6.1. Differential Geometry#

References
Basics

Let \(\mathbf{X}(q^k)\) a parametrization of points in a \(n\)-dimensional space, \(k = 1:n\).

Natural basis, \(\mathbf{b}_k := \frac{\partial \mathbf{X}}{\partial q^k}\). Vectors of the natural basis can be used to write vectors and tensor fields as a linear combination of them, or in their components w.r.t. that basis

\[\mathbf{v} = v^k \mathbf{b}_k \quad , \quad \mathbf{A} = A^{kl} \mathbf{b}_k \otimes \mathbf{b}_l \ .\]

Metric tensor, The covariant compoenents are \(\mathbf{b}_k \cdot \mathbf{b}_l =: g_{kl}\). Some properties: \(g_{k}^{\ l} = \delta_{k}^{l}\), \(g_{ab} g^{bc} = \delta_{a}^{c}\).

Contravariant basis, \(\{ \mathbf{b}^k \}\), s.t. \(\mathbf{b}_k \cdot \mathbf{b}^l = \delta_k^l\). It’s easy to prove that \(\mathbf{b}_k = g_{kl} \mathbf{b}^l\) (just scalar product with \(\mathbf{b}^j\)…), and the law for raising or lower indices holds, \(A^{ij} = g^{ik} A_{k}^{\ j}\),…

Derivatives of the basis vectors. The components of the derivative \(\frac{\partial \mathbf{b}_i}{\partial q^k} = \Gamma_{ik}^l\mathbf{b}_l\) are defined as the Christoffel symbols of the second type. As these are second-order derivatives, for Schwartz theorem about mixed partial derivatives, the symmetry \(\Gamma_{ij}^k = \Gamma_{ji}^k\) immediately follows. The derivative of the vectors of the contravariant basis follows from

\[\begin{split}\begin{aligned} 0 & = \partial_{k} \left( \mathbf{b}_l \cdot \mathbf{b}^m \right) = \\ & = \partial_{k} \mathbf{b}_l \cdot \mathbf{b}^m + \mathbf{b}_l \cdot \partial_k \mathbf{b}^m = \\ & = \Gamma_{kl}^{n} \underbrace{ \mathbf{b}_n \cdot \mathbf{b}^m}_{ = \delta_{n}^m} + \mathbf{b}_l \cdot \partial_k \mathbf{b}^m \ , \end{aligned}\end{split}\]

and thus \(\partial_k \mathbf{b}^m = - \Gamma_{kl}^{m} \mathbf{b}^l\).

Derivatives of the metric tensor. Using the definition of the covariant components of the metric tensor and the derivatives of the vectors of the natural basis, it’s easy to prove

\[\partial_c g_{ab} = \Gamma_{ac}^{d} g_{db} + \Gamma_{bc}^{d} g_{ad}\]

Using the property \(g_{ab} g^{bc} = \delta_{a^c}\), from its derivatives \(\partial_d\),

\[\partial_d g_{ab} g^{bc} + g_{ab} \partial_d g^{bc} = 0 \ ,\]

it follows that \(\partial_d g^{ec} = - g^{ea} \partial_d g_{ab} g^{bc}\).

Relations between the Christoffel symbols and the derivatives of the metric tensor.

\[\begin{split}\begin{aligned} \partial_c g_{ab} & = \Gamma_{ac}^{d} g_{db} + \Gamma_{bc}^{d} g_{ad} \\ \partial_a g_{bc} & = \Gamma_{ba}^{d} g_{dc} + \Gamma_{ca}^{d} g_{bd} \\ \partial_b g_{ca} & = \Gamma_{cb}^{d} g_{da} + \Gamma_{ab}^{d} g_{cd} \\ \end{aligned}\end{split}\]

so that, summing the first 2 equations and subracting the last one,

(6.1)#\[\Gamma_{ac}^{b} = \frac{1}{2} g^{bd} \left( \partial_c g_{ad} + \partial_a g_{dc} - \partial_{d} g_{ac} \right)\]

