Monopole:
When the gravitational tidal field of a source changes with time, those changes propagate out from the source at speed 𝑐. These changing tidal fields constitute gravitational radiation. If the changes are ongoing or oscillatory, they are gravitational waves, as shown below:
By analogy with electromagnetic dipole radiation, we can say the following things about gravitational radiation:
To make the last point more quantitative, though, we must delve a bit further into the nature of electromagnetism and gravity, and point out some of the differences between them. Ultimately these differences come down once again to the principle of equivalence: that the gravitational "charge" of a body is the same as its inertia.
First, note that if electric charge were not conserved, we could create pulses of longitudinal electromagnetic radiation, falling off as 1/𝑟, simply by creating or destroying charge in some location. This is called monopole radiation since at large distances it depends only on the electric "monopole", or total charge, of a system. However, conservation of total electric charge prevents this from occuring. In other words (using overdots to represent time derivatives ∂/∂𝑡):
However, the first-order spatial moment of a charge distribution, 𝑷 = ∑𝑄𝑖 𝒔𝑖, is not a conserved quantity: we can change it freely by moving a charge around, or by separating pairs of positive and negative charges. This is called the electric dipole moment, since it is the dominant moment of an electric dipole (i.e. a pair of balanced positive and negative charges). As shown ealier, we get 1/𝑟 transverse electromagnetic radiation, or dipole radiation, when ∂2𝑷/∂𝑡2 = ∑𝑄𝑖 𝒂𝑖 ≠ 0:
Now consider the case of gravitation. Once again, conservation of mass prevents monopole radiation. However, conservation of momentum (Newton's third law) means that we always have ∂2𝑷/∂𝑡2 = ∑𝑀𝑖 𝒂𝑖 = 0 for an isolated system:
To get the 1/𝑟 radiation field, there should be a time-varying second-order moment, or quadrupole moment of the mass distribution, denoted 𝐈 = ∑𝑀𝑖 𝒔𝑖⊗𝒔𝑖. (The notation ⊗ denotes a tensor product of two vectors, making 𝐈 a rank-2 tensor. This is not actually the full expression for 𝐈; see here for the mathematical details. For now, all we need to know is that 𝐈, and its magnitude 𝐼, are quadratic in 𝒔.) The strength of the quadrupole field then scales as:
Note that the 1/𝑟 radiation term depends on the fourth time derivative of the quadrupole moment. In general this will contain terms of the form 𝑀𝑠𝑎̈, 𝑀𝑣𝑎̇, and 𝑀𝑎2. For the case of oscillatory motion, like the animation at the top of this page, with masses 𝑀 moving over distances 𝑠 with frequency 𝑓, all these terms scale as 𝑀𝑓 4𝑠2. The amplitude of gravitational radiation therefore scales as:
Also note from the animation that the field configuration repeats when the system has undergone half a rotation, meaning that quadrupole radiation has a frequency that is twice the rotation frequency of the source.
The tidal field 𝑔′ is the physically measurable part of gravitational phenomena: it represents an observable relative acceleration or force between two displaced "test masses". However, when discussing gravitational waves, the most common parameter describing the amplitude is a dimensionless strain:
What does this quantity mean? Remember that 𝑔′ is a gravity gradient, so 𝑔′×𝑑 gives the difference in gravity, i.e. the differential acceleration, between two objects separated by a small displacement 𝑑. Two time integrals of acceleration give us the change in this displacement as a function of time. Thus ℎ is twice the fractional change in displacement between two nearby masses due to the gravitational wave. This change in displacement occurs in the plane transverse to the direction of radiation, and causes a stretch along one axis and a squeeze along the orthogonal axis: this is illustrated below, showing how a ring of freely-floating masses would be disturbed by a passing gravitational wave. The net distortion is twice as much as a stretching or squeezing alone, which is the reason for the factor of 2 in the equations for ℎ.
Note that ℎ is not itself directly observeable. A constant ℎ, or an ℎ that varies linearly with time, is exactly equivalent to starting the masses at slightly different positions, or with a slight relative velocity. Only the second and higher derivatives of ℎ produce accelerations that would indicate the presence of gravitational radiation.
From the above scaling for 𝑔′ we get ℎ ∼ 𝐺𝑀𝑠2/𝑐2λ2𝑟, or:
The first term is roughly the size of a black hole of mass 𝑀, so the distance 𝑟 to the system must be (much) greater. Similarly, 𝑣/𝑐 gives the internal speeds of masses in the system in units of the speed of light, which must be less than (usually much less than) one. Thus ℎ approaches unity when one is standing in the immediate vicinity of black holes moving around each other at close to lightspeed, and is less for any other circumstance.
In particular, the length scale of a "typical" black hole 10× as massive as our Sun is 14 km, and a pair of such objects achive relative speeds ∼ 𝑐 just before they collide, which might occur on a yearly basis within a volume of radius 6×1021 km (200 megaparsecs). So the strongest waves we expect to observe passing the Earth will have ℎ ≲ 10−21. This is enough to distort the shape of the Earth by 10−14 m, or about 0.1% of the size of an atom. By contrast, the (nonradiative) tidal field of the Moon raises a tidal bulge of about 1 m on the Earth's oceans.
For more quantitative discussion of gravitational radiation, click here.
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