## Wednesday, May 24, 2017

### Pauli-decomposition

any complex $2\times2$-matrix $A_{\mu\nu}$ (for instance, the lensing Jacobian) can be decomposed in terms of 3 Pauli-matrices $\sigma^{(n)}_{\mu\nu}$ and the unit matrix $\sigma^{(0)}_{\mu\nu}$,

A_{\mu\nu} = \sum_{n=0}^3 a_n\sigma^{(n)}_{\mu\nu}.

can you show that the coefficients are given by $a_n = (A_{\mu\nu}\sigma^{(n)}_{\nu\mu})/2$ and that the set of matrices is a complete basis?

## Wednesday, May 17, 2017

### derivative of a constant

the partial derivative $\partial_\mu\phi$ of a constant field $\phi$ is zero... but what about the covariant derivative $\nabla_\mu\phi$?

## Wednesday, May 10, 2017

### relativistic waves

a free massless scalar field $\varphi$ is described by the Lagrange density $\mathcal{L} = \partial_\mu\varphi\partial^\mu\varphi$, leading to an equation of motion $\partial_\mu\partial^\mu\varphi=0$, which is solved by plane waves travelling at the speed of light $c$. can you explain in what way electromagnetic (or gravitational) waves differ from scalar waves in this respect? where do their wave properties and propagation come from?

## Wednesday, May 3, 2017

### Schwarzwald relativity

imagine a traditional clock (perhaps a cuckoo-clock from the black forest, driven by a pendulum) inside a gravitational potential: does it run slow because of relativistic time dilation or fast because the oscillation time scales like $\propto g^{-1/2}$?