De Felice F., Bini D. Classical Measurements in Curved Space-Times
Подождите немного. Документ загружается.
4.1 Orthonormal frames 63
we have
a
(C)
(fw,U,u)
= ν
d ˆν
ˆ
d
dτ
(U,u)
e
ˆ
d
+
ζ
(fw)
+ ζ
(sc)
×
u
ν. (4.25)
Tetrads {u = e
ˆ
0
,e
ˆa
} may be further specified by particular transport laws for
the triad {e
ˆa
} along the world line of u itself, as we will show next.
Fermi-Walker frames
A vector field X is said to be Fermi-Walker transported along the congruence C
u
when its Fermi-Walker derivative along C
u
is equal to zero, i.e., from (2.83)
D
(fw,u)
X
dτ
u
= ∇
u
X +[a(u) ∧u] X =0. (4.26)
If X = u, then the above relation holds identically. If X = e
ˆa
we have instead
D
(fw,u)
e
ˆa
dτ
u
= ∇
u
e
ˆa
− a(u)
ˆa
u
= a(u)
ˆa
u + C
(fw)
ˆ
b
ˆa
e
ˆ
b
− a(u)
ˆa
u
= C
(fw)
ˆ
b
ˆa
e
ˆ
b
=0. (4.27)
Since {e
ˆ
b
} is a basis in LRS
u
, then C
(fw)
ˆ
b
ˆa
= 0. In this case the tetrad frame is
termed Fermi-Walker.
Absolute Frenet-Serret frames
A Frenet-Serret frame {E
ˆα
} (α =0, 1, 2, 3) along a time-like curve with unit
tangent vector U is defined as a solution of the following equations:
DE
ˆ
0
dτ
U
= κE
ˆ
1
,
DE
ˆ
1
dτ
U
= κE
ˆ
0
+ τ
1
E
ˆ
2
,
DE
ˆ
2
dτ
U
= −τ
1
E
ˆ
1
+ τ
2
E
ˆ
3
,
DE
ˆ
3
dτ
U
= −τ
2
E
ˆ
2
,
(4.28)
where E
ˆ
0
= U and
κ =Γ
ˆ
1
ˆ
0
ˆ
0
,τ
1
= C
(fw)
ˆ
2
ˆ
1
,τ
2
= C
(fw)
ˆ
3
ˆ
2
(4.29)
are respectively the magnitude of the acceleration and the first and second tor-
sions of the world line. τ
1
and τ
2
are the components of the Frenet-Serret angular
velocity ω
(FS)
= τ
1
E
ˆ
3
+ τ
2
E
ˆ
1
of the spatial triad {E
ˆa
} with respect to a Fermi-
Walker frame defined along U . The frame {E
ˆα
} (with dual W
ˆα
) is termed an
absolute Frenet-Serret frame.
64 Special frames
In compact form we have
DE
ˆα
dτ
U
= E
ˆ
β
C
ˆ
β
ˆα
, (4.30)
with
C = C
K
+ C
T
. (4.31)
Here
C
K
= κ(E
ˆ
1
⊗ W
ˆ
0
+ E
ˆ
0
⊗ W
ˆ
1
), (4.32)
and
C
T
= τ
ˆ
1
(E
ˆ
1
⊗ W
ˆ
2
+ E
ˆ
2
⊗ W
ˆ
1
)+τ
2
(E
ˆ
2
⊗ W
ˆ
3
− E
ˆ
3
⊗ W
ˆ
2
), (4.33)
W
ˆα
being the dual frame of E
ˆα
. Therefore
C
K
= −κ[W
ˆ
0
∧ W
ˆ
1
]=−[a(U) ∧ U]
, (4.34)
and we have the following relation for the Fermi-Walker derivative along U of a
generic vector X:
D
(fw,U
)X
dτ
U
=
DX
dτ
U
− C
K
X. (4.35)
This form of the Fermi-Walker derivative along a time-like world line will be used
to generalize to the case of a null world line (see, for instance, Bini et al., 2006).
Relative Frenet-Serret frames
Let us consider again a curve γ with tangent vector field U and a family of
observers u represented by the congruence C
u
of its integral curves. These cross
the world line of U and at each of its points one defines a vector representing
the (unique) 3-velocity of the particle U, as measured locally by the observer u,
namely
ν(U, u)=νˆν(U, u), (4.36)
recalling that U = γ[u + νˆν(U, u)]; clearly ν(U, u) ∈ LRS
u
. We shall now con-
struct, along the world line of U, a frame in LRS
u
following a Frenet-Serret
procedure.
