two objects orbiting each other in a quasi-Keplerian planar orbit (basically, as a planet would orbit the Sun) '''''will''''' radiate.
| |
| * A spinning non-axisymmetric planetoid — say with a large bump or dimple on the equator — '''''will''''' radiate.
| |
| * A [[supernova]] '''''will''''' radiate except in the unlikely event that the explosion is perfectly symmetric.
| |
| * An isolated non-spinning solid object moving at a constant velocity '''''will not''''' radiate. This can be regarded as a consequence of the principle of [[Momentum#Conservation of linear momentum|conservation of linear momentum]].
| |
| * A spinning disk '''''will not''''' radiate. This can be regarded as a consequence of the principle of [[Angular momentum#Conservation of angular momentum|conservation of angular momentum]]. However, it ''will'' show [[gravitomagnetism|gravitomagnetic]] effects.
| |
| * A spherically pulsating spherical star (non-zero monopole moment or [[mass]], but zero quadrupole moment) '''''will not''''' radiate, in agreement with [[Birkhoff's theorem (relativity)|Birkhoff's theorem]].
| |
| | |
| More technically, the third time derivative of the [[quadrupole moment]] (or the ''l''-th time derivative of the ''l''-th [[multipole expansion|multipole moment]]) of an isolated system's [[stress-energy tensor]] must be nonzero in order for it to emit gravitational radiation. This is analogous to the changing dipole moment of charge or current necessary for electromagnetic radiation.
| |
| | |
| === Power radiated by orbiting bodies ===
| |
| [[Image:orbit2.gif|thumb|200px|Two stars of dissimilar mass are in [[circular orbits]]. Each revolves about their common [[center of mass]] (denoted by the small red cross) in a circle with the larger mass having the smaller orbit.]]
| |
| | |
| [[Image:orbit1.gif|thumb|200px|Two stars of similar mass are in circular orbits about their center of mass]]
| |
| | |
| [[Image:orbit5.gif|thumb|200px|Two stars of similar mass are in highly [[elliptical orbit]]s about their center of mass ]]
| |
| | |
| Gravitational waves carry energy away from their sources and, in the case of orbiting bodies, this is associated with an inspiral or decrease in orbit. Imagine for example a simple system of two masses — such as the Earth-Sun system — moving slowly compared to the speed of light in circular orbits. Assume that these two masses orbit each other in a circular orbit in the <math>x</math>-<math>y</math> plane. To a good approximation, the masses follow simple [[Planetary orbit|Keplerian orbits]]. However, such an orbit represents a changing quadrupole moment. That is, the system will give off gravitational waves.
| |
| | |
| Suppose that the two masses are <math>m_1</math> and <math>m_2</math>, and they are separated by a distance <math>r</math>. The power given off (radiated) by this system is:
| |
| | |
| :<math>P = \frac{\mathrm{d}E}{\mathrm{d}t} = - \frac{32}{5}\, \frac{G^4}{c^5}\, \frac{(m_1m_2)^2 (m_1+m_2)}{r^5}</math> ,<ref name="Gravitational Radiation">[http://www.eftaylor.com/exploringblackholes/GravWaves100707V2.pdf Gravitational Radiation]</ref>
| |
| | |
| where ''G'' is the [[gravitational constant]], ''c'' is the speed of light in vacuum and where the negative sign means that power is being given off by the system, rather than received. For a system like the Sun and Earth, <math>r</math> is about 1.5{{e|11}} m and <math>m_1</math> and <math>m_2</math> are about 2{{e|30}} and 6{{e|24}} kg respectively. In this case, the power is about 200 watts. This is truly tiny compared to the [[Solar_constant#Solar_constant|total electromagnetic radiation given off by the Sun]] (roughly 3.86{{e|26}} watts).
| |
| | |
| In theory, the loss of energy through gravitational radiation could eventually drop the Earth into the Sun. However, the total energy of the Earth orbiting the Sun ([[kinetic energy]] plus [[gravitational potential energy]]) is about 1.14{{e|36}} [[joules]] of which only 200 joules per second is lost through gravitational radiation, leading to a decay in the orbit by about 1{{e|-15}} meters per day or roughly the diameter of a [[proton]]. At this rate, it would take the Earth approximately 1{{e|13}} times more than the current [[age of the Universe]] to spiral onto the Sun. This estimate overlooks the decrease in ''r'' over time, but the majority of the time the bodies are far apart and only radiating slowly, so the difference is unimportant in this example. In only a few billion years, [[Earth#Future|the Earth is]] predicted to be swallowed by the Sun in the red giant stage of its life.
| |
| | |
| A more dramatic example of radiated gravitational energy is represented by two solar mass neutron stars orbiting at a distance from each other of 1.89{{e|8}} m (only 0.63 [[light-second]]s apart). [The Sun is 8 light minutes from the Earth.] Plugging their masses into the above equation shows that the gravitational radiation from them would be 1.38{{e|28}} watts, which is about 100 times more than the Sun's electromagnetic radiation.
| |
| | |
| === Orbital decay from gravitational radiation ===
| |
| {{see also|Two-body problem in general relativity}}
| |
| | |
| Gravitational radiation robs the orbiting bodies of energy. It first circularizes their orbits and then gradually shrinks their radius. As the energy of the orbit is reduced, the distance between the bodies decreases, and they rotate more rapidly. The overall angular momentum is reduced however. This reduction corresponds to the angular momentum carried off by gravitational radiation. The rate of decrease of distance between the bodies versus time is given by:<ref name="Gravitational Radiation"/>
| |
| | |
| :<math>\frac{\mathrm{d}r}{\mathrm{d}t} = - \frac{64}{5}\, \frac{G^3}{c^5}\, \frac{(m_1m_2)(m_1+m_2)}{r^3}\ </math>,
| |
| | |
| where the variables are the same as in the previous equation.
| |
| | |
| The orbit decays at a rate proportional to the inverse third power of the radius. When the radius has shrunk to half its initial value, it is shrinking eight times faster than before. By [[Kepler's laws of planetary motion|Kepler's Third Law]], the new rotation rate at this point will be faster by <math>\sqrt{8}=2.828</math>, or nearly three times the previous orbital frequency. As the radius decreases, the power lost to gravitational radiation increases even more. As can be seen from the previous equation, power radiated varies as the inverse fifth power of the radius, or 32 times more in this case.
| |
| | |
| If we use the previous values for the Sun and the Earth, we find that the Earth's orbit shrinks by 1.1{{e|-20}} meter per second. This is 3.5{{e|-13}} m per year, which is about 1/300 the diameter of a [[hydrogen atom]]. The effect of gravitational radiation on the size of the Earth's orbit is negligible over the age of the universe. This is not true for closer orbits.
| |
| | |
| A more practical example is the orbit of a [[Solar analog|Sun-like star]] around a heavy [[black hole]]. Our [[Milky Way]] has a, potential, 4 million solar-mass black hole at its center in [[Sagittarius A]]. Such [[supermassive black holes]] are being found in the center of almost all [[galaxy|galaxies]]. For this example take a 2 million solar-mass black hole with a solar-mass star orbiting it at a radius of 1.89{{e|10}} m (63 light-seconds). The mass of the black hole will be 4{{e|36}} kg and its [[gravitational radius]] will be 6{{e|9}} m. The orbital period will be 1,000 seconds, or a little under 17 minutes. The solar-mass star will draw closer to the black hole by 7.4 meters per second or 7.4 km per orbit. A collision will not be long in coming.
