284 Exercises
(c) In a rotating Minkowski space-time, consider the ZAMO family of
observers n
= −dt or n
= ∂
t
+Ω∂
φ
. Show that they are geodesic
(a(n) = 0), vorticity-free (ω(n) = 0, by definition), and expansion-free
(θ(n) = 0).
(d) In a rotating Minkowski space-time, consider the family of circularly
rotating orbits U =Γ(∂
t
+ ζ∂
φ
).
1. Show that Γ
−2
=1− r
2
(ζ −Ω)
2
.
2. Show that for null circular orbits ζ = ζ
±
= −Ω ± 1/r.
65. Consider a rotating Minkowski space-time,
ds
2
= −γ
−2
dt
2
+2r
2
Ωdtdφ + r
2
dφ
2
+ dr
2
+ dz
2
,
where γ
−2
=1− Ω
2
r
2
, and the family of static observers m = γ∂
t
.
(a) Show that the projected metric onto LRS
m
is given by
P (m)=dr ⊗dr + γ
2
r
2
dφ ⊗dφ + dz ⊗ dz.
(b) Show that the Ricci scalar associated with this 3-metric is given by
R = −6Ω
2
γ
4
.
(c) Consider the r − φ part of this 3-metric,
P (m)|
z=const.
= dr ⊗ dr + γ
2
r
2
dφ ⊗dφ.
1. Show that the Ricci scalar associated with this 2-metric is again
given by
R = −6Ω
2
γ
4
.
2. Show that the embedding of this 2-metric is completely Minkowskian
(h
RR
= γ
−6
< 1) and that the surface Z = Z(R) can be explicitly
obtained in terms of elliptic functions (Bini, Carini, and Jantzen,
1997a; 1997b).
66. Using the purely imaginary 4-form
E
α
1
...α
4
= iη
α
1
...α
4
,
E
α
1
...α
4
= −iη
α
1
...α
4
instead of η for the duality operation on 2-form index pairs defines the
“hook” duality operation, with symbol ˘. For example, for a 2-form F this
leads to ˘ F = i
∗
F and hence to ˘ ˘ S = S. One can then find eigen-2-forms
of the operation ˘ with eigenvalues ±1, called self-dual (+) and anti-self-
dual (−), respectively.