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The strength to do this comes from the chemical forces in her muscles, which come from elec- Gravitational waves can be emitted by black holes, • tromagnetic interactions. which are described in the article General Rel- In fact, the electromagnetic force between the elec- ativity and Gravitation. Indeed, gravita- tron and the proton in a hydrogen atom is 2 1039 times tional waves provide only way to make direct ob- bigger than the gravitational force between× them. The servations of these objects. Since there is now reason that gravity can nevertheless dominate on the strong indirect evidence that giant black holes in- cosmic scale is that opposite electrical charges cancel habit the centers of many (or even most) galax- each other, while the gravitational forces of all the par- ies (see Supermassive black holes in AGN), ticles add. and since smaller ones are common in the Galaxy Another fact about gravity that was known to New- (see Black Hole Candidates in X-Ray Bi- ton is what is now called the equivalence principle (see naries), there is great interest in making direct General Relativity and Gravitation). This is observations of them. the principle that all bodies accelerate in the same way Gravitational waves can come from extraordinar- in a gravitational field, so that the trajectory that a • ily early in the history of the Universe. The elec- freely falling body (a body influenced only by gravity) tromagnetic radiation from the Big Bang is called follows in a given gravitational field depends only on its the Cosmic Microwave Background. Obser- starting position and velocity, not on what it is made vations of it describe the Universe at it was about of. 105 years after the Big Bang. Studies of cosmo- Imagine now a machine made in some way to detect logical Nucleosynthesis give information about gravitational waves. Whatever the method of detec- what the Universe was like as little as 3 minutes tion, a wave needs somehow to alter the internal state after the Big Bang. Gravitational waves, if they of the detector. If the wave carries a gravitational field can be detected, would picture the Universe when that is completely uniform across the detector, then by it was only perhaps 10−24 seconds old, just at the the equivalence principle all of the parts of the detector end of Inflation. will accelerate together, and its state will not change at all. To detect a gravitational wave, the machine must Gravitational radiation is the last fundamental measure the non-uniformities of the gravitational field • prediction of Einstein’s general relativity that has across a detector. not yet been directly verified. If another theory of These non-uniformities are called tidal forces, be- gravity is correct, then differences could in prin- cause they produce the stretching effects that raise tides ciple show up in the properties of gravitational on the Earth. Gravitational waves are traveling tidal waves, such as their polarization. In principle, forces. there must be a better theory of gravity, since Newton’s theory of gravity had no gravitational general relativity is not a quantum theory, a defi- waves. For Newton, if a gravitational field changed in ciency that theoretical physicists today are work- some way, that change took place instantaneously ev- ing hard to remedy. The majority belief today is erywhere in space. This is not a wave. Let us consider that there should be a unified theory of the fun- what we mean by the term “wave” in ordinary language. damental forces, in which gravitation is related to Imagine a child’s rubber duck floating in a bath tub the other forces. Evidence for the nature of this half full of water. If a child presses down on the duck relation could show up in observations of grav- very gently, until is is nearly submerged, then the level itational waves, particularly those from the Big of the water will rise everywhere in a nearly uniform Bang. way, and this is not called a wave. If instead he drops the duck, then the disturbance rises around the base These motivations and their implications are devel- of the duck rapidly, moves away from it, and eventually oped in the following sections. Each section begins with reaches the walls. This is a wave. Wave motion requires an introduction to the physical ideas and then develops a finite speed for the propagation of disturbances. If the some of the mathematical details. disturbance is very slow, as for the floating duck, then the wavelength is very long, and near the site of the The physics of gravitational radiation: disturbance the wave motion is not noticeable. We say weakness and strength we are in the “near zone”. But when we are more than a wavelength away, then we see waves, and this is the The starting point for understanding gravitational ra- “wave zone” or “far zone”. For the dropped duck, we see diation is Newtonian gravity. The weakness of gravity that waves because their wavelengths are shorter than is evident. If a child picks up a book, she defeats the the size of the bath. In general relativity, the speed cumulative gravitational pull of the entire planet Earth of gravity is the speed of light. Because of this finite
2 speed, gravity must exhibit wave effects. ple detector just by monitoring the distance between Many of Newton’s contemporaries were unhappy two nearby freely falling particles. If they are genuinely that his theory of gravity was based on instantaneous free, then any changes in their separations would indi- “action at a distance”, but Newton’s theory fit the ob- cate the passage of a gravitational wave. Because this servational facts. If gravity had a finite speed of propa- measures a tidal effect, the bigger the separation of the gation, there was no evidence for it in the solar system. particles, the bigger will be the change in their separa- Interestingly, the brilliant 18th century French math- tion, at least for particles that are separated by less than ematician and physicist Laplace tried out a variation a gravitational wavelength. Most modern gravitational on Newton’s theory in which gravity was represented wave detectors are designed to be as big as cost and by something “flowing” out of its source with a finite practicality allows. These are described in the article speed. He reasoned that a planet like the Earth, moving Gravitational Radiation Detection on Earth through this fluid of gravity, would experience friction and in Space. and gradually spiral in towards the Sun. Although gravitational radiation is well understood Laplace could show that the observational limits on in theoretical terms in general relativity, the complex- this inspiral even in his day were so stringent that the ity and non-linearity of Einstein’s equations means that speed of gravity in his model needed to be huge com- calculations are often difficult. In the historical de- pared to the speed of light. He did not find this result velopment of general relativity, between 1915 and the attractive and took the theory no further. (Laplace also 1950’s and 1960’s, these mathematical difficulties cre- explored the notion of what we now call a black hole, ated confusion over the physical nature of gravitational which for him was a region where gravity was strong radiation, and in particular over whether they carried enough to trap light.) energy away from the source. Improved mathematical It is interesting that today, observations of the two techniques finally resolved the matter in favor of the neutron stars in the binary system PSR1913+16 spi- simple physical picture presented here, but this picture raling together as they orbit one another provides the would not be complete without the strong mathemati- most convincing evidence that gravitational waves ex- cal underpinning that now exists. ist and are as described in general relativity. (See below The question of energy in gravitational waves is still and the article Binary Stars as a Probe of Gen- a delicate one. There is no question that waves carry eral Relativity.) Laplace had the right effect, but energy (and momentum) away from their sources. Nev- the wrong theory. This evidence is described in the next ertheless, it is not possible in