8.1 Curved Space and the Riemann Tensor
111
The tensor R
α ηβγ is called the Riemann curvature tensoror simply the Riemann
tensor; it is constructed from the connections and their derivatives and will be
calculated and defined explicitly below.
We will prove Theorem 2 and define the Riemann tensor by direct algebraic
manipulation. First denote the covariant derivative as
T
α β = ξ
α ;β = ξ
α ,β +
α
βη ξ
η
.
(8.5)
Then by the definition of covariant tensor derivatives
ξ
α ;β;γ = T
α β;γ = T
α β,γ +
α
τ γ T
τ β −
λ
βγ T
α λ
=
ξ
α ,β,γ +
α
βη,γ ξ
η
+
α
βη ξ
η ,γ
+
α
τ γ
ξ
τ ,β +
τ
βη ξ
η
−
λ
βγ T
α λ .
(8.6)
Clearly ξ
α ;γ ;β is given by the same expression with β and γ reversed. From (8.6) it
is easy to write the difference; we find
ξ
α ;β;γ − ξ
α ;γ ;β =
α
βη,γ ξ
η
−
α
γ η,β ξ
η
+
α
βη ξ
η ,γ −
α
γ η ξ
η ,β +
α
τ γ ξ
τ ,β −
α
τβ ξ
τ ,γ
+
α
τ γ
τ
βη ξ
η
−
α
τβ
τ
γ η ξ
η
.
(8.7)
But the terms in the square bracket cancel, and we are left with
ξ
α ;β;γ − ξ
α ;γ ;β = R
α ηβγ ξ
η
,
R
α ηβγ ≡
α
βη,γ −
α
γ η,β +
α
τ γ
τ
βη −
α
τβ
τ
γ η .
(8.8)
This proves the theorem and defines the very important Riemann tensor. Notice that
it is built with only the connections and their derivatives, and of course it does not
depend on the vector ξ
η : it is a purely geometrical object in that it is constructed from
only the metric tensor. Also note that it is indeed a tensor by (8.8) and the quotient
theorem. The Riemann tensor may look a bit formidable at first since it is fourth rank
and is composed of many terms, but its importance makes it worth study.
From the above Theorem 2 we may restate Theorem 1 in a beautiful new way.
Theorem 3 A Euclidean or pseudo-Euclidean space has a zero Riemann tensor.
This is now obvious, being merely a restatement of Theorem 1.
We finally have come to the definition of a curved versus a flat space. We call a
space flat if the Riemann tensor is zero, and curved if the Riemann tensor is not zero.
This clearly fits our needs for a general and useful definition of flat space, since we
see by Theorem 3 that Euclidean 2 and 3-space are flat, as is the Minkowski space
of special relativity.
111
The tensor R
α ηβγ is called the Riemann curvature tensoror simply the Riemann
tensor; it is constructed from the connections and their derivatives and will be
calculated and defined explicitly below.
We will prove Theorem 2 and define the Riemann tensor by direct algebraic
manipulation. First denote the covariant derivative as
T
α β = ξ
α ;β = ξ
α ,β +
α
βη ξ
η
.
(8.5)
Then by the definition of covariant tensor derivatives
ξ
α ;β;γ = T
α β;γ = T
α β,γ +
α
τ γ T
τ β −
λ
βγ T
α λ
=
ξ
α ,β,γ +
α
βη,γ ξ
η
+
α
βη ξ
η ,γ
+
α
τ γ
ξ
τ ,β +
τ
βη ξ
η
−
λ
βγ T
α λ .
(8.6)
Clearly ξ
α ;γ ;β is given by the same expression with β and γ reversed. From (8.6) it
is easy to write the difference; we find
ξ
α ;β;γ − ξ
α ;γ ;β =
α
βη,γ ξ
η
−
α
γ η,β ξ
η
+
α
βη ξ
η ,γ −
α
γ η ξ
η ,β +
α
τ γ ξ
τ ,β −
α
τβ ξ
τ ,γ
+
α
τ γ
τ
βη ξ
η
−
α
τβ
τ
γ η ξ
η
.
(8.7)
But the terms in the square bracket cancel, and we are left with
ξ
α ;β;γ − ξ
α ;γ ;β = R
α ηβγ ξ
η
,
R
α ηβγ ≡
α
βη,γ −
α
γ η,β +
α
τ γ
τ
βη −
α
τβ
τ
γ η .
(8.8)
This proves the theorem and defines the very important Riemann tensor. Notice that
it is built with only the connections and their derivatives, and of course it does not
depend on the vector ξ
η : it is a purely geometrical object in that it is constructed from
only the metric tensor. Also note that it is indeed a tensor by (8.8) and the quotient
theorem. The Riemann tensor may look a bit formidable at first since it is fourth rank
and is composed of many terms, but its importance makes it worth study.
From the above Theorem 2 we may restate Theorem 1 in a beautiful new way.
Theorem 3 A Euclidean or pseudo-Euclidean space has a zero Riemann tensor.
This is now obvious, being merely a restatement of Theorem 1.
We finally have come to the definition of a curved versus a flat space. We call a
space flat if the Riemann tensor is zero, and curved if the Riemann tensor is not zero.
This clearly fits our needs for a general and useful definition of flat space, since we
see by Theorem 3 that Euclidean 2 and 3-space are flat, as is the Minkowski space
of special relativity.