| |
| | |
| Assume that a pair of solar-mass [[neutron stars]] are in circular orbits at a distance of 1.89{{e|8}} m (189,000 km). This is a little less than 1/7 the diameter of the Sun or 0.63 [[light-second]]s. Their orbital period would be 1,000 seconds. Substituting the new mass and radius in the above formula gives a rate of orbit decrease of 3.7{{e|-6}} m/s or 3.7 mm per orbit. This is 116 meters per year and is not negligible over cosmic time scales.
| |
| | |
| Suppose instead that these two neutron stars were orbiting at a distance of 1.89{{e|6}} m (1890 km). Their period would be 1 second and their orbital velocity would be about 1/50 of the speed of light. Their orbit would now shrink by 3.7 meters per orbit. A collision is imminent. A runaway loss of energy from the orbit results in an ever more rapid decrease in the distance between the stars. They will eventually merge to form a black hole and cease to radiate gravitational waves. This is referred to as the [[inspiral]].
| |
| | |
| The above equation can not be applied directly for calculating the lifetime of the orbit, because the rate of change in radius depends on the radius itself, and is thus non-constant with time. The lifetime can be computed by integration of this equation (see next section).
| |
| | |
| === Orbital lifetime limits from gravitational radiation ===
| |
| | |
| Orbital lifetime is one of the most important properties of gravitational radiation sources. It determines the average number of binary stars in the universe that are close enough to be detected. Short lifetime binaries are strong sources of gravitational radiation but are few in number. Long lifetime binaries are more plentiful but they are weak sources of gravitational waves. LIGO is most sensitive in the frequency band where two neutron stars are about to merge. This time frame is only a few seconds. It takes luck for the detector to see this blink in time out of a million year orbital lifetime. It is predicted that such a merger will only be seen once per decade or so.
| |
| | |
| The lifetime of an orbit is given by:<ref name="Gravitational Radiation"/>
| |
| | |
| :<math>t= \frac{5}{256}\, \frac{c^5}{G^3}\, \frac{r^4}{(m_1m_2)(m_1+m_2)}\ </math>,
| |
| | |
| where r is the initial distance between the orbiting bodies. This equation can be derived by integrating the previous equation for the rate of radius decrease. It predicts the time for the radius of the orbit to shrink to zero. As the orbital speed becomes a significant fraction of the speed of light, this equation becomes inaccurate. It is useful for inspirals until the last few milliseconds before the merger of the objects.
| |
| | |
| Substituting the values for the mass of the Sun and Earth as well as the orbital radius gives a very large lifetime of 3.44{{e|30}} seconds or 1.09{{e|23}} years (that is approximately 10<sup>13</sup> times larger than the [[age of the universe]]). The actual figure would be slightly less than that. The Earth will break apart from [[Roche limit|tidal forces]] if it orbits closer than a few radii from the Sun. This would form a ring around the Sun and instantly stop the emission of gravitational waves.
| |
| | |
| If we use a 2 million solar mass black hole with a solar mass star orbiting it at 1.89{{e|10}} meters, we get a lifetime of 6.50{{e|8}} seconds or 20.7 years.
| |
| | |
| Assume that a pair of solar mass neutron stars with a diameter of 10 kilometers are in circular orbits at a distance of 1.89{{e|8}} m (189,000 km). Their lifetime is 1.30{{e|13}} seconds or about 414,000 years. Their orbital period will be 1,000 seconds and it could be observed by [[Laser Interferometer Space Antenna|LISA]] if they were not too far away. A far greater number of white dwarf binaries exist with orbital periods in this range. White dwarf binaries have masses on the order of our Sun and diameters on the order of our Earth. They cannot get much closer together than 10,000 km before they will merge and cease to radiate gravitational waves. This results in the creation of either a neutron star or a black hole. Until then, their gravitational radiation will be comparable to that of a neutron star binary. LISA is the only gravitational wave experiment that is likely to succeed in detecting such types of binaries.
| |
| | |
| If the orbit of a neutron star binary has decayed to 1.89{{e|6}}m (1890 km), its remaining lifetime is 130,000 seconds or about 36 hours. The orbital frequency will vary from 1 revolution per second at the start and 918 revolutions per second when the orbit has shrunk to 20 km at merger. The gravitational radiation emitted will be at twice the orbital frequency. Just before merger, the inspiral can be observed by LIGO if the binary is close enough. LIGO has only a few minutes to observe this merger out of a total orbital lifetime that may have been billions of years. The chance of success with LIGO as initially constructed is quite low despite the large number of such mergers occurring in the universe, because the sensitivity of the instrument does not 'reach' out to enough systems to see events frequently. No mergers have been seen in the few years that initial LIGO has been in operation, and it is thought that a merger should be seen about once per several tens of years of observing time with initial LIGO.<ref name=Abadie2010>{{cite journal | author1=LIGO Scientific Collaboration | author2=Virgo Collaboration | title= Predictions for the rates of compact binary coalescences observable by ground-based gravitational-wave detectors | journal=Classical and Quantum Gravity | volume=27 | page=17300 | year=2010 | arxiv=1003.2480 |bibcode = 2010CQGra..27q3001A |doi = 10.1088/0264-9381/27/17/173001 }}</ref> The upgraded Advanced LIGO detector, with a ten times greater sensitivity, 'reaches' out 10 times further -- encompassing a volume 1000 times greater, and seeing 1000 times as many candidate sources. Thus, the expectation is that detections will be made at the rate of tens per year.
| |
| | |
| ===Wave amplitudes from the Earth–Sun system===
| |
| We can also think in terms of the amplitude of the wave from a system in circular orbits. Let <math>\theta </math> be the angle between the perpendicular to the plane of the orbit and the line of sight of the observer. Suppose that an observer is outside the system at a distance <math>R</math> from its center of mass. If R is much greater than a wavelength, the two polarizations of the wave will be
| |
| | |
| :<math>h_{+} = -\frac{1}{R}\, \frac{G^2}{c^4}\, \frac{2 m_1 m_2}{r} (1+\cos^2\theta) \cos\left[2\omega(t - R)\right],</math>
| |
| :<math>h_{\times} = -\frac{1}{R}\, \frac{G^2}{c^4}\, \frac{4 m_1 m_2}{r}\, (\cos{\theta})\sin \left[2\omega(t-R)\right].</math>
| |
| | |
| Here, we use the constant [[angular velocity]] of a circular orbit in Newtonian physics:
| |
| | |
| :<math>\omega=\sqrt{G(m_1+m_2)/r^3}.</math>
| |
| | |
| For example, if the observer is in the <math>x</math>-<math>y</math> plane then <math>\theta=\pi/2</math>, and <math>\cos (\theta) = 0</math>, so the <math>h_\times</math> polarization is always zero. We also see that the frequency of the wave given off is twice the rotation frequency. If we put in numbers for the Earth-Sun system, we find: | |
| | |
| :<math>h_{+} =-\frac{1}{R}\, \frac{G^2}{c^4}\, \frac{4m_1 m_2}{r} = -\frac{1}{R}\, 1.7\times 10^{-10}\, \mathrm{meters}.</math>
| |
| | |
| In this case, the minimum distance to find waves is ''R'' ≈ 1 [[light-year]], so typical amplitudes will be ''h'' ≈ 10<sup>−26</sup>. That is, a ring of particles would stretch or squeeze by just one part in 10<sup>26</sup>. This is well under the detectability limit of all conceivable detectors.