general relativity to local- section. ize the energy in the radiation to regions smaller than In general relativity, Einstein used the principle of about a wavelength. Indeed the equivalence principle equivalence as the basis for a geometrical description of shows that “point” particles feel nothing, no matter gravity. In the four-dimensional world of space-time, how strong the wave. The wave only acts by stretching the trajectory of a particle falling freely in a gravita- space-time, producing a tidal distortion in the separa- tional field is a certain fixed curve. Its direction at any tions between particles (see the discussion of polariza- point depends on the velocity of the particle. The equiv- tion below). alence principle implies that there is a preferred set of For this reason, energy is localized only in regions, curves in space-time: at any point, pick any direction, not at points. It is nevertheless real energy: the nonlin- and there is a unique curve in that direction that will be earity of general relativity allows waves to create grav- the trajectory of any particle starting with that veloc- itation themselves. Recent numerical simulations have ity. These trajectories are thus properties of space-time shown that focussed gravitational waves can actually itself. form black holes, trapping themselves. If the waves are Moreover, if there were no gravitational field, the weak, they enter the focussing region and re-emerge. If trajectories would be simple straight lines. Even in a they are strong enough, they enter and never leave. gravitational field, a small freely falling particle does not “feel” any acceleration: its internal state is the same as Gravitational waves in a quasi-Newtonian model if there were no gravity. Therefore Einstein postulated that a gravitational field made space-time curved, and It is possible to calculate the approximate size of the that the preferred trajectories were locally straight lines effect of a given gravitational wave by beginning with that simply changed direction as they moved through Newtonian gravity and adding waves to it. In Newto- the curved space-time, in much the same way as a great nian gravity the gravitational field produced by a mass circle on a sphere changes direction relative to other M at a distance r is given by great circles as one goes along it. For weak gravitational φ = GM/r, (1) fields of slowly moving bodies, Einstein’s theory reduces − to Newton’s in the first approximation. where G is Newton’s gravitational constant. The field For gravitational waves, one could make a very sim- of a gravitational wave must be a ripple on this, which
3 means a small change that oscillates in space and time. detectors such events have been high on the list of pos- A suitable form for a change that propagates at the sible sources of gravitational waves. speed of light in the z-direction with an angular fre- To solve Equation (5), one takes r and z as con- quency ω is: stants, and uses the smallness of the right-hand-side, which implies that the changes in ℓ are tiny compared GM δφ = ǫ sin[ω(z/c t)], (2) to ℓ itself. On the right-hand side one can therefore − r − replace ℓ by ℓ0, the initial value of ℓ, and then sim- where ǫ is a dimensionless number that would be ex- ply integrate twice in time to get (for an initial value pected to be small compared to 1. Its size is the subject dℓ/dt = 0) of the next main section. The field δφ produces an acceleration in the z- ℓ(t) ℓ0 GM − = ǫ 2 sin[ω(z/c t)]. (6) direction that depends on its z-derivative. Both 1/r ℓ0 − rc − and the sin() term depend on z. The derivative of 1/r 2 The right-hand-side of Equation (6) is identical to that will be proportional to 1/r , which is how the acceler- of Equation (2). This is an important conclusion which ation falls off in Newton’s theory (where φ is the only fits neatly with Einstein’s geometrical conception of field). But the derivative of the sin() term does not gravity: the size of a gravitational wave gives directly change the 1/r; rather, it essentially just multiplies δφ the stretching of the distance between nearby free parti- by ω/c. At sufficiently large distances from the source, 2 cles. It is conventional to call this h/2 and refer to h as this term will dominate the 1/r term and the acceler- the gravitational wave potential ation produced by the wave will be: ℓ(t) ℓ0 GM h := 2 − =2δφ. (7) δaz = ǫω cos[ω(z/c t)]. (3) ℓ0 − rc − Note that this term would not be present in the x- and The amplitude of the oscillations of h is y-derivatives, so these components of the acceleration GM are much smaller in this quasi-Newtonian model of a h 2ǫ . (8) ∼ rc2 gravitational wave. It is evident from this that a detector must be able Effect on a simple detector to measure changes in its own size that are smaller than one part in 1021 to have a reasonable chance of making The tidal part of this acceleration, for a detector that astronomical observations. The extraordinary small- has size ℓ in the z-direction, is to a first approximation ness of this effect also explains why ordinary objects in the Universe are transparent to gravitational waves. d 2 GM ℓ δaz = ǫω ℓ sin[ω(z/c t)]. (4) As the waves pass through them, they disturb them so · dz rc2 − little (parts per 1021 typically) that the transfer of en- If a detector consists of two freely falling particles with ergy to the object and any back-reaction effects of this this relative acceleration, the equation of motion for on the wave are negligibly small. their separation ℓ will be 2 Energy flux carried by waves d ℓ GM = ǫω2ℓ sin[ω(z/c t)], (5) dt2 rc2 − The energy in the waves can also be estimated from these equations and general physical principles. Quite The dimensionless coefficient ǫ(GM/rc2) is typically generally, in classical field theories, the energy flux of very small. Even if ǫ is of order 1, the other number is, a propagating sinusoidal plane wave is proportional to with reasonable values for M and R, the square of the time-derivative of the fundamental −1 field. In electromagnetism, the Poynting flux is propor- GM −21 M r 2 =2.4 10 . tional to the square of the time-derivative of the vector rc × M⊙ 20 Mpc potential. The mass and distance scales here are those appropriate In general relativity, the flux is therefore propor- to a neutron star in the nearest large cluster of galax- tional to the square of the time-derivative of h(t). The ies, the Virgo Cluster. (The distance unit is based on proportionality constant must be built only out of c, G, the astronomers’ parsec, denoted pc, which is about and pure numbers. To get the right units, it must be 3 3 1016 m. The unit Mpc is a megaparsec.) It is be- proportional to c /G; to get the pure number, a calcula- lieved× that several neutron stars are formed in the Virgo tion in general relativity is required: 1/32π for a linearly Cluster each year in supernova explosions. Ever since polarized wave. (Polarization is described later.) This the beginning of the development of gravitational wave
