Formulae and details

The following formulae are quantitatively accurate expressions for gravitational radiation from sources that are only mildly relativistic (v/c much less than 1). They should be all you need to get started with computations of gravitational radiation.

Quadrupole formulae

Gravitational waves are generated by a time-varying quadrupole moment I. This is a rank-2 three-dimensional tensor: that is, it consists of 9 components Iij, where each of i and j run over the three spatial dimensions. All tensor equations on this page are written as component equations, assuming a Cartesian coordinate system. Note that the symbol I is often used to denote the moment of inertia of a system. Both the quadrupole moment and the moment of inertia are rank-2 tensors representing the second-order moment of mass with respect to position, but with different projections of their diagonal (i=j) and off-diagonal (i≠j) components. In the following derivations, where these distinctions matter, superscripts will be used to denote the particular projection.

Specifically, gravitational waves are transverse (they produce only forces perpendicular to their direction of propagation), and traceless (these forces are "shearing" forces that do not cause any overall expansion or contraction). Therefore they depend only on the "transverse traceless", or TT, projection of the quadrupole moment.

Consider a position vector s within the source (a body or system of bodies). The corresponding second-order position tensor is a rank-2 tensor with components Sij=sisj. Now suppose we are viewing the source from a location r, where r is much larger than s. Then the transverse traceless part of this tensor has components:

where n is a unit vector in the r direction, sr = sn, s² = ss, and δij is the identity tensor: δij = 1 if i = j and 0 otherwise. You can verify that the resulting tensor is symmetric (Sij = Sji for all i,j), transverse (ΣiSijni = 0 for all j), and traceless (ΣiSii = 0).

The transverse traceless quadrupole moment of a source is then given by either of the following equations, depending on whether we treat the source as a set of point particles with masses M(p) or as a continuous mass distribution with density ρ(s) over some volume:

The instantaneous gravitational wave strain and tidal field gradient are then given by:

where the (t-r/c) time dependence means that the field now depends on the configuration of the system at a time r/c in the past: this is known as the retarded time. The assumptions here are that:

The first inequality means that we do not need to worry about wave generation by or propagation through the expanding universe, which would require a more in-depth treatment by general relativity. The second inequality ensures that we are dominated by the transverse 1/r radiative fields, and not the near-zone static tidal fields. The last inequality ensures that the velocities in the source are much smaller than c, so that this 1/r radiation is dominated by the quadrupole moment and not higher moments; it also ensures that the gravitational binding energy of the system is small, so that the mass-energy of the gravitational field itself does not contribute nonlinearly to the radiation.


Some properties of gravitational radiation are easier to see by introducing a particular Cartesian coordinate system: where the z-axis points in the same direction r as the radiation is propagating, and the x and y coordinates are in the orthogonal (transverse) directions. Then we have:

where h+ and h× are called the "plus" and "cross" amplitudes for that particular choice of x and y coordinates. These are the "linear" polarization components of gravitational radiation: pure plus or cross radiation is typically produced by systems whose motions are confined to one axis, or to a plane that is viewed edge-on, as shown below.

Equivalently, gravitational radiation at a specified frequency can be decomposed into left- and right-handed circular polarizations, though this representation is less common (and usually less convenient). Pure circularly-polarized radiation is typical of rotating systems viewed along the axis of rotation, as depicted below.

Waves from rotating sources

Many sources of gravitational waves are nearly uniformly rotating systems, such as spinning rigid bodies or objects in nearly circular orbits, where the frequency fr of rotation can be treated as nearly constant over many cycles. Let e1, e2, and e3 be the principal axes of the source, and have it rotating about e3. The strongest radiation is along this rotation axis, with amplitude:

(Since we are dealing with principle axes, it doesn't matter whether Iii represents the quadrupole moment or the moment of inertia: the principle axis components are the same.)

Now suppose we are viewing the source from some other direction ez, and let &iota = arccos(e3ez) be the inclination to the line of sight. To get the x and y axes in the transverse plane, define ex ∝ ez×e3 to lie along the intersection of the plane of rotation to the transverse plane, and ey = ez×ex to give a right-handed coordinate system. Then the waveforms of the two polarizations we observe are:

where Φ(t-r/c) is the instantaneous angle between the ex and e1 axes at (retarded) time t-r/c. Note that the dependence on 2Φ rather than Φ means that the gravitational wave frequency is twice the rotation frequency fr, and the wavelength is λ = c/2fr.

If the rotating source consists of a pair of (small) bodies in a circular orbit, then the frequency and quadrupole moment are explicitly related to the total mass M = m1+m2 and reduced mass μ = m1m2/M of the system (via Kepler's law), giving:

and of course Φ=Φ0+2πfr t.

Energy transport

Like electromagnetic radiation, gravitational radiation carries energy and momentum in the direction of propagation n. The energy flux (i.e. the energy dE crossing an area dA orthogonal to n in a time dt) is given by:

The momentum flux is n/c times the energy flux: that is, one can think of the energy as being carried by a stream of particles (gravitons) moving at speed c in the direction of propagation.

If we integrate this flux over a sphere around the source, we get the total luminosity, or energy emitted per unit time. The result cannot depend on the TT quadrupole moment, since "tranverse" can only refer to a specific direction of propagation. Instead it depends just on the traceless quadrupole moment IT, whose components are:

The luminosity is then given by:

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Sections marked with provide optional additional mathematical detail.

Start: Gravitational waves demystified
Analogy: Electromagnetic fields
Electromagnetic field of an accelerated charge
  Derivation of the radiative electromagnetic field
Electromagnetic waves
Gravitational tidal field
  Equivalence between dipole and tidal field
Gravitaional waves
  Formulae and details
Differences between gravitational and electromagnetic radiation
Gravitational wave spectrum