tag:blogger.com,1999:blog-19025291575744287302017-07-23T10:12:16.728+02:00cosmology question of the weekCQW is an educational resource for theoretical physics and astrophysics, field theory, relativity and cosmology. we post a new question every wednesday for students to get and for teachers to stay in shape.CQWnoreply@blogger.comBlogger251125tag:blogger.com,1999:blog-1902529157574428730.post-66552819036123689642017-07-19T00:00:00.000+02:002017-07-19T00:00:36.233+02:00horizons in relativityan 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?<br /><br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-70377813933692295792017-07-12T00:00:00.000+02:002017-07-12T00:00:19.787+02:00tidal shear invariantsthe 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?spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-2750048230763076492017-07-05T00:00:00.000+02:002017-07-05T00:00:14.850+02:00clocks in accelerationcan 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?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-47804618724907241552017-06-28T00:00:00.000+02:002017-06-28T00:00:03.051+02:00120 - 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?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-74217285239440842692017-06-21T00:00:00.000+02:002017-06-21T00:00:11.293+02:00120 - 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?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-92099917371151015282017-06-14T00:00:00.000+02:002017-06-14T00:00:32.372+02:00magic of Pauli-matricesin 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?<br /><br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-84097052231293866442017-06-07T00:00:00.000+02:002017-06-07T00:00:54.358+02:00Pauli and Euclidcan 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?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-47775439986095556762017-05-31T00:00:00.000+02:002017-05-31T00:00:05.725+02:00Pauli and Lorentzcan 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?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-11699092686843056802017-05-24T00:00:00.000+02:002017-05-24T00:00:04.133+02:00Pauli-decompositionany 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 $\sigma^{(0)}_{\mu\nu}$,<br />\begin{equation}<br />A_{\mu\nu} = \sum_{n=0}^3 a_n\sigma^{(n)}_{\mu\nu}.<br />\end{equation}<br />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?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-89088775722376573972017-05-17T00:00:00.000+02:002017-05-17T00:00:31.234+02:00derivative of a constantthe partial derivative $\partial_\mu\phi$ of a constant field $\phi$ is zero... but what about the covariant derivative $\nabla_\mu\phi$?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-76750248538448716292017-05-10T00:00:00.000+02:002017-05-10T00:00:08.313+02:00relativistic wavesa 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?spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-50362683930708119842017-05-03T00:00:00.000+02:002017-05-03T00:00:08.631+02:00Schwarzwald relativityimagine 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 $\propto g^{-1/2}$?spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-352049172441115772017-04-26T00:00:00.000+02:002017-04-26T00:00:11.986+02:00quantify curvaturethe curvature can be quantified with the invariant curvature $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ or with the Ricci-scalar $g^{\mu\nu}g^{\rho\sigma}R_{\rho\mu\sigma\nu}$. do they carry the same information? can you think of situations where the Ricci-scalar is zero but the invariant curvature is not?spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-6314796557286748242017-04-23T10:41:00.000+02:002017-04-23T10:41:09.885+02:00duality and the propagation of wavesthe propagation direction $\vec{k}$, the electric field $\vec{E}$ and the magnetic field $\vec{B}$ of an electromagnetic wave form a right-handed system: is the same true if one applies duality, i.e. if one interchanges $\vec{E}\rightarrow \vec{B}$ and $\vec{B}\rightarrow -\vec{E}$?<br /><br />this post is in celebration of $2^{17}$ views.<br /><br />CQWnoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-50761385815182501252017-04-19T00:00:00.000+02:002017-04-19T00:00:00.161+02:00dark energy in the labwould it be possible to determine properties of cosmological fluids in a laboratory experiment?spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-41219242078216583062017-04-12T00:00:00.000+02:002017-04-12T00:00:32.147+02:00Rydberg satelliteswould it be necessary to impose angular momentum quantisation for the orbits of satellites around Earth? what would be the radial distance between successive orbits?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-57390220542780261392017-04-05T00:00:00.000+02:002017-04-05T00:00:01.192+02:00entropy densitycan you estimate the entropy density of the universe and compare to the energy density?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-1183566395183210742017-03-29T00:00:00.000+02:002017-03-29T00:00:08.933+02:00measurement of the Planck-constantcould one use naturally occuring Planck-spectra for determining the Planck-constant? what would be a viable experiment and would the Sun or the CMB provide a better measurement?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-22543164128416721182017-03-22T00:00:00.000+01:002017-03-22T00:00:05.245+01:0040, 20, 10imagine that one measures the 40 components of the Christoffel-symbol by observing geodesic motion: in what way are the 20 components of curvature fixed? and how does that determine the 10 components of the metric?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-23795340213130493582017-03-15T00:00:00.000+01:002017-03-15T00:00:24.750+01:00intuition from the geodesic equationwhat would be your argument why the Christoffel-symbol $\Gamma^\rho_{\mu\nu}$ needs to be symmetric in the lower two indices $\mu$ and $\nu$?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-20250841578443794112017-03-08T00:00:00.000+01:002017-03-08T00:00:10.935+01:00Lorentz force and accelerationplease explain why the 6 entries of the Maxwell field tensor $F_{\mu\nu}$ determine 4 components of 4-acceleration, and how it is possible to infer all 6 entries from experimenting with a charge.<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-25730033552527406512017-03-01T00:00:00.000+01:002017-03-01T00:00:11.147+01:00Christoffel-symbols and geodesic motionwhy does one need 40 entries of the Christoffel symbol to determine the 4 components of 4-acceleration? can one, from observation of geodesic motion, infer all 40 components?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-40688137613332307062017-02-22T00:00:00.000+01:002017-02-22T00:00:29.414+01:00charge asymmetry, part 2if the charge of electrons was not the exact opposite of the charge of protons, the rings of Saturn would act as a current loop, resulting in a magnetic field. could you estimate the magnitude of the magnetic field (in terms of its dipole moment)?spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-73332100840699137052017-02-15T00:00:00.000+01:002017-02-15T00:00:09.236+01:00gravitational field of the hot suncould you estimate by how much the gravitational field of the sun is larger because of the high internal energy compared to simply the rest mass? can one write down a correction factor as a function of temperature $T$?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-72923817033810958572017-02-08T00:00:00.000+01:002017-02-08T00:00:02.621+01:00spin-down of the Crab-pulsarthe Crab-pulsar is a rapidly spinning magnetic neutron star, which induces very large electric fields in the surrounding Crab-nebula, where the resulting electric currents are dissipated by Ohm losses. could you estimate the time-scale of this process and judge if it's a viable model for explaining the spin-down?<br /><br />spirounoreply@blogger.com0