The first vector of this frame is chosen as the unit space-like direction ˆν(U, u).
The remaining two vectors are the relative normal ˆη
(fw,U,u)
which, apart from the
sign, is defined as the normalized P (u)-projected covariant derivative of ˆν(U, u)
along U and its cross product with ˆν(U, u), namely the relative binormal
ˆ
β
(fw,U,u)
=ˆν(U, u) ×
u
ˆη
(fw,U,u)
.Let{E
(fw,U,u)
ˆa
} = {ˆν(U, u), ˆη
(fw,U,u)
,
ˆ
β
(fw,U,u)
}
be this frame.
4.1 Orthonormal frames 65
Projecting the covariant derivatives along U of these vectors onto LRS
u
and
re-parameterizing the derivatives with respect to the spatial arc length
d
(U,u)
= γ(U, u)ν(U, u)dτ
U
, (4.37)
namely
D
(fw,U,u)
dτ
U
E
(fw,U,u)
ˆa
= γν
D
(fw,U,u)
d
(U,u)
E
(fw,U,u)
ˆa
, (4.38)
leads to the relative Frenet-Serret equations:
D
(fw,U,u)
d
(U,u)
ˆν(U, u)=k
(fw,U,u)
ˆη
(fw,U,u)
, (4.39)
D
(fw,U,u)
d
(U,u)
ˆη
(fw,U,u)
= −k
(fw,U,u)
ˆν(U, u)+τ
(fw,U,u)
ˆ
β
(fw,U,u)
, (4.40)
D
(fw,U,u)
d
(U,u)
ˆ
β
(fw,U,u)
= −τ
(fw,U,u)
ˆη
(fw,U,u)
. (4.41)
Here the coefficients k
(fw,U,u)
and τ
(fw,U,u)
are respectively the relative Fermi-
Walker curvature and torsion of the world line of U as measured by u. The frame
E
(fw,U,u)
ˆa
is termed a relative Frenet-Serret frame.
Equation (4.39) does not allow one to fix the signs of the curvature k
(fw,U,u)
and of the frame vector ˆη
(fw,U,u)
individually. Once these signs are fixed, although
arbitrarily, the third vector
ˆ
β
(fw,U,u)
=ˆν
(U,u)
×
u
ˆη
(fw,U,u)
is chosen so as to make
the frame right-handed, and this fixes the sign of the torsion.
The relative Frenet-Serret equations can be written in a more compact form as
D
(fw,U,u)
d
(U,u)
E
(fw,U,u)
ˆa
= ω
(fw,U,u)
×
u
E
(fw,U,u)
ˆa
, (4.42)
where
ω
(fw,U,u)
= τ
(fw,U,u)
ˆν(U, u)+k
(fw,U,u)
ˆ
β
(fw,U,u)
(4.43)
defines the relative Frenet-Serret angular velocity of the spatial frame.
The world line with 4-velocity U is said to be u-relatively straight if the Fermi-
Walker relative curvature vanishes, k
(fw,U,u)
=0;andu-relatively flat if the
Fermi-Walker relative torsion vanishes, τ
(fw,U,u)
= 0. When the relative curva-
ture vanishes identically the relative Frenet-Serret frame can still be defined by
adopting an appropriate limiting procedure. On the other hand, when it vanishes
only at an isolated point, one must allow it to have either sign when it is non-zero
in order to extend the relative normal continuously through this point.
Comoving relative Frenet-Serret frame
Consider again a test particle with 4-velocity U being the target of an observer u.
The centripetal acceleration of the particle is measured by the observer u in
66 Special frames
his own LRS
u
, while the centrifugal acceleration is an “inertial acceleration”
measured by the particle U in its own LRS
U
. In Newtonian mechanics these rest
frames coincide, the time being absolute; therefore one can simply consider the
centripetal acceleration as opposite to the centrifugal one.
In general relativity the observer and the test particle in relative motion have
different rest frames, so a comparison between the two accelerations is only pos-
sible with a suitable geometrical representation of the centrifugal acceleration of
the test particle in the rest frame of the observer.
