Here we provide a geometric derivation of the radiative 1/*r*
piece of the electromagnetic field.

First, we assume that the charge velocity is always much less than
*c*, so that we can ignore relativistic effects such as
"squeezing" of the field lines. This means that the inner
moving-field part of a field line will be parallel to its outer
static-field part. The diagram below uses
*v*≈*c*/2 to make the geometry more obvious, but
relativistic squeezing has been ignored, and our derivation will take
the limit as *v* becomes much less than *c*.

Second, we will consider the radiative field at an angle
*θ* from the axis of the charge's motion, and at a distance
*ct* much greater than *c*Δ*t*, where *t* is
the time that the shell reaches that distance and Δ*t* is
the duration that the charge was accelerating. This means that the
shell is very thin compared to its radius, the field lines in the
shell are very close to transverse, and the distance from the charge's
initial to its current position is very close to *vt*.

Now consider the green strip indicated on the diagram. All of the
field lines within the shaded blue region will pass orthogonally
through that strip. The angular span *α* of field lines
that will intersect this strip can be computed from the triangle with
sides *ct* and *vt*:

Taking *v* much less than *c* gives
*α*=(*v*/*c*)sin*θ*.

Now at a radius *r*=*ct*, these field lines would
normally be spread over a linear distance *rα*, but
instead they have been concentrated into a strip of width
*c*Δ*t*. This represents an enhancement of the
radiative transverse field over the static field by a factor
*rα*/*c*Δ*t*, giving:

(More formally, we can rotate the whole diagram about the axis of
motion, show that the solid angle subtended by the blue region is
2*π*(*v*/*c*)sin²*θ*, and the area
of the circular green strip is
2*πrc*Δ*t*sin*θ*, and obtain the same
result.)

But (*v*/Δ*t*)sin*θ* is just the
transverse component of the charge's average acceleration,
*a*⊥, so we have:

Now of course the field lines in the shell are not purely
transverse: they move radially outward a distance cΔ*t*
while sweeping transversely a distance *rα*. So the radial
component is a factor *c*Δ*t*/*rα* less
than the transverse component. But this means that the radial
component is just given by the standard formula for a static field,
while the formula above gives the transverse component of the overall
field.

It turns out that this formula is correct even if the charge is
doing something other than a simple impulsive acceleration, as long as
its velocity is always much less than *c*. Specifically, the
transverse field at a distance *r* depends on the instantaneous
acceleration of the charge at a time *r*/*c* in the past
(called the *retarded time*):

If the source of the field is a collection of charges close
together, we can simply add their fields. By "close together" we mean
that the shortest timescales of their motion is much greater than the
light travel time between them, but much less than the light travel
time to the observer. This allows us to define a common origin for
the source, a common (retarded) time coordinate *t'=t-r/c* at
that origin that is valid over the entire source, and a common
transverse direction. This gives:

where * P*=Σ

where the first formula is suitable for a continuous charge
distribution *ρ*(* s*), and the second for a
collection of discrete charges

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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

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