why is the tail of a comet curved? and could you estimate how fast the curvature of the tail could change?

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.

## Wednesday, December 21, 2016

## Wednesday, December 14, 2016

### Poynting-flux and magnetic charges

in 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?

## Wednesday, December 7, 2016

### electrodynamics with non-conserved charges

following up on last week's question: could one construct a theory of electrodynamics with non-conserved charges, i.e. where $\partial^\mu\jmath_\mu\neq 0$? what consequences would that have?

## Wednesday, November 30, 2016

### gravitational fields for non-conservative systems

Einstein's field equation links the Einstein-tensor $G_{\mu\nu}$ to the energy-momentum tensor $T_{\mu\nu}$, which is divergence-free, $\nabla^\mu T_{\mu\nu}=0$. energy-momentum conservation is usually established by considering fields whose Lagrange-density $\mathcal{L}$ does not have an explicit dependence on time or position, leading to the definition of $T_{\mu\nu}$ with the corresponding continuity. in what way would fields $\phi$ with $\mathcal{L}(\phi,\nabla^\mu\phi,x^\mu)$ source a gravitational field?

## Wednesday, November 23, 2016

## Wednesday, November 16, 2016

### dissipation time scale

there are many structures visible in Jupiter's atmosphere. can you estimate a dissipation time scale for the turbulent motion?

## Wednesday, November 9, 2016

### motion or not?

can you show that a cosmological fluid in a FLRW-universe it at rest in all freely falling frames, but shows recession motion?

## Tuesday, November 8, 2016

### implications of a confirmed $\Lambda$

this post is in celebration of $2^{16}$ views!

imagine that future experiments have constrained the dark energy eos-parameters to be $w_0=-1$ and $w_a=0$ at high precision: would that imply that we have confirmed that the covariant divergence $\nabla^\mu g_{\mu\nu}$ of the metric $g_{\mu\nu}$ is zero, implying a metric connection, at least under the FLRW-symmetries?

imagine that future experiments have constrained the dark energy eos-parameters to be $w_0=-1$ and $w_a=0$ at high precision: would that imply that we have confirmed that the covariant divergence $\nabla^\mu g_{\mu\nu}$ of the metric $g_{\mu\nu}$ is zero, implying a metric connection, at least under the FLRW-symmetries?

## Wednesday, November 2, 2016

### cosmology works?

a freely-falling observer experiences the metric $g_{\mu\nu}$ of generated by any energy-momentum distribution $T_{\mu\nu}$ as being of Minkowskian shape $\eta_{\mu\nu}$. the observer would conclude that there is no curvature and consequently, no gravitational fields and that the energy-momentum is vanishing.

but isn't that what we're doing in cosmology? as a freely-falling observer we aim to determine the energy-momentum of all cosmological fluids $T_{\mu\nu}$ expressed in terms of their density parameters $\Omega$ and their equations of state $w$. but how can one measure the energy-momentum-tensor of a gravitating matter distribution with a gravitational experiment?

but isn't that what we're doing in cosmology? as a freely-falling observer we aim to determine the energy-momentum of all cosmological fluids $T_{\mu\nu}$ expressed in terms of their density parameters $\Omega$ and their equations of state $w$. but how can one measure the energy-momentum-tensor of a gravitating matter distribution with a gravitational experiment?

## Wednesday, October 26, 2016

### spherical gravitational wave

is it possible to have a spherically symmetric gravitational wave around an object?

## Wednesday, October 19, 2016

### electricity and gravity

why is there a cosmological constant (even) in the classical Poisson-equation, $\Delta\Phi = 4\pi G\rho + \lambda$, but no such term in the wave-equation of electrodynamics, $\Box A^\mu = 4\pi\jmath^\mu$?

## Wednesday, October 12, 2016

### $CPT$ and relativity

with earlier CQWs and the idea that in
relativity there's no fundamental difference between space and time
coordinates (apart from the signature of the metric), could you
formulate a combined $PT$-invariance of the field equation, where
$\vec{x}\rightarrow-\vec{x}$ and $t\rightarrow-t$? what's the relation to charges of the gravitational field (i.e. masses) and a combined $CPT$-symmetry?