4 gives relativity are a system of many coupled, nonlinear, par- tial differential equations. But in four circumstances the 3 2 1 c dh emission mechanisms are understood in some detail: F = (9) gw 32π G dt Small-amplitude pulsations of relativistic stars 2 2 • −5 f h −2 and black holes. Normally gravitational radiation = 1.6 10 22 W m , × 100 Hz 10− carries away energy and damps pulsations away, but in rotating stars the opposite may happen: for a wave with frequency f = ω/2π. For comparison, the radiated loss of angular momentum may allow reflected sunlight from Jupiter has a flux on Earth of the star to spin down to an energetically more fa- 2.3 10−7 W m−2, almost 100 times smaller than that × −22 vored state, in which case the perturbation will of a gravitational wave with an amplitude of 10 ! grow, at least until nonlinear effects intervene. Discovered by S. Chandrasekhar and now called Deficiencies of the quasi-Newtonian model the Chandrasekhar-Friedman-Schutz (CFS) insta- The calculation and equations in this section have been bility, it is thought to limit the rotation speed of framed within a modified Newtonian model of gravity young neutron stars (see below). Black holes also with a propagation speed of c, and one would expect emit gravitational radiation when they are dis- some differences from general relativity. The most im- turbed, e.g. by something falling into them, but portant difference is in the direction in which the tidal they are not unstable: they always settle down forces act. In the simple model, wave accelerations act into a steady state again. in the z-direction, which was the direction of propaga- Radiation from “test” objects orbiting black tion of the wave. This is called a longitudinal wave. • holes. If the mass of the object is small enough In general relativity, gravitational waves are trans- then the total gravitational field may be treated verse waves: if the wave propagates in the z-direction as a linear perturbation of the exactly known field then the tidal forces act only in the xy-plane. We will of a black hole (the Schwarzschild or, with rota- discuss later the exact form that their action in this tion, the Kerr solution). These studies give in- plane takes. Remarkably, the rest of the formulas above sight into the general problem of gravitational ra- are good approximations even in general relativity, pro- diation, and they also predict gravitational wave- vided ǫ is calculated correctly, as described in the next forms that might be observed by space-based ob- section. servatories looking at compact stars falling into the giant black holes in the centers of galaxies. The emission of gravitational waves (See below and the article Gravitational Ra- diation Detection on Earth and in Space.) The previous section described the propagation of gravi- tational waves, their interaction with detectors, and the Weak gravitational fields and slow motion. Such energy they carry. This section deals with the strength • weakly relativistic sources are studied in the post- with which waves are emitted by astronomical bodies. Newtonian approximation, which includes higher- In Newtonian gravity there is a fundamental the- order corrections to Newtonian gravity from gen- orem, proved by Newton, that the gravitational field eral relativity. This is analogous to the slow- outside a spherical body is not only spherical, but it is motion multipole approximation that is so pow- the same as that of a point mass located at the origin of erful in the study of electromagnetic radiation. the body. It has the form given in Equation (1). In par- Most realistic gravitational-wave sources can be ticular, the field is independent of the size of the body, studied to some approximation this way. as long as we consider only points outside it. This is Collisions of black holes and neutron stars. These true even if the star pulsates in a spherical manner. • This theorem is essentially the same in general rel- events, which are expected to be observed by ativity, and is known as Birkhoff’s Theorem. Outside a gravitational wave detectors (see below), must be spherical body the field is the same as that of a black modeled by solving the full set of Einstein equa- hole of the same mass as the body (the Schwarzschild tions on a powerful computer. Techniques to do metric), even if the body is pulsating spherically. But this are advancing rapidly, and simulations of re- if the pulsation is nonspherical, then the outside field alistic mergers of stars and black holes from in- will change. In general relativity the changes generally spiraling orbits can be expected to yield useful propagate as a wave. So gravitational waves will be results in the first years of the 21st century. emitted by nonspherical motions. In general the calculation of the emitted waves is extremely difficult, since the field equations of general
5 Quadrupole approximation is close. The fundamental quantity is the spatial ten- sor (matrix) Q , the second moment of the mass (or The post-Newtonian approximation has so far been the jk charge) distribution: most powerful of these methods, and it yields the most insight into the emission mechanisms. Its fundamental 3 Qjk = ρxj xkd x. (11) result is the quadrupole formula, which gives the first Z approximation to the radiation emitted by a weakly rel- A gravitational wave in general relativity is represented ativistic system. by a matrix h rather than a single scalar h, and its The quadrupole formula is analogous to the dipole jk source (in the quadrupole approximation) is Q . formula of electromagnetism. In this language, jk As in electromagnetism, the amplitude of the radi- monopole means spherical, which emits no radiation. ation is proportional to the second time-derivative of This is also true in electromagnetism, where it is linked Q , and it falls off inversely with the distance r from to conservation of charge. The “monopole moment” in jk the source. A factor of G/c4 is needed in order to get a electromagnetism is the total charge of a system, and dimensionless amplitude h, and a factor of 2 to be con- since that does not change, there can be no spherical sistent with the definition in Equation (8). The result radiation. for h is: Again in electromagnetism, the dipole moment is jk 2G d2Q defined as the integral h = jk . (12) jk rc4 dt2 3 General relativity describes waves with a matrix be- di = ρxid x, Z cause gravity is geometry, and the effects of gravity are represented by the stretching of space-time. This ma- where ρ is the charge density and xi is a Cartesian co- trix contains that distortion information. Here is the ordinate. If this integral is time-dependent, then the information about the transverse action of the waves amplitude of the electromagnetic waves will be propor- that the quasi-Newtonian model of the last section did tional to its first time-derivative ddi/dt, and the radiated not get right. energy will be proportional (as we remarked earlier) to the square of the time derivative of this amplitude, i.e. Simple estimates to d2d /dt2 2. i | i | PIn the post-Newtonian approximation to general rel- If the motion inside the source is highly non-spherical, 2 2 ativity, the calculation goes remarkably similarly. The then a typical component of d Qjk/dt will (from Equa- 2 2 monopole moment is now the total mass-energy, which tion (11)) have magnitude MvN.S., where vN.S. is the is the dominant source of the gravitational field for non-spherical part of the squared velocity inside the non-relativistic bodies, and which is constant as long source. So one way of approximating any component as the radiation is weak. (Radiation will carry away of Equation (12) is energy, but in the post-Newtonian approximation that 2 is a higher-order effect.) The dipole moment is given 2GMvN.S. h 4 . (13) by the same equation as above, but with ρ interpreted ∼ rc as the density of mass-energy. Comparing this with Equation (8) we see that the ratio However, here general relativity departs from elec- ǫ of the wave to the Newtonian potential is simply tromagnetism. The time-derivative of the dipole mo- 2 ment is, since the mass-energy is conserved, just the vN.S. ǫ 2 . integral of the velocity vi: ∼ c By the virial theorem for self-gravitating bodies, this 3 d˙i = ρvid x. (10) will not be larger than Z 2 ǫ < φint/c , (14) But this is the total momentum in the system, and (to lowest order) this is constant. Therefore, there is no en- where φint is the maximum value of the Newtonian ergy radiated due to dipole effects in general relativity. gravitational potential inside the system. This provides The gravitational field far from the source does con- a convenient bound in practice. It should not be taken tain a dipole piece if d˙i is non-zero, but this is constant to be more accurate than that. 2 because it reflects the fact that the source has non-zero For a neutron star source one has φint 0.2c . If total momentum and is therefore moving through space. the star is in the Virgo cluster, then the upper∼ limit on To find genuine radiation in general relativity one the amplitude of the radiation from such a source is 5 must go one step beyond the dipole approximation to 10−22. This has been the goal of detector development× the quadrupole terms. These are also studied in elec- for decades, to make detectors that can observe waves tromagnetism, and the analogy with relativity again at or below an amplitude of 10−21.
