De Felice F., Bini D. Classical Measurements in Curved Space-Times
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2.1 The space-time 13
contraction) or
(left contraction). If S and T are two arbitrary tensor fields, the
left contraction S
T denotes a contraction between the rightmost contravariant
index of S and the leftmost covariant index of T (i.e. S
...α
...
T
...
α...
), and the right
contraction S
T denotes a contraction between the rightmost covariant index
of S and the leftmost contravariant index of T (i.e. S
...
...α
T
α...
...
), assuming in each
case that such indices exist. If B is a
1
1
-tensor field, namely
B = B
α
β
e
α
⊗ ω
β
, (2.11)
and it acts on a vector field X by right contraction, we have
B
X =[B X]
α
e
α
= B
α
β
X
β
e
α
. (2.12)
Similarly, if A and B are two
1
1
-tensor fields, their right contraction is given by
A
B =[A B]
α
β
e
α
⊗ ω
β
=[A
α
γ
B
γ
β
] e
α
⊗ ω
β
. (2.13)
The identity transformation is represented by the unit tensor or Kronecker
delta tensor δ:
δ = δ
α
β
e
α
⊗ ω
β
. (2.14)
The trace of a
1
1
-tensor is defined as
Tr A = A
α
α
. (2.15)
Any
1
1
-tensor can be decomposed into a pure-trace and a trace-free (TF) part
A = A
(TF)
+
1
4
[Tr A]δ. (2.16)
Contraction over p indices of two general tensors is symbolically described as
(right p-contraction)
S
p
T ≡ S
α...
β
1
...β
p
T
β
1
...β
p
...
(2.17)
or (left p-contraction)
S
p
T ≡ S
α... β
1
...β
p
T
β
1
...β
p
...
. (2.18)
Change of frame
Given a non-degenerate
1
1
-tensor field A, i.e. such that the determinant
det(A
α
β
) is everywhere non-vanishing, one can prove that the elements of the
inverse matrix are the components of a tensor A
−1
which is termed the inverse
tensor of A:
A
−1
A = δ,
A
−1
α
γ
A
γ
β
= δ
α
β
. (2.19)
14 The theory of relativity: a mathematical overview
Non-degenerate
1
1
-tensor fields induce frame transformations as
¯e
α
= A
−1
e
α
=
A
−1
β
α
e
β
, ¯ω
α
= ω
α
A = A
α
β
ω
β
. (2.20)
The space-time metric
Let (g
αβ
) be a symmetric non-degenerate matrix of signature +2 and denote by
(g
αβ
) its inverse. Then
g=g
αβ
ω
α
⊗ ω
β
, g
−1
= g
αβ
e
α
⊗ e
β
(2.21)
locally define a pseudo-Riemannian metric tensor (g) and its inverse (g
−1
) satis-
fying
g
−1
g=δ, g
αγ
g
γβ
= δ
α
β
. (2.22)
Using the notation g = |det(g
αβ
)| one finds det(g
αβ
)=−g and
d ln g = g
αβ
dg
αβ
=Tr[g
−1
dg]. (2.23)
Let us choose an orientation on M, namely an everywhere non-zero 4-form
˜
O;
a frame {e
α
} is termed oriented if
˜
O(e
0
,e
1
,e
2
,e
3
) > 0.
Since the metric is non-degenerate, it determines an isomorphism between the
tangent and cotangent spaces at each point of the manifold which in index-
notation corresponds to “raising” and “lowering” indices. For a vector field X
and a 1-form θ one has
X
=g X, X
α
= g
αβ
X
β
,
θ
=g
−1
θ, θ
α
= g
αβ
θ
β
,
(2.24)
where, as is customary, we use the sharp () and flat () notation for an arbitrary
tensor to mean the tensor obtained by raising or lowering, respectively, all of the
indices which are not already of the appropriate type.
