Now what happens if a charge starts out at rest, and then is suddenly accelerated to some constant velocity?

The field should initially be that of a stationary charge:
observers have no way of knowing that it will suddenly start moving.
Even after it starts moving, distant observers will take time to
realize this: information about the sudden change in motion cannot
reach them any faster than the maximum speed *c* allowed by
relativity. (This speed is commonly called "the speed of light",
though it is actually the *maximum* speed of light, or of any
other physical particle or wave.)

Meanwhile, once the charge reaches a uniform velocity, observers close to it should simply see the ordinary field of a moving charge: the fact that it used to be "stationary" is not permanently imprinted on the charge. This leads to the following field:

Close in we have the field of a moving charge, and farther out we
have the field of a stationary charge. Between these two regions is a
spherical shell of stretched field lines connecting the two fields.
This shell carries the information about the charge's sudden surge of
acceleration: it expands at speed *c*, but has a constant
thickness equal to *c*Δ*t*, where Δ*t* is
the duration of the acceleration.

The stretched field lines in this shell are what we call
*electromagnetic radiation*. Two properties are immediately
obvious from the diagram:

- The fields in electromagnetic radiation are not radial, but
*transverse*(i.e. perpendicular to the radius). - Far from the source, the field lines of the radiation are much more tightly packed than the "backgound" of the stationary or uniformly moving source.

To make that last point more quantitative, note that the field
lines of a stationary charge spread radially. At a distance *r*
from the source, a sphere with area 4*πr*² intersects all
field lines perpendicularly, so the field line density (field
strength) goes down as 1/*r*²:

where *Q* is the charge and 1/4*πε*0 is Coulomb's electric constant.

By contrast, in the radiation shell, the field lines are largely
transverse. A circular strip bounded by the inner and outer edges of
the shell will intersect some fraction of the field lines: as the
shell expands, the strip's radius increases but its width
*c*Δ*t* does not, while it continues to cut across the
same set of field lines. The field line density (field strength)
therefore goes down as 1/*r*. Furthermore, it can readily be shown that the
fraction of field lines that pass through any given strip goes as
*v*⊥/*c*, where
*v*⊥ is the charge's final velocity
component perpendicular to the strip, i.e. perpendicular to the radial
line. (This formula changes when one includes the relativistic
"squeezing" of the moving field, but is correct for speeds much less
than *c*.) Thus the transverse field in the shell is:

where *a*⊥ is the component of the
charge's *acceleration* perpendicular to the radial line. So,
far from the originating sources, the radiative fields will be far
stronger than the stationary fields.

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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