6 Polarization of gravitational waves The matrix nature of the wave amplitude comes from '+' general relativity and has no Newtonian analog. In or- der to find the effect of the waves on the separation 0.2 h/2 of two free particles (the idealized detector), one has t -0.2 to start with hjk as given by Equation (12) or by any other calculation, and then do three things: 'x'
1. Project the matrix hjk onto a plane perpendic- ular to the direction of travel of the wave. In the simple case considered above, where the wave was traveling in the z-direction, this means leav- Figure 1: Polarization of gravitational waves. The cen- ing the components hxx, hxy, hyy alone and setting the remaining{ components to} zero. It is ter line gives the wave as a function of time, with an then a two-dimensional matrix in the transverse amplitude of h = 0.2, and the top and bottom lines plane. show to scale the distortions produced by two polariza- tions with this amplitude. 2. Remove the two-dimensional trace of the pro- T T jected matrix. Call the resulting matrix hjk , where TT stands for Transverse-Traceless. In of 1/5 comes from a careful calculation in general rela- tivity. The result is the gravitational wave luminosity the example this means subtracting (hxx +hyy)/2 in the quadrupole approximation: from both hxx and hyy. Then there are only two T T T T independent components left, hxy = hyx and T T T T ...... 2 hxx = hyy . G 1 − Lgw = 5 QjkQjk Q , (16) 5c − 3 Xj,k 3. To find the change in the separation of two par- ticles that have an initial separation given by the vector ℓ , let the matrix hT T act on it: where Q is the trace of the matrix Qjk. Its squared k jk third derivative must be subtracted in order to ensure that spherical motions do not radiate. δℓ = hT T ℓ . (15) j jk k This equation will be used in the next section to Xk estimate the back-reaction effect on a system that emits It is clear that any longitudinal component of the gravitational radiation. separation ℓj between the particles is unaffected by the wave (in the example, this is the z-separation), and that Emission estimates there are two degrees of freedom (the two independent Until observations of gravitational waves are success- components of hT T ) to move particles in the plane per- jk fully made, one can only make intelligent guesses about pendicular to the propagation direction. These two de- most of the sources that will be seen. There are many grees of freedom are the two polarizations of the wave. that could be strong enough to be seen by the early Fig. 1 shows the conventional definition of the two detectors: binary stars, supernova explosions, neutron independent polarizations, from which any other can be stars, the early Universe. The estimates in this section made by superposition. What is shown is the effect of are accurate only to within factors of order 1. The es- a wave on a ring of free particles in a plane transverse timates correctly show how the important observables to the wave. The first line shows a wave with h = 0, xy scale with the properties of the systems. conventionally called the “+” polarization. The bottom line shows a wave with hxx =0, the “ ” polarization. × Man-made gravitational waves Luminosity in gravitational waves One source can be ruled out: man-made gravitational radiation. Imagine creating a wave generator with the The energy carried by the gravitational wave must be following extreme properties. It consists of two masses proportional to the square of the time-derivative of the of 103 kg each (a small car) at opposite ends of a beam wave amplitude, so it will depend on the sum of the 3 3 10 m long. At its center the beam pivots about an squares of the components d Qjk/dt . The energy flux 2 axis. This centrifuge rotates 10 times per second. All falls off as 1/r , but when integrated over a sphere of the velocity is non-spherical, so v2 in Equation (13) radius r to get the total luminosity, the dependence on N.S. is about 105 m2 s−2. The frequency of the waves will r goes away, as it should. The luminosity contains a fac- 5 actually be 20 Hz, since the mass distribution of the tor G/c on dimensional grounds, and a further factor
7 2 system is periodic with a period of 0.05 s, only half have vN.S. = GM/4R. The gravitational-wave ampli- the rotation period. The wavelength of the waves will tude from Equation (13) can then be written therefore be 1.5 107 M, about the diameter of the earth. In order to× detect gravitational waves, not near- 1 GM GM hbinary 2 2 . (18) zone Newtonian gravity, the detector must be at least ∼ 2 rc Rc one wavelength from the source. Then the amplitude Compare this to the implications of putting Equa- h can be deduced from Equation (13): h 5 10−43. ∼ × tion (14) into Equation (8). This is far too small to contemplate detecting! The gravitational-wave luminosity of such a system is, by a calculation analogous to that for bumps on neu- Radiation from a spinning neutron star tron stars, Some likely gravitational wave sources behave like the 5 1 c5 GM centrifuge, only on a grander scale. Suppose a neutron Lbinary 2 . star of radius R spins with a frequency f and has an ∼ 80 G Rc irregularity, a bump of mass m on its otherwise axially In this equation there appears the important constant symmetric shape. Then the bump will emit gravita- c5/G = 3.6 1052 W, a number with the dimen- tional radiation (again at frequency 2f because it spins sions of luminosity× built only from fundamental con- about its center of mass, so it actually has mass excesses stants. By comparison, the luminosity of the Sun is on two sides of the star), and the non-spherical velocity only 3.8 1026 W. Close binaries can therefore radiate will be just vN.S. = 2πRf. The radiation amplitude more energy× in gravitational waves than in light. will be, from Equation (13), The radiation of energy by the orbital motion causes 2 2 the orbit to shrink. The shrinking will make any ob- h 2(2πRf/c) Gm/rc , (17) bump ∼ served gravitational waves increase in frequency with and the luminosity, from Equation (16) (assuming time. This is called a chirp. The timescale for this is roughly 4 comparable components of Qjk contribute to −4 20GM GM the sum), t = Mv2/L . (19) chirp binary ∼ c3 Rc2 5 6 2 4 Lbump (G/5c )(2πf) m R . ∼ The binary pulsar system – verifying gravita- The radiated energy would presumably come from the tional waves rotational energy of the star. This would lead to a spin- down of the star on a timescale This orbital shrinking has already been observed in the Hulse-Taylor Pulsar system, containing the radio 1 − −3 1 2 5 −1 Gm v pulsar PSR1913+16 and an unseen neutron star in a tspindown = mv /Lbump f 2 . 