3 Curvature

#math

Derivative Operators

Definition

A derivative operator $\nabla$ on a manifold $M$ is a map which takes each smooth (or merely differentiable) tensor field of type $(k, l)$ to a smooth tensor field (so we are assigning a tensor to each point on the manifold) of type $(k, l + 1)$ and satisfies the following 5 properties:

Linearity: For all $A, B \in T (k, l)$ and $α, β \in R$ ,
Leibniz Rule:

\nabla_{c} (α A_{b_{1}, . . ., b_{l}}^{a_{1}, . . ., a_{k}} + β B_{b_{1}, . . ., b_{l}}^{a_{1}, . . ., a_{k}}) = α \nabla_{c} (A_{b_{1}, . . ., b_{l}}^{a_{1}, . . ., a_{k}}) + β \nabla_{c} (B_{b_{1}, . . ., b_{l}}^{a_{1}, . . ., a_{k}})

Commutativity with contraction
For all $f \in F$ and all $t^{a} \in V_{p}$ ,

t (f) = t^{a} \nabla_{a} f

Torsion free: For all $f \in F$ ,

\nabla_{a} \nabla_{b} f = \nabla_{b} \nabla_{a} f

As an example of a derivative operator:

Definition

Let $ψ$ be a coordinate system and let ${\frac{\partial}{\partial x^{μ}}}$ and ${d x^{μ}}$ be the associated coordinate bases. We can define a derivative operator $\partial_{a}$ , called the ordinary derivative as follows: For any smooth tensor field $T_{b_{1}, . . ., b_{l}}^{a_{1}, . . ., a_{k}}$ we take its components in this coordinate basis denoted by $T_{ν_{1}, . . ., ν_{l}}^{μ_{1}, . . ., μ_{k}}$ . We define $\partial_{c} T_{b_{1}, . . ., b_{l}}^{a_{1}, . . ., a_{k}}$ to be the tensor whose components are the partial derivatives $\partial T_{ν_{1}, . . ., ν_{l}}^{μ_{1}, . . ., μ_{k}} / \partial x^{σ}$ .

Of course a different choice of coordinate system $ψ^{'}$ will yield a different derivative operator $\partial_{a}^{'}$ , that is the components of the tensor $\partial_{c} T^{a_{1}, . . ., a_{k}}_{b_{1}, . . ., b_{l}}$ in the new (primed) coordinates will not be equal.

Note on ordinary derivatives

Remember that $\partial T^{μ_{1}, . . ., μ_{k}}_{ν_{1}, . . ., ν_{l}} / \partial x^{σ}$ depends on the coordinate representation. For example $f (x, y) = x y$ and we take $v = \partial / \partial y$ as our vector. Then clearly $v (f) = y^{α} \partial_{α} f = \partial f / \partial y = x$ but note that this is for the $x - y$ coordinate space, i.e. $\nabla_{a} (f) = (\partial / \partial y) (f)$

If we work in the $r - θ$ coordinate space then we get instead $\tilde{\nabla_{a}} f = (s i n (θ) \partial / \partial r + \frac{1}{r} c o s (θ) \partial / \partial θ) f$ but since $f (r, θ) = r^{2} s i n (θ) c o s (θ)$ you would find that the action (result of the derivative) agrees, it just looks different.

By condition (4) any two derivative operators $\nabla_{a}, \tilde{\nabla_{a}}$ must agree on scalar fields. Let $ω_{b}$ be an arbitrary dual vector field and $f$ an arbitrary scalar field, by the Leibniz Rule we get that:

{\tilde{\nabla}}_{a} (f ω_{b}) - \nabla_{a} (f ω_{b}) = f \times ({\tilde{\nabla}}_{a} (ω_{b}) - \nabla_{a} (ω_{b}))

Now at some point $p$ , $\nabla_{a} (ω_{b})$ and ${\tilde{\nabla}}_{a} (ω_{b})$ each depend on how $ω_{b}$ changes as one moves away from $p$ . However notice that if you have another dual vector field $ω_{b}^{'}$ that agrees with $ω_{b}$ on the value of $p$ then since they are dual vectors we can find a basis representation:

ω_{b}^{'} - ω_{b} = \sum_{α = 1}^{n} f_{(α)} μ_{b}^{α}

with all $f_{(α)}$ vanishing at p. Hence we get that:

