can you show that the density inside a sphere which contracts according to a Hubble-law $\upsilon(r)\propto -r$ remains homogeneous? what's the relation between density $\rho$ and time $t$?

we wish all our readers and followers merry x-mas and a happy and successful year 2016. we hope to see you again in january.

## Wednesday, December 23, 2015

## Wednesday, December 16, 2015

### Schwarzschild radius of electrons

if electrons are truly pointlike particles, would it be a problem for the field lines emanating from the charge to cross the Schwarzschild radius?

## Wednesday, December 9, 2015

### replacing Maxwell's equations

Maxwell's equations describe the sourcing and the time evolution of electric and magnetic fields. could you replace them with 1. charge conservation, 2. the Coulomb-law and 3. the Lorentz transformation for fields? would there be something missing?

## Wednesday, December 2, 2015

### gravitational wave interferometers

CQW celebrates its fourth anniversary!

imagine a Michelson-Morley-type interferometer for the detection of gravitational waves: are there directions in which it is insensitive to linearly polarised gravitational waves? what about circularly polarised gravitational waves?

imagine a Michelson-Morley-type interferometer for the detection of gravitational waves: are there directions in which it is insensitive to linearly polarised gravitational waves? what about circularly polarised gravitational waves?

## Wednesday, November 25, 2015

### Hubble nucleus

This week's CQW is a guest post by Phillip Helbig:

Read the question then, without calculating or estimating anything, first make a quick guess as to the result. Then work out the result (an order of magnitude or two is close enough).

Neglect gravity and other types of interactions and imagine the entire observable universe being compressed into a ball (rather like a movie of the expanding universe played in reverse). There are about a hundred billion stars per galaxy on average, and at least a hundred billion galaxies in the observable universe. Or think of the Hubble Deep Field, which has the angular size of a small rice grain held at arm's length, full of galaxies - and this was intentionally chosen because it was apparently empty! The sky is about 25 million times larger.

What is the size of the ball when the density is equal to the density of nuclear matter?

Read the question then, without calculating or estimating anything, first make a quick guess as to the result. Then work out the result (an order of magnitude or two is close enough).

Neglect gravity and other types of interactions and imagine the entire observable universe being compressed into a ball (rather like a movie of the expanding universe played in reverse). There are about a hundred billion stars per galaxy on average, and at least a hundred billion galaxies in the observable universe. Or think of the Hubble Deep Field, which has the angular size of a small rice grain held at arm's length, full of galaxies - and this was intentionally chosen because it was apparently empty! The sky is about 25 million times larger.

What is the size of the ball when the density is equal to the density of nuclear matter?

## Wednesday, November 18, 2015

### let it rip!

what property of dark energy would lead to an infinite scale factor in a finite time in the future? would a cosmological constant be able to do this?

## Wednesday, November 11, 2015

### no big bang

can you construct a cosmological model with an asymptotically constant scale factor in the past, such that the age of the universe would be infinite and no big bang would have occurred? what would be a stable construction?

## Wednesday, November 4, 2015

### vacuum fluctuations

what's wrong with this argument? a moving observer would see quantum fluctuations in e.g. the electromagnetic field blueshifted from the forward and redshifted from the backward direction and would be able to determine the velocity by the amplitude of the motion dipole.

## Wednesday, October 28, 2015

### conditions for cosmic horizons

This week's CQW is a guest post by Phillip Helbig:

An important concept in classical cosmology (for the purposes of this post defined as cosmology within the framework of Friedmann-Lemaître cosmological models) is that of horizons. The particle horizon is the spherical surface from which radiation from the big bang is just reaching us now or, alternatively, due to symmetry, the sphere which is now just reached by radiation emitted from our location at the big bang. This is the same as the "observable universe". The event horizon is the sphere beyond which radiation emitted now will never reach us or, alternatively, due to symmetry, the sphere which will ever just be reached by radiation emitted from our location now. (Note for experts: I am discussing spatial horizons defined at the current cosmic time. In the literature, one also finds discussion of horizons defined in space-time. Somewhat confusing is when the particle horizon is described as a two-dimensional surface in space and the event horizon as a three dimensional surface in space-time. There is no inconsistency, but one must be careful when comparing different papers on this topic.)

The particle horizon exists if the integral

\begin{equation}

\int_{0}^{t_0} \frac{\mathrm{d}t}{R(t)}

\end{equation}

is finite; $t$ is cosmic time ($t=0$ corresponds to the big bang if there is one), $t_0$ is the current epoch, and $R$ is the scale factor.

The event horizon exists if the integral

\begin{equation}

\int_{t_0}^{\infty} \frac{\mathrm{d}t}{R(t)}

\end{equation}

is finite. One can of course work out $R(t)$ for given values of the cosmological parameters $\Omega$ (density parameter) and $\Lambda$ (cosmological constant), compute the integral, and check whether it is finite in order to determine whether the corresponding horizon exists for the cosmological model in question.

However, there are simple qualitative descriptions which give the necessary and sufficient conditions for each type of horizon. What are they?