| |
| | |
| ===Radiation from other sources===
| |
| Although the waves from the Earth-Sun system are minuscule, astronomers can point to other sources for which the radiation should be substantial. One important example is the [[PSR B1913+16|Hulse-Taylor binary]] — a pair of stars, one of which is a [[binary pulsar|pulsar]].<ref>[http://arxiv.org/PS_cache/astro-ph/pdf/0407/0407149v1.pdf Relativistic Binary Pulsar B1913+16: Thirty Years of Observations and Analysis]</ref> The characteristics of their orbit can be deduced from the [[Doppler shift]]ing of radio signals given off by the pulsar. Each of the stars has a mass about 1.4 times that of the Sun and the size of their orbit is about 1/75 of the Earth-Sun orbit. This means the distance between the two stars is just a few times larger than the diameter of our own Sun. The combination of greater masses and smaller separation means that the energy given off by the Hulse-Taylor binary will be far greater than the energy given off by the Earth-Sun system — roughly 10<sup>22</sup> times as much.
| |
| | |
| The information about the orbit can be used to predict just how much energy (and angular momentum) should be given off in the form of gravitational waves. As the energy is carried off, the stars should draw closer to each other. This effect is called an '''[[inspiral]]''', and it can be observed in the pulsar's signals. The measurements on the Hulse-Taylor system have been carried out over more than 30 years. It has been shown that the gravitational radiation predicted by general relativity allows these observations to be matched within 0.2 percent. In 1993, [[Russell Alan Hulse|Russell Hulse]] and [[Joseph Hooton Taylor Jr.|Joe Taylor]] were awarded the [[Nobel Prize in Physics]] for this work, which was the first indirect evidence for gravitational waves. Unfortunately, the orbital lifetime of this binary system before merger is about 1.84 billion years. This is a substantial fraction of the age of the universe.
| |
| | |
| Inspirals are very important sources of gravitational waves. Any time two compact objects (white dwarfs, neutron stars, or [[binary black hole|black holes]]) are in close orbits, they send out intense gravitational waves. As they spiral closer to each other, these waves become more intense. At some point they should become so intense that direct detection by their effect on objects on Earth or in space is possible. This direct detection is the goal of several large scale experiments.<ref>[http://pr.caltech.edu/periodicals/EandS/articles/LXXI3/blackholeslayout-web.pdf Crashing Black Holes]</ref>
| |
| | |
| The only difficulty is that most systems like the Hulse-Taylor binary are so far away. The amplitude of waves given off by the Hulse-Taylor binary as seen on Earth would be roughly ''h'' ≈ 10<sup>−26</sup>. There are some sources, however, that astrophysicists expect to find with much larger amplitudes of ''h'' ≈ 10<sup>−20</sup>. At least eight other binary pulsars have been discovered.<ref>[http://relativity.livingreviews.org/Articles/lrr-2005-7/download/lrr-2005-7Color.pdf Binary and Millisecond Pulsars]</ref>
| |
| | |
| ==Astrophysics and gravitational waves==
| |
| {{unsolved|physics|Can gravitational waves be detected experimentally?}}
| |
| | |
| [[File:wavy.gif|360px|thumb|Two-dimensional representation of gravitational waves generated by two [[neutron star]]s orbiting each other.]]
| |
| | |
| During the past century, [[astronomy]] has been revolutionized by the use of new methods for observing the universe. Astronomical observations were originally made using [[visible light]]. [[Galileo Galilei]] pioneered the use of telescopes to enhance these observations. However, visible light is only a small portion of the [[electromagnetic spectrum]], and not all objects in the distant universe shine strongly in this particular band. More useful information may be found, for example, in radio wavelengths. Using [[radio telescopes]], astronomers have found [[pulsars]], [[quasars]], and other extreme objects that push the limits of our understanding of physics. Observations in the [[microwave]] band have opened our eyes to the [[Cosmic microwave background radiation|faint imprints]] of the [[Big Bang]], a discovery [[Stephen Hawking]] called the "greatest discovery of the century, if not all time". Similar advances in observations using [[gamma ray]]s, [[x-ray]]s, [[ultraviolet light]], and [[infrared light]] have also brought new insights to astronomy. As each of these regions of the spectrum has opened, new discoveries have been made that could not have been made otherwise. Astronomers hope that the same holds true of gravitational waves.
| |
| | |
| Gravitational waves have two important and unique properties. First, there is no need for any type of matter to be present nearby in order for the waves to be generated by a binary system of uncharged black holes, which would emit no electromagnetic radiation. Second, gravitational waves can pass through any intervening matter without being scattered significantly. Whereas light from distant stars may be blocked out by [[interstellar dust]], for example, gravitational waves will pass through essentially unimpeded. These two features allow gravitational waves to carry information about astronomical phenomena never before observed by humans.
| |
| | |
| The sources of gravitational waves described above are in the low-frequency end of the gravitational-wave spectrum (10<sup>−7</sup> to 10<sup>5</sup> Hz). An astrophysical source at the high-frequency end of the gravitational-wave spectrum (above 10<sup>5</sup> Hz and probably 10<sup>10</sup> Hz) generates{{Clarify|date=May 2009}} relic gravitational waves that are theorized to be faint imprints of the Big Bang like the cosmic microwave background (see [[gravitational wave background]]).<ref>L. P. Grishchuk (1976), "Primordial Gravitons and the Possibility of Their Observation," Sov. Phys. JETP Lett. 23, p. 293.</ref> At these high frequencies it is potentially possible that the sources may be "man made"<ref name = "HI"/> that is, gravitational waves generated and detected in the laboratory.<ref>Braginsky, V. B., Rudenko and Valentin, N. Section 7: "Generation of gravitational waves in the laboratory," ''Physics Report'' (Review section of ''Physics Letters''), 46, No. 5. 165–200, (1978).</ref><ref name= "BLW06">Li, Fangyu, Baker, R. M L, Jr., and Woods, R. C., "Piezoelectric-Crystal-Resonator High-Frequency Gravitational Wave Generation and Synchro-Resonance Detection," in the proceedings of ''Space Technology and Applications International Forum (STAIF-2006)'', edited by M.S. El-Genk, American Institute of Physics Conference Proceedings, Melville NY 813: 2006.</ref>
| |
| | |
| ===Energy, momentum, and angular momentum carried by gravitational waves===
| |
| Waves familiar from other areas of physics such as water waves, sound waves, and electromagnetic waves are able to carry [[energy]], [[momentum]], and [[angular momentum]]. By carrying these away from a source, waves are able to rob that source of its energy as well as its linear and angular momentum. Gravitational waves perform the same function. Thus, for example, a binary system loses angular momentum as the two orbiting objects spiral towards each other—the angular momentum is radiated away by gravitational waves.
