4
Bateman Waves in the Linear Approximation
Abstract
In Minkowskian space-time in which points are labelled with rectangular Cartesian coordinates x, y, z and time t the so-called t-lines correspond to constant
values of x, y, z. These world lines constitute a time-like geodesic congruence
which is twist-free, shear-free and expansion-free. They play an important role
in encoding in a bivector field on Minkowskian space-time the information
contained in the electric and magnetic 3-vectors on three dimensional Euclidean
space. A striking illustration of this is found in the study of electromagnetic
radiation and the derivation of Bateman electromagnetic waves. The counterpart
of the latter in the case of gravitational waves in the linear approximation using
the gauge invariant and covariant approach demonstrates that the existence of the
gravitational waves is due to perturbations in the shear of the t-lines.
4.1
Electromagnetic Radiation
As coordinates x i (i = 0, 1, 2, 3) in Minkowskian space-time we take x 0 = t as the
time coordinate and x 1 = x, x 2 = y, x 3 = z as rectangular Cartesian coordinates,
writing for short x α = (x, y, z) (α = 1, 2, 3). The Minkowskian metric tensor
in these coordinates has components η ij = diag(1, −1, −1, −1). Indices on the
components of tensors will be raised and lowered using η ij and η ij respectively
with η ij defined by η ij η jk = δ
i
k . A vacuum Maxwell field in three dimensional
Euclidean space is described by a pair of 3-vectors E, B representing the electric
and magnetic fields respectively. When viewed as a field on Minkowskian spacetime the information in this pair of 3-vectors is encoded in a real bivector field with
components F ij = −F ji . The relationship between the pair of vector fields and the
tensor field is made using the time-like geodesic congruence on Minkowskian spacetime consisting of the integral curves of the vector field u i = δ
i
0 . These parallel
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. A. Hogan, D. Puetzfeld, Frontiers in General Relativity, Lecture Notes
in Physics 984, https://doi.org/10.1007/978-3-030-69370-1_4
69
Bateman Waves in the Linear Approximation
Abstract
In Minkowskian space-time in which points are labelled with rectangular Cartesian coordinates x, y, z and time t the so-called t-lines correspond to constant
values of x, y, z. These world lines constitute a time-like geodesic congruence
which is twist-free, shear-free and expansion-free. They play an important role
in encoding in a bivector field on Minkowskian space-time the information
contained in the electric and magnetic 3-vectors on three dimensional Euclidean
space. A striking illustration of this is found in the study of electromagnetic
radiation and the derivation of Bateman electromagnetic waves. The counterpart
of the latter in the case of gravitational waves in the linear approximation using
the gauge invariant and covariant approach demonstrates that the existence of the
gravitational waves is due to perturbations in the shear of the t-lines.
4.1
Electromagnetic Radiation
As coordinates x i (i = 0, 1, 2, 3) in Minkowskian space-time we take x 0 = t as the
time coordinate and x 1 = x, x 2 = y, x 3 = z as rectangular Cartesian coordinates,
writing for short x α = (x, y, z) (α = 1, 2, 3). The Minkowskian metric tensor
in these coordinates has components η ij = diag(1, −1, −1, −1). Indices on the
components of tensors will be raised and lowered using η ij and η ij respectively
with η ij defined by η ij η jk = δ
i
k . A vacuum Maxwell field in three dimensional
Euclidean space is described by a pair of 3-vectors E, B representing the electric
and magnetic fields respectively. When viewed as a field on Minkowskian spacetime the information in this pair of 3-vectors is encoded in a real bivector field with
components F ij = −F ji . The relationship between the pair of vector fields and the
tensor field is made using the time-like geodesic congruence on Minkowskian spacetime consisting of the integral curves of the vector field u i = δ
i
0 . These parallel
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. A. Hogan, D. Puetzfeld, Frontiers in General Relativity, Lecture Notes
in Physics 984, https://doi.org/10.1007/978-3-030-69370-1_4
69