## Wednesday, October 5, 2016

### solutions to general relativity

is it possible to find a solution to Einstein's field equation for an arbitrary distribution of matter? in what way is the problem different from electrodynamics?

## Wednesday, July 27, 2016

### CQW holiday project: Oort's cloud

the solar system is encapsulated in Oorts cloud, a vast cloud of small rocks beyond the orbit of Pluto. can you set up a model for the obscuration of the Sun from this cloud? by how much would the optical magnitude of the Sun decrease?

with that, we wish all our readers a nice summer vacation and hope to see you again in october. if you happen to have an interesting question, please contact us under cosmologyquestionoftheweek"[AT]"gmail"[DOT]"com.

with that, we wish all our readers a nice summer vacation and hope to see you again in october. if you happen to have an interesting question, please contact us under cosmologyquestionoftheweek"[AT]"gmail"[DOT]"com.

## Wednesday, July 20, 2016

### gravity from a variational principle

in classical mechanics and electrodynamics one obtains a second-order equation of motion from squares of the first derivatives, either of the position with time or of the derivatives of the potentials. gravity seems to be different: the Ricci-scalar in the Einstein-Hilbert action contains already second derivatives: can you explain this difference?

## Wednesday, July 13, 2016

### parity of gravitational fields

can you show parity invariance of Einstein's field equation, i.e. the invariance of the equation if $\vec{x}\rightarrow-\vec{x}$? can you think of possible implications for solutions?

## Wednesday, July 6, 2016

### time reversal in Einstein's field equation

can you show that Einstein's field equation is time-reversible? from that, can you say something about the existence of advanced potentials in gravity?

## Wednesday, June 29, 2016

### degree of symmetry

can you show that Schwarzschild black holes and FLRW-models in cosmology have the same degree of symmetry? what's the corresponding quantity in cosmology to the Schwarzschild radius $r_s=2GM/c^2$?

## Wednesday, June 22, 2016

### similarity in cosmology

following up on last week's question: imagine a FLRW-model with only one fluid characterised by a density parameter $\Omega$. could you find in a second model with a different $\Omega$ a time with the same expansion rate and acceleration as the first model?

## Wednesday, June 15, 2016

### similarity in modern physics

can you construct a similarity transform that relates the Schwarzschild-metrics of black holes with different mass to each other? what would be a sensible parameter? would it be possible to formulate an addition theorem for masses and gravitational fields using that relation, which could replace linear superposition for a nonlinear field theory?

## Wednesday, June 8, 2016

### local field equations and nonlocal components

the Poisson-equation of electrostatics (or Newtonian gravity) links the second derivatives of the potential $\Phi$ to the local charge (or matter) density $\rho$, by requiring $\Delta\Phi=4\pi(G)\rho$. this is nice, because $\Delta\Phi$ is the trace of the Hessian $\partial_i\partial_j\Phi$, making it invariant under rotations and therefore generates a spherically symmetric potential. but what fixes the off-diagonal components? would there be a possibility to measure them directly?

## Wednesday, June 1, 2016

### homogeneity of random fields

can you give an argument why the covariance $\langle\delta(k)\delta(k^\prime)^*\rangle$ of a homogeneous random field $\delta(x)$ does not change if one introduces a shift in the coordinates $x\rightarrow x+y$?

## Wednesday, May 25, 2016

### equivalence between gravity and inertia

equivalence between gravitational and inertial forces implies that all inertial forces are identical to gravity and in fact generated by a transformation of the metric field of a matter distribution relative to which one accelerates a frame of reference. can you outline a setup where one replaces e.g. the Coriolis-force by a gravitational field?

## Wednesday, May 18, 2016

### interstellar (the movie)

the astronauts in the movie "interstellar" use rotation of their space ship to generate artificial gravity by means of a centrifugal force. would there be a way to distinguish the two? if inertial and gravitational forces are indistinguishable, how would you explain that the artifical gravity onboard the space ship is in fact actual gravity, i.e. what generates this gravitational field? if space was completely devoid of matter, would that still work?

## Wednesday, May 11, 2016

### underdetermined electrodynamics?

following up on last week's post: is there a similar problem in electrodynamics because one sources 6 components of the field tensor from just 4 components of the charge current density?