6.1.1. Gradient, covariant derivative and directional derivative#

Gradient of a scalar field

\[\nabla f = \mathbf{b}^k \partial_k f \ .\]

Gradient of a vector field

\[\nabla \mathbf{v} = \mathbf{b}^k \mathbf{b}_i \left( \partial_k v^i + \Gamma_{kl}^{i} v^l \right) \ .\]
Proof
\[\begin{split}\begin{aligned} \nabla \mathbf{v} & = \mathbf{b}^k \dfrac{\partial}{\partial q^k} \left( v^i \mathbf{b}_i \right) = \\ & = \mathbf{b}^k \mathbf{b}_i \partial_k v^i + \mathbf{b}^k \mathbf{b}_l \Gamma_{ik}^{l} v^i = \\ & = \mathbf{b}^k \mathbf{b}_i \left( \partial_k v^i + \Gamma_{kl}^{i} v^l \right) \ . \end{aligned}\end{split}\]

Gradient of a 2-nd order tensor field

\[\nabla \mathbf{A} = \mathbf{b}^k \mathbf{b}_i \mathbf{b}_j \left( \partial_k A^{ij} + \Gamma_{kl}^{i} A^{lj} + \Gamma_{kl}^{j} A^{il} \right) \ = \mathbf{b}^k \mathbf{b}_i \mathbf{b}_j \nabla_k A^{ij}.\]
Proof

…

6.1.2. Curvature Tensor#

todo. Meaning of this definition

…

Definition through the action on an arbitrary vector field, whose components are

\[\left\{ \mathbf{v} \cdot \mathbf{R} \right\}_{\sigma \eta \nu} = v_{\xi} R^{\xi}_{\ \ \sigma \eta \nu} = \left( \nabla_{\eta} \nabla_{\nu} - \nabla_{\nu} \nabla_{\eta} \right) v_{\sigma}\]
First term
\[\begin{split}\begin{aligned} \nabla \nabla \mathbf{v} & = \mathbf{b}^{\eta} \partial_{\eta} \left[ \mathbf{b}^{\nu} \partial_{\nu} \left( v_{\sigma} \mathbf{b}^{\sigma} \right) \right] = \\ & = \mathbf{b}^{\eta} \partial_{\eta} \left[ \mathbf{b}^{\nu} \mathbf{b}^{\sigma} \left( \partial_{\nu} v_{\sigma} - \Gamma_{\nu \sigma}^{\xi} v_{\xi} \right) \right] = \\ & = \mathbf{b}^{\eta} \mathbf{b}^{\nu} \mathbf{b}^{\sigma} \left[ - \Gamma_{\eta \nu}^{\mu} (\dots)_{\mu \sigma} - \Gamma_{\eta \sigma}^{\mu} (\dots)_{\nu \mu} + \partial_{\eta \nu} v_{\sigma} - \partial_{\eta} \left( \Gamma_{\nu \sigma}^{\xi} v_{\xi} \right) \right] = \\ & = \mathbf{b}^{\eta} \mathbf{b}^{\nu} \mathbf{b}^{\sigma} \left[ - \Gamma_{\eta \nu}^{\mu} \left( \partial_{\mu} v_{\sigma} - \Gamma_{\mu \sigma}^{\xi} v_{\xi} \right) - \Gamma_{\eta \sigma}^{\mu} \left( \partial_{\nu} v_{\mu} - \Gamma_{\mu \nu}^{\xi} v_{\xi} \right) + \partial_{\eta \nu} v_{\sigma} - \partial_{\eta} \Gamma_{\nu \sigma}^{\xi} v_{\xi} - \Gamma_{\nu \sigma}^{\xi} \partial_{\eta} v_{\xi} \right] = \\ & = \mathbf{b}^{\eta} \mathbf{b}^{\nu} \mathbf{b}^{\sigma} \nabla_{\eta \nu} v_{\sigma} \ . \end{aligned}\end{split}\]
\(\ \nabla_{\sigma \eta} v_{\nu} - \nabla_{\eta \sigma} v_{\nu} \ \)

