De Felice F., Bini D. Classical Measurements in Curved Space-Times
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3.4 Adapted frames 43
where C
(fw)
ba
are termed Fermi-Walker structure functions.
1
From (3.60) relation
(3.52) can be written as
∇
u
e
a
= a(u)
a
e
0
+ C
(fw)
b
a
e
b
, (3.61)
implying that
P (u)∇
u
e
a
= C
(fw)
b
a
e
b
. (3.62)
Let us now prove that C
(fw)
b
a
can also we written as
C
(fw)
b
a
= C
(lie)
b
a
− k(u)
b
a
, (3.63)
where
C
(lie)
b
a
≡ ω
b
(£(u)
u
e
a
), (3.64)
recalling that ω
b
is the dual basis; Eq. (3.64) implies that
P (u)£
u
e
a
= £(u)
u
e
a
= C
(lie)
b
a
e
b
. (3.65)
In terms of the structure functions of the frame, we have by definition
£(u)
u
e
a
≡ P (u)£
u
e
a
= P (u)[u, e
a
]=P (u) C
α
0a
e
α
= C
b
0a
e
b
, (3.66)
where we have used the relations P (u)e
0
=0andP (u)e
b
= e
b
; therefore
C
(lie)
b
a
= C
b
0a
. (3.67)
Using (3.45) we obtain
£
u
e
a
= ∇
u
e
a
−∇
e
a
u = ∇
u
e
a
+ k(u)
c
a
e
c
, (3.68)
so, projecting both sides orthogonally to u yields
£(u)
u
e
a
= P (u)∇
u
e
a
+ k(u)
c
a
e
c
, (3.69)
which leads to (3.63).
Using (3.45), one obtains
∇
e
a
e
0
=Γ
γ
0a
e
γ
= −k(u)
b
a
e
b
,
∇
e
a
e
b
=Γ
γ
ba
e
γ
= −k(u)
ba
e
0
+Γ
c
ba
e
c
. (3.70)
Summarizing, we have
Γ
a
00
= a(u)
a
, Γ
0
a0
= a(u)
a
, Γ
b
a0
= C
(fw)
b
a
,
Γ
b
0a
= −k(u)
b
a
, Γ
0
ba
= −k(u)
ba
.
(3.71)
1
A more correct notation for the Fermi-Walker structure functions should be C
(fw,u,e
c
)
ba
,
i.e. a symbol which includes explicit dependence on the frame. Clearly when the frame
is fixed the simplified notation proves useful.
44 Space-time splitting
We can also express the structure functions in terms of kinematical quantities.
In fact, from the definition
e
α
C
α
βγ
=[e
β
,e
γ
]=∇
e
β
e
γ
−∇
e
γ
e
β
(3.72)
we have
e
α
C
α
0b
= ∇
u
e
b
−∇
e
b
u
= a(u)
b
u +[C
(fw)
c
b
+ k(u)
c
b
]e
c
= a(u)
b
u + C
(lie)
c
b
e
c
, (3.73)
so that
C
0
0b
= a(u)
b
,C
c
0b
= C
(lie)
c
b
. (3.74)
Similarly
e
α
C
α
bc
= ∇
e
b
e
c
−∇
e
c
e
b
=2ω(u)
bc
u +2Γ
d
[cb]
e
d
, (3.75)
so that
C
0
bc
=2ω(u)
bc
,C
d
bc
=2Γ
d
[cb]
. (3.76)
We recall that the structure functions satisfy the Jacobi identities
[[e
α
,e
β
],e
ρ
]+[[e
β
,e
ρ
],e
α
]+[[e
ρ
,e
α
],e
β
]=0, (3.77)
which implies the following differential relations:
e
[ρ
(C
σ
αβ]
) −C
σ
ν[ρ
C
ν
αβ]
=0. (3.78)
Splitting such relations using an observer-adapted frame gives
0=−∇(u)
(lie)
ω(u)
ab
+ ∇(u)
[a
a(u)
b]
,
0=−∇(u)
(lie)
C
d
ab
+2e
[a
C
(lie)
f
b]
+2a(u)
[a
C
(lie)
f
b]
,
0=−∇(u)
[r
ω(u)
ba]
+ a(u)
[r
ω(u)
ba]
,
0=−e
[r
C
c
ab]
+ C
c
s[r
C
s
ab]
+2C
(lie)
c
[r
ω(u)
ab]
. (3.79)
The first of these relations can also be written as
[∇(u)
(lie)
+Θ(u)]ω(u)
s
−
1
2
[curl
u
a(u)]
s
=0, (3.80)
where we have used the property
∇(u)
(lie)
η(u)
abc
=Θ(u)η(u)
abc
. (3.81)
3.4 Adapted frames 45
The third equation becomes
− div
u
ω(u)+a(u) · ω(u) = 0 (3.82)
after contraction of both sides with η(u)
rba
.
