Wednesday, June 21, 2017

120 - gasp!

compute the number of bits that can be stored by a sphere with the Hubble-radius $\chi_H$ and compare that number to the factor by which a naive estimate of the dark energy density is larger than the actual density $\rho_\mathrm{crit}$... surprised?

Wednesday, June 14, 2017

magic of Pauli-matrices

in the last two weeks CQW asked about the generation of rotations and Lorentz-transforms from the off-diagonal Pauli-matrices $\sigma^{(1)}$ and $\sigma^{(2)}$. what type of transformation would be generated by $\sigma^{(0)}$ and $\sigma^{(3)}$? what types of transformations would be commutative?


Wednesday, June 7, 2017

Pauli and Euclid

can you show that rotations $R_{\mu\nu}$ are generated by the Pauli-matrix $\sigma^{(2)}_{\mu\nu}$ with the angle $i\omega$ as an additive parameter?

Wednesday, May 31, 2017

Pauli and Lorentz

can you show that Lorentz-transforms $\Lambda_{\mu\nu}$ are generated by the Pauli-matrix $\sigma^{(1)}_{\mu\nu}$ with the rapidity $\omega$, $\tanh\omega=\beta$, as an additive parameter?

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?