2 ∼ 4π Rc c binary orbit. Discovered in 1974 by R Hulse and J Taylor, it has established that gravitational radiation It is felt that neutron star crusts are not strong enough is correctly described by general relativity. For their to support asymmetries with a mass of more than about discovery, Hulse and Taylor received the 1993 Nobel −5 m 10 M⊙, and from this one can estimate the like- ∼ Prize for Physics. lihood that the observed spindown timescales of Pul- The key to the importance of this binary system sars are due to gravitational radiation. In most cases, is that all of the important parameters of the system it seems that gravitational wave losses cannot be the can be measured before one takes account of the orbital main spindown mechanism. shrinking. This is because a number of post-Newtonian But lower levels of radiation would still be observ- effects on the arrival time of pulses at the Earth, such able by detectors under construction, and this may be as the precession of the position of the periastron and coming from a number of stars. In particular, there the time-dependent gravitational redshift of the pulsar is a class of neutron stars in X-Ray Binary Stars. period as it approaches and recedes from its companion, They are accreting, and it is possible that accretion are measured in this system. They fully determine the will create some kind of mass asymmetry or else lead to masses and separation of the stars and the inclination a rotational instability of the CFS type in the r-modes and eccentricity of their orbit. From these numbers, (see below). In either case, the stars could turn out be without any free parameters, it is possible to compute long-lived sources of gravitational waves. the shrinking timescale predicted by general relativity. The observed rate matches the predicted rate to within Radiation from a binary star system the observational errors of less than 1%. The stars are in an eccentric orbit (e = 0.615) and Another “centrifuge” is a binary star system. Two stars both have masses of 1.4M⊙. The orbital semimajor of the same mass M in a circular orbit of radius R
8 axis is about 7 108 m. Equation (19) assumes a circu- template has to be multiplied into the data stream at lar orbit and gives× a shrinking timescale of 6 1010 y. each distinguishable arrival time. This is then a cor- This is an overestimate, however, partly because× it is relation of the template with the data stream. Nor- in any case a rough approximation, and partly because mally this is done efficiently using Fast Fourier Trans- the timescale is very sensitive to eccentricity. With the form methods. observed eccentricity, a careful calculation shows that Second, the expected signal usually depends on a the expected shrinking timescale is around 4 108 y, number of unknown parameters. For example, the ra- consistent with observations. × diation from a binary system depends on the chirp mass , and it might arrive with an arbitrary phase. There- M Chirping binaries fore, many related templates must be separately applied to the data to cover the whole family of signals. For a circular equal-mass binary, the orbital shrinking Third, matched filtering enhances the signal only if timescale and the frequency of the orbit determine both the template stays in phase with the signal for the whole M and R. If in addition a gravitational wave detector data set. If they go out of phase, the method begins to measures the wave amplitude hbinary, then the distance reduce the signal-to-noise ratio. For long-duration sig- r to the binary system can be determined. nals, such as for low-mass neutron-star coalescing bina- Remarkably, this conclusion holds even for bina- ries or continuous-wave signals from neutron stars (see ries with unequal masses. In such a case, the measur- below), this requires the analysis of large data sets, and able mass is the chirp mass of the binary, defined as 2 5 often forces the introduction of additional parameters := µ3/5M / , where µ is the reduced mass of the bi- M T to allow for small effects that can make the signal drift nary system and MT its total mass. Then the distance out of phase with the template. It also means that the r is still measurable from the chirp rate, frequency, and method works well only if there is a good prediction of amplitude. In other words, a chirping binary is a stan- the form of the signal. dard candle in astronomy. Post-Newtonian corrections Because the first signals will be weak, matched fil- to the orbit, if observed in the waveform, can determine tering will be used wherever possible. As a simple rule the individual masses of the stars and even their spins. of thumb, the detectability of a signal depends on its effective amplitude heff, defined as Recognizing weak signals heff = h Ncycles, (20) For ground-based detectors, all expected signals have p amplitudes that are close to or even below the instru- where Ncycles is the number of cycles in the waveform mental noise level in the detector output. Such signals that are matched by the template. can nevertheless be detected with confidence if their For example, the effective amplitude of the radia- waveform matches an expected waveform. The pattern tion from a bump on a neutron star (Equation (17))will recognition technique that will be used by detector sci- be hbump√2fTobs, where Tobs is the observation time. entists is called matched filtering. In order to detect this radiation, detectors may need Matched filtering works by multiplying the output of to observe for long periods, say 4 months, during which the detector by a function of time (called the template) they accumulate billions of cycles of the waveform. Dur- that represents an expected waveform, and summing ing this time, the star may spin down by a detectable (integrating) the result. If there is a signal matching amount, and the motion of the Earth introduces large the waveform buried in the noise then the output of the changes in the apparent frequency of the signal, so filter will be higher than expected for pure noise. matched filtering needs to be done with care and preci- A simple example of such a filter is the Fourier trans- sion. form, which is a matched filter for a constant-frequency Another example is a binary system followed to co- signal. The noise power in the data stream is spread out alescence, i.e. where the chirp time in Equation (19) is over the spectrum, while the power in the signal is con- less than the observing time. For neutron-star binaries centrated in a single frequency. This makes the signal observed by ground-based detectors this will always be easier to recognize. The improvement of the signal-to- the case (see the next section), so the effective ampli- noise ratio for the amplitude of the signal is propor- tude is roughly tional to the square-root of the number of cycles of the −1 4 GM GM / wave contained in the data. This is well-known for the h h f t , (21) chirp ∼ bin gw chirp ∼ rc2 Rc2 Fourier transform, and it is generally true for matched p filtering. where for fgw one must use twice the orbital frequency Matched filtering can make big demands on com- GM/R3/4π. This may seem a puzzling result, be- putation, for several reasons. First, the arrival time of causep it says that the effective amplitude of the signal a short-duration signal is generally not known, so the gets smaller as the stars get closer. But this just means
