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.

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

where * n* is a unit vector in the

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

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 frequency *fr* of
rotation can be treated as nearly constant over many cycles. Let
* e*1,

(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

where Φ(*t-r/c*) is the instantaneous angle between the
* ex* and

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* = *m*1+*m*2 and reduced mass
*μ* = *m*1*m*2/*M* of the system (via Kepler's law), giving:

and of course Φ=Φ0+2*πfr t*.

Like electromagnetic radiation, gravitational radiation carries
energy and momentum in the direction of propagation * n*.
The energy flux (i.e. the energy

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

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 **I***T*, 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