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?

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 $\propto g^{-1/2}$?

Wednesday, April 26, 2017

quantify curvature

the 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?

Sunday, April 23, 2017

duality and the propagation of waves

the 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 $\vec{B}\rightarrow -\vec{E}$?

this post is in celebration of $2^{17}$ views.

Wednesday, April 19, 2017

dark energy in the lab

would it be possible to determine properties of cosmological fluids in a laboratory experiment?

Wednesday, April 12, 2017

Rydberg satellites

would it be necessary to impose angular momentum quantisation for the orbits of satellites around Earth? what would be the radial distance between successive orbits?

Wednesday, April 5, 2017

entropy density

can you estimate the entropy density of the universe and compare to the energy density?

Wednesday, March 29, 2017

measurement of the Planck-constant

could 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?

Wednesday, March 22, 2017

40, 20, 10

imagine 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?

Wednesday, March 15, 2017

intuition from the geodesic equation

what would be your argument why the Christoffel-symbol $\Gamma^\rho_{\mu\nu}$ needs to be symmetric in the lower two indices $\mu$ and $\nu$?

Wednesday, March 8, 2017

Lorentz force and acceleration

please 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.

Wednesday, March 1, 2017

Christoffel-symbols and geodesic motion

why 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?

Wednesday, February 22, 2017

charge asymmetry, part 2

if 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)?

Wednesday, February 15, 2017

gravitational field of the hot sun

could 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 $T$?

Wednesday, February 8, 2017

spin-down of the Crab-pulsar

the 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?

Wednesday, February 1, 2017

electromagnetic duality

could 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 $\vec{B}\rightarrow -\vec{E}$?

Wednesday, January 25, 2017

gravity and retardation

the 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?

Saturday, January 21, 2017

forces in relativity

relativistic forces are always velocity dependent but velocity dependent forces are not necessarily relativistic... is that true?

this post is in celebration of $10^5$ views!

Wednesday, January 18, 2017

very heavy and very light

can 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?

Wednesday, January 11, 2017

forces on dust particles

can 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?

we wish all CQW-readers a successful year 2017 full of cosmological discoveries.

Wednesday, January 4, 2017

funny way of writing Maxwell's equations

can 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?