MTH3008 Weekly Problems 8

Original Documents: Problem Sheet / My Handwritten Solutions / [[mth3008 weekly problem sheet 8 solutions.pdf|Provided Solutions]]

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8.1. Christoffel Symbols in Orthogonal Coordinates

Question

The Christoffel symbols of the first kind ${\Gamma }_{ijk}$ can be written in terms of the metric tensor $g_{ij}$ as ${\Gamma }_{ijk}=\frac{1}{2}\left(\frac{\partial g_{ik}}{\partial x^j}+\frac{\partial g_{ij}}{\partial x^k}-\frac{\partial g_{kj}}{\partial x^i}\right).$

This expression simplifies for orthogonal coordinate systems (that is, when $g_{ij}=0$ for $i\ne j$).

Find the simplified expressions for ${\Gamma }_{ijk}$ in the following cases:

1. $i=j=k$;
2. $i=j\ne k$;
3. $i\ne j=k$;
4. $i,j,k$ are all distinct.

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8.2. Christoffel Symbols in Spherical Coordinates

Question

We consider spherical coordinates $(x^1,x^2,x^3)=(r,\phi ,\theta )$, where the metric tensor has components $g_{11}=1,g_{22}=r^2,g_{33}=r^2{\sin }^2\phi ,$ and all other components of $g_{ij}$ are zero.

Find all Christoffel symbols of the first kind ${\Gamma }_{ijk}$ and the corresponding Christoffel symbols of the second kind ${\Gamma }^i{}_{jk}$.

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8.3. Exotic Polar-Type Coordinates in 2D

Question

Consider the 2D coordinates $(x^1,x^2)=(\rho ,\theta )$ in the plane, where the position vector $\mathbf{r}$ is given by $\mathbf{r}={\rho }^2\cos (2\theta ){\mathbf{i}}_1+{\rho }^2\sin (2\theta ){\mathbf{i}}_2,$ with ${\mathbf{i}}_1,{\mathbf{i}}_2$ the usual Cartesian basis vectors in ${\mathbb{R}}^2$.

1. Find the corresponding coordinate basis vectors ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$.
2. Compute the covariant metric tensor $g_{ij}$ and present your answer as a $2\times 2$ matrix. Using that ${\mathbf{e}}_1,{\mathbf{e}}_2$ form an orthogonal basis, determine the contravariant metric tensor $g^{ij}$.
3. Find all Christoffel symbols of the first kind ${\Gamma }_{ijk}$.
4. Find all Christoffel symbols of the second kind ${\Gamma }^i{}_{jk}$.

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8.4. Time-Dependent 3D Coordinate System

Question

Consider the coordinates $(x^1,x^2,x^3)=(x,y,t)$ in ${\mathbb{R}}^3$, where the position vector $\mathbf{r}$ is given by $\mathbf{r}=(x^2-y^2){\mathbf{i}}_1+(2xy){\mathbf{i}}_2+t{\mathbf{i}}_3,$ with ${\mathbf{i}}_1,{\mathbf{i}}_2,{\mathbf{i}}_3$ the usual Cartesian basis vectors.

1. Find the corresponding coordinate basis vectors ${\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3$.
2. Compute the covariant metric tensor $g_{ij}$ and the contravariant metric tensor $g^{ij}$, and give your final answers as $3\times 3$ matrices.
3. Find all Christoffel symbols of the first kind ${\Gamma }_{ijk}$.
4. Find all Christoffel symbols of the second kind ${\Gamma }^i{}_{jk}$.

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8.5. Tensorial Nature of the Covariant Derivative

Question

The transformation law for the Christoffel symbols of the second kind is ${\Gamma }^{i^{\prime }}{}_{j^{\prime }{k}^{\prime }}=L^{i^{\prime }}{}_{\ell }{L}^m{}_{j^{\prime }}{L}^n{}_{k^{\prime }}{\Gamma }^{\ell }{}_{mn}+L^{i^{\prime }}{}_m{L}^n{}_{k^{\prime }}\frac{\partial L^m{}_{j^{\prime }}}{\partial x^n},$ where $L^{i^{\prime }}{}_{\ell }$ and $L^m{}_{j^{\prime }}$ are the components of the coordinate transformation.

Show that the covariant derivative of a covariant vector is a second-rank covariant tensor.

Hint: You may use the identities $L^m{}_{k^{\prime }}=\frac{\partial x^m}{\partial x^{k^{\prime }}}$ and $L^r{}_{j^{\prime }}{L}^{j^{\prime }}{}_{\ell }={\delta }^r{}_{\ell }$ that have already been established.

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