Let's start with the usual high-school description of an electric field around an isolated electric charge. A positive charge produces a field that points radially away from it, whose strength falls off as the inverse square of distance. Pictorially this is shown by drawing "field lines" originating at the source: the direction of the field is given by the direction of the lines, and the strength of the field is given by density of lines through a surface perpendicular to them. This field is shown below.
Now consider a charge that is moving uniformly with constant velocity. According to special relativity, an observer moving with the same velocity should see just the electric field of a stationary charge: the field should point radially away from the location of the charge in that moving reference frame, with a strength (field line density) that falls off as the inverse square of the distance measured in that reference frame.
This means that in the "stationary" reference frame, the field lines will still remain connected to and point radially from the moving charge: effectively the change in reference frame transforms the spray of field lines as if they were a physical object moving along with the charge. This is shown below.
(This diagram assumes a charge moving at 0.5 times the speed of light, and includes a slight horizontal "squeezing" of the field lines due to relativistic length contraction. However, this squeezing is not essential to any of the subsequent discussion of electromagnetic radiation.)
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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