Wednesday, July 19, 2017

horizons in relativity

an accelerated object disappears behind a horizon as well as an object at the Schwarzschild radius of a black hole: is there a difference between the two cases? is there an analogous effect in cosmology?

Wednesday, July 12, 2017

tidal shear invariants

the gravitational shear field could be quantified by the trace $\sum_{i}\partial^2_{ii}\Phi$, which is proportional to the density $\rho$ due to Poisson's equation $\Delta\Phi=4\pi G\rho$ or by the quantity $\sum_{ij}\partial^2_{ij}\Phi\partial^2_{ji}\Phi$. in what way do the two quantifiers differ? can you construct more of them? what would be their relation to the Ricci-scalar $R$ or the invariant curvature $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ in general relativity?

Wednesday, July 5, 2017

clocks in acceleration

can you explain how different types of clocks (string pendulum, spring pendulum, quartz oscillator, $LC$-circuit, atomic clock, Einstein-clock) are affected by an accelerated frame of reference? how is symmetry established between acceleration of the clock and acceleration of the observer, and in what way are the situations equivalent to a gravitational field?

Wednesday, June 28, 2017

120 - gasp! gasp!

the gravitational entropy $S$ in a FLRW-universe can be estimated to be proportional to the ratio between the Hubble-length and the Planck-length, all squared. can you write down an expression for $S$ only in terms of the constants of Nature?

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?