Aberration in Gravito-Electromagnetism
469
field) in terms of the instantaneous time is to find t r as a function of t. We will then
identify a small parameter in which we can perform a Taylor expansion to obtain
a manageable expression. 1 A useful mathematical result is the Lagrange inversion
theorem.
We define the dimensionless retarded time z as
z ≡
ct r
R 0
,
(11)
with R 0 , the distance between the center of the oscillator and the detector. To make
apparent the small parameter, we consider a motion of the form
R(z) = R 0 (1 + αf (z)),
(12)
where f (z) describes the motion around R 0 in z and α is a small parameter. We
define a dimensionless instantaneous time y in a similar way to the retarded time,
y ≡
ct
R 0
− 1 then z = y − αf (z).
(13)
The important point to note is the −1 in the definition of y it represents the time
it takes for the signal to propagate the distance R 0 . We then have z equals y plus
a small perturbation that depends on z. However, we must in the end express our
results in terms of y + 1 rather than just y as y + 1 is the actual instantaneous time
(apart from multiplicative factors).
The Lagrange inversion theorem 2 says if the equation in z
z = y + ηφ(z)
has one root in the interior of a contour C, then any function g(z) analytic on and
inside C can be expanded in a power series in η by the formula
g(z) = g(y) +
∞
n=1
η n
n!
d n−1
dy n−1
g
(y)φ
n (y)
.
We will use the theorem with φ(z) replaced by R(z) and η being replaced by −α.
The motion of the mass on the left, O 1 , is taken to be
R(z) = −r(z) = R 0 (1 + α sin(vz)) ˆ
x,
(14)
1 For a non-relativistic motion, surely t r will be equal to t plus a constant and some small
perturbation that depends on t r .
2 For details see, for example, the book on modern analysis of Whittaker and Watson [5].
469
field) in terms of the instantaneous time is to find t r as a function of t. We will then
identify a small parameter in which we can perform a Taylor expansion to obtain
a manageable expression. 1 A useful mathematical result is the Lagrange inversion
theorem.
We define the dimensionless retarded time z as
z ≡
ct r
R 0
,
(11)
with R 0 , the distance between the center of the oscillator and the detector. To make
apparent the small parameter, we consider a motion of the form
R(z) = R 0 (1 + αf (z)),
(12)
where f (z) describes the motion around R 0 in z and α is a small parameter. We
define a dimensionless instantaneous time y in a similar way to the retarded time,
y ≡
ct
R 0
− 1 then z = y − αf (z).
(13)
The important point to note is the −1 in the definition of y it represents the time
it takes for the signal to propagate the distance R 0 . We then have z equals y plus
a small perturbation that depends on z. However, we must in the end express our
results in terms of y + 1 rather than just y as y + 1 is the actual instantaneous time
(apart from multiplicative factors).
The Lagrange inversion theorem 2 says if the equation in z
z = y + ηφ(z)
has one root in the interior of a contour C, then any function g(z) analytic on and
inside C can be expanded in a power series in η by the formula
g(z) = g(y) +
∞
n=1
η n
n!
d n−1
dy n−1
g
(y)φ
n (y)
.
We will use the theorem with φ(z) replaced by R(z) and η being replaced by −α.
The motion of the mass on the left, O 1 , is taken to be
R(z) = −r(z) = R 0 (1 + α sin(vz)) ˆ
x,
(14)
1 For a non-relativistic motion, surely t r will be equal to t plus a constant and some small
perturbation that depends on t r .
2 For details see, for example, the book on modern analysis of Whittaker and Watson [5].