Spatial-Fermi-Walker and spatial-Lie temporal derivatives
In the previous subsection we introduced the Fermi-Walker structure functions
C
(fw)
b
a
, as follows:
P (u)∇
u
e
a
= C
(fw)
b
a
e
b
. (3.83)
A similar relation was defined for the Lie derivative projected along u,whichwe
also termed a spatial-Lie temporal derivative (see (3.27)):
£(u)
u
e
a
= P (u)£
u
e
a
= C
(lie)
b
a
e
b
= C
b
0a
e
b
. (3.84)
It is useful to handle both these operations with a unified notation
2
∇(u)
(tem)
e
a
= C
(tem)
b
a
e
b
, (3.85)
where “tem” refers either to “fw” or to “lie”, with the following definitions:
∇(u)
(fw)
e
a
≡ P (u)∇
u
e
a
, ∇(u)
(lie)
e
a
≡ P (u)£
u
e
a
= £(u)
u
e
a
. (3.86)
Therefore if X is a vector field orthogonal to u, i.e. X ·u =0,wehave
∇(u)
(tem)
X = ∇(u)
(tem)
(X
a
e
a
)=
dX
a
dτ
u
e
a
+ X
a
C
(tem)
b
a
e
b
=
dX
b
dτ
u
+ X
a
C
(tem)
b
a
e
b
=(∇(u)
(tem)
X
b
)e
b
. (3.87)
The operation ∇(u)
(fw)
= P (u)∇
u
is termed a spatial-Fermi-Walker temporal
derivative. It can be extended to non-spatial fields. If we apply this operation to
the vector field u itself we have
∇(u)
(fw)
u = P (u)∇
u
u = a(u), (3.88)
and if X = fu we have
∇(u)
(fw)
X = ∇(u)
(fw)
(fu)=fa(u),
∇(u)
(lie)
X = ∇(u)
(lie)
(fu)=0. (3.89)
2
This notation has been introduced by R. T. Jantzen, who also considered two more spatial
temporal derivatives, corotating Fermi-Walker and Lie
. Details can be found in Jantzen,
Carini, and Bini (1992).
46 Space-time splitting
Hence the temporal derivatives so defined through their action on purely spatial
and purely temporal fields can now act on any space-time field.
