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 $\sigma^{(0)}_{\mu\nu}$,
\begin{equation}
A_{\mu\nu} = \sum_{n=0}^3 a_n\sigma^{(n)}_{\mu\nu}.
\end{equation}
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 $\propto g^{-1/2}$?