9 that the signal will be more detectable if it is picked up earlier, since Equation (21) assumes that the signal is Gravitational Dynamics followed right to coalescence. If one picks up the sig- 1013 nal at earlier times, then there are more cycles of the 1012 y chirp line Binar waveform to filter for, and this naturally gives a bet- 1011 106 M BH binary 10 -4 Hz o 10 f = 10 ter signal-to-noise ratio. This gives an advantage to 106 M BH burst 109 y lifetime = 1 yr o detectors that can operate at lower frequencies. This (m) Sun Space band Binar 108 f = 1 Hz
ius R close NS-NS binary has been an important consideration in the design of 107
rad le line Earth band modern detectors. (See Gravitational Radiation 106 NS-NS coalescence Black ho 105 4 Hz Detection on Earth and in Space.) 15 Mo BH f = 10 NS In general, the sensitivity of detectors will be lim- 104 3 ited not just by detector technology, but also by the 10 102 duration of the observation, the quality of the signal 10-1 100 101 102 103 104 105 106 107 108 predictions, and even by the availability of computer mass M (solar masses) processing power for the data analysis.
Figure 2: Mass-radius plot for gravitational wave Astronomical sources of gravitational sources. waves
4 Estimating the frequency mass objects. Nothing over a mass of about 10 M⊙ can The signals for which the best waveform predictions are radiate above 1 Hz. available have narrowly defined frequencies. In some A number of typical relativistic objects are placed cases the frequency is dominated by an existing motion, in the diagram: a neutron star, a binary pair of neu- such as the spin of a pulsar. But in most cases the tron stars that spirals together as they orbit, some black frequency will be related to the natural frequency for a holes. Two other interesting lines are drawn. The lower self-gravitating body, defined as (dashed) line is the 1-year coalescence line, where the orbital shrinking timescale in Equation (19) is less than f0 = Gρ/¯ 4π, (22) one year. The upper (solid) line is the 1-year chirp line: p if a binary lies below this line then its orbit will shrink whereρ ¯ is the mean density of mass-energy in the enough to make its orbital frequency increase by a mea- source. This is of the same order as the binary orbital surable amount in one year. (In a one-year observation frequency and the fundamental pulsation frequency of one can in principle measure changes in frequency of the body. 1 yr−1, or 3 10−8 Hz.) × The frequency is determined by the size R and mass It is clear from the figure that any binary system M of the source, takingρ ¯ = 3M/4πR3. For a neutron that is observed from the ground will coalesce within an star of mass 1.4M⊙ and radius 10 km, the natural fre- observing time of one year. Since pulsar statistics sug- 5 quency is f0 =1.9 kHz. For a black hole of mass 10M⊙ gest that this happens less than once every 10 years 2 and radius 2GM/c = 30 km, it is f0 = 1 kHz. And for in our Galaxy, ground-based detectors must be able to 6 a large black hole of mass 2.5 10 M⊙, such as the one register these events in a volume of space containing at the center of our Galaxy,× this goes down in inverse at least 106 galaxies in order to have a hope of see- proportion to the mass to f0 = 4 mHz. ing occasional coalescences. When detectors reach this Fig. 2 shows the mass-radius diagram for likely sensitivity (sometime in the first decade of the 21st cen- sources of gravitational waves. Three lines of constant tury), then astronomers will be able to use the observed 4 natural frequency are plotted: f0 = 10 Hz, f0 = 1 Hz, chirping binaries as standard candles to measure dis- −4 and f0 = 10 Hz. These are interesting frequencies tance scales in the Universe. from the point of view of observing techniques: grav- 4 itational waves between 1 and 10 Hz are accessible Radiation from neutron-star normal modes to ground-based detectors, while lower frequencies are observable only from space. (See the article Gravi- In Fig. 2 there is a dot for the typical neutron-star. The tational Radiaton Detection on Earth and in corresponding frequency is the fundamental vibrational Space.) Also shown is the line marking the black-hole frequency of such an object. In fact, neutron stars have boundary. This has the equation R = 2GM/c2. There a rich spectrum of non-radial normal modes, which fall are no objects below this line. This line cuts through into several families: f-, g-, p-, w-, and r-modes have all the ground-based frequency band in such a way as to been studied. If their gravitational wave emissions can restrict ground-based instruments to looking at stellar- be detected, then the details of their spectra would be a sensitive probe of their structure and of the Equa-
10 tion of State of Neutron Stars, in much the same Black holes and gravitational waves way that Helioseismology probes the interior of the Black holes are regions of spacetime within which every- Sun. This is a challenge to ground-based detectors, thing is trapped: light cannot escape, nor can anything which cannot yet make sensitive observations as high else that moves slower than light. The boundary of as 10 kHz. this region is called the event horizon. This boundary is a dynamical surface. If any mass-energy falls into Radiation from gravitational collapse the hole, the area of the horizon increases. In addition, The event that forms a neutron star is the gravitational the horizon will generally wobble when this happens. collapse that produces in a supernova. It is difficult These wobbles settle down quickly, emitting gravita- to predict the waveform or amplitude expected from tional waves, and leaving a smooth (and slightly larger) this event, because we have no observational evidence horizon afterwards. about how nonspherical the collapse event might be in Undisturbed black holes are time-independent and a typical supernova: the collapse is hidden deep within smooth. In fact, according to general relativity the ex- the star. So we can only guess. For example, a gravi- ternal gravitational field of such a black hole and the tational wave burst might be broad-band, centered on size and shape of its horizon are fully determined by 1 kHz, or it might be a few cycles of radiation at a fre- only three numbers: the total mass, electric charge quency anywhere between 100 Hz and 10 kHz, chirping and angular momentum of the black hole. This black- up or down. The amplitude could be large, in which a hole uniqueness theorem is remarkable, considering how good fraction of the energy released by the collapse is much variety there can be in the material that collapsed radiated in gravitational waves, or it could be negligi- to form the black hole and that may have subsequently bly small. It is indeed ironic that, although detecting fallen in. supernovae was the initial goal of detector development Observations of the gravitational waves emitted by a when it started 4 decades ago, little more is known to- wobbling horizon or by a particle in orbit around a black day about what to expect than scientists knew then. hole have the potential to test the uniqueness theorem and thereby to verify the predictions of general relativ- Radiation from r-modes ity about the strongest possible gravitational fields. Astronomers now recognize that there is an abun- Hot neutron stars that rotate faster than about 100- dance of black holes in the universe. Observations of 200 Hz appear to be unstable to the