| |
| | |
| The waves can also carry off linear momentum, a possibility that has some interesting implications for [[astrophysics]].<ref name="Merritt2004">{{Cite journal
| |
| | last = Merritt
| |
| | first = D.
| |
| | author-link = David Merritt
| |
| | title = Consequences of Gravitational Wave Recoil
| |
| | journal = The Astrophysical Journal Letters
| |
| | volume = 607
| |
| | issue = 1
| |
| | pages = L9–L12
| |
| | bibcode = 2004ApJ...607L...9M
| |
| | date = May 2004
| |
| | doi = 10.1086/421551
| |
| | last2 = Milosavljevi
| |
| | first2 = Milo
| |
| | last3 = Favata
| |
| | first3 = Marc
| |
| | last4 = Hughes
| |
| | first4 = Scott A.
| |
| | last5 = Holz
| |
| | first5 = Daniel E.
| |
| | postscript = <!--None-->
| |
| | arxiv = astro-ph/0402057
| |
| | display-authors = 1 }}
| |
| </ref> After two supermassive black holes coalesce, emission of linear momentum can produce a "kick" with amplitude as large as 4000 km/s. This is fast enough to eject the coalesced black hole completely from its host galaxy. Even if the kick is too small to eject the black hole completely, it can remove it temporarily from the nucleus of the galaxy, after which it will oscillate about the center, eventually coming to rest.<ref>{{Cite journal
| |
| | last = Gualandris
| |
| | first = A.
| |
| | last2 = Merritt
| |
| | first2 = D. et al.
| |
| | author2-link = David Merritt
| |
| | title = Ejection of Supermassive Black Holes from Galaxy Cores
| |
| | journal = The Astrophysical Journal
| |
| | volume = 678
| |
| | issue = 2
| |
| | pages = 780–797
| |
| | bibcode = 2008ApJ...678..780G
| |
| | date = May 2008
| |
| | doi = 10.1086/586877
| |
| | postscript = <!--None-->
| |
| | display-authors = 2
| |
| | author3 = <Please add first missing authors to populate metadata.>
| |
| | arxiv = 0708.0771 }}
| |
| </ref> A kicked black hole can also carry a star cluster with it, forming a [[hyper-compact stellar system]].<ref name="MSK09">{{Cite journal
| |
| | last1 = Merritt | first1 = D.
| |
| | author-link = David Merritt
| |
| | last2 = Schnittman | first2 = J. D.
| |
| | last3 = Komossa | first3 = S.
| |
| | title = Hypercompact Stellar Systems Around Recoiling Supermassive Black Holes
| |
| | journal = The Astrophysical Journal
| |
| | volume = 699
| |
| | issue = 2
| |
| | pages = 1690–1710
| |
| | bibcode = 2009ApJ...699.1690M
| |
| | year = 2009
| |
| | doi = 10.1088/0004-637X/699/2/1690
| |
| | postscript = <!--None--> | arxiv = 0809.5046 }}
| |
| </ref> Or it may carry gas, allowing the recoiling black hole to appear temporarily as a "[[HE0450-2958|naked quasar]]".
| |
| The [[quasar]] [[SDSS J0927+2943|SDSS J092712.65+294344.0]] is believed to contain a recoiling supermassive black hole.<ref name="kom">{{Cite journal | last = Komossa | first = S. | last2 = Zhou | first2 = H. | last3 = Lu | first3 = H. | title = A Recoiling Supermassive Black Hole in the Quasar SDSS J092712.65+294344.0? | journal = The Astrophysical Journal
| |
| | volume = 678| pages = L81–L84 | date = May 2008 | year = 2008 | bibcode = 2008ApJ...678L..81K | doi = 10.1086/588656 | issue = 2 | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}} | arxiv = 0804.4585 }}</ref>
| |
| | |
| == Detecting gravitational waves ==
| |
| | |
| === Difficulties in detection ===
| |
| | |
| As of October 18, 2013, gravitational waves have not been ''directly'' observed. This knowledge gap is primarily due to the massive presence of noise in the low frequencies where antennas currently operate. Gravitational waves are expected to have frequencies <math>10^{-16}\, \mathrm{Hz}<f<10^4\, \mathrm{Hz}</math>
| |
| <ref name=" arXiv:gr-qc/9506086">{{cite journal | url=http://arxiv.org/abs/gr-qc/9506086 | title=Gravitational Waves | author=Thorne, Kip S. | journal=Cornell University Library | year=1995}}</ref>
| |
| | |
| ===Ground-based interferometers===
| |
| {{Main|Gravitational wave detector}}
| |
| Though the Hulse-Taylor observations were very important, they give only ''indirect'' evidence for gravitational waves. A more conclusive observation would be a ''direct'' measurement of the effect of a passing gravitational wave, which could also provide more information about the system that generated it. Any such direct detection is complicated by the [[Orders of magnitude (length)|extraordinarily small]] effect the waves would produce on a detector. The amplitude of a spherical wave will fall off as the inverse of the distance from the source (the <math>1/R</math> term in the formulas for <math>h</math> above). Thus, even waves from extreme systems like merging binary black holes die out to very small amplitude by the time they reach the Earth. Astrophysicists expect that some gravitational waves passing the Earth may be as large as ''h'' ≈ 10<sup>−20</sup>, but generally no bigger.{{Citation needed|date=February 2009}}
| |
| | |
| A simple device theorised to detect the expected wave motion is called a [[Weber bar]] — a large, solid bar of metal isolated from outside vibrations. This type of instrument was the first type of gravitational wave detector. Strains in space due to an incident gravitational wave excite the bar's [[resonant frequency]] and could thus be amplified to detectable levels. Conceivably, a nearby supernova might be strong enough to be seen without resonant amplification. With this instrument, [[Joseph Weber]] claimed to have detected daily signals of gravitational waves. His results, however, were contested in 1974 by physicists [[Richard Garwin]] and [[David Douglass]]. Modern forms of the Weber bar are still operated, [[cryogenically]] cooled, with [[SQUID|superconducting quantum interference devices]] to detect vibration. Weber bars are not sensitive enough to detect anything but extremely powerful gravitational waves.<ref>For a review of early experiments using Weber bars, see {{cite journal | first=J. | last=Levine | title=Early Gravity-Wave Detection Experiments, 1960–1975 | journal= Physics in Perspective (Birkhäuser Basel) | volume=6 | issue= 1 | pages=42–75 |date=April 2004 | doi=10.1007/s00016-003-0179-6 | bibcode = 2004PhP.....6...42L }}</ref>
| |
| | |
| [[MiniGRAIL]] is a spherical gravitational wave antenna using this principle. It is based at [[Leiden University]], consisting of an exactingly machined 1150 kg sphere cryogenically cooled to 20 mK.<ref>[http://www.minigrail.nl/AboutMiniGRAIL/AboutMiniGRAIL-index.html Gravitational Radiation Antenna In Leiden]</ref> The spherical configuration allows for equal sensitivity in all directions, and is somewhat experimentally simpler than larger linear devices requiring high vacuum. Events are detected by measuring [[Multipole moments|deformation of the detector sphere]]. MiniGRAIL is highly sensitive in the 2–4 kHz range, suitable for detecting gravitational waves from rotating neutron star instabilities or small black hole mergers.<ref name='MiniGRAIL_2000'>{{Cite journal | first=Arlette | last=de Waard| coauthors=Luciano Gottardi, and Giorgio Frossati| contribution=Spherical Gravitational Wave Detectors: cooling and quality factor of a small CuAl6% sphere| title=Marcel Grossman meeting on General Relativity| publisher= | place=Rome| pages= | date=Italy| year=2000| id= | contribution-url= | format=PDF| postscript=<!--None--> }}</ref>
| |
| | |
| [[Image:Ligo.svg|thumb|right|350px|A schematic diagram of a laser interferometer.]]