Frame components of the Riemann tensor
From the definition
e
α
R
α
βγδ
=[∇
e
γ
, ∇
e
δ
]e
β
− C
σ
γδ
∇
e
σ
e
β
, (3.90)
we have
e
α
R
α
0b0
=[∇
e
b
, ∇
u
]u −C
σ
b0
∇
e
σ
u (3.91)
= ∇
e
b
(a(u)
c
e
c
) −∇
u
(∇
e
b
u) −C
0
b0
a(u)
c
e
c
− C
c
b0
∇
e
c
u
=
[∇(u)
b
+ a(u)
b
]a(u)
c
+ ∇(u)
(fw)
k(u)
c
b
− [k(u)
2
]
c
b
e
c
,
where
[k(u)
2
]
c
b
≡ k(u)
c
s
k(u)
s
b
,
so that
R
c
0b0
=[∇(u)
b
+ a(u)
b
]a(u)
c
+ ∇(u)
(fw)
k(u)
c
b
− [k(u)
2
]
c
b
, (3.92)
where
∇(u)
(fw)
k(u)
c
b
= ∇
u
k(u)
c
b
+ C
(fw)
c
f
k(u)
f
b
− C
(fw)
f
b
k(u)
c
f
. (3.93)
Similarly
e
α
R
α
bcd
=[∇
e
c
, ∇
e
d
]e
b
− C
0
cd
∇
u
e
b
− C
f
cd
∇
e
f
e
b
(3.94)
= ∇
e
c
(∇
e
d
e
b
) −∇
e
d
(∇
e
c
e
b
)+2ω(u)
cd
∇
u
e
b
−C
f
cd
∇
e
f
e
b
= ∇
e
c
(−k(u)
bd
u +Γ
f
bd
e
f
) −∇
e
d
(−k(u)
bc
u +Γ
f
bc
e
f
)
+2ω(u)
cd
(a(u)
b
u + C
(fw)
f
b
e
f
)
−C
f
cd
(−k(u)
bf
u +Γ
r
bf
e
r
), (3.95)
that is,
e
α
R
α
bcd
=[−e
c
(k(u)
bd
) −Γ
s
bd
k(u)
sc
+ e
d
(k(u)
bc
)+Γ
s
bc
k(u)
sd
−2ω(u)
cd
a(u)
b
+ C
s
cd
k(u)
bs
]u
+[2e
[c
(Γ
f
|b|d]
)+ 2Γ
f
s[c
Γ
s
|b|d]
− 2k(u)
b[c
k(u)
f
d]
−2ω
cd
C
(fw)
f
b
− C
s
cd
Γ
f
bs
]e
f
. (3.96)
Analyzing the time and space components of the Riemann tensor relative to the
basis, we have
R
0
bcd
= −2[∇(u)
[c
k(u)
|b|d]
+ ω(u)
cd
a(u)
b
] (3.97)
3.4 Adapted frames 47
and
R
f
bcd
= R
(fw)
f
bcd
− 2k(u)
b[c
k(u)
f
d]
= R
(fw)
f
bcd
+2k(u)
f
[c
k(u)
|b|d]
, (3.98)
where
R
(fw)
f
bcd
=2e
[c
Γ
f
|b|d]
+2Γ
s
b[c
Γ
f
|s|d]
− C
s
cd
Γ
f
bs
−2ω(u)
cd
C
(fw)
f
b
. (3.99)
This tensor is termed a Fermi-Walker spatial Riemann tensor;
3
it can be written
in invariant form as follows
R
(fw)
(u)(X, Y )Z = {[∇(u)
X
, ∇(u)
Y
]Z −∇(u)
[X,Y ]
}Z
−2ω(u)(X, Y )∇(u)
(fw)
Z, (3.100)
where X, Y ,andZ are spatial fields with respect to u and we note that
[X, Y ]=P(u)[X, Y ] −2ω
(X, Y ) u. (3.101)
The Fermi-Walker spatial Riemann tensor does not have all the symmetries of
a three-dimensional Riemann tensor. For instance it does not satisfy the Ricci
identities. In fact we have
0=R
f
[bcd]
= R
(fw)
f
[bcd]
− 2k(u)
[bc
k(u)
f
d]
. (3.102)
Recalling that the antisymmetric part of a
0
3
-tensor X
bcd
can be written as
X
[bcd]
=
1
3
X
[bc]d
+ X
[cd]b
+ X
[db]c
=
1
3
X
b[cd]
+ X
c[db]
+ X
d[bc]
, (3.103)
it follows that (3.102) can be written more conveniently as
R
(fw)
f
[bcd]
=2k(u)
f
[b
ω(u)
cd]
. (3.104)
From the latter one can construct a new Riemann tensor with all the necessary
symmetries. Ferrarese (1965) has shown that the symmetry-obeying Riemann
tensor, denoted by R
(sym)
ab
cd
, is related to the Fermi-Walker Riemann tensor
(3.100)by
R
(sym)
ab
cd
= R
(fw)
ab
cd
− 2ω(u)
ab
ω(u)
cd
− 4θ(u)
[a
[c
ω(u)
b]
d]
= R
ab
cd
+2k(u)
b
[c
k(u)
a
d]
− 2ω(u)
ab
ω(u)
cd
−4θ(u)
[a
[c
ω(u)
b]
d]
, (3.105)
3
A Lie spatial Riemann tensor can be defined similarly, replacing the Fermi-Walker structure
functions C
(fw)
f
b
with the corresponding Lie structure functions C
(lie)
f
b
according to (3.63).
