2
Bivector Formalism in General Relativity
Abstract
The use of bivectors, or skew symmetric tensors, is a powerful research tool
in general relativity. Such objects arise naturally in the context of electromagnetism. We therefore introduce them via electromagnetic test fields on arbitrary
space-times. All details are provided including an important application to
electromagnetic radiation due to Ivor Robinson. This is followed by the extension
of bivector theory to gravitational fields. An explicit illustration of their use in
relation to Kerr space-time is given.
2.1
Bivectors and Electromagnetic Fields
A source-free electromagnetic test field on a flat or curved background space-time is
described by a skew-symmetric tensor with real components F ab = −F ba satisfying
Maxwell’s source-free field equations
F
ab ;b = 0 and
∗ F
ab ;b = 0 ,
(2.1)
where
∗ F ab =
1
2
η abcd F
cd ,
(2.2)
are the components of the dual of F ab . The covariant derivative with respect to
the Riemannian connection calculated with the metric tensor g ab is indicated by
a semi-colon, η abcd =
√
−g g abcd , with g = det(g ab ) and abcd the Levi-Civita
permutation symbol in four dimensions, is the covariant permutation tensor and
indices are raised with g ab , where g ab g bc = δ a
c , as usual. From the properties of
η abcd (see Appendix A) it is useful to note that the dual of the dual leads back to the
© 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_2
19
Bivector Formalism in General Relativity
Abstract
The use of bivectors, or skew symmetric tensors, is a powerful research tool
in general relativity. Such objects arise naturally in the context of electromagnetism. We therefore introduce them via electromagnetic test fields on arbitrary
space-times. All details are provided including an important application to
electromagnetic radiation due to Ivor Robinson. This is followed by the extension
of bivector theory to gravitational fields. An explicit illustration of their use in
relation to Kerr space-time is given.
2.1
Bivectors and Electromagnetic Fields
A source-free electromagnetic test field on a flat or curved background space-time is
described by a skew-symmetric tensor with real components F ab = −F ba satisfying
Maxwell’s source-free field equations
F
ab ;b = 0 and
∗ F
ab ;b = 0 ,
(2.1)
where
∗ F ab =
1
2
η abcd F
cd ,
(2.2)
are the components of the dual of F ab . The covariant derivative with respect to
the Riemannian connection calculated with the metric tensor g ab is indicated by
a semi-colon, η abcd =
√
−g g abcd , with g = det(g ab ) and abcd the Levi-Civita
permutation symbol in four dimensions, is the covariant permutation tensor and
indices are raised with g ab , where g ab g bc = δ a
c , as usual. From the properties of
η abcd (see Appendix A) it is useful to note that the dual of the dual leads back to the
© 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_2
19