| |
| A more sensitive class of detector uses laser [[interferometry]] to measure gravitational-wave induced motion between separated 'free' masses.<ref>The idea of using laser interferometry for gravitational wave detection was first mentioned by Gerstenstein and Pustovoit 1963 Sov. Phys.–JETP 16 433. Weber mentioned it in an unpublished laboratory notebook. [[Rainer Weiss]] first described in detail a practical solution with an analysis of realistic limitations to the technique in R. Weiss (1972). "Electromagetically Coupled Broadband Gravitational Antenna". Quarterly Progress Report, Research Laboratory of Electronics, MIT 105: 54.</ref> This allows the masses to be separated by large distances (increasing the signal size); a further advantage is that it is sensitive to a wide range of frequencies (not just those near a resonance as is the case for Weber bars). Ground-based interferometers are now operational. Currently, the most sensitive is [[LIGO]] — the Laser Interferometer Gravitational Wave Observatory. LIGO has three detectors: one in [[Livingston, Louisiana]]; the other two (in the same vacuum tubes) at the [[Hanford site]] in [[Richland, Washington]]. Each consists of two [[Fabry–Pérot interferometer|light storage arms]] that are 2 to 4 kilometers in length. These are at 90 degree angles to each other, with the light passing through 1m diameter vacuum tubes running the entire 4 kilometers. A passing gravitational wave will slightly stretch one arm as it shortens the other. This is precisely the motion to which an interferometer is most sensitive.
| |
| | |
| Even with such long arms, the strongest gravitational waves will only change the distance between the ends of the arms by at most roughly 10<sup>−18</sup> meters. LIGO should be able to detect gravitational waves as small as <math>h \sim 5\times 10^{-20}</math>. Upgrades to LIGO and other detectors such as [[Virgo interferometer|Virgo]], [[GEO 600]], and [[TAMA 300]] should increase the sensitivity still further; the next generation of instruments (Advanced LIGO and Advanced Virgo) will be more than ten times more sensitive. Another highly sensitive interferometer ([[LCGT]]) is currently in the design phase. A key point is that a tenfold increase in sensitivity (radius of 'reach') increases the volume of space accessible to the instrument by one thousand times. This increases the rate at which detectable signals should be seen from one per tens of years of observation, to tens per year.<ref name=Abadie2010 />
| |
| | |
| Interferometric detectors are limited at high frequencies by [[shot noise]], which occurs because the lasers produce photons randomly; one analogy is to rainfall—the rate of rainfall, like the laser intensity, is measurable, but the raindrops, like photons, fall at random times, causing fluctuations around the average value. This leads to noise at the output of the detector, much like radio static. In addition, for sufficiently high laser power, the random momentum transferred to the test masses by the laser photons shakes the mirrors, masking signals at low frequencies. Thermal noise (e.g., [[Brownian motion]]) is another limit to sensitivity. In addition to these 'stationary' (constant) noise sources, all ground-based detectors are also limited at low frequencies by [[seismic]] noise and other forms of environmental vibration, and other 'non-stationary' noise sources; creaks in mechanical structures, lightning or other large electrical disturbances, etc. may also create noise masking an event or may even imitate an event. All these must be taken into account and excluded by analysis before a detection may be considered a true gravitational wave event.
| |
| | |
| Space-based interferometers, such as [[Laser Interferometer Space Antenna|LISA]] and [[DECIGO]], are also being developed. LISA's design calls for three test masses forming an equilateral triangle, with lasers from each spacecraft to each other spacecraft forming two independent interferometers. LISA is planned to occupy a solar orbit trailing the Earth, with each arm of the triangle being five million kilometers. This puts the detector in an [[Interplanetary medium|excellent vacuum]] far from Earth-based sources of noise, though it will still be susceptible to shot noise, as well as artifacts caused by [[cosmic ray]]s and [[solar wind]].
| |
| | |
| There are currently two detectors focusing on detection at the higher end of the gravitational wave spectrum (10<sup>−7</sup> to 10<sup>5</sup> Hz): one at [[University of Birmingham]], England, and the other at [[Istituto Nazionale di Fisica Nucleare|INFN]] Genoa, Italy. A third is under development at [[Chongqing University]], China. The Birmingham detector measures changes in the polarization state of a [[microwave]] beam circulating in a closed loop about one meter across. Two have been fabricated and they are currently expected to be sensitive to periodic spacetime strains of <math>h\sim{2 \times 10^{-13}/\sqrt{\mathrm{Hz}}} </math>, given as an [[Spectral density|amplitude spectral density]]. The INFN Genoa detector is a resonant antenna consisting of two coupled spherical [[superconducting]] harmonic oscillators a few centimeters in diameter. The oscillators are designed to have (when uncoupled) almost equal resonant frequencies. The system is currently expected to have a sensitivity to periodic spacetime strains of <math>h\sim{2 \times 10^{-17}/\sqrt{\mathrm{Hz}}} </math>, with an expectation to reach a sensitivity of <math>h\sim{2 \times 10^{-20}/\sqrt{\mathrm{Hz}}} </math>. The Chongqing University detector is planned to detect relic high-frequency gravitational waves with the predicted typical parameters ?<sub>g</sub> ~ 10<sup>10</sup> Hz (10 GHz) and ''h'' ~ 10<sup>−30</sup>-10<sup>−31</sup>.
| |
| | |
| ===Using pulsar timing arrays===
| |
| | |
| Pulsars are rapidly rotating stars. A pulsar emits beams of radio waves that, like lighthouse beams, sweep through the sky as the pulsar rotates. The signal from a pulsar can be detected by radio telescopes as a series of regularly spaced pulses, essentially like the ticks of a clock. Gravitational waves affect the time it takes the pulses to travel from the pulsar to a telescope on Earth. A [[pulsar timing array]] uses [[millisecond pulsar]]s to seek out perturbations due to gravitational waves in measurements of pulse arrival times at a telescope, in other words, to look for deviations in the clock ticks. In particular, pulsar timing arrays can search for a distinct pattern of correlation and anti-correlation between the signals over an array of different pulsars (resulting in the name “pulsar timing array"). Although pulsar pulses travel through space for hundreds or thousands of years to reach us, pulsar timing arrays are sensitive to perturbations in their travel time of much less than a millionth of a second.
