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 waves:
To make the last point more quantitative, though, we have to 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, we 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 = ΣQisi, is not a conserved quantity: we can change it freely by moving a charge around, or by separating pairs of balanced positive and negative charges. This is also called the electric dipole moment, since it is the first and most significant nonzero 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 = ΣQiai ≠ 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 = ΣMiai = 0 for an isolated system:
To get our 1/r radiation field, we must have a time-varying second-order moment, or "quadrupole" moment of the mass distribution, denoted I = ΣMisi⊗si. (The notation ⊗ denotes a tensor product of two vectors, indicating that I is a rank-2 tensor. However, this distinction is not so important; what is important 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 Ms∂²a/∂t², Mv∂a/∂t, 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 4s 2. The amplitude of gravitational radiation therefore scales as:
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 instantaneous 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.
Remember 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/λ²r, or:
The first term is roughly the size of a black hole of mass M, so the distance r to the system must clearly be much greater. Similarly, v/c is the ratio of the speeds of masses in the system to the speed of light, which must be less than (usually much less than) unity. Thus h approaches unity when one is standing in the immediate vicinity of black holes moving about at 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 14km, and such objects achive speeds around c only when they collide, which might occur on a yearly basis within a volume of radius 6×1020km (20 megaparsecs). So the strongest waves we expect to observe passing the Earth will have h ∼ 10-20 or less. This is enough to distort the shape of the Earth by 10-13 metres, or about 1% of the size of an atom. By contrast, the (nonradiative) tidal field of the Moon raises a tidal bulge of about 1 metre on the Earth's oceans.
For more quantitative discussion of gravitational radiation, click here.
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additional mathematical detail.
Start: Gravitational waves demystified
Analogy: Electromagnetic fields
Electromagnetic field of an accelerated charge
Derivation of the radiative electromagnetic field
Gravitational tidal field
Equivalence between dipole and tidal field
Formulae and details
Differences between gravitational and electromagnetic radiation
Gravitational wave spectrum