If u is a vector field and C
u
is the congruence of its integrable curves, at each
point where the particle U crosses a curve of C
u
the vectors u and U define a boost
of (u, ν(U, u)) into (U, −ν(u, U)) and leaves the orthogonal 2-space LRS
u
∩LRS
U
invariant, as discussed in Section 3.3. Let
ˆ
V(u, U)=−ˆν(u, U)=γ(νu +ˆν(U, u)) (4.44)
be the negative of the relative velocity of u with respect to U as in (3.114).
The projection into the rest space of U of the covariant derivative of any
vector field X defined on the world line of U and lying in LRS
U
leads to the
usual Fermi-Walker derivative along U:
D
(fw,U )
X
dτ
U
= P (U )
DX
dτ
U
= γν
D
(fw,U )
X
d
(U,u)
,X∈ LRS
U
. (4.45)
One can construct a Frenet-Serret-like frame starting from
ˆ
V(u, U) by adding
two new vectors, both lying in LRS
U
,
ˆ
N
(fw,u,U )
being the normal and
ˆ
B
(fw,u,U )
the binormal of the world line of U. They are solutions of the Frenet-Serret
relations
D
(fw,U )
d
(U,u)
ˆ
V(u, U)=K
(fw,u,U )
ˆ
N
(fw,u,U )
,
D
(fw,U )
d
(U,u)
ˆ
N
(fw,u,U )
= −K
(fw,u,U )
ˆ
V(u, U)+T
(fw,u,U )
ˆ
B
(fw,u,U )
,
D
(fw,U)
d
(U,u)
ˆ
B
(fw,u,U )
= −T
(fw,u,U )
ˆ
N
(fw,u,U )
. (4.46)
Let us now prove that the following relation holds:
K
(fw,u,U )
ˆ
N
(fw,u,U )
= γk
(fw,U,u)
ˆη
(fw,U,u)
+ˆν(U, u) ×
u
[ˆν(U, u) ×
u
F
(G)
(fw,U,u)
]. (4.47)
First recall that the Fermi-Walker derivative along U of a vector orthogonal to
U reduces to the projection on LRS
U
of its covariant derivative along U itself.
Hence, differentiating (4.44) with respect to the parameter (U, u), we obtain
4.1 Orthonormal frames 67
D
(fw,U)
ˆ
V(u, U)
d
(U,u)
=
1
γν
P (U)
D
dτ
U
[γ(νu +ˆν(U, u))]
=
1
γν
P (U)
d(γν)
dτ
U
u + γν
Du
dτ
U
+ˆν(U, u)
dγ
dτ
U
+ γ
Dˆν(U, u)
dτ
U
=
1
γν
P (U)
dγ
νdτ
U
u −γνF
(G)
(fw,U,u)
+ˆν(U, u)
dγ
dτ
U
+ γ
Dˆν(U, u)
dτ
U
=
1
γν
P (U)
dγ
dτ
U
u
ν
+ˆν(U, u)
− γνF
(G)
(fw,U,u)
+ γ
Dˆν(U, u)
dτ
U
,
(4.48)
wherewehaveused(3.156) and the relation
dγ
dτ
U
= γ
3
ν
dν
dτ
U
. (4.49)
The first term in square brackets in the last line of (4.48) is along U and therefore
vanishes after contraction with P (U); hence we have
D
(fw,U)
ˆ
V(u, U)
d
(U,u)
=
1
γν
P (U)
−γνF
(G)
(fw,U,u)
+ γ
Dˆν(U, u)
dτ
U
. (4.50)
The last term on the right-hand side of (4.48) can be written as
Dˆν(U, u)
dτ
U
= P (u)
Dˆν(U, u)
dτ
U
− uu ·
Dˆν(U, u)
dτ
U
= P (u)