Unit volume 4-form
The Levi-Civita permutation symbols
α
1
...α
4
and
α
1
...α
4
are totally antisym-
metric with
0123
=1=
0123
, (2.25)
so that they vanish unless α
1
...α
4
is a permutation of 0 ...3, in which case their
value is the sign of the permutation. The components of the unit volume 4-form
are related to the Levi-Civita symbols by
η
α
1
α
2
α
3
α
4
= g
1/2
α
1
α
2
α
3
α
4
,η
α
1
α
2
α
3
α
4
= −g
−1/2
α
1
α
2
α
3
α
4
. (2.26)
2.1 The space-time 15
Generalized Kronecker deltas
The following identities define the generalized Kronecker deltas and relate them
to the Levi-Civita alternating symbols and to the unit volume 4-form η:
δ
α
1
...α
4
β
1
...β
4
=
α
1
...α
4
β
1
...β
4
= −η
α
1
...α
4
η
β
1
...β
4
,
δ
α
1
...α
p
β
1
...β
p
=
1
(4 −p)!
δ
α
1
...α
p
γ
p+1
...γ
4
β
1
...β
p
γ
p+1
...γ
4
= −
1
(4 −p)!
η
α
1
...α
p
γ
p+1
...γ
4
η
β
1
...β
p
γ
p+1
...γ
4
. (2.27)
A familiar alternative definition is given by
δ
α
1
...α
p
β
1
...β
p
= det
⎛
⎜
⎝
δ
α
1
β
1
··· δ
α
1
β
p
.
.
.
.
.
.
δ
α
p
β
1
··· δ
α
p
β
p
⎞
⎟
⎠
= p!δ
α
1
[β
1
···δ
α
p
β
p
]
. (2.28)
Note that the second of the relations (2.27) holds in a four-dimensional Rieman-
nian manifold with signature +2; in a three-dimensional Euclidean manifold one
has instead
δ
α
1
...α
p
β
1
...β
p
=
1
(3 −p)!
η
α
1
...α
p
γ
p+1
...γ
3
η
β
1
...β
p
γ
p+1
...γ
3
. (2.29)
Symmetrization and antisymmetrization
Given an object with only covariant or contravariant indices, or a subset of only
covariant or contravariant indices from those of a mixed object, one can always
project out the purely symmetric and purely antisymmetric parts. For example
for a
0
p
-tensor field one has
[ALT S]
α
1
...α
p
= S
[α
1
...α
p
]
=
1
p!
δ
β
1
...β
p
α
1
...α
p
S
β
1
...β
p
,
[SYM S]
α
1
...α
p
= S
(α
1
...α
p
)
=
1
p!
σ
S
σ(α
1
...α
p
)
, (2.30)
where the sum is over all the permutations σ of {α
1
...α
p
}.
If T is a
0
2
-tensor, its symmetric and antisymmetric parts are given by
T
(μν)
=
1
2
(T
μν
+ T
νμ
) (2.31)
T
[μν]
=
1
2
(T
μν
− T
νμ
) . (2.32)
16 The theory of relativity: a mathematical overview
Exterior product
The exterior or wedge product of p 1-forms ω
α
i
(i =1,...p) is defined as
ω
α
1
∧···∧ω
α
p
≡ p! ω
[α
1
⊗···⊗ω
α
p
]
= δ
α
1
...α
p
β
1
...β
p
ω
β
1
⊗···⊗ω
β
p
= ω
α
1
...α
p
, (2.33)
where notation (2.8) has been used.
From Eq. (2.33), the wedge product of a p-form S with a q-form T has the
following expression
[S ∧ T ]
α
1
...α
p+q
=
(p + q)!
p!q!
S
[α
1
...α
p
T
α
p+1
...α
p+q
]
, (2.34)
so that the relation
S ∧ T =(−1)
pq
T ∧ S (2.35)
holds identically.
Hodge duality operation
The Hodge duality operation associates with a p-form S the (4 −p)-form
∗
S with
components
[
∗
S]
α
p+1
...α
4
=
1
p!
S
α
1
...α
p
η
α
1
...α
p
α
p+1
...α
4
, (2.36)
i.e. with η at the right-hand side of the p-form S and with a proper contraction
of the first indices of η (right duality convention). From the above definition one
finds that a p-form S satisfies the identity
∗∗
S ≡ (−1)
p−1
S. (2.37)
The duality operation applied to the wedge product of a p-form S and a q-form
T (with q ≥ p) implies the following identity:
∗
(S ∧
∗
T )=(−1)
1+(4−q)(q−p)
1
p!
