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 cΔt, where Δt 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:
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 4πr² intersects all field lines perpendicularly, so the field line density (field strength) goes down as 1/r²:
where Q is the charge and 1/4π&epsilon0 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 cΔt 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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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