## Wednesday, May 4, 2016

### underdetermined general relativity?

the Riemann curvature tensor has in 4 dimension 20 entries, but the field equation of general relativity only links the 10 entries of the Ricci-tensor to the energy-momentum tensor. what determines the remaining 10 entries of the Riemann curvature?

## Wednesday, April 27, 2016

### divergence-free tensors

can you find apart from the Einstein-tensor and the metric tensor new tensors whose covariant divergence vanishes? would they be viable terms for the field equation?

## Wednesday, April 20, 2016

### eternal sunshine

can you estimate the life-time of the Sun with the result from previous week, assuming that the source of energy is fusion? What about Jules Verne's idea that the Sun was powered by meteors hitting the surface or by burning coal (to be found in "Round the Moon")?

## Wednesday, April 13, 2016

### more power, R2!

can you estimate the power density of the Sun, i.e. the amount of power produced by fusion in each cubic meter? what measurements would you need to carry out, and what assumptions do you make?

## Wednesday, April 6, 2016

### moonwalk

can you use mechanical similarity to explain why astronauts on the moon appear to move in slow motion or as if they were under water?

## Wednesday, March 30, 2016

### Zeno and cosmology

can you give an intuitive argument why in a FLRW-cosmology with only a cosmological constant $\Lambda$ distant objects appear at a constant angular size irrespective of their redshift?

## Wednesday, March 23, 2016

### equivalence

if gravity is equivalent to an inertial force, are then all inertial forces gravitational? if so, where does the bulging of the Earth come from, which is usual attributed to centrifugal acceleration due to its rotation?

## Wednesday, March 16, 2016

### $\pi$ in the sky

one needs the value of $\pi$ for quite a number of computations in FLRW-cosmologies: can you list them and say to what accuracy $\pi$ is actually needed?

bonus question: which of the approximations of $\pi$ would work for these particular purposes?

(this post in in celebration of 14.Mar.2016 just 2 days ago)

bonus question: which of the approximations of $\pi$ would work for these particular purposes?

(this post in in celebration of 14.Mar.2016 just 2 days ago)

## Wednesday, March 9, 2016

### scale of homogeneity

can you estimate on what scale one would need to observe cosmological redshift such that peculiar motion is small enough compared to cosmological recession?

## Wednesday, March 2, 2016

### non-ideal fluids

FLRW-cosmologies assume aside homogeneity and isotropy of the metric ideal fluids - but how would non-ideal fluids change FLRW-solutions?

## Wednesday, February 24, 2016

### speed of light and speed of gravity

can you explain why electromagnetic and gravitational waves travel at the same speed?

## Wednesday, February 17, 2016

### negative cosmological constant

could $\Lambda$ be negative? what consequences would that have for the evolution of FLRW-universes and would it be conceivable from a theoretical point of view?

## Wednesday, February 10, 2016

### curved or straight?

please list as many experiments as possible that would allow you to distinguish between curvature and dark energy with an equation of state of $w=-1/3$.

## Wednesday, February 3, 2016

### correlation function

can you show that the integral $\int\mathrm{d}^nx\:C(r)$ of the correlation function $C(r)$ as a function of distance $r$, $r^2=\sum_{i=1}^n x_i^2$, of a random field in $n$ dimensions vanishes?

## Wednesday, January 27, 2016

### Lorentz boosted resistor

last post in the series "relativistically moving electric circuits": a resistor heats up due to an electric current moving through. but could you boost into a frame in which the electrons are at rest, in which case there would be no heat generation? how would you reconcile the two observations with Lorentz-transforms?

## Wednesday, January 20, 2016

### Lorentz boosted capacitor

imagine to observe a charged capacitor from a moving frame of reference. can you explain inituitively how the electric field is Lorentz-transformed?

## Wednesday, January 13, 2016

### Lorentz boosted coil

can you explain how the magnetic field of a moving coil would be transformed into an electric field if the coil is in motion relative to an observer?

## Wednesday, January 6, 2016

### redshift drift

welcome back to CQW in 2016 with a question about redshifting:

can you construct a cosmological model in which the redshift of distant objects stays constant? would that mean that the last-scattering surface of the CMB has a constant distance?

can you construct a cosmological model in which the redshift of distant objects stays constant? would that mean that the last-scattering surface of the CMB has a constant distance?

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