tag:blogger.com,1999:blog-19025291575744287302017-05-24T00:00:03.987+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.comBlogger243125tag: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.com0tag:blogger.com,1999:blog-1902529157574428730.post-65814400275539289682017-02-01T00:00:00.000+01:002017-02-01T00:00:18.081+01:00electromagnetic dualitycould you provide an argument (both for the equations of motion and the Lagrange-density) why in empty space electrodynamics is invariant under the replacement $\vec{E}\rightarrow\vec{B}$ and $\vec{B}\rightarrow -\vec{E}$?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-5570595469324604252017-01-25T00:00:00.000+01:002017-01-25T00:00:01.069+01:00gravity and retardationthe field equation of general relativity is invariant under time-reversal (and parity inversion). in what way is retardation of the gravitational field generated by a matter distribution is preferred over advanced potentials?<br /><br />spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-13173733597821131742017-01-21T19:07:00.001+01:002017-01-21T19:07:09.294+01:00forces in relativityrelativistic forces are always velocity dependent but velocity dependent forces are not necessarily relativistic... is that true?<br /><br />this post is in celebration of $10^5$ views!<br /><br />CQWnoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-55209634933262075592017-01-18T00:00:00.000+01:002017-01-18T00:00:11.520+01:00very heavy and very lightcan you explain why the ratio between the Hubble mass (i.e. the mass inside the Hubble volume today for a critical universe) and the Planck mass is about $10^{60}$? why are stars roughly in the middle (on a logarithmic scale)? or even more puzzling: why's the ratio between Hubble and Planck-mass about equal to the ratio between stellar masses and the masses of nucleons?<br /><br />spirounoreply@blogger.com2tag:blogger.com,1999:blog-1902529157574428730.post-70319455695039085262017-01-11T00:00:00.000+01:002017-01-11T00:00:13.435+01:00forces on dust particlescan you compute the ratio between radiation pressure and gravity acting on a dust particle in the solar system? could one relate the two forces to the mass of the Sun and establish a minimal mass from which on there would be no dust and hence no planetary system?<br /><br />we wish all CQW-readers a successful year 2017 full of cosmological discoveries.spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-91573604502029271842017-01-04T00:00:00.000+01:002017-01-04T00:00:03.668+01:00funny way of writing Maxwell's equationscan you reformulate the Maxwell-equations in terms of the complex-valued field $\vec{X} = \vec{E}+\mathrm{i}\vec{B}$? would there be potentials $\Phi$ and $\vec{A}$ for $\vec{X}$, and can one even find a covariant formulation?<br /><br />CQWnoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-11983794070505792742016-12-21T00:00:00.000+01:002016-12-21T00:00:29.656+01:00comets and their tailswhy is the tail of a comet curved? and could you estimate how fast the curvature of the tail could change?<br /><br />with this very christmassy question we wish all followers of CQW a merry x-mas and a happy and successful year 2017. we hope to see you again in january.spirounoreply@blogger.com0tag:blogger.com,1999:blog-1902529157574428730.post-84925212571146962142016-12-14T00:00:00.000+01:002016-12-14T00:00:20.639+01:00Poynting-flux and magnetic chargesin the derivation of the Poynting-theorem one needs to substitute a term $\vec{B}\mathrm{div}\vec{B}$, which is equal to zero due to the fact that the magnetic field is purely rotational, $\mathrm{div}\vec{B} = 0$. insisting to introduce magnetic charges by requiring $\mathrm{div}\vec{B} = 4\pi\rho_\mathrm{mag}$, would that lead to a violation of energy-momentum conservation? are there other arguments, e.g. from the covariance of electrodynamics, why this is not possible?<br /><br />spirounoreply@blogger.com0