Rotational Inertia:
Traceless Quadrupole:
Transverse Traceless:
The following formulae are quantitatively accurate expressions for gravitational radiation from sources that are only mildly relativistic (𝑣/𝑐 ≪ 1). They should be all you need to get started with computations of gravitational radiation.
Gravitational waves are generated by a time-varying quadrupole moment: a second-order spatial moment of the mass distribution. We represent any such moment as a rank-2 tensor (or matrix) 𝐈, which consists of 9 components 𝐼𝑖𝑗, where each of 𝑖 and 𝑗 run over the three spatial dimensions (𝑥, 𝑦, 𝑧). All tensor equations on this page are written as component equations, assuming a Cartesian coordinate system.
Note that there are several such second-order moments used in different areas of physics. Most commonly, the symbol 𝐈 is used to represent the rotational moment of inertia, which is not the same as the quadrupole moment that appears in multipole representations of the gravitational field. E.g. the external gravitational field of a sphere depends only on its total mass (monopole moment), not its radius. Changing the size of the sphere will definitely affect its moment of inertia, but will not change its mass quadrupole.
To make our notation explicit, we use superscripts to identify the different types (projections) of second-order mass moments: 𝐈𝑅 for the rotational inertia, 𝐈𝑇 for the traceless quadrupole moment. A third projection that is relevant to gravitational radiation is the transverse traceless quadrupole moment 𝐈𝑇𝑇, whose components are orthogonal to some direction of propagation 𝒏. This is because gravitational waves themselves are transverse (producing forces perpendicular to their direction of propagation) and traceless (producing "shearing" forces that cause no overall expansion or contraction). Emission of gravitational waves generally requires a time-varying 𝐈𝑇, but gravitational wave emission in a particular direction depends on the time-varying 𝐈𝑇𝑇 for that direction 𝒏.
Let 𝒔 be a position vector within a mass distribution with density ρ(𝒔), and 𝒏 be a unit vector pointing towards an observer. The components of the different second-order mass moments are:
where 𝑠𝑛 = 𝒔·𝒏 = ∑𝑘 𝑠𝑘𝑛𝑘, 𝑠2 = 𝒔·𝒔 = ∑𝑘 𝑠2𝑘, and δ𝑖𝑗 is the identity matrix (δ𝑖𝑗 = 1 if 𝑖=𝑗, 0 otherwise).
All these formulae are explicitly symmetric in 𝑖↔𝑗. It is straightforward to show that (𝑠𝑖𝑠𝑗 − ⅓ 𝑠2δ𝑖𝑗), and hence 𝐼𝑇𝑖𝑗, is traceless: ∑𝑖 𝐼𝑇𝑖𝑖 = 0. Slightly more involved algebra will show that 𝐼𝑇𝑇𝑖𝑗 is both traceless and transverse: ∑𝑖 𝐼𝑇𝑇𝑖𝑗 𝑛𝑖 = 0 for all 𝑗. The same formulae can be used for a set of discrete masses 𝑀(𝑝) by replacing the integral ∭ρ(𝒔) 𝑑3𝒔 with a sum ∑(𝑝) 𝑀(𝑝).
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 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 middle inequality ensures that we are dominated by the transverse 1/𝑟 radiative fields, and not the non-propagating near-zone tidal fields. The last inequality implies that the velocities in the source are much smaller than 𝑐, so that this 1/𝑟 radiation will usually be dominated by the quadrupole moment and not higher moments; it also ensures that the gravitational binding energy of the system is small compared to its rest mass, so that the mass-energy of the gravitational field itself does not contribute 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 𝒏 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.
Many sources of gravitational waves are nearly uniformly rotating systems, such as spinning rigid bodies or objects in nearly circular orbits, where the angular frequency of rotation ω𝑟 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:
Clearly the system must be non-axisymmetric (𝐼11≠𝐼22) in order to radiate, since only this will produce a changing quadrupole moment. (The tensor 𝐈 in this expression may be 𝐈𝑇, 𝐈𝑇𝑇, or 𝐈𝑅. Since we are working in a principle-axis basis where 𝐼𝑖𝑗 is diagonal and 𝒏 = 𝒆3, we have 𝐼11 − 𝐼22 = ∭ρ(𝒔)[ 𝑠21 − 𝑠22 ]𝑑3𝒔 for both 𝐈𝑇 and 𝐈𝑇𝑇. Using 𝐈𝑅 gives the opposite sign but does not change ℎ0.)