Dˆν(U, u)
dτ
U
+ uˆν(U, u) ·
Du
dτ
U
= P (u)
Dˆν(U, u)
dτ
U
− uˆν(U, u) ·F
(G)
(fw,U,u)
= γνk
(fw,U,u)
ˆη
(fw,U,u)
− uˆν(U, u) ·F
(G)
(fw,U,u)
. (4.51)
The projection orthogonal to U of this quantity is
P (U)
Dˆν(U, u)
dτ
U
= γνk
(fw,U,u)
ˆη
(fw,U,u)
− ˆν(U, u) ·F
(G)
(fw,U,u)
[P (U)u], (4.52)
noting that
P (U)ˆη
(fw,U,u)
=ˆη
(fw,U,u)
+ U
U · ˆη
(fw,U,u)
=ˆη
(fw,U,u)
, (4.53)
68 Special frames
since U = γ[u + νˆν(U, u)] and ˆν(U, u) · ˆη
(fw,U,u)
=0.Fromtheaboveitthen
follows that
D
(fw,U)
ˆ
V(u, U)
d
(U,u)
= −P (U )F
(G)
(fw,U,u)
+ γk
(fw,U,u)
ˆη
(fw,U,u)
−
ˆν(U, u) · F
(G)
(fw,U,u)
ν
(u −γU)
= γk
(fw,U,u)
ˆη
(fw,U,u)
− F
(G)
(fw,U,u)
− Uγνˆν(U, u) ·F
(G)
(fw,U,u)
−
ˆν(U, u) · F
(G)
(fw,U,u)
ν
(u −γU)
= γk
(fw,U,u)
ˆη
(fw,U,u)
− F
(G)
(fw,U,u)
− ˆν(U, u) · F
(G)
(fw,U,u)
γνU +
u −γU
ν
= γk
(fw,U,u)
ˆη
(fw,U,u)
− F
(G)
(fw,U,u)
+ˆν(U, u) · F
(G)
(fw,U,u)
ˆν(U, u).
Finally,
D
(fw,U)
ˆ
V(u, U)
d
(U,u)
= γk
(fw,U,u)
ˆη
(fw,U,u)
+ˆν(U, u) ×
u
ˆν(U, u) ×
u
F
(G)
(fw,U,u)
, (4.54)
which completes the proof.
The cross product of (4.47) with
ˆ
V(u, U) yields
K
(fw,u,U )
ˆ
V(u, U) ×
U
ˆ
N
(fw,u,U )
= γk
(fw,U,u)
ˆ
V(u, U) ×
U
ˆη
(fw,U,u)
+(ˆν(U, u) · F
(G)
(fw,U,u)
)
ˆ
V(u, U) ×
U
ˆν(U, u)
−
ˆ
V(u, U) ×
U
F
(G)
(fw,U,u)
, (4.55)
that is
K
(fw,u,U )
ˆ
B
(fw,u,U )
= γk
(fw,U,u)
ˆ
V(u, U) ×
U
ˆη
(fw,U,u)
+(ˆν(U, u) · F
(G)
(fw,U,u)
)
ˆ
V(u, U) ×
U
ˆν(U, u)
−
ˆ
V(u, U) ×
U
F
(G)
(fw,U,u)
. (4.56)
To proceed further it is necessary to evaluate the cross product in LRS
U
of
a vector X(U), for example, which belongs to LRS
U
, with a vector Y (u), for
example, which belongs to LRS
u
. The result is the following:
X(U) ×
U
Y (u)=γ {[P(u, U)X(U ) ×
u
Y (u)]
−(ν · P (u, U)X(U))(ν ×
u
Y )
−u[P(u, U)X(U ) ·(ν ×
u
Y (u))]}. (4.57)
4.2 Null frames 69
In our case X(U)=
ˆ
V(u, U)and
P (u, U)
ˆ
V(u, U)=γˆν(U, u) ; (4.58)
therefore we have
ˆ
V(u, U) ×
U
Y (u)=γ
2
[ˆν(U, u) ×
u
Y (u)] −γ
2
ν
2
[ˆν(U, u) ×
u
Y (u)]
=ˆν(U, u) ×
u
Y (u). (4.59)
From (4.56) it then follows that
K
(fw,u,U )
ˆ
B
(fw,u,U )
= γk
(fw,U,u)
ˆν(U, u) ×
u
ˆη
(fw,U,u)
− ˆν(U, u) ×
u
F
(G)
(fw,U,u)
= γk
(fw,U,u)
ˆ
β
(fw,U,u)
− ˆν(U, u) ×
u
F
(G)
(fw,U,u)
. (4.60)
The spatial triad
{E
(fw,u,U )
ˆa
} = {
ˆ
V(u, U),
ˆ
N
(fw,u,U )
,
ˆ
B
(fw,u,U )
=
ˆ
V(u, U) ×
U
ˆ
N
(fw,u,U )
} (4.61)
is termed a comoving relative Frenet-Serret frame.