| |
|
| |
| Globally there are three active pulsar timing array projects. The [http://nanograv.org North American Nanohertz Gravitational Wave Observatory] uses data collected by the Arecibo Radio Telescope and [[Green Bank Telescope]]. The [http://www.atnf.csiro.au/research/pulsar/ppta/ Parkes Pulsar Timing Array] at the [[Parkes Observatory|Parkes radio-telescope]] has been collecting data since March 2005. The [http://www.epta.eu.org/ European Pulsar Timing Array] uses data from the four largest telescopes in Europe: the [[Lovell Telescope]], the [[Westerbork Synthesis Radio Telescope]], the [[Effelsberg Telescope]] and the [[Nancay Radio Telescope]]. (Upon completion the Sardinia Radio Telescope will be added to the EPTA also.) These three projects have begun collaborating under the title of the [[International Pulsar Timing Array]] project.
| |
| | |
| ===Einstein@Home===
| |
| {{Main|Einstein@Home}}
| |
| In some sense, the easiest signals to detect should be constant sources. Supernovae and neutron star or black hole mergers should have larger amplitudes and be more interesting, but the waves generated will be more complicated. The waves given off by a spinning, aspherical neutron star would be "[[monochrome|monochromatic]]"—like a [[pure tone]] in [[acoustics]]. It would not change very much in amplitude or frequency.
| |
| | |
| The [[Einstein@Home]] project is a [[distributed computing]] project similar to [[SETI@home]] intended to detect this type of simple gravitational wave. By taking data from LIGO and GEO, and sending it out in little pieces to thousands of volunteers for parallel analysis on their home computers, Einstein@Home can sift through the data far more quickly than would be possible otherwise.<ref>[http://einstein.phys.uwm.edu/ Einstein@Home<!-- Bot generated title -->]</ref>
| |
| | |
| ==Mathematics==
| |
| [[Einstein's equations]] form the fundamental law of general relativity. The curvature of spacetime can be expressed mathematically using the [[metric tensor]] — denoted <math>g_{\mu \nu} \,</math>. The metric holds information regarding how distances are measured in the space under consideration. Because the propagation of gravitational waves through space and time change distances, we will need to use this to find the solution to the wave equation.
| |
| | |
| Spacetime curvature is also expressed with respect to a [[covariant derivative]], <math>\nabla \,</math>, in the form of the [[Einstein tensor]], <math>G_{\mu \nu}</math>. This curvature is related to the [[stress-energy tensor]], <math>T_{\mu\nu}</math>, by the key equation
| |
| | |
| :<math>G_{\mu \nu} = \frac{8\pi G_N}{c^4} T_{\mu \nu},</math>
| |
| | |
| where <math>G_N</math> is Newton's [[gravitational constant]], and <math>c</math> is the speed of light. We assume [[geometrized units]], so <math>G_N = 1 = c</math>.
| |
| | |
| With some simple assumptions, Einstein's equations can be rewritten to show explicitly that they are [[wave equation]]s. To begin with, we adopt some coordinate system, like <math>(t,r,\theta,\phi) \,</math>. We define the "flat-space metric" <math>\eta_{\mu\nu} \,</math> to be the quantity that — in this coordinate system — has the components we would expect for the flat space metric. For example, in these spherical coordinates, we have
| |
| :<math>
| |
| \eta_{\mu \nu} =
| |
| \begin{bmatrix}
| |
| -1 & 0 & 0 & 0 \\
| |
| 0 & 1 & 0 & 0 \\
| |
| 0 & 0 & r^2 & 0 \\
| |
| 0 & 0 & 0 & r^2 \sin^2\theta
| |
| \end{bmatrix}.</math>
| |
| This flat-space metric has no physical significance; it is a purely mathematical device necessary for the analysis. Tensor indices are raised and lowered using this "flat-space metric".
| |
| | |
| Now, we can also think of the physical metric <math>g_{\mu \nu}</math> as a [[matrix (mathematics)|matrix]], and find its [[determinant]], <math>\det g</math>. Finally, we define a quantity
| |
| | |
| :<math>\bar{h}^{\alpha \beta} \equiv \eta^{\alpha \beta} - \sqrt{|\det g|} g^{\alpha \beta} \,</math> .
| |
| | |
| This is the crucial field, which will represent the radiation. It is possible (at least in an [[asymptotically flat spacetime]]) to choose the coordinates in such a way that this quantity satisfies the "de Donder" gauge conditions (conditions on the coordinates):
| |
| | |
| :<math>\nabla_\beta\, \bar{h}^{\alpha \beta} = 0,</math>
| |
| | |
| where <math>\nabla</math> represents the flat-space derivative operator. These equations say that the [[divergence]] of the field is zero. The linear Einstein equations can now be written<ref name="RMP80">{{cite journal | last=Thorne | first=Kip | title=Multipole expansions of gravitational radiation | journal=Reviews of Modern Physics | volume=52 |date=April 1980 | issue=2 | doi=10.1103/RevModPhys.52.299 | pages=299 | bibcode=1980RvMP...52..299T}}</ref> as
| |
| | |
| :<math>\Box \bar{h}^{\alpha \beta} = -16\pi \tau^{\alpha \beta} \,</math> ,
| |
| | |
| where <math>\Box = -\partial_t^2 + \Delta \,</math> represents the flat-space [[d'Alembertian]] operator, and <math>\tau^{\alpha \beta} \,</math> represents the stress-energy tensor plus quadratic terms involving <math>\bar{h}^{\alpha \beta} \,</math>. This is just a wave equation for the field with a source, despite the fact that the source involves terms quadratic in the field itself. That is, it can be shown that solutions to this equation are waves traveling with velocity 1 in these coordinates.
| |
| | |
| ===Linear approximation===
| |
| The equations above are valid everywhere — near a black hole, for instance. However, because of the complicated source term, the solution is generally too difficult to find analytically. We can often assume that space is nearly flat, so the metric is nearly equal to the <math>\eta^{\alpha \beta} \,</math> tensor. In this case, we can neglect terms quadratic in <math>\bar{h}^{\alpha \beta} \,</math>, which means that the <math>\tau^{\alpha \beta} \,</math> field reduces to the usual stress-energy tensor <math>T^{\alpha \beta} \,</math>. That is, Einstein's equations become
| |
| | |
| :<math>\Box \bar{h}^{\alpha \beta} = -16\pi T^{\alpha \beta} \,</math> .
| |
| | |
| If we are interested in the field far from a source, however, we can treat the source as a point source; everywhere else, the stress-energy tensor would be zero, so
| |
| | |
| :<math>\Box \bar{h}^{\alpha \beta} = 0 \,</math> .