{\tilde{\nabla}}_{a} (ω_{b}^{'} - ω_{b}) - \nabla_{a} (ω_{b}^{'} - ω_{b}) = \sum_{α = 1}^{n} f_{(α)} {{\tilde{\nabla}}_{a} μ_{b}^{(α)} - \nabla_{a} μ_{b}^{(α)}} = 0

That is ${\tilde{\nabla}}_{a} ω_{b}^{'} - \nabla_{a} ω_{b}^{'} = {\tilde{\nabla}}_{a} ω_{b} - \nabla_{a} ω_{b}$ which means that the difference only depends on the value of $ω_{b}$ at $p$ ! Thus we ${\tilde{\nabla}}_{a} - \nabla_{a}$ defines a map from dual vectors at $p$ to tensors of type $(0, 2)$ . Consequently this means that ${\tilde{\nabla}}_{a} - \nabla_{a}$ defines a tensor of type $(1, 2)$ at $p$ , which would be denoted by $C^{c}_{a b}$ . Thus we get:

\nabla_{a} ω_{b} = {\tilde{\nabla}}_{a} ω_{b} - C^{c}_{a b} ω_{c}

Which displays the possible disagreements of the actions of $\nabla_{a}$ and ${\tilde{\nabla}}_{a}$ on dual vector fields. Since derivative operators are torsion free $C^{c}_{a b}$ is symmetric:

C^{c}_{a b} = C^{c}_{b a}

In general for a tensor $T \in T (k, l)$ we find:

\nabla_{a} T^{b_{1} . . . b_{k}}_{c_{1} . . . c_{l}} = {\tilde{\nabla}}_{a} T^{b_{1} . . . b_{k}}_{c_{1} . . . c_{l}} + \sum_{i} C^{b_{i}}_{a d} T^{b_{1} . . . d . . . b_{k}}_{c_{1} . . . c_{l}} + \sum_{j} C^{d}_{a c_{j}} T^{b_{1} . . . b_{k}}_{c_{1} . . . d . . . c_{l}}

Note to clarify the sum pattern.

$d$ replaces the position of $b_{i}$ (or $c_{j}$ )

The most important application of the previous equation that arises from the case where ${\tilde{\nabla}}_{a}$ is an ordinary derivative operator $\partial_{a}$ :

\nabla_{a} t^{b} = \partial_{a} t^{b} + Γ_{a c}^{b} t^{c}

$Γ_{a c}^{b}$ is called the Christoffel symbol. Note this symbol depends on the local coordinate system, if we change the coordinate system the symbol changes and are not related to each other (for example by means of tensor transformations).

Definition

Let $\nabla_{a}$ be a derivative operator, and $C$ is a curve with tangent $t^{a}$ . A vector $v^{a}$ give at each point on a curve is said to be parallelly transported as one moves along the curve if the equation:

t^{a} \nabla_{a} v^{b} = 0

If we have a metric $g_{a b}$ on a manifold, a natural choice of a derivative operator is picked out if we demand the inner product $g_{a b} v^{a} w^{b}$ remain unchanged if parallelly transported along any curve. This imposes the following condition:

\partial_{a} g_{b c} = 0

which defines a unique derivative operator.

Theorem

Let $g_{a b}$ be a metric. Then there exists a unique derivative operator $\nabla_{a}$ satisfying $\nabla_{a} g_{b c} = 0$ .
Of particular note the Christoffel symbol for this unique ordinary derivative opertor is:

Γ_{a b}^{c} = \frac{1}{2} g^{c d} {\partial_{a} g_{b d} + \partial_{b} g_{a d} - \partial_{d} g_{a b}}

thus the coordinate basis components of the Christoffel symbol are:

Γ_{μ υ}^{ρ} = \sum_{σ} \frac{1}{2} g^{ρ σ} {\frac{\partial g_{υ σ}}{\partial x^{μ}} + \frac{\partial g_{μ σ}}{\partial x^{υ}} - \frac{\partial g_{υ υ}}{\partial x^{σ}}}

Curvature

Definition

Let $w_{c}$ be a dual vector and $\nabla_{a}$ be a derivative operator. We can then define $R_{a b c}^{d}$ to be the tensor field such that:

\nabla_{a} \nabla_{b} w_{c} - \nabla_{b} \nabla_{a} w_{c} = R_{a b c}^{d} w_{d}

$R_{a b c}^{d}$ is called the Riemann curvature tensor.

The Riemann curvature tensor is directly related to the failure of a vector to return to its initial value when parallelly transported along a small curve.