Exploiting symmetry properties of Christoffel symbols,

todo!!! change indices to match the definition of the curvature tensor

\[\begin{split}\begin{aligned} \nabla_{\sigma \eta} v_{\nu} - \nabla_{\eta \sigma} v_{\nu} & = - \underbrace{\Gamma_{\sigma \eta}^{\xi} \partial_{\xi} v_{\nu}}_{1} + \underbrace{\Gamma_{\sigma \eta}^{\mu} \Gamma_{\mu \nu}^{\xi} v_{\xi}}_{2} - \underbrace{\Gamma_{\sigma \nu }^{\xi} \partial_{\eta} v_{\xi}}_{3} + \Gamma_{\sigma \nu }^{\mu} \Gamma_{\mu \eta}^{\xi} v_{\xi} + \underbrace{\partial_{\sigma \eta} v_{\nu}}_{4} - \partial_{\sigma} \Gamma_{\eta \nu}^{\xi} v_{\xi} - \underbrace{\Gamma_{\eta \nu}^{\xi} \partial_{\sigma} v_{\xi}}_{5} + \\ & - \left[ - \underbrace{\Gamma_{\eta \sigma}^{\xi} \partial_{\xi} v_{\nu}}_{1} + \underbrace{\Gamma_{\eta \sigma}^{\mu} \Gamma_{\mu \nu}^{\xi} v_{\xi}}_{2} - \underbrace{\Gamma_{\eta \nu }^{\xi} \partial_{\sigma} v_{\xi}}_{5} + \Gamma_{\eta \nu }^{\mu} \Gamma_{\mu \sigma}^{\xi} v_{\xi} + \underbrace{\partial_{\eta \sigma} v_{\nu}}_{4} - \partial_{\eta} \Gamma_{\sigma \nu}^{\xi} v_{\xi} - \underbrace{\Gamma_{\sigma \nu}^{\xi} \partial_{\eta} v_{\xi} }_{3} \right] = \\ & = \left[ \partial_{\eta} \Gamma_{\sigma \nu}^{\xi} - \partial_{\sigma} \Gamma_{\eta \nu}^{\xi} + \Gamma_{\sigma \nu}^{\mu} \Gamma_{\mu \eta}^{\xi} - \Gamma_{\eta \nu}^{\mu} \Gamma_{\mu \sigma}^{\xi} \right] v_{\xi} \ . \end{aligned}\end{split}\]

Thus, it follows that

\[R^{\xi}_{\ \ \sigma \eta \nu} = \partial_{\eta} \Gamma_{\sigma \nu}^{\xi} - \partial_{\nu} \Gamma_{\eta \sigma}^{\xi} + \Gamma_{\sigma \nu}^{\mu} \Gamma_{\mu \eta}^{\xi} - \Gamma_{\eta \sigma}^{\mu} \Gamma_{\mu \nu}^{\xi}\]
\[R_{\phi \sigma \eta \nu} = g_{\phi \xi} R^{\xi}_{\ \ \sigma \eta \nu} = g_{\phi \xi} \left( \partial_{\eta} \Gamma_{\sigma \nu}^{\xi} - \partial_{\nu} \Gamma_{\eta \sigma}^{\xi} + \Gamma_{\sigma \nu}^{\mu} \Gamma_{\mu \eta}^{\xi} - \Gamma_{\eta \sigma}^{\mu} \Gamma_{\mu \nu}^{\xi}\right)\]

6.1.2.1. Ricci’s tensor#

todo Meaning

Ricci’s tensor is defined as the contraction of the first and third indices of the curvature tensor with the metric tensor,

\[R_{\sigma \nu} := g^{\alpha \beta} R_{\alpha \sigma \beta \nu} = \underbrace{g^{\alpha \beta} g_{\alpha \gamma}}_{\delta_{\gamma}^{\beta}} R^{\gamma}_{\ \ \sigma \beta \nu} = R^{\mu}_{\ \ \sigma \mu \nu}\]
\[R_{\sigma \nu} = \partial_{\xi} \Gamma^{\xi}_{\sigma \nu} - \partial_{\nu} \Gamma^{\xi}_{\xi \sigma} + \Gamma_{\sigma \nu}^{\mu} \Gamma_{\mu \xi}^{\xi} - \Gamma_{\xi \nu}^{\mu} \Gamma^{\xi}_{\mu \sigma}\]