48 Space-time splitting
or
R
ab
cd
= R
(sym)
ab
cd
+2θ(u)
a
[c
θ(u)
b
d]
+2ω
ab
ω
cd
+2ω(u)
a
[c
ω(u)
b
d]
, (3.106)
where (3.98) has been used.
In the special case of a Born-rigid congruence of observers u, i.e. θ(u)=0,we
find that (3.105) can be written as
R
(sym)
ab
cd
= R
(fw)
ab
cd
− 2ω(u)
ab
ω(u)
cd
= R
ab
cd
+2ω(u)
b
[c
ω(u)
a
d]
− 2ω(u)
ab
ω(u)
cd
, (3.107)
whereas in the special case of a vorticity-free congruence, i.e. ω(u)=0,wehave
R
(sym)
ab
cd
= R
(fw)
ab
cd
= R
ab
cd
+2θ(u)
b
[c
θ(u)
a
d]
. (3.108)
Together with the spatial symmetric Riemann tensor R
(sym)
ab
cd
we also intro-
duce the spatial symmetric Ricci tensor, R
(sym)
a
b
= R
(sym)
ca
cb
,aswellasthe
associated scalar R
(sym)
= R
(sym)
a
a
.
3.5 Comparing families of observers
Let u and U be two unitary time-like vector fields. Define the relative spatial
velocity of U with respect to u as
ν(U, u)
α
= −(u
σ
U
σ
)
−1
P (u)
α
β
U
β
= −(u
σ
U
σ
)
−1
[U
α
+(u
ρ
U
ρ
)u
α
] . (3.109)
The space-like vector field ν(U, u)
α
, orthogonal to u, can also be written as
ν(U, u)
α
= ||ν(U, u)||ˆν(U, u)
α
, (3.110)
where ˆν(U, u) is the unitary vector giving the direction of ν(U, u)intherest
frame of u. Similar formulas hold for the relative velocity of u with respect to U;
hence the 4-velocities of the two observers admit the following decomposition:
U = γ(U, u)[u + ν(U, u)]
= γ(U, u)[u + ||ν(U, u)||ˆν(U, u)], (3.111)
and
u = γ(u, U)[U + ν(u, U )]
= γ(u, U)[U + ||ν(u, U)||ˆν(u, U)]. (3.112)
Both the spatial relative velocity vectors have the same magnitude,
||ν(U, u)|| =[ν(U, u)
α
ν(U, u)
α
]
1/2
=[ν(u, U)
α
ν(u, U )
α
]
1/2
.
3.5 Comparing families of observers 49
The common gamma factor is related to that magnitude by
γ(U, u)=γ(u, U )=[1−||ν(U, u)||
2
]
−1/2
= −U
α
u
α
; (3.113)
hence we recognize it as the relative Lorentz factor. It is convenient to abbreviate
γ(U, u)byγ and ||ν(U, u)|| by ν when their meaning is clear from the context
and there are no more than two observers involved.
