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:

- Whereas static fields have both radial and transverse components, the radiative fields are purely transverse.
- Whereas static fields fall off as 1/
*r*³, the radiative fields fall off only as 1/*r*, and soon completely dominate over the static fields.

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* = Σ

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
*∂*²* P*/

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** = Σ*Mi si*⊗

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∂*²*a*/*∂t*²,
*Mv∂a*/*∂t*, and *Ma*². 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* 4*s* 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* ∼ *GMs*²/*λ*²*r*,
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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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