emission of gravita- various kinds have located black holes in X-ray binary tional radiation through amplification of their r-modes systems in the Galaxy and in the centers of galaxies. by the CFS mechanism. In stars colder than about These two classes of black holes have very differ- 8 10 K, viscosity may be strong enough to damp out this ent masses. Stellar black holes typically have masses instability. This instability may explain why only old, of around 10M⊙, and are thought to have been formed recycled, cold pulsars are seen at higher rotation rates. by the gravitational collapse of the center of a large, It also suggests that the formation of a rapidly rotating evolved Red Giant Star, perhaps in a supernova ex- neutron star may be followed by a period of steady grav- plosion. Massive black holes in galactic centers seem to 6 10 itational radiation as the star emits angular momentum have masses between 10 and 10 M⊙, but their history and spins down to its stability limit. If as few as 10% and method of formation are not yet understood. of all the neutron stars formed since Star Formation Both kinds of black hole can radiate gravitational began (at a redshift of perhaps 4) went through such waves. According to Fig. 2, stellar black hole radia- a spindown, then they may have produced a detectable tion will be in the ground-based frequency range, while random background of gravitational radiation. galactic holes are detectable only from space. The ra- Interestingly, the r-modes are disturbances primar- diation from a black hole typically is strongly damped, ily of the fluid velocity; they have little density pertur- lasting only a few cycles about the frequency, which for bation. Their name comes from their similarity to the a spherical black hole is given by Equation (22) with Rossby waves of oceanography. The gravitational ra- R =2GM/c2: diation they emit is not primarily mass-quadrupole (as −1 in Equation (12)), but rather mass-current-quadrupole, M fBH 10 Hz. the analog of magnetic quadrupole radiation in elec- ∼ M⊙ tromagnetism. This is the wave counterpart of what is called gravitomagnetism, which is responsible for the Stellar-mass black holes Lense-Thirring effect: an extra precession of a spinning gyroscope as it orbits a rotating body like the Earth Radiation from stellar black holes is expected mainly caused by the spin-spin gravitational coupling of the from coalescing binary systems, when one or both of gyroscope to the Earth. the components is a black hole. Although such systems are thought to be rarer than systems of two neutron
11 stars, the larger mass of the black hole makes the system universe. This would give an event rate of several per visible from a greater distance. By measuring the chirp year. If the supermassive black holes were formed from mass (as discussed above) observers will recognize that smaller holes in a hierarchical merger scenario, then the they have a black-hole system. It is very possible that event rate could be hundreds or thousands per year. It the first observations of binaries by interferometers will is likely that only space-based observations of gravita- be of black holes. tional waves will answer these questions. When a two-black-hole binary coalesces, there A second scenario for the production of radiation should be a burst of gravitational radiation that will by massive black holes is the swallowing of a stellar- depend in detail on the masses and spins of the objects. mass black hole or a neutron star by the large hole. Numerical simulations of such events will be needed to Massive black holes exist in the middle of dense star interpret this signal, and possibly even to extract it from clusters. The tidal disruption of main-sequence or gi- the instrumental noise of the detector. The research ant stars that stray too close to the hole is thought to field of numerical relativity is making rapid progress, provide the gas that powers the quasar phenomenon. and it can be expected to produce informative simula- These clusters will also contain a good number of neu- tions in the first few years of the 21st century, using the tron stars and stellar-mass black holes. They are too largest and fastest computers available at that time. compact to be disrupted by the hole even if they fall directly into it. Massive and supermassive black holes Such captures therefore emit a gravitational wave signal that may be approximated by studying the mo- Gravitational radiation is expected from supermassive tion of a “point mass” near a black hole. It will again black holes in two ways. In one scenario, two massive emit a chirp of radiation, but in this case the orbit may black holes spiral together in a much more powerful be very eccentric. The details of the waveform encode version of the coalescence we have just discussed. The information about the geometry of space-time near the frequency is much lower, but the amplitude is higher. hole. In particular, it may be possible to measure the Equation (21) implies that the effective signal ampli- mass and spin of the hole and thereby to test the unique- tude is almost linear in the masses of the holes, so that 6 ness theorem for black holes. The event rate is not a signal from two 10 M⊙ black holes will have an ampli- 5 very dependent on the details of galaxy formation, and tude 10 times bigger than the signal from two 10M⊙ is probably high enough for many detections per year holes at the same distance. Even allowing for differ- from a space-based detector. ences in technology, space-based detectors will be able to study such events with a very high signal-to-noise ratio no matter where in the universe they occur. Gravitational waves from the Big Bang Observations of coalescing massive black-hole bina- Gravitational waves have traveled almost unimpeded ries will therefore provide strong tests of the validity of through the universe since they were generated at times general relativity in the regime of strong gravitational as early as 10−24 s after the Big Bang. Observing them fields, provided that numerical simulations can match would provide important constraints on theories of In- the accuracy of the observations by that time. flation and high-energy physics. The event rate for such coalescences is not easy to Inflation is an attractive scenario for the early uni- predict: it could be zero, but it may be large. It seems verse because it makes the large-scale homogeneity of that the central core of most galaxies may contain a the universe easier to understand. It also provides 6 black hole of at least 10 M⊙. This is known to be true a mechanism for producing initial density perturba- for our galaxy and for a number of others nearby. Su- tions large enough to evolve into galaxies as the uni- 9 permassive black holes (up to a few times 10 M⊙) are verse expands. These perturbations are accompanied believed to power Quasi-Stellar Objects and Ac- by gravitational-field perturbations that travel through tive Galaxies. There is some evidence that the mass the universe, redshifting in the same way that photons of the central black hole is proportional to the mass of do. Today these perturbations should form a random the core of the host galaxy. background of gravitational radiation. If black holes are formed with their galaxies, in a sin- The perturbations arise by parametric amplification gle spherical gravitational collapse event, and if noth- of quantum fluctuations in the gravitational wave field ing happens to them after that, then coalescences will that existed before inflation began. The huge expansion never be seen. But it is believed that Galaxy Forma- associated with inflation puts energy into these fluctu- tion probably occurred through the merger of smaller ations, converting them into real gravitational waves 6 units, sub-galaxies of masses upwards of 10 M⊙. If with classical amplitudes. these units had their own black holes, then the mergers If inflation did not occur, then the perturbations would have resulted in the coalescence of many of the that led to galaxies must have arisen in some other way, holes on a timescale shorter than the present age of the and it is possible that this alternative mechanism also