| |
| | |
| Now, this is the usual homogeneous wave equation — one for each component of <math>\bar{h}^{\alpha \beta} \,</math>. Solutions to this equation are well known. For a wave moving away from a point source, the radiated part (meaning the part that dies off as <math>1/r \,</math> far from the source) can always be written in the form <math>A(t-r,\theta,\phi)/r \,</math>, where <math>A \,</math> is just some function. It can be shown<ref>{{cite book | author=C. W. Misner, K. S. Thorne, and J. A. Wheeler | title=Gravitation | publisher=W. H. Freeman and Co. | year=1973}}</ref> that — to a linear approximation — it is always possible to make the field traceless. Now, if we further assume that the source is positioned at <math>r=0</math>, the general solution to the wave equation in spherical coordinates is
| |
| | |
| :<math>
| |
| \begin{array}{lcl}
| |
| \bar{h}^{\alpha \beta} & = &
| |
| \frac{1}{r}\, \begin{bmatrix}
| |
| 0 & 0 & 0 & 0 \\
| |
| 0 & 0 & 0 & 0 \\
| |
| 0 & 0 & A_{+}(t-r,\theta,\phi) & A_{\times}(t-r,\theta,\phi) \\
| |
| 0 & 0 & A_{\times}(t-r,\theta,\phi) & -A_{+}(t-r,\theta,\phi)
| |
| \end{bmatrix} \\
| |
| \\
| |
| & \equiv &
| |
| \begin{bmatrix}
| |
| 0 & 0 & 0 & 0 \\
| |
| 0 & 0 & 0 & 0 \\
| |
| 0 & 0 & h_{+}(t-r,r,\theta,\phi) & h_{\times}(t-r,r,\theta,\phi) \\
| |
| 0 & 0 & h_{\times}(t-r,r,\theta,\phi) & -h_{+}(t-r,r,\theta,\phi)
| |
| \end{bmatrix}
| |
| \end{array}
| |
| \,</math>
| |
| | |
| where we now see the origin of the two polarizations.
| |
| | |
| ===Relation to the source===
| |
| If we know the details of a source — for instance, the parameters of the orbit of a binary — we can relate the source's motion to the gravitational radiation observed far away. With the relation
| |
| | |
| :<math>\Box \bar{h}^{\alpha \beta} = -16\pi \tau^{\alpha \beta} \,</math> ,
| |
| | |
| we can write the solution in terms of the tensorial [[Green's function]] for the d'Alembertian operator:<ref name="RMP80"/>
| |
| | |
| :<math>
| |
| \bar{h}^{\alpha \beta} (t,\vec{x}) =
| |
| -16\pi \int\, G^{\alpha \beta}_{\gamma \delta} (t,\vec{x};t',\vec{x}')\, \tau^{\gamma \delta}(t',\vec{x}')\, \mathrm{d}t'\, \mathrm{d}^3x'
| |
| </math> .
| |
| | |
| Though it is possible to expand the Green's function in tensor [[spherical harmonics]], it is easier to simply use the form
| |
| | |
| :<math>G^{\alpha \beta}_{\gamma \delta} (t,\vec{x};t',\vec{x}') = \frac{1}{4\pi} \delta_{\gamma}^\alpha\, \delta_{\delta}^\beta\, \frac{\delta(t\pm|\vec{x}-\vec{x}'|-t')} {|\vec{x}-\vec{x}'|}</math> ,
| |
| | |
| where the positive and negative signs correspond to ingoing and outgoing solutions, respectively. Generally, we are interested in the outgoing solutions, so
| |
| | |
| :<math>
| |
| \bar{h}^{\alpha \beta} (t,\vec{x}) =
| |
| -4 \int\, \frac{\tau^{\alpha \beta}(t-|\vec{x}-\vec{x}'|,\vec{x}')}{|\vec{x}-\vec{x}'|}\, \mathrm{d}^3x'
| |
| </math> .
| |
| | |
| If the source is confined to a small region very far away, to an excellent approximation we have:
| |
| | |
| :<math>
| |
| \bar{h}^{\alpha \beta} (t,\vec{x}) \approx
| |
| -\frac{4}{r}\, \int\, \tau^{\alpha \beta}(t-r,\vec{x}')\, \mathrm{d}^3x'
| |
| </math> ,
| |
| | |
| where <math>r=|\vec{x}|</math> .
| |
| | |
| Now, because we will eventually only be interested in the spatial components of this equation (time components can be set to zero with a coordinate transformation), and we are integrating this quantity — presumably over a region of which there is no boundary — we can put this in a different form. Ignoring divergences with the help of [[Stokes' theorem]] and an empty boundary, we can see that
| |
| | |
| :<math>
| |
| \int\, \tau^{i j}(t-r,\vec{x}')\, \mathrm{d}^3x'
| |
| =
| |
| \int\, x'^i x'^j \nabla_k \nabla_l \tau^{k l} (t-r,\vec{x}')\, \mathrm{d}^3x'
| |
| </math> ,
| |
| | |
| Inserting this into the above equation, we arrive at
| |
| | |
| :<math>
| |
| \bar{h}^{i j} (t,\vec{x}) \approx
| |
| -\frac{4}{r}\, \int\, x'^i x'^j \nabla_k \nabla_l \tau^{k l} (t-r,\vec{x}')\, \mathrm{d}^3x'
| |
| </math> ,
| |
| | |
| Finally, because we have chosen to work in coordinates for which <math>\nabla_\beta\, \bar{h}^{\alpha \beta} = 0</math>, we know that <math>\nabla_\beta\, \tau^{\alpha \beta} = 0</math>. With a few simple manipulations, we can use this to prove that
| |
| | |
| :<math>\nabla_0 \nabla_0 \tau^{00} = \nabla_j \nabla_k \tau^{jk}</math> .
| |
| | |
| With this relation, the expression for the radiated field is
| |
| | |
| :<math>
| |
| \bar{h}^{i j} (t,\vec{x}) \approx
| |
| -\frac{4}{r}\, \frac{\mathrm{d}^2}{\mathrm{d}t^2}\, \int\, x'^i x'^j \tau^{0 0} (t-r,\vec{x}')\, \mathrm{d}^3x'
| |
| </math> .
| |
| | |
| In the linear case, <math>\tau^{00} = \rho</math>, the density of mass-energy.
| |
| | |
| To a very good approximation, the density of a simple binary can be described by a pair of delta-functions, which eliminates the integral. Explicitly, if the masses of the two objects are <math>M_1</math> and <math>M_2</math>, and the positions are <math>\vec{x}_1</math> and <math>\vec{x}_2</math>, then
| |
| | |
| :<math>\rho(t-r,\vec{x}') = M_1 \delta^3(\vec{x}'-\vec{x}_1(t-r)) + M_2 \delta^3(\vec{x}'-\vec{x}_2(t-r))</math> .
| |
| | |
| We can use this expression to do the integral above:
| |
| | |
| :<math>
| |
| \bar{h}^{i j} (t,\vec{x}) \approx
| |
| -\frac{4}{r}\, \frac{\mathrm{d}^2}{\mathrm{d}t^2}\, \left\{ M_1 x_1^i(t-r) x_1^j(t-r) + M_2 x_2^i(t-r) x_2^j(t-r) \right\}
| |
| </math> .
