20
2 Bivector Formalism
original skew-symmetric tensor in the sense that
∗∗ F ab =
1
2
η abcd
∗ F
cd
=
1
4
η abcd η
cdpq F pq = −F ab .
(2.3)
If a time-like vector field with components u a is chosen such that u a u a = 1 then
u a is the 4-velocity of an observer. Relative to this observer the electromagnetic
field described by F ab consists of an electric vector E a = F ab u b and a magnetic
vector H a = ∗ F ab u b . Both of these vectors are orthogonal to u a and thus each
has three independent components. There is a one to one correspondence between
the six independent components of the electric and magnetic vectors and the six
independent components of F ab . A knowledge of F ab , given u a , clearly determines
E a and H a . The converse is also true in the sense that a knowledge of E a and H a ,
given u a , determines F ab . This follows from the formula
F ab = E a u b − E b u a + η abcd u
c H
d ,
(2.4)
which the reader may like to deduce. In this sense a pair of vectors E a , H a is
encoded in the skew-symmetric tensor F ab . For this reason such an F ab is referred
to as a (real) bivector. We can write Maxwell’s equations (2.1) in the complex form
F
ab ;b = 0 ,
(2.5)
with F ab = F ab + i ∗ F ab . Thus F ab = −F ba is a complex bivector which satisfies
∗
F ab = −i F ab .
(2.6)
There are two quadratic invariants, F ab F ab and F ab
∗ F ab , which can be
constructed from the Maxwell bivector F ab . The reader can verify that the invariant
∗ F ab
∗ F ab is proportional to F ab F ab . In terms of the electric and magnetic vectors
introduced above these invariants are given by
1
2
F ab F
ab
= E a E
a
− H a H
a and
1
2
F ab
∗ F
ab
= 2 E a H
a .
(2.7)
Hence if the electric and magnetic vectors are equal in magnitude and orthogonal to
each other then these invariants both vanish. In this case the electromagnetic field
is a pure electromagnetic radiation field. The simplest such case one can imagine
is that of plane electromagnetic waves. Such waves are light waves and thus travel
with the speed of light. The histories of the plane wave fronts are null hypersurfaces
in space-time. Such hypersurfaces have equations of the form
ϕ(x
a ) = constant ,
(2.8)
2 Bivector Formalism
original skew-symmetric tensor in the sense that
∗∗ F ab =
1
2
η abcd
∗ F
cd
=
1
4
η abcd η
cdpq F pq = −F ab .
(2.3)
If a time-like vector field with components u a is chosen such that u a u a = 1 then
u a is the 4-velocity of an observer. Relative to this observer the electromagnetic
field described by F ab consists of an electric vector E a = F ab u b and a magnetic
vector H a = ∗ F ab u b . Both of these vectors are orthogonal to u a and thus each
has three independent components. There is a one to one correspondence between
the six independent components of the electric and magnetic vectors and the six
independent components of F ab . A knowledge of F ab , given u a , clearly determines
E a and H a . The converse is also true in the sense that a knowledge of E a and H a ,
given u a , determines F ab . This follows from the formula
F ab = E a u b − E b u a + η abcd u
c H
d ,
(2.4)
which the reader may like to deduce. In this sense a pair of vectors E a , H a is
encoded in the skew-symmetric tensor F ab . For this reason such an F ab is referred
to as a (real) bivector. We can write Maxwell’s equations (2.1) in the complex form
F
ab ;b = 0 ,
(2.5)
with F ab = F ab + i ∗ F ab . Thus F ab = −F ba is a complex bivector which satisfies
∗
F ab = −i F ab .
(2.6)
There are two quadratic invariants, F ab F ab and F ab
∗ F ab , which can be
constructed from the Maxwell bivector F ab . The reader can verify that the invariant
∗ F ab
∗ F ab is proportional to F ab F ab . In terms of the electric and magnetic vectors
introduced above these invariants are given by
1
2
F ab F
ab
= E a E
a
− H a H
a and
1
2
F ab
∗ F
ab
= 2 E a H
a .
(2.7)
Hence if the electric and magnetic vectors are equal in magnitude and orthogonal to
each other then these invariants both vanish. In this case the electromagnetic field
is a pure electromagnetic radiation field. The simplest such case one can imagine
is that of plane electromagnetic waves. Such waves are light waves and thus travel
with the speed of light. The histories of the plane wave fronts are null hypersurfaces
in space-time. Such hypersurfaces have equations of the form
ϕ(x
a ) = constant ,
(2.8)