Time

time

-Two polarizations-

Figure 12.7. Ring of test masses in free space oscillating when a monochromatic (single frequency) gravitational wave is passing into or out of the page. Time runs downward (arrows). The particle oscillations are transverse to the propagation direction as in electromagnetic radiation. The two linear polarizations are shown. Two circular polarizations not shown are linear combinations of the linear polarizations.

where l is the length across the ring and A l is the change in the length due to the wave. The strains associated with the two polarizations shown in Fig. 7 are denoted as h + and hx respectively.

Quadrupole radiation

The gravitational radiation predicted by general relativity is quadrupole (or highermoment) radiation. This describes the angular distribution of the emitted radiation. It arises from a mass distribution with an accelerating quadrupole moment. Dipole radiation is not possible because, in part, there is only one sign of gravitational mass; one can not assemble a dipole distribution of gravitational mass because one would need + and — masses.

A mass has a quadrupole moment if it looks like a discus or a football, as well as a host of other non-symmetrical shapes. A sphere has no quadrupole moment. The discus and football (without the pointed ends or sharp edges) approximate the shapes of the oblate spheroid and the prolate spheroid respectively. The degree of departure from a spherical shape is described by the quadrupole moment tensor Q.

The tensor Q has nine components Qjk where j and k each take on values 1, 2, and 3 which may correspond to the x, y, z axes of a Cartesian coordinate system. For a set of discrete masses, a representative component of Q is where each term of the summation refers to one of the masses, r is the distance of the object from the chosen origin (often the center of mass), Xj and xk are the j and k position components (e.g., the y and z components) of mass element mA, and 8 jk is the Kronecker delta symbol which equals unity if j = k and zero if j = k. Note that the units are those of the moment of inertia, kg m2.

It is instructive to construct from (8) the 3 x 3 quadrupole tensor for several simple mass distributions such as a dumbbell or four masses in a square pattern. For a symmetric distribution, e.g., six masses symmetrically placed on the positive and negative legs of the three axes, the expression (8) yields zero for all nine components. Similarly a sphere would yield zero values. In contrast prolate and oblate distributions yield non-zero components. We give an example below.

It actually only takes five numbers to describe the "quadrupoleness" of a mass distribution. Quadrupoles can be described with spherical harmonics with index 1 = 2 and there are five of them (21 + 1 = 5). Our tensor is symmetric in the off-diagonal elements and hence three of the nine values are redundant. Also the trace of the matrix (sum of the diagonal elements) is constrained to a fixed value (zero for the normalization of (8)) which makes another component redundant. The five remaining independent values are those needed. (Do not confuse the " 1" in this paragraph with that for the "length" elsewhere in this chapter.)

A rapid oscillation of the shape of the mass, and therefore its quadrupole moment, e.g., from prolate to oblate to prolate, etc., will generate gravitational waves. It is the second time derivative of the tensor Q that yields (not derived here) the strain tensor h with components hjk at a distance R (m) from the oscillating mass. Thus, without proof, and for SI units,

The derivative of a tensor is obtained simply by taking the derivative of each of its components. This tensor contains the information needed to extract the strength of the two polarizations h+ and hx observed at some arbitrary distance and direction. This "acceleration" of Q is the analog of the acceleration of electric dipoles which yield electromagnetic radiation.

(kg m2; component jk of (12.8) quadrupole tensor)

2G d2Qjk 1

c4 dt2 R

44 d2 Qjk 1

strain tensor)

Merger of neutron-star binary

The small value of the coefficient in (9) means that the values of h are very small unless very massive bodies are considered, that is astronomical bodies. Consider two neutron stars spiraling around one another in a binary system that is in its death throes (Fig. 8a). Let them be in circular orbits almost in contact with one another so their surfaces almost touch. (Tidal forces would begin to distort the stars a few orbits prior to this.) Application of Kepler's third law yields an orbital period of only ~1.0 ms.

One can refer to the orbits of the two stars as a "binary circular orbit" (in the singular) because momentum conservation demands that the orbits of the two stars about the center of mass have identical eccentricities. Otherwise the center of mass would not be stationary (or uniformly moving).

Variable quadrupole moment

The quadrupole moment tensor at time t for this case is easily constructed from (8) if we approximate the mass distribution as two point masses each of mass m in a circular binary orbit of angular frequency m (rad/s) in the x, y plane. Each mass is at a distance r from the origin (Fig. 8a); the origin is taken to be the center of mass. The separation of the stars is a = 2r.

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