72
4 Observer Space-Time Formalism
σ + :=
1 0
0 −1
,
σ × :=
0 1
1 0
,
(4.7.27)
W + := E + + B × ,
W × := E × − B + ,
(4.7.28)
and
E +,× := σ
AB
+,× E AB ,
B +,× := σ
AB
+,× B AB .
(4.7.29)
We next express these equations in terms of physically-interpretable quantities. We
first split J into its determinant and conformal parts
J ≡ D J c ,
det J c ≡ 1 ,
(4.7.30)
and then express the latter through sl(2, R) generators
J c := exp
s ◦ ε + s + σ + + s × σ ×
≡ 1 cosh S + (s ◦ ε + s + σ + + s × σ × )
sinh S
S
,
(4.7.31)
where
ε AB ≡ ε AB ,
S :=
−s 2
◦ + s
2
+ + s
2
× .
(4.7.32)
In particular, the eigenvalues of J now read
λ ± [ J] = De
±S
,
(4.7.33)
and we also define the complex combination
s := s + + is × ,
S ≡
−s 2
◦ + |s| 2 .
(4.7.34)
Thus, for a source located at γ(ζ, ϑ) with 4-velocity e
μ
0 (γ(ζ, ϑ)), D(ζ, ϑ) is the
angular diameter distance to it, s ◦ (ζ, ϑ) is the angle by which the observed image
has been rotated with respect to the Sachs basis, while the s(ζ, ϑ) parametrizes the
shear deformation of that image with respect to the Sachs basis. Compared to the
usual parametrizations of J c (see e.g. [18, 22, 24]), the advantage of the grouptheoretically motivated one we chose in Eq. (4.7.31) is that it is independent of the
order in which the rotation and shear effects are considered, since they are both
described through generators of the corresponding Lie algebra.
Let us now decompose K . The fact that J c ∈ SL(2, R), implies that
Q c := (∂ ζ J c ) J
−1
c ,
(4.7.35)
is an element of sl(2, R)
Tr Q c ≡ 0 .
(4.7.36)
4 Observer Space-Time Formalism
σ + :=
1 0
0 −1
,
σ × :=
0 1
1 0
,
(4.7.27)
W + := E + + B × ,
W × := E × − B + ,
(4.7.28)
and
E +,× := σ
AB
+,× E AB ,
B +,× := σ
AB
+,× B AB .
(4.7.29)
We next express these equations in terms of physically-interpretable quantities. We
first split J into its determinant and conformal parts
J ≡ D J c ,
det J c ≡ 1 ,
(4.7.30)
and then express the latter through sl(2, R) generators
J c := exp
s ◦ ε + s + σ + + s × σ ×
≡ 1 cosh S + (s ◦ ε + s + σ + + s × σ × )
sinh S
S
,
(4.7.31)
where
ε AB ≡ ε AB ,
S :=
−s 2
◦ + s
2
+ + s
2
× .
(4.7.32)
In particular, the eigenvalues of J now read
λ ± [ J] = De
±S
,
(4.7.33)
and we also define the complex combination
s := s + + is × ,
S ≡
−s 2
◦ + |s| 2 .
(4.7.34)
Thus, for a source located at γ(ζ, ϑ) with 4-velocity e
μ
0 (γ(ζ, ϑ)), D(ζ, ϑ) is the
angular diameter distance to it, s ◦ (ζ, ϑ) is the angle by which the observed image
has been rotated with respect to the Sachs basis, while the s(ζ, ϑ) parametrizes the
shear deformation of that image with respect to the Sachs basis. Compared to the
usual parametrizations of J c (see e.g. [18, 22, 24]), the advantage of the grouptheoretically motivated one we chose in Eq. (4.7.31) is that it is independent of the
order in which the rotation and shear effects are considered, since they are both
described through generators of the corresponding Lie algebra.
Let us now decompose K . The fact that J c ∈ SL(2, R), implies that
Q c := (∂ ζ J c ) J
−1
c ,
(4.7.35)
is an element of sl(2, R)
Tr Q c ≡ 0 .
(4.7.36)