| |
| | |
| Using mass-centered coordinates, and assuming a circular binary, this is
| |
| | |
| :<math>
| |
| \bar{h}^{i j} (t,\vec{x}) \approx
| |
| -\frac{4}{r}\, \frac{M_1 M_2}{R}\, n^i(t-r) n^j(t-r)
| |
| </math> ,
| |
| | |
| where <math>\vec{n} = \vec{x}_1 / |\vec{x}_1|</math>. Plugging in the known values of <math>\vec{x}_1(t-r)</math>, we obtain the expressions given above for the radiation from a simple binary.
| |
| | |
| ==References==
| |
| {{reflist|3}}
| |
| | |
| ==See also==
| |
| * [[General relativity]]
| |
| * [[Big Bang Observer]] (BBO), proposed successor to LISA
| |
| * [[Deci-hertz Interferometer Gravitational wave Observatory|DECIGO]] "Deci-hertz Interferometer Gravitational wave Observatory", the planned laser interferometric detector in space
| |
| * [[Gravitational field]]
| |
| * [[Gravitomagnetism]]
| |
| * [[Graviton]]
| |
| * [[Gravitational wave astronomy|Graviton astronomy]]
| |
| * [[Hawking radiation]], for gravitationally induced electromagnetic radiation from black holes
| |
| * [[HM Cancri]]
| |
| * [[Laser Interferometer Space Antenna|LISA]] the proposed Laser Interferometer Space Antenna
| |
| * [[LIGO]], [[VIRGO]], [[GEO 600]], and [[TAMA 300]] — Gravitational wave detectors
| |
| * [[Linearised Einstein field equations]]
| |
| * [[Orbital resonance]]
| |
| * [[Peres metric]]
| |
| * [[pp-wave spacetime]], for an important class of exact solutions modelling gravitational radiation
| |
| * [[Spin-flip]], a consequence of gravitational wave emission from binary [[supermassive black hole]]s
| |
| * [[Sticky bead argument]], for a physical way to see that gravitational radiation should carry energy
| |
| * [[Tidal force]]
| |
| | |
| ==Further reading==
| |
| * Chakrabarty, Indrajit, "[http://arxiv.org/pdf/physics/9908041 Gravitational Waves: An Introduction]". arXiv:physics/9908041 v1, Aug 21, 1999.
| |
| * Landau, L. D. and Lifshitz, E. M., The Classical Theory of Fields (Pergamon Press),(1987).
| |
| * Will, Clifford M., ''[http://www.livingreviews.org/lrr-2006-3/ The Confrontation between General Relativity and Experiment]''. Living Rev. Relativity 9 (2006) 3.
| |
| * Peter Saulson, "Fundamentals of Interferometric Gravitational Wave Detectors", World Scientific, 1994.
| |
| * J. Bicak, W.N. Rudienko, "Gravitacionnyje wolny w OTO i probliema ich obnarużenija", Izdatielstwo Moskovskovo Universitieta, 1987.
| |
| * A. Kułak, "Electromagnetic Detectors of Gravitational Radiation", PhD Thesis, Cracow 1980 (In Polish).
| |
| * P. Tatrocki, "On intuitive description of graviton detector", www.philica.com .
| |
| * P. Tatrocki, "Can the LIGO, VIRGO, GEO600, AIGO, TAMA, LISA detectors really detect?", www.philica.com .
| |
| | |
| ==Bibliography==
| |
| * [[Michael Berry (physicist)|Berry, Michael]], ''Principles of cosmology and gravitation'' (Adam Hilger, Philadelphia, 1989). ISBN 0-85274-037-9
| |
| * [[Harry Collins|Collins, Harry]], ''Gravity's Shadow: the search for gravitational waves'', University of Chicago Press, 2004.
| |
| * [[Jim Peebles|P. J. E. Peebles]], ''Principles of Physical Cosmology'' (Princeton University Press, Princeton, 1993). ISBN 0-691-01933-9.
| |
| * [[John Archibald Wheeler|Wheeler, John Archibald]] and Ciufolini, Ignazio, ''Gravitation and Inertia'' (Princeton University Press, Princeton, 1995). ISBN 0-691-03323-4.
| |
| *Woolf, Harry, ed., ''Some Strangeness in the Proportion'' (Addison–Wesley, Reading, Massachusetts, 1980). ISBN 0-201-09924-1.
| |
| | |
| ==External links==
| |
| {{Commonscat|Gravitational waves}}
| |
| * {{britannica|242499|Gravitational waves}}
| |
| * {{In Our Time|Gravitational Waves|b007h8gv|Gravitational_Waves}}
| |
| * [http://brownbag.lisascience.org The LISA Brownbag] – Selection of the most significant e-prints related to LISA science
| |
| * [http://www.astroparticle.org/ Astroparticle.org]. To know everything about astroparticle physics, including gravitational waves
| |
| * [http://elmer.tapir.caltech.edu/ph237/CourseMaterials.html Caltech's Physics 237-2002 Gravitational Waves by Kip Thorne] '''Video plus notes:''' Graduate level but does not assume knowledge of General Relativity, Tensor Analysis, or Differential Geometry; Part 1: Theory (10 lectures), Part 2: Detection (9 lectures)
| |
| * [http://www.astronomycast.com/astronomy/ep-71-gravitational-waves www.astronomycast.com January 14, 2008 Episode 71: Gravitational Waves]
| |
| * [http://www.ligo.caltech.edu/ Laser Interferometer Gravitational Wave Observatory]. LIGO Laboratory, operated by the [[California Institute of Technology]] and the [[Massachusetts Institute of Technology]]
| |
| * [http://www.ligo.org/ The LIGO Scientific Collaboration]
| |
| * [http://www.ligo.caltech.edu/einstein.ram Einstein's Messengers] – The LIGO Movie by [[National Science Foundation|NSF]]
| |
| * [http://einstein.phys.uwm.edu/ Home page for Einstein@Home project], a distributed computing project processing raw data from LIGO Laboratory, searching for gravitational waves
| |
| * [http://archive.ncsa.uiuc.edu/Cyberia/NumRel/GravWaves.html The National Center for Supercomputing Applications] – a numerical relativity group
| |
| * [http://www.black-holes.org/gwa1.html Caltech Relativity Tutorial] – A basic introduction to gravitational waves, and astrophysical systems giving off gravitational waves
| |
| * [http://arxiv.org/abs/gr-qc/0211084 Resource Letter GrW-1: Gravitational waves] – a list of books, journals and web resources compiled by Joan Centrella for research into gravitational waves
| |
| * [http://www.rri.res.in/htmls/tp/GravRad-grim.pdf Mathematical and Physical Perspectives on Gravitational Radiation] – written by B F Schutz of the [[Max Planck Institute]] explaining the significance and background of some key concepts in gravitational radiation
| |
| * [http://www.sciencebits.com/BlackHoleSimulation Binary BH Merger] – estimating the radiated power and merger time of a BH binary using dimensional analysis
| |
| | |
| {{Gravitational wave observatories}}
| |
| {{Relativity}}
| |
| | |
| {{DEFAULTSORT:Gravitational Wave}}
| |
| [[Category:Binary stars]]
| |
| [[Category:Black holes]]
| |
| [[Category:Effects of gravitation]]
| |
| [[Category:Gravitational waves]]
| |
| | |
| [[he:כבידה#גלי כבידה]]
| |