Let us notice here that by substituting (3.111)into(3.112) we obtain the
following relation:
− ˆν(u, U)=γ[ˆν(U, u)+νu], (3.114)
which together with
U = γ[u + νˆν(U, u)] (3.115)
yields a unique relative boost B(U, u)fromu to U, namely
B(U, u)u = U
= γ[u + νˆν(U, u)]
B(U, u)ˆν(U, u)=−ˆν(u, U)
= γ[ˆν(U, u)+νu]. (3.116)
The inv
erse relations hold by interchanging U with u. The boost acts as the
identity on the intersection of their local rest spaces LRS
u
∩ LRS
U
.
Maps between LRSs
The spatial measurements of two observers in relative motion can be compared
by relating their respective LRSs. Let U and u be two such observers and LRS
U
and LRS
u
their LRSs. There exist several maps between these LRSs, as we now
demonstrate.
Combining the projection operators P (U)andP(u) one can form the following
“mixed projection” maps:
(i) P (U, u)fromLRS
u
into LRS
U
, defined as
P (U, u)=P (U)P (u):LRS
u
→ LRS
U
, (3.117)
with its inverse:
P (U, u)
−1
: LRS
U
→ LRS
u
(3.118)
(ii) P (u, U)fromLRS
U
into LRS
u
, defined as
P (u, U)=P (u)P (U):LRS
U
→ LRS
u
, (3.119)
with its inverse:
P (u, U)
−1
: LRS
u
→ LRS
U
. (3.120)
50 Space-time splitting
Note that P (U, u) = P (u, U)
−1
shown as, from their representations,
P (U, u)=P (u)+γνU ⊗ ˆν(U, u),
P (U, u)
−1
= P (U )+νU ⊗ ˆν(u, U),
P (u, U)=P (U)+γνu ⊗ ˆν(u, U ),
P (u, U)
−1
= P (u)+νu ⊗ ˆν(U, u). (3.121)
Let us deduce each of the above relations.
(a) Relation (3.121)
1
is easily verified. Consider a vector X ∈ LRS
u
; then, by
definition,
P (U, u)X = P (U)P (u)X = P (U)X = X + U(U · X) ; (3.122)
but
U · X = γ(u · X + νˆν(U, u) · X)=γνˆν(U, u) ·X.
Hence
P (U, u)X = X + γνUˆν(U, u) · X =[P (u)+γνU ⊗ ˆν(U, u)]
X, (3.123)
which completes the proof.
(b) Let us now verify (3.121)
2
. Consider a vector X = P (U, u)Y ∈ LRS
U
, with
Y ∈ LRS
u
; then, by using the relation u = γ(U + νˆν(u, U)) and the property
Y · u =0,wehave
Y · U = −νY · ˆν(u, U).
Hence
Y = P (U)Y − U(U · Y )=P (U, u)Y + νU[Y · ˆν(u, U)]
= P (U, u)Y + νU[P (U, u)Y · ˆν(u, U)]. (3.124)
Substituting Y = P (U, u)
−1
X gives
P (U, u)
−1
X = P (U)X + νU[X · ˆν(u, U )]
=[P (U)+νU ⊗ ˆν(u, U)]
X, (3.125)
which completes the proof.
(c) Relations (3.121)
3
and (3.121)
4
are straightforwardly verified by exchanging
U with u in (3.121)
1
and (3.121)
2
respectively.
One can then show that
P (U, u)ˆν(U, u)=−γ ˆν(u, U ),
P (u, U)
−1
ˆν(U, u)=−
1
γ
ˆν(u, U ). (3.126)
Note that, from the property (3.3) of the projection operator P (U), we have
P (U, u)=P (U)
P (u)=P (U) P (U, u)=P (U, u) P (u). (3.127)
3.5 Comparing families of observers 51
This is a mixed tensor which is spatial with respect to u in its covariant index and
with respect to U in its contravariant index.
4
Moreover the following relations
hold:
P (U)=P (U, u)
P (U, u)
−1
,
P (u)=P (U, u)
−1
P (U, u). (3.128)
Boost maps
Similarly to what we have done in combining projection maps, the boost B(U, u)
induces an invertible map between the local rest spaces of the given observers
defined as
B
(lrs)
(U, u) ≡ P (U )B(U, u)P (u):LRS
u
→ LRS
U
. (3.129)
It also acts as the identity on the intersection of their subspaces LRS
U
∩LRS
u
.