Now suppose we are viewing the source from some other direction 𝒏 = 𝒆𝑧, and let ι = arccos(𝒆3 · 𝒆𝑧) be the inclination to the line of sight. We define 𝑥 and 𝑦 axes in the transverse plane such that 𝒆𝑥 ∝ 𝒆3 × 𝒆𝑧 points along the intersection of rotational and transverse planes, and 𝒆𝑦 = 𝒆𝑧 × 𝒆𝑥 gives a right-handed coordinate system. Then the waveforms of the two observed polarizations are:
where Φ(𝑡 − 𝑟/𝑐) is the instantaneous angle between the 𝒆𝑥 and 𝒆1 axes at (retarded) time 𝑡 − 𝑟/𝑐. For constant or nearly-constant frequency we can simply write Φ = Φ0 + ω𝑟 𝑡. Note that the waves' dependence on 2Φ rather than Φ means that the gravitational wave angular frequency ω𝑔𝑤 is twice the rotation angular frequency ω𝑟, hence:
If the rotating source consists of a pair of bodies 𝑚1, 𝑚2 in a circular orbit with separation 𝑅, then the quadrupole moment of the system is 𝐼11 = μ𝑅2 where μ = 𝑚1𝑚2/𝑀 is the reduced mass and 𝑀 = 𝑚1+𝑚2 is the total mass of the system. We can use Kepler's law 𝐺𝑀 = ω2𝑟 𝑅3 to eliminate 𝑅:
where the last form introduces the chirp mass ℳ = μ3/5𝑀2/5 = (𝑚1𝑚2)3/5/𝑀1/5 (so named for reasons explained below). This has the interesting property that, at any given frequency, systems with the same chirp mass produce the same quadrupole radiation, even if the individual masses and separations are very different.
Like electromagnetic radiation, gravitational radiation carries energy and momentum in the direction of propagation 𝒏. The energy flux density, or irradiance (i.e. the energy 𝑑𝐸 crossing an area 𝑑𝐴 orthogonal to 𝒏 in a time 𝑑𝑡) is given by:
The momentum flux density is 𝒏/𝑐 times the energy flux density: 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. For electromagnetic radiation, the momentum flux density is often called the radiation pressure, though this supposes that the momentum stream could be substantially absorbed or reflected by a barrier – a situation that never occurs with gravitational radiation.
If we integrate the energy flux density over a sphere around the source, we get the total energy flux, i.e. energy emitted per unit time (sometimes called radiant power, radiosity, or luminosity). 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:
For a system undergoing uniform (or nearly uniform) rotation about a principal axis 𝒆3, this takes on the form:
For a system of two masses in Keplerian orbit we once again get an interesting result that the energy flux depends only on the frequency and chirp mass ℳ:
Two masses in Keplerian orbit have a total energy 𝐸 = −𝐺𝑀μ/2𝑅 (negative because they are bound with respect to masses at infinite separation). As gravitational waves carry away energy, the masses become more tightly bound, spiraling in to tighter and faster orbits. The orbital (and gravitational-wave) frequency sweeps up in a "chirp" characterized by the chirp mass ℳ. Kepler's law gives us 𝐸 = −½μ (𝐺𝑀ω𝑟)2/3; differentiating with respect to 𝑡 and setting equal to the negative of the gravitational-wave flux gives:
Integrating to get ω𝑟(𝑡), and again to get the orbital phase angle Φ(𝑡):
where we have written the time dependence explicitly as a time to coalescence 𝑡𝑐 when 𝑅 → 0 and ω𝑟 → ∞. Note that even as the frequency goes to infinity, the orbital phase (and number of orbits) remains finite. In reality the frequency never goes to infinity, because actual objects coalesce at nonzero orbital radius: either due the physical size of the objects, or (for sufficiently compact objects) due to destabilizing relativistic effects.
In particular, considering the case of an extreme mass ratio inspiral μ ≪ 𝑀, the orbital evolution timescale ω𝑟/ ̇ω𝑟 is always much longer than the dynamic (orbital) timescale 1/ω𝑟, by at least a factor ∼ 𝑀/μ even for highly-relativistic orbits. The objects will spiral in gradually until they reach an innermost stable circular orbit (ISCO), at which point they abruptly plunge and merge within one orbit. The critical orbit turns out to be:
For non-extreme mass ratio inspirals, the orbital evolution and dynamic timescales become comparable as the bodies approach the instability point, and the transition from "inspiral" to "plunge" is smoothed out rather than occuring at a definite 𝑅ISCO. Notheless the above formulae are a useful guide for where the inspiral signal cuts off. In this case, however, it will be followed by an oscillation or ringdown of the newly-formed black hole, which will emit gravitational waves with frequencies ω𝑔𝑤 between ∼ ⅓ and ∼ few × 𝑐3/𝐺𝑀.
An individual spinning body emits gravitational waves if it is non-axisymmetric, e.g. due to rigid deformation. Such an object is typically modeled as a triaxial ellipsoid spinning about a principal axis 𝒆3, with ellipticity ε in its equatorial cross-section (note that this is not the same as spin-flattening, which produces an axisymmetric oblate spheroid with a circular equatorial cross-section ε = 0). Ellipticity ε in the rotational plane gives |𝐼11 − 𝐼22| = ε 𝐼𝑅33 where 𝐼𝑅33 is the rotational inertia (not the quadrupole moment) about the spin axis. For objects with non-ellipsoidal shapes this can be taken as a defining equation for their effective ellipticity. Thus:
The radiated energy flux in this case is drawn not from gravitational binding energy, but from the rotational energy 𝐸 = ½ 𝐼𝑅33 ω2𝑟, so gravitational-wave emission slows the spin of the object. Differentiating over 𝑡 and setting equal to the negative radiated power gives the spindown rate:
Other processes besides gravitational-wave emission can drive spindown (and a few can cause spinup, though this is rare). For a known object with a measured spindown, assuming that most or all of the spindown is due to gravitational radiation, we can combine the above equations to get a spindown limit on ℎ0:
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