6.1.2.2. Curvature scalar#

todo Meaning

\[R := g^{\sigma \nu} R_{\sigma \nu} = R^{\sigma}_{\ \ \sigma} \ .\]
\[R = g^{\sigma \nu} R_{\sigma \nu} = g^{\sigma \nu} \left( \partial_{\xi} \Gamma^{\xi}_{\sigma \nu} - \partial_{\nu} \Gamma^{\xi}_{\xi \sigma} + \Gamma_{\sigma \nu}^{\mu} \Gamma_{\mu \xi}^{\xi} - \Gamma_{\xi \nu}^{\mu} \Gamma^{\xi}_{\mu \sigma} \right)\]

6.1.2.3. Properties#

  • \(R_{abcd} = - R_{abdc}\)

Proof

By direct inspection of the expression of the components of the curvature tensor,

\[R_{\phi \sigma \eta \nu} = g_{\phi \xi} R^{\xi}_{\ \ \sigma \eta \nu} = g_{\phi \xi} \left( \partial_{\eta} \Gamma_{\sigma \nu}^{\xi} - \partial_{\nu} \Gamma_{\eta \sigma}^{\xi} + \Gamma_{\sigma \nu}^{\mu} \Gamma_{\mu \eta}^{\xi} - \Gamma_{\eta \sigma}^{\mu} \Gamma_{\mu \nu}^{\xi}\right)\]

and the symmetry of the Christoffel symbols, \(\Gamma_{ab}^c = \Gamma_{ba}^c\).

  • \(R_{abcd} = - R_{bacd}\)

  • \(R_{abcd} + R_{acdb} + R_{adbc} = 0\)

  • \(R_{abcd} = R_{cdab}\)

  • \(R_{abcd;e} + R_{abde;c} + R_{abec;d} = 0\) (Second Bianchi identity)

Proof
\[\begin{split}\begin{aligned} R_{\phi \sigma \eta \nu; \chi} & = \left( g_{\phi \xi} R^{\xi}_{\ \ \sigma \eta \nu} \right)_{;\chi} = \\ & = g_{\phi \xi} \left( \partial_{\eta} \Gamma_{\sigma \nu}^{\xi} - \partial_{\nu} \Gamma_{\eta \sigma}^{\xi} + \Gamma_{\sigma \nu}^{\mu} \Gamma_{\mu \eta}^{\xi} - \Gamma_{\eta \sigma}^{\mu} \Gamma_{\mu \nu}^{\xi}\right)_{; \chi} = \\ & = \end{aligned}\end{split}\]

todo

6.1.2.3.1. Divergence of Ricci’s tensor#

\[ \nabla_{\mu} R^{\mu}_{\ \ \nu} = \frac{1}{2} \nabla_{\nu} R \ . \]
Proof
\[\begin{aligned} R_{\sigma \nu} = R^{\xi}_{\ \ \sigma \xi \nu} = \partial_{\xi} \Gamma_{\sigma \nu}^{\xi} - \partial_{\sigma} \Gamma_{\xi \nu}^{\xi} + \Gamma_{\sigma \nu}^{\mu} \Gamma_{\mu \xi}^{\xi} - \Gamma_{\xi \nu}^{\mu} \Gamma_{\mu \sigma}^{\xi} \end{aligned}\]
\[R^{\mu}_{\ \ \nu} = g^{\mu \sigma} R_{\sigma \nu}\]
\[\begin{split}\begin{aligned} \nabla_{\mu} R^{\mu}_{\ \ \nu} & = \partial_\mu R^{\mu}_{\ \ \nu} + \Gamma_{\mu \sigma}^{\mu} R^{\sigma}_{\ \ \nu} - \Gamma_{\sigma \nu}^{\mu} R^{\sigma}_{\ \ \mu} = \\ & = \\ \end{aligned}\end{split}\]