S
p
T. (2.38)
With p =0andq = 1 this reduces to (2.37), while when p = q one has
∗
(S ∧
∗
T )=(−1)
1
p!
S
p
T =
∗
η
1
p!
S
p
T, (2.39)
since
∗
η =(−1). Removing the duality operation from each side leads to
S ∧
∗
T =
1
p!
(S
p
T ) η. (2.40)
2.2 Derivatives on a manifold 17
2.2 Derivatives on a manifold
Differentiation on a manifold is essential to derive physical laws and define phys-
ical measurements. It stems from appropriate criteria for comparison between
the algebraic structures at any two points of the manifold. Clearly differentiation
allows one to define new quantities, as we shall see next.
Covariant derivative and connection
The covariant derivative ∇
e
γ
S of an arbitrary
p
q
-tensor S along the e
γ
frame
direction is a
p
q+1
-tensor with components
[∇
e
γ
S]
α
1
...α
p
β
1
...β
q
≡∇
γ
S
α
1
...α
p
β
1
...β
q
(2.41)
= e
γ
S
α
1
...α
p
β
1
...β
q
+Γ
α
1
δγ
S
δ...
β
1
...
+ ···−Γ
δ
β
1
γ
S
α
1
...
δ...
−···.
Here the coefficients Γ
α
γδ
are the components of a linear connection on M
defined as
∇
e
α
e
β
=Γ
γ
βα
e
γ
↔∇
e
α
ω
β
= −Γ
β
γα
ω
γ
. (2.42)
The covariant derivative satisfies the following rules. For any pair of vector fields
X and Y and of real functions f and h,wehave
∇
fX+hY
= f∇
X
+ h∇
Y
. (2.43)
From this it follows that
∇
X
S = X
α
∇
e
α
S ≡ X
α
∇
α
S. (2.44)
A connection is said to be compatible with the metric if the latter is constant
under covariant differentiation, that is
0=∇
γ
g
αβ
= e
γ
(g
αβ
) −g
δβ
Γ
δ
αγ
− g
αδ
Γ
δ
βγ
= e
γ
(g
αβ
) −Γ
βαγ
− Γ
αβγ
, (2.45)
where we set
Γ
αβγ
≡ g
ασ
Γ
σ
βγ
. (2.46)
Given two vector fields X and Y , consider the covariant derivative of Y in the
direction of X as
∇
X
Y =
X
α
e
α
(Y
δ
)+Γ
δ
βα
X
α
Y
β
e
δ
. (2.47)
From (2.47) and (2.5) we deduce the following relation
∇
X
Y −∇
Y
X =[X, Y ] −[2Γ
δ
[βα]
− C
δ
αβ
]X
α
Y
β
e
δ
. (2.48)
The quantity
T (X, Y )=∇
X
Y −∇
Y
X −[X, Y ] (2.49)
18 The theory of relativity: a mathematical overview
is a
1
3
-tensor termed torsion. In the theory of relativity the torsion is assumed
to be identically zero. Hence, X and Y being arbitrary, the following relation
holds
Γ
δ
[βα]
=
1
2
C
δ
αβ
. (2.50)
From this, Eq. (2.45) can be inverted, yielding
Γ
α
βγ
=
1
2
g
αδ
[e
β
(g
δγ
)+e
γ
(g
βδ
) −e
δ
(g
γβ
)+C
δγβ
+ C
βδγ
− C
γβδ
] , (2.51)
where we set C
αβγ
= g
ασ
C
σ
βγ
. In the event that the structure functions C
αβγ
are all zero – in which case the bases e
α
are termed holonomous – the components
of the linear connection are known as Christoffel symbols.
