12
2 Motivation
where η denotes the Minkowski metric. Such a basis ˆ
e a is known as a “tetrad” or
“vierbein”. When interpreting it as an observer frame, the time-like element ˆ
e 0 represents the 4-velocity of the observer, while the space-like elements {ˆ e i }
3
i=1 provide a
Cartesian basis of the observer rest-frame. The condition (2.1.1) does not determine
the ˆ
e a entirely, as it is invariant under internal Lorentz transformations of that basis
ˆ
e a →
b
a ˆ
e b .
(2.1.2)
The tetrad indices a, b, c, . . . should therefore be understood as forming the vector
representation of the Lorentz group and we displace them using the Minkowski metric
η ab . Given the observer frame interpretation of ˆ
e a this freedom should be expected.
Indeed, the angular parametrization of the observed sky is defined only up to a
rotation, which explains the SO(3) ⊂ SO(1, 3) subgroup acting on the spatial frame
ˆ
e i → R
j
i ˆ
e j . As for the boosts, they alter the observer’s 4-velocity ˆ
e 0 →
a
0 ˆ
e a , thus
allowing us to relate different observers. In particular, note that boosts span the full
interior of the tangent light-cone in T ˆ
P M, so we have access to all possible observers,
with all possible spatial frame orientations.
Let us now consider an arbitrary coordinate system x
μ
∈ {t, x, y, z} around ˆ
P,
so that we can decompose ˆ
e a ≡ ˆ
e
μ
a ∂ μ and (2.1.1) becomes ˆ
g μν ˆ
e
μ
a ˆ
e
ν
b ≡ η ab . Denoting
by ˆ
e
a
μ the coefficients of the inverse matrix of ˆ
e
μ
a , we then have that ˆ
e
a
μ ≡ η
ab
ˆ
g μν ˆ
e
ν
b
and that ˆ
e
a
μ dx
μ
∈ T
∗
ˆ
P
M is the corresponding tetrad basis of the cotangent space. The
components of a tensor T that are measured by the observer ˆ
e a at ˆ
P are then given
by the projection on that basis
ˆ
T
b 1 ...b n
a 1 ...a m
:= ˆ
e
μ 1
a 1
. . . ˆ
e
μ m
a m
ˆ
e
b 1
ν 1
. . . ˆ
e
b n
ν n
ˆ
T
ν 1 ...ν n
μ 1 ...μ m
.
(2.1.3)
Under a Lorentz transformation (2.1.2), which physically modifies the observer, the
components (2.1.3) mix accordingly in the corresponding tensor representation of
the Lorentz group. Thus, exactly as in special relativity, but now locally at ˆ
P (i.e.
in T ˆ
P M), measurement is observer-dependent. On the other hand, the quantities
(2.1.3) are invariant under coordinate transformations, since all coordinate-induced
indices are fully contracted and the tensor field is evaluated at a definite point ˆ
P.
Thus, as one should demand in generally-covariant theories, physical observables
are independent of the way we choose to parametrize space-time.
The prototypical example of observables are those lying in the photon
4-momentum ˆ
k at ˆ
P. Expressing it in the tetrad basis ˆ
k
a
:= ˆ
e
a
μ k
μ , the light-like condition becomes η ab ˆ
k
a ˆ
k
b
≡ 0, so the ˆ
k
a numbers can be parametrized as in (1.0.1).
We have that ˆ
ω := ˆ
k
0 is the photon frequency, while (ϑ, ϕ) is the angular position in
the sky of the correspond light source, as measured by the observer ˆ
e a . Thus, the ˆ
k
information, expressed in the basis ˆ
e a , provides the numbers that the corresponding
observer uses to parametrize light spectra ( ˆ
ω) and the night sky (ϑ, ϕ). In particular,
the derivative with respect to ϑ, ϕ will lead to the construction of deviation quantities
such as the Jacobi map.
2 Motivation
where η denotes the Minkowski metric. Such a basis ˆ
e a is known as a “tetrad” or
“vierbein”. When interpreting it as an observer frame, the time-like element ˆ
e 0 represents the 4-velocity of the observer, while the space-like elements {ˆ e i }
3
i=1 provide a
Cartesian basis of the observer rest-frame. The condition (2.1.1) does not determine
the ˆ
e a entirely, as it is invariant under internal Lorentz transformations of that basis
ˆ
e a →
b
a ˆ
e b .
(2.1.2)
The tetrad indices a, b, c, . . . should therefore be understood as forming the vector
representation of the Lorentz group and we displace them using the Minkowski metric
η ab . Given the observer frame interpretation of ˆ
e a this freedom should be expected.
Indeed, the angular parametrization of the observed sky is defined only up to a
rotation, which explains the SO(3) ⊂ SO(1, 3) subgroup acting on the spatial frame
ˆ
e i → R
j
i ˆ
e j . As for the boosts, they alter the observer’s 4-velocity ˆ
e 0 →
a
0 ˆ
e a , thus
allowing us to relate different observers. In particular, note that boosts span the full
interior of the tangent light-cone in T ˆ
P M, so we have access to all possible observers,
with all possible spatial frame orientations.
Let us now consider an arbitrary coordinate system x
μ
∈ {t, x, y, z} around ˆ
P,
so that we can decompose ˆ
e a ≡ ˆ
e
μ
a ∂ μ and (2.1.1) becomes ˆ
g μν ˆ
e
μ
a ˆ
e
ν
b ≡ η ab . Denoting
by ˆ
e
a
μ the coefficients of the inverse matrix of ˆ
e
μ
a , we then have that ˆ
e
a
μ ≡ η
ab
ˆ
g μν ˆ
e
ν
b
and that ˆ
e
a
μ dx
μ
∈ T
∗
ˆ
P
M is the corresponding tetrad basis of the cotangent space. The
components of a tensor T that are measured by the observer ˆ
e a at ˆ
P are then given
by the projection on that basis
ˆ
T
b 1 ...b n
a 1 ...a m
:= ˆ
e
μ 1
a 1
. . . ˆ
e
μ m
a m
ˆ
e
b 1
ν 1
. . . ˆ
e
b n
ν n
ˆ
T
ν 1 ...ν n
μ 1 ...μ m
.
(2.1.3)
Under a Lorentz transformation (2.1.2), which physically modifies the observer, the
components (2.1.3) mix accordingly in the corresponding tensor representation of
the Lorentz group. Thus, exactly as in special relativity, but now locally at ˆ
P (i.e.
in T ˆ
P M), measurement is observer-dependent. On the other hand, the quantities
(2.1.3) are invariant under coordinate transformations, since all coordinate-induced
indices are fully contracted and the tensor field is evaluated at a definite point ˆ
P.
Thus, as one should demand in generally-covariant theories, physical observables
are independent of the way we choose to parametrize space-time.
The prototypical example of observables are those lying in the photon
4-momentum ˆ
k at ˆ
P. Expressing it in the tetrad basis ˆ
k
a
:= ˆ
e
a
μ k
μ , the light-like condition becomes η ab ˆ
k
a ˆ
k
b
≡ 0, so the ˆ
k
a numbers can be parametrized as in (1.0.1).
We have that ˆ
ω := ˆ
k
0 is the photon frequency, while (ϑ, ϕ) is the angular position in
the sky of the correspond light source, as measured by the observer ˆ
e a . Thus, the ˆ
k
information, expressed in the basis ˆ
e a , provides the numbers that the corresponding
observer uses to parametrize light spectra ( ˆ
ω) and the night sky (ϑ, ϕ). In particular,
the derivative with respect to ϑ, ϕ will lead to the construction of deviation quantities
such as the Jacobi map.