12 produced gravitational waves. One candidate is cos- sources (binary white dwarf systems and r-mode spin- mic defects, including cosmic strings and cosmic tex- down, as discussed above) could obscure a cosmological ture. (See Topological Defects in Cosmology.) background at these levels. Although observations at present seem to rule cosmic defects out as a candidate for galaxy formation, cos- Predicted spectrum of cosmological radiation mic strings may nevertheless have produced observable gravitational waves. The simplest models of inflation suggest that the spec- If inflation did not occur, there could also be a ther- trum of the gravitational wave background should be mal background of gravitational waves at a tempera- flat, so that Ωgw is independent of frequency over a ture similar to that of the cosmological microwave back- very large range of frequencies. In this case, the ob- ground, but this radiation would have such a high fre- served fluctuations in the cosmic microwave background quency that it would not be detectable by any known radiation set a limit on gravitational radiation at ultra- or proposed technique. low frequencies, and this constrains the energy density in the observable range (0.1 mHz to 10 kHz) to below The random background will be detectable as a noise −13 in the detector that competes with instrumental noise. about 10 of closure. This will be too small to be seen In a single detector, such as the first space-based de- by the current and planned detectors on the ground or tector, this noise must be larger than the instrumental in space. noise to be detected, and one must have great confi- But there is a great deal of room in these models for dence in the detector in order to claim that the observed other spectra. The period before inflation may produce noise is external. This is how the cosmic microwave initial conditions for the phase of parametric amplifi- background was originally discovered in a radio tele- cation that give large amounts of radiation in the ob- scope. servable frequency range. One family of models based If there are two detectors, then one can multiply in superstring theory has a spectrum that rises at high the outputs of the two detectors together and sum (in- frequencies. If a cosmological background from inflation tegrate). In this way, the random wave field in one de- or from cosmic defects can be observed, it will contain tector acts like a matched filtering template, matching important clues to the nature of the theory that unifies the random field in the other detector. This allows the gravitation with the rest of quantum physics. detection of noise that is below the instrumental noise of the individual detectors. For this to work, the two Conclusions detectors must be close enough together to experience The first few years of the 21st century should see the the same random wave field. In practice, the sensitivity first direct detections of gravitational radiation and the of this method falls off rapidly with separation if the opening of the field of gravitational wave astronomy. detectors are more than a wavelength apart. Beyond that, over a period of a decade or more, one may expect observations to yield important and use- Measure of the strength of random gravitational ful information about binary systems, stellar evolution, waves neutron stars, black holes, strong gravitational fields, When describing the strength of a random wave field, and cosmology. it is not appropriate to measure the amplitude of any If gravitational wave astronomy follows the example single component. Rather, the r.m.s. amplitude of the of other fields, like X-Ray Astronomy and Radio field is the observable quantity. It is common to use Astronomy, then at some level of sensitivity it will an equivalent measure, the energy density ρgw(f) in begin to discover sources that were completely unex- the radiation field as a function of frequency f. For a pected. Many scientists think the chance of this hap- cosmological field, what is relevant is to normalize this pening early is very good, since the processes that pro- energy density to the critical density ρc required to close duce gravitational waves are so different from those that the universe. It is thus conventional to define produce the electromagnetic radiation on which most present knowledge of the universe is based, and since dρgw/ρc Ωgw := . (23) more than 90% of the matter in the universe is dark d ln f and interacts with visible matter only through gravita- tion. This is roughly the fraction of the closure energy density Present and planned detectors are known not to be in random gravitational waves between the frequency f ideal for some kinds of gravitational wave sources. Sen- and 2.718f. sitive measurements of a cosmological background of Current and planned detectors may reach a sensi- 9 10 radiation from the big bang may not be possible with tivity of Ω 10− at 1 mHz and 10− at 40 Hz, gw these instruments if the spectrum follows the predic- but there is a possibility∼ that backgrounds due to other tions of “standard” inflation theory. Most of the nor-
13 mal mode oscillations of neutron stars will be very hard to detect, because the radiation is weak and at a high frequency, but the science there is compelling: neutron- star seismology may be the only way to probe the in- teriors of neutron stars and understand these complex and fascinating objects. Detector technology will con- tinually improve, and these sources provide important long-term goals for this field. There will clearly be much to do after the first observations are successfully made.
Bibliography Folkner, W M, ed., 1998 Laser Interferometer Space Antenna - AIP Conference Proceeding 456 (Woodbury, NY: American Institute of Physics) Marck J-A, Lasota J-P, eds., 1996 Relativistic Gravita- tion and Gravitational Radiation (Cambridge and New York: Cambridge University Press) Thorne, K S 1994 Black Holes and Time Warps: Ein- stein’s Outrageous Legacy (New York: W. W. Norton and Co) Wald, R M, ed., 1998 Black Holes and Relativistic Stars (Chicago: University of Chicago Press)
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