Curvature
The curvature or Riemann tensor is defined as
R(e
γ
,e
δ
)e
β
=
[∇
e
γ
, ∇
e
δ
] −∇
[e
γ
,e
δ
]
e
β
= R
α
βγδ
e
α
. (2.52)
Hence the components in a given frame are
R
α
βγδ
= e
γ
(Γ
α
βδ
) −e
δ
(Γ
α
βγ
) −C
γδ
Γ
α
β
+Γ
α
γ
Γ
βδ
− Γ
α
δ
Γ
βγ
. (2.53)
This tensor is explicitly antisymmetric in its last pair of indices and so defines a
1
1
-tensor-valued 2-form Ω:
Ω=e
α
⊗ ω
β
⊗ Ω
α
β
= e
α
⊗ ω
β
⊗
1
2
R
α
βγδ
ω
γδ
. (2.54)
Covariant derivative and curvature
A remarkable property of the covariant derivative is that it does not commute
with itself. In fact, given a vector X and using coordinate components (C
γ
αβ
=0)
we have
∇
[α
∇
β]
X
γ
=
1
2
R
γ
σαβ
X
σ
, (2.55)
while given a 1-form ω we have
∇
[α
∇
β]
ω
γ
=
1
2
ω
σ
R
σ
γβα
. (2.56)
Relations (2.55) and (2.56) can be generalized to arbitrary tensors; for a
1
1
-
tensor T
ρ
σ
we have
∇
[α
∇
β]
T
ρ
σ
=
1
2
T
μ
σ
R
μ
ρβα
+
1
2
R
σ
μαβ
T
ρ
μ
. (2.57)
2.2 Derivatives on a manifold 19
Applying the above relation to the metric tensor we have
∇
[α
∇
β]
g
ρσ
=
1
2
g
μσ
R
μ
ρβα
+
1
2
g
ρμ
R
μ
σβα
≡ 0. (2.58)
Hence
R
σρβα
+ R
ρσβα
=0, (2.59)
implying that the totally covariant (or contravariant) Riemann tensor is antisym-
metric with respect to the first pair of indices as well.
Ricci and Bianchi identities
The Ricci identities are given by
3R
α
[βγδ]
= R
α
βγδ
+ R
α
γδβ
+ R
α
δβγ
=0, (2.60)
whereas the Bianchi identities are given by
3∇
[
R
γδ]β
α
= ∇
R
γδβ
α
+ ∇
δ
R
γβ
α
+ ∇
γ
R
δβ
α
=0. (2.61)
Finally, if we make the algebraic sum of the Ricci identities (2.60)foreachper-
mutation of the indices of a totally covariant Riemann tensor,
R
α[βγδ]
+ R
δ[αβγ]
− R
γ[δαβ]
− R
β[γδα]
=0, (2.62)
it follows that
R
αβγδ
= R
γδαβ
, (2.63)
i.e. the Riemann tensor is symmetric under exchange of the two pairs of indices.
Ricci tensor and curvature scalar
From the symmetries of the curvature tensor its first contraction generates the
Ricci curvature tensor
R
αβ
= R
γ
αγβ
, (2.64)
which is symmetric due to the Ricci identities. Its trace defines the curvature
scalar
R ≡ R
α
α
= R
γα
γα
. (2.65)
20 The theory of relativity: a mathematical overview
Weyl and Einstein tensors
The Weyl tensor is a suitable combination of the curvature tensor, the Ricci
tensor, and the curvature scalar, as follows:
C
αβ
γδ
= R
αβ
γδ
− 2R
[α
[γ
δ
β]
δ]
+
R
6
δ
αβ
γδ
= R
αβ
γδ
− 2δ
[α
[γ
S
β]
δ]
, (2.66)
where
S
β
α
= R
β
α
−
1
6
Rδ
β
α
. (2.67)
The Weyl tensor is trace-free, i.e. C
α
βαδ
= 0, and is invariant under conformal
transformations of the metric.