In compact form the comoving relative Frenet-Serret relations become
D
(fw,U)
d
(U,u)
E
(fw,u,U )
ˆa
=Ω
(fw,u,U )
×
U
E
(fw,u,U )
ˆa
, (4.62)
where Ω
(fw,u,U )
= T
(fw,u,U )
ˆ
V(u, U)+K
(fw,u,U )
ˆ
B
(fw,u,U )
defines the comoving rel-
ative angular velocity of this new frame with respect to a Fermi-Walker spatial
frame. Again no assumption is made about the choice of signs for the curvature
and torsion in this general discussion.
4.2 Null frames
Null frames include null vectors, and are particularly convenient for treating null
fields such as electromagnetic and gravitational radiation. These frames can be
complex or real.
Complex (Newman-Penrose) null frames
A (complex, null) Newman-Penrose frame {l, n, m, ¯m} (Newman and Penrose,
1962) can be associated with an orthonormal frame {e
ˆα
} in the following way:
l =
1
√
2
[e
ˆ
0
+ e
ˆ
1
],n=
1
√
2
[e
ˆ
0
− e
ˆ
1
],
m =
1
√
2
[e
ˆ
2
+ ie
ˆ
3
], ¯m =
1
√
2
[e
ˆ
2
− ie
ˆ
3
], (4.63)
with l·n = −1andm· ¯m = 1. The four vectors l, n, m, ¯m are null vectors (l·l =0,
n · n =0,m · m =0, ¯m · ¯m = 0); two of them are real (l, n) and the other two
70 Special frames
are complex conjugate (m, ¯m). The frame components of the metric are then
equal to
(g
ab
)=(g
ab
)=
⎛
⎜
⎜
⎝
0 −100
−1000
0001
0010
⎞
⎟
⎟
⎠
. (4.64)
Real null frames
Instead of a complex null frame it is useful to introduce a real null frame made up
of two null vectors and two space-like vectors. Choosing the spatial vectors orthog-
onal to each other and normalized to unity, one constructs a quasi-orthogonal real
null frame. Denoting such a frame as {E
α
} (α =1, 2, 3, 4), the standard relation
with an orthonormal frame {e
ˆα
} is the following:
E
1
=
1
√
2
(e
ˆ
0
+ e
ˆ
2
),
E
2
= e
ˆ
1
,
E
3
=
1
√
2
(e
ˆ
0
− e
ˆ
2
),
E
4
= e
ˆ
3
. (4.65)
The frame components of the metric are given by
(g
ab
)=(E
a
· E
b
)=
⎛
⎜
⎜
⎝
00−10
0100
−1000
0001
⎞
⎟
⎟
⎠
=(g
ab
). (4.66)
Let us denote by Π and Π
the vector spaces generated by (E
1
,E
3
) and (E
2
,E
4
),
respectively. The null vectors E
1
and E
3
are defined up to a multiplicative
factor, while the spatial vectors E
2
and E
4
can be arbitrarily rotated in the
2-plane Π
. Therefore there arises a set of equivalent frames, each being a suit-
able Frenet-Serret frame along a null curve.
This frame satisfies the following set of Frenet-Serret evolution equations:
DE
1
dλ
= −KE
2
,
DE
2
dλ
= T
1
E
1
−KE
3
,
DE
3
dλ
= T
1
E
2
+ T
2
E
4
,
DE
4
dλ
= T
2
E
1
, (4.67)
where λ is an arbitrary parameter along E
1
and the quantities K, T
1
,andT
2
play
roles analogous to the Frenet-Serret curvature and torsions in the time-like case.
The connection matrix
DE
a
dλ
= E
b
C
b
a
(4.68)
4.2 Null frames 71
has components
(C
a
b
)=
⎛
⎜
⎜
⎝
0 T
1
0 T
2
−K 0 T
1
0
0 −K 00
00T
2
0
⎞
⎟
⎟
⎠
, (C
ab
)=
⎛
⎜
⎜
⎝
0 T
1
0 T
2
−T
1
0 K 0
0 −K 00
−T
2
000
⎞
⎟
⎟
⎠
. (4.69)
The invariants of the matrix are given by
I
1
=
1
2
C
ab
C
ab
=2KT
1
,I
2
=
1
2
C
ab
∗
C
ab
=2KT
2
, (4.70)
where
(
∗
C
ab
)=
1
2
η
abcd
C
cd
=
⎛
⎜
⎜
⎝
0 T
2
0 −T
1
−T
2
00 0
000−K
T
1
0 K 0
⎞
⎟
⎟
⎠
. (4.71)
Finally, one can evaluate the four complex eigenvalues of the matrix (C
a
b
),
λ
1,2
= ±ω ≡ [−KT
1
+ K(T
2
1
+ T
2
2
)
1/2
]
1/2
,
λ
3,4
= ±iχ ≡ i[KT
1
+ K(T
2
1
+ T
2
2
)
1/2
]
1/2
, (4.72)
which in turn define two non-negative quantities ω and χ.
