Overview of Gravitational RadiationAs direct detection of gravitational radiation draws nearer, it is useful to consider what suchdetections will teach us about the universe. The first such detection, of course, will be of immediatesignificance because it will be a direct confirmation of a dramatic prediction of general relativity:to paraphrase John Wheeler, that spacetime tells sources how to move, and moving sources tellspacetime how to ripple.Beyond this first detection, gravitational wave detections will pass into the realm of astronomy,allowing new observational windows onto some of the most dynamic phenomena in the universe.These include merging neutron stars and black holes, supernova explosions, and possibly echoesfrom the very early history of the universe as a whole. They are also anticipated to provide thecleanest tests of predictions of general relativity in the realm of strong gravity.However, there are important differences from standard astronomy. In electromagnetic obser-vations, in every waveband there are sources so strong that they can be detected without knowinganything about the source. You don’t need to understand nuclear fusion in order to see the Sun!In contrast, as we will see, most of the expected sources of gravitational radiation are so weakthat sophisticated statistical techniques are required to detect them at all. These techniques in-volve matching templates of expected waveforms against the observed data stream. Maximumsensitivity therefore requires a certain understanding of what the sources look like, hence of thecharacteristics of those sources. In addition, when detections occur, it will be important to putthem into an astrophysical context so that the implications of the discoveries are evident.In the next four lectures you will get a survey of sources of gravitational radiation. As anaside, it is useful to remember that historically the most interesting sources discovered with anew telescope or satellite have often been unexpected, and this is also possible with gravitationalradiation. However, you can’t sell a large project by appealing entirely to the unknown, so weshould at least describe what we can imagine at this point!Before discussing types of sources, though, we need to have some general perspective on howgravitational radiation is generated and how strong it is. We will begin by discussing radiation ina general context.By definition, a radiation field must be able to carry energy to infinity. If the amplitude of thefield a distance r from the source in the direction (θ, φ) is A(r, θ, φ), the flux through a sphericalsurface at r is F (r, θ, φ) ∝ A2(r, θ, φ). If for simplicity we assume that the radiation is sphericallysymmetric, A(r, θ, φ) = A(r), this means that the luminosity at a distance r is L(r) ∝ A2(r)4πr2.Note, though, that when one expands the static field of a source in moments, the slowest-decreasingmoment (the monopole) decreases like A(r) ∝ 1/r2, implying that L(r) ∝ 1/r2and hence noenergy is carried to infinity. This tells us two things, regardless of the nature of the radiation (e.g.,electromagnetic or gravitational). First, radiation requires time variation of the source. Second,the amplitude must scale as 1/r far from the source.– 2 –We can now explore what types of variation will produce radiation. We’ll start with electro-magnetic radiation, and expand in moments. For a charge density ρe(r), the monopole moment isRρe(r)d3r. This is simply the total charge Q, which cannot vary, hence there is no electromagneticmonopolar radiation. The next static moment is the dipole moment,Rρe(r)rd3r. There is no ap-plicable conservation law, so electric dipole radiation is possible. One can also look at the variationof currents. The lowest order such variation (the “magnetic dipole”) isRρe(r)r × v(r)d3r. Onceagain this can vary, so magnetic dipole radiation is possible. The lower order moments will typicallydominate the field unless their variation is reduced or eliminated by some special symmetry.Now consider gravitational radiation. Let the mass-energy density be ρ(r). The monopolemoment isRρ(r)d3r, which is simply the total mass-energy. This is constant, so there cannotbe monopolar gravitational radiation. The static dipole moment isRρ(r)rd3r. This, however, isjust the center of mass-energy of the system. In the center of mass frame, therefore, this momentdoes not change, so there cannot be electric dipolar radiation in this frame (or any other, sincethe existence of radiation is frame-independent). The equivalent of the magnetic dipolar momentisRρ(r)r × v(r)d3r. This, however, is simply the total angular momentum of the system, so itsconservation means that there is no magnetic dipolar gravitational radiation either. The nextstatic moment is quadrupolar: Iij=Rρ(r)rirjd3r. This is not conserved, therefore there can bequadrupolar gravitational radiation.This allows us to draw general conclusions about the type of motion that can generate gravi-tational radiation. A spherically symmetric variation is only monopolar, hence it does not produceradiation. No matter how violent an explosion or a collapse (even into a black hole!), no grav-itational radiation is emitted if spherical symmetry is maintained. In addition, a rotation thatpreserves axisymmetry (without contraction or expansion) does not generate gravitational radia-tion because the quadrupolar and higher moments are unaltered. Therefore, for example, a neutronstar can rotate arbitrarily rapidly without emitting gravitational radiation as long as it maintainsaxisymmetry.This immediately allows us to focus on the most promising types of sources for gravitationalwave emission. The general categories are: binaries, continuous wave sources (e.g., rotating starswith nonaxisymmetric lumps), bursts (e.g., asymmetric collapses), and stochastic sources (i.e.,individually unresolved sources with random phases; the most interesting of these would be abackground of gravitational waves from the early universe). We will discuss each of these insubsequent lectures.For now, however, it will be useful to reconsider gravitational waves from the standpointof dynamic changes in spacetime. We will begin by reviewing some relevant aspects of generalrelativity that we covered earlier in the class.To characterize the warping of spacetime, we define the metric tensor gαβas follows. Supposethat there are two events A and B, determined by four