The one-fold contraction of the Bianchi identities (using the trace-free property
of the Weyl tensor (2.66)) reduces to
0=∇
[
R
γδ]
αβ
δ
γ
α
= ∇
α
R
δ
αβ
+2∇
[
R
δ]
β
= ∇
α
C
δ
αβ
+ ∇
[
R
β
δ]
−
R
6
δ
β
δ]
. (2.68)
The two-fold contraction of the Bianchi identities leads to
0=∇
[
R
αβ
γδ]
δ
γ
α
δ
δ
β
. (2.69)
Defining the Einstein tensor as
G
δ
≡ R
δ
−
1
2
δ
δ
R, (2.70)
relation (2.69) implies identically that
∇
δ
G
δ
=0. (2.71)
Cotton tensor
The Cotton tensor (Cotton, 1899; Eisenhart, 1997) is defined as
R
β
δ
≡ 2∇
[
R
β
δ]
−
R
6
δ
β
δ]
=2∇
[
S
β
δ]
. (2.72)
Hence the divergence of the Weyl tensor can be written in a more compact
form as
∇
α
C
δ
αβ
= −
1
2
R
β
δ
. (2.73)
2.2 Derivatives on a manifold 21
The property of the Einstein tensor of being divergence-free leads to the trace-free
property of the Cotton tensor:
R
β
αβ
= ∇
β
G
β
α
=0. (2.74)
The totally antisymmetric part of the fully covariant (or contravariant) Cotton
tensor is zero due to the symmetry of the Ricci and metric tensors, that is
3R
[αβγ]
= R
αβγ
+ R
βγα
+ R
γαβ
=0. (2.75)
Finally from (2.71) it follows that the divergence of the Cotton tensor is also
zero:
∇
β
R
β
δ
=0. (2.76)
Absolute derivative along a world line
Given a curve γ parameterized by λ and denoting by ˙γ the corresponding tangent
vector field, the absolute derivative of any tensor field S(λ) defined along the
curve is
DS
dλ
= ∇
˙γ
S. (2.77)
Given a local coordinate system {x
α
}, the curve is described by the functions
γ
α
(λ)=x
α
(γ(λ)) ≡ x
α
(λ), so that
[˙γ(λ)]
α
=
dx
α
dλ
(λ).
For a vector field X defined on γ, its absolute derivative takes the form
DX
dλ
=
dX
α
(λ)
dλ
+Γ
α
γβ
(λ)˙γ
β
(λ)X
γ
(λ)
e
α
. (2.78)
Parallel transport and geodesics
A tensor field S defined on a curve γ with parameter λ is said to be parallel
transported along the curve if its absolute derivative along γ is zero:
DS
dλ
=0. (2.79)
A curve γ whose tangent vector ˙γ satisfies the equation
D ˙γ
dλ
(λ)=f(λ)˙γ(λ), (2.80)
22 The theory of relativity: a mathematical overview
where f(λ) is a function defined on γ, is termed geodesic. It is always possible to
re-parameterize the curve with σ(λ) such that Eq. (2.80) becomes
D
˙
γ
dσ
(σ)=0, (2.81)
where
˙
γ
α
=
dx
α
dσ
.
In this case the parameter σ is said to be affine and it is defined up to linear
transformations. From Eq. (2.81) we see that a characteristic feature of an affine
geodesic is that the tangent vector is parallel transported along it.
Fermi-Walker derivative and transport
Let us introduce a new parameter s along a non-null curve γ such that the tangent
vector of the curve u
α
= dx
α
/ds is unitary, that is, u · u = ±1. If the curve is
time-like then u · u = −1 and the parameter s is termed proper time, as already
noted. The curvature vector (or acceleration) of the curve is defined as
a(u)=
Du
ds
. (2.82)
Consider a vector field X(s)onγ. We define the Fermi-Walker derivative of X
along γ as the vector field D
(fw,u)
X/ds having components
D
(fw,u)
ds
X
α
≡
DX
ds
α
± [a(u)
α
(u ·X) −u
α
(a(u) ·X)]
=
DX
ds
α
± [a(u) ∧u]
α
γ
X
γ
, (2.83)
where the signs ± should be chosen according to the causal character of the curve:
+ for a time-like curve and − for a space-like one.
The generalization to a tensor field T ∈
1
1
, for example, is the following:
D
(fw,u)
ds
T
α
β
≡
DT
ds
α
β
± ([a(u) ∧u]
α
γ
T
γ
β
−[a(u) ∧ u]
γ
β
T
α
γ
) . (2.84)
The tensor field T is said to be Fermi-Walker transported along the curve if
its Fermi-Walker derivative is identically zero.