Let us consider the decompositions
C = C
K
+ C
T
,
C
= C
K
+ C
T
(4.73)
with
C
K
= −K(E
2
⊗ W
1
+ E
3
⊗ W
2
),
C
K
= K(E
2
∧ E
3
), (4.74)
and
C
T
=[T
1
(E
1
⊗ W
2
+ E
2
⊗ W
3
)+T
2
(E
1
⊗ W
4
+ E
4
⊗ W
3
)],
C
T
=
T
2
1
+ T
2
2
E
1
∧
T
1
E
2
+ T
2
E
4
T
2
1
+ T
2
2
, (4.75)
W
a
being the dual frame of E
a
. The curvature part C
K
of the connection matrix
generates a null rotation by an angle |K| in the time-like hyperplane of E
1
, E
2
,
E
3
, which leaves the null vector E
3
fixed (C
K
E
3
= 0). The torsion part C
T
of the connection matrix generates a null rotation by an angle
T
2
1
+ T
2
2
in the
time-like hyperplane of E
1
, E
3
,(T
2
1
+ T
2
2
)
−1/2
(T
1
E
2
+ T
2
E
4
), which leaves the
null vector E
1
fixed (C
T
E
1
= 0).
72 Special frames
Along a Killing trajectory the Frenet-Serret scalars are all constant, leading to
considerable simplification of the above formulas. This is the situation for null
circular orbits in stationary axisymmetric space-times, for example.
There still remains the problem that the whole Frenet-Serret frame machinery
is still subject to re-parameterization, under which only the relative ratios of the
curvature and torsions are invariant. The curvature is just the norm of the second
derivative,
K =
D
2
x
dλ
2
·
D
2
x
dλ
2
1/2
≥ 0, (4.76)
and it transforms as the square of the related rate of change of the parameters
λ →
ˆ
λ, that is
K =
ˆ
K
d
ˆ
λ
dλ
2
. (4.77)
Thus dΛ=K
1/2
dλ =
ˆ
K
1/2
d
ˆ
λ is invariant (when non-zero), like the arc length
in the non-null case, and can be used to introduce a preferred unit curvature
parameterization when the curvature is everywhere non-vanishing, defined up
to an additive constant like the arc length in the null case. This was already
introduced by Vessiot for the Riemannian case in 1905 (Vessiot, 1905), who called
Λthepseudo-arc length.
The full transformation of all the Frenet-Serret quantities is easily calculated.
Letting Λ
= dΛ/dλ and using a hatted notation for the new quantities, one finds
that the frame vectors undergo a boost and a null rotation, with the invariance
of the last vector confirming the invariant nature of the osculating hyperplane
for which it is the unit normal:
ˆ
E
1
=(Λ
)
−1
E
1
,
ˆ
E
2
= E
2
+ ζE
1
,
ˆ
E
3
=Λ
E
3
+ ζ
ζ
2
E
1
+ E
2
,
ˆ
E
4
= E
4
, (4.78)
where ζ =Λ
/(KΛ
), and the new Frenet-Serret quantities are given by
ˆ
K =(Λ
)
−2
K,
ˆ
T
1
= T
1
+ ζ
+
K
2
ζ
2
,
ˆ
T
2
= T
2
. (4.79)
The pseudo-arc length parameterization is obtained by setting Λ
= K
1/2
(so that
ζ = K
/(2K
2
)), and the new form of the Frenet-Serret equations, once hats are
dropped, can be obtained from (4.67) simply by making the substitutions K→1
and λ → Λ. Note that, in the case of constant curvature, Λ
= 0 (and ζ = 0);
this is just a simple affine parameter transformation and both torsions remain
unchanged, while the frame undergoes a simple boost constant rescaling.