Because the boost is an isometry, exchanging the role of U and u in (3.129) leads
to the inverse boost:
B
(lrs)
(U, u)
−1
≡ B
(lrs)
(u, U):LRS
U
→ LRS
u
. (3.130)
Before giving explicit representations of B
(lrs)
(U, u) and its inverse B
(lrs)
(U, u)
−1
we note that any map between the local rest spaces of two observers may be
expressed only in terms of quantities which are spatial with respect to those
observers.
The representations of the boost and its inverse can be given in terms of the
associated tensors
B
(lrs)
u
(U, u),B
(lrs)
U
(U, u),B
(lrs)
u
(u, U),B
(lrs)
U
(u, U), (3.131)
defined by
B
(lrs)
u
(U, u)=P(U, u)
−1
B
(lrs)
(U, u),
B
(lrs)
U
(U, u)=B
(lrs)
(U, u) P (U, u)
−1
, (3.132)
with the corresponding expressions for the inverse boost obtained simply by
exchanging the roles of U and u, and with
B
(lrs)
(U, u)=B
(lrs)
U
(U, u) P (U, u)=P (U, u) B
(lrs)
u
(U, u). (3.133)
The explicit expression for B
(lrs)
u
(U, u), for example, is given by
B
(lrs)
u
(U, u)=P(u)+
1 −γ
γ
ˆν(U, u) ⊗ ˆν(U, u). (3.134)
4
This is a kind of two-point tensor that Schouten (1954) termed a connecting tensor.
52 Space-time splitting
This can be shown as follows. Let X ∈ LRS
u
; then
B
(lrs)
u
(U, u)X = P (U, u)
−1
[B
(lrs)
(U, u)X]
= P (U, u)
−1
[B
(lrs)
(U, u)X
||
ˆν(U, u)+X
⊥
], (3.135)
where X
||
= X · ˆν(U, u)andX
⊥
= X −X
||
ˆν(U, u), that is
X = X
||
ˆν(U, u)+X
⊥
. (3.136)
We have also used the fact that the boost reduces to the identity for vectors not
belonging to the boost plane, as X
⊥
. In this case the boost plane is spanned by
the vectors u and ˆν(U, u). Taking into account (3.116), that is,
B
(lrs)
(U, u)ˆν(U, u)=−ˆν(u, U ), (3.137)
as well as the linearity of the boost map, we have
B
(lrs)
u
(U, u)X = P (U, u)
−1
[X −X
||
ˆν(u, U ) − X
||
ˆν(U, u)]
= P (U, u)
−1
X − X
||
P (U, u)
−1
[ˆν(u, U)+ˆν(U, u)]
= X +
1 −γ
γ
X
||
ˆν(U, u), (3.138)
where P (U, u)
−1
X = X because X ∈ LRS
u
:
P (U, u)
−1
X = P (U, u)
−1
P (u)X = P (U, u)
−1
P (U)P (u)X
= P (U, u)
−1
P (U, u)X = P (u)X = X. (3.139)
Hence P (U, u)
−1
ˆν(U, u)=ˆν(U, u), because ν(U, u) belongs to LRS
u
.Moreover,
from (3.126), by exchanging the roles of U and u, we find
P (U, u)
−1
ˆν(u, U )=−
1
γ
ˆν(U, u)=
1
γ
B(u, U)ˆν(u, U)
=
1
γ
B(U, u)
−1
ˆν(u, U ). (3.140)
Therefore
B
(lrs)
u
(U, u)X = X −X
||
−
1
γ
+1
ˆν(U, u)
= X −(X · ˆν(U, u))
−
1
γ
+1
ˆν(U, u)
=
P (u) −
γ − 1
γ
ˆν(U, u) ⊗ ˆν(U, u)
X, (3.141)
which is equivalent to (3.134).