112
8 Curved Space and Gravity
The converse of Theorem 3 is also true, that if the Riemann tensor is zero we can
find a coordinate system in which the metric tensor is globally constant. The proof is
a bit tedious, so we will not give it here but refer the reader to Adler (1975). Instead
we will devote our time to another interesting geometric property that is equivalent
to flatness. We state this as Theorem 4.
Theorem 4 We can set up a constant vector field (that is one with zero covariant
derivative) by parallel displacement from some initial vector at an initial point if and
only if the space is flat, that is if the Riemann tensor is zero.
This is a very restrictive and perhaps surprising theorem. The proof involves doing
the construction explicitly and is straight-forward. Begin with a vector V
α at some
arbitrary point in the space, and parallel displace it along some curve C to a point
labeled x
λ , to produce the field V
α
x
λ
. If this is to be a unique and well-defined
field then it cannot depend on which curve between the initial point and x
λ one uses;
a different curve C
would do as well. The covariant derivative of this field must be
zero by construction; this is easy to see, since by definition
V
α ;β dx
β
=
V
α ,β +
α
βγ V
γ
dx
β
= dV
α
+
α
βγ V
γ dx
β
.
(8.9)
Since we set the field up by parallel displacement
dV
α
= −
α
βγ V
γ dx
β
, so V
α ;β = 0.
(8.10)
Because of this we may express the ordinary derivative of the field in terms of
connections as
V
α ,γ = −
α
βγ V
β
.
(8.11)
But for a well-defined field the order of the ordinary second derivatives does not
matter, V
α ,γ ,δ = V
α ,δ,γ , so from (8.11)
α
βγ V
β
,δ
=
α
βδ V
β
,γ
,
α
βγ,δ V
β
+
α
βγ V
β ,δ =
α
βδ,γ V
β
+
α
βδ V
β ,γ .
(8.12)
Using (8.11) to simplify this we write
α
βγ,δ V
β
−
α
βγ
β
δτ V
τ
=
α
βδ,γ V
β
+
α
βδ
β
γ τ V
τ
,
(8.13)
and relabel indices to see that
α
γβ,δ −
α
δβ,γ +
α
δτ
τ
γβ −
α
γ τ
τ
δβ
V
β
= R
α βγ δ V
β
= 0.
(8.14)
Since the vector may have any value we see that the Riemann tensor must vanish;
conversely, if the Riemann tensor vanishes the construction goes through, so the
8 Curved Space and Gravity
The converse of Theorem 3 is also true, that if the Riemann tensor is zero we can
find a coordinate system in which the metric tensor is globally constant. The proof is
a bit tedious, so we will not give it here but refer the reader to Adler (1975). Instead
we will devote our time to another interesting geometric property that is equivalent
to flatness. We state this as Theorem 4.
Theorem 4 We can set up a constant vector field (that is one with zero covariant
derivative) by parallel displacement from some initial vector at an initial point if and
only if the space is flat, that is if the Riemann tensor is zero.
This is a very restrictive and perhaps surprising theorem. The proof involves doing
the construction explicitly and is straight-forward. Begin with a vector V
α at some
arbitrary point in the space, and parallel displace it along some curve C to a point
labeled x
λ , to produce the field V
α
x
λ
. If this is to be a unique and well-defined
field then it cannot depend on which curve between the initial point and x
λ one uses;
a different curve C
would do as well. The covariant derivative of this field must be
zero by construction; this is easy to see, since by definition
V
α ;β dx
β
=
V
α ,β +
α
βγ V
γ
dx
β
= dV
α
+
α
βγ V
γ dx
β
.
(8.9)
Since we set the field up by parallel displacement
dV
α
= −
α
βγ V
γ dx
β
, so V
α ;β = 0.
(8.10)
Because of this we may express the ordinary derivative of the field in terms of
connections as
V
α ,γ = −
α
βγ V
β
.
(8.11)
But for a well-defined field the order of the ordinary second derivatives does not
matter, V
α ,γ ,δ = V
α ,δ,γ , so from (8.11)
α
βγ V
β
,δ
=
α
βδ V
β
,γ
,
α
βγ,δ V
β
+
α
βγ V
β ,δ =
α
βδ,γ V
β
+
α
βδ V
β ,γ .
(8.12)
Using (8.11) to simplify this we write
α
βγ,δ V
β
−
α
βγ
β
δτ V
τ
=
α
βδ,γ V
β
+
α
βδ
β
γ τ V
τ
,
(8.13)
and relabel indices to see that
α
γβ,δ −
α
δβ,γ +
α
δτ
τ
γβ −
α
γ τ
τ
δβ
V
β
= R
α βγ δ V
β
= 0.
(8.14)
Since the vector may have any value we see that the Riemann tensor must vanish;
conversely, if the Riemann tensor vanishes the construction goes through, so the