Chapter 6
Appendix
6.1 Observer Frames Are Orthonormal
Consider an observer with 4-velocity ˆ
u performing measurements in an infinitesimal neighborhood ˆ
I of her position ˆ
P. She uses a local coordinate system x
μ to
parametrize ˆ
I and therefore the basis ∂ μ to decompose tensors at ˆ
P. In particular, she parametrizes evolution with her proper time, meaning that the norm of the
time-coordinate intervals dt ≡ dx
0 is unity
ˆ
g
−1
(dt, dt) ≡ ˆ
g
tt
= −1 .
(6.1.1)
Secondly, she parametrizes the measuring apparatus with spatial coordinates x
α ,
where α ∈ {x, y, z}, and associates a physical distance to some interval dx
α using
the Euclidean scalar product
ˆ
g
−1
dx
α
, dx
β
≡ ˆ
g
αβ
= δ
αβ
.
(6.1.2)
Finally, she is at rest with respect to these coordinates ˆ
u
α
= 0 and, since t is her
proper time, we have
ˆ
u
μ
= δ
μ
t .
(6.1.3)
Now conditions (6.1.1) and (6.1.2) on the inverse metric at ˆ
P imply for the metric
ˆ
g tt = −
1
1 + ˆ
g tα ˆ
g tα ,
ˆ
g tα =
ˆ
g
tα
1 + ˆ
g tβ ˆ
g tβ ,
ˆ
g αβ = δ αβ −
ˆ
g
tα
ˆ
g
tβ
1 + ˆ
g tγ ˆ
g tγ .
(6.1.4)
On the other hand, condition (6.1.3) implies
− 1 ≡ ˆ
g
ˆ
u, ˆ
u
= ˆ
g tt ,
(6.1.5)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
E. Mitsou and J. Yoo, Tetrad Formalism for Exact Cosmological Observables,
SpringerBriefs in Physics, https://doi.org/10.1007/978-3-030-50039-9_6
133