An important concept in classical cosmology (for the purposes of this post defined as cosmology within the framework of Friedmann-Lemaître cosmological models) is that of horizons. The particle horizon is the spherical surface from which radiation from the big bang is just reaching us now or, alternatively, due to symmetry, the sphere which is now just reached by radiation emitted from our location at the big bang. This is the same as the "observable universe". The event horizon is the sphere beyond which radiation emitted now will never reach us or, alternatively, due to symmetry, the sphere which will ever just be reached by radiation emitted from our location now. (Note for experts: I am discussing spatial horizons defined at the current cosmic time. In the literature, one also finds discussion of horizons defined in space-time. Somewhat confusing is when the particle horizon is described as a two-dimensional surface in space and the event horizon as a three dimensional surface in space-time. There is no inconsistency, but one must be careful when comparing different papers on this topic.)

The particle horizon exists if the integral

\begin{equation}

\int_{0}^{t_0} \frac{\mathrm{d}t}{R(t)}

\end{equation}

is finite; $t$ is cosmic time ($t=0$ corresponds to the big bang if there is one), $t_0$ is the current epoch, and $R$ is the scale factor.

The event horizon exists if the integral

\begin{equation}

\int_{t_0}^{\infty} \frac{\mathrm{d}t}{R(t)}

\end{equation}

is finite. One can of course work out $R(t)$ for given values of the cosmological parameters $\Omega$ (density parameter) and $\Lambda$ (cosmological constant), compute the integral, and check whether it is finite in order to determine whether the corresponding horizon exists for the cosmological model in question.

However, there are simple qualitative descriptions which give the necessary and sufficient conditions for each type of horizon. What are they?

## Wednesday, October 21, 2015

### falling faster

can freely falling bodies overtake each other on the same trajectory? what would Newton and what would Einstein give as a reason?

## Wednesday, October 14, 2015

### cosmologies without beginning

what would be observable consequences in a cosmological model where $a=0$ is not reached in a finite past? could you tweak the cosmological parameters in such a way that there's no CMB or no BBN?

## Wednesday, October 7, 2015

### gravity (the movie)

two astronauts had to abandon the space shuttle and are adrift in orbit around the Earth, holding their hands. what force is acting between the astronauts due to the difference in orbital periods?

## Wednesday, July 29, 2015

### CQW holiday project: gravitational field of a cube

show that the lowest two non-vanishing multipole components of the gravitational field of a cubical planet are the monopole and the octupole. what's the next-lowest non-vanishing multipole?

(a solution without a multipole expansion and the added bonus question on the shape of the surface of a lake on such a planet is discussed in the paper "the gravitational field of a cube" by J.M. Chappell, A. Iqbal, D. Abbott and M. Chappell)

with that, we wish all readers of CQW a nice summer vacation and hope to see you again in october. if you came across an interesting question that would fit our blog, you're invited to appear as a guest author: please contact us for that purpose under cosmologyquestionoftheweek"[AT]"gmail"[DOT]"com.

(a solution without a multipole expansion and the added bonus question on the shape of the surface of a lake on such a planet is discussed in the paper "the gravitational field of a cube" by J.M. Chappell, A. Iqbal, D. Abbott and M. Chappell)

with that, we wish all readers of CQW a nice summer vacation and hope to see you again in october. if you came across an interesting question that would fit our blog, you're invited to appear as a guest author: please contact us for that purpose under cosmologyquestionoftheweek"[AT]"gmail"[DOT]"com.

## Wednesday, July 22, 2015

### temperature measurement

imagine a device which is able to measure the thermal radiation power $S(\nu)$ at a given frequency $\nu$. would it be possible to determine unambiguously the temperature of a blackbody by carrying out measurements at two frequencies $\nu_1$ and $\nu_2$?

## Wednesday, July 15, 2015

### wrong sign in the induction equation

imagine that the induction equation would read $\mathrm{rot}\vec{E} = +\partial_{ct}\vec{B}$ instead of $\mathrm{rot}\vec{E} = -\partial_{ct}\vec{B}$. can you think of as many consequences as possible this would have?

## Wednesday, July 8, 2015

### magnetic charges and energy conservation

imagine to extend Maxwell electrodynamics with a magnetic charge density $\rho_m$. would the resulting set of equations still obey energy-momentum conservation? what changes would this yield in the Poynting-law?

## Wednesday, July 1, 2015

### too many Maxwell equations?

why are there 2 scalar and 2 vectorial Maxwell equations in electrodynamics, which provide 8 relations in total for only two vector fields, i.e. 6 field components? is the system of equations overdetermined?

## Wednesday, June 24, 2015

### critical density in stationary universes

imagine that the scale factor does not change, because we're at the point of maximum extend of a recollapsing universe. would it be possible to define a critical density?

## Wednesday, June 17, 2015

### conservation laws in relativity

why can you formulate energy-momentum conservation in general relativity in a differential form but not in all cases in integral form?

## Wednesday, June 10, 2015

### $\Lambda$ in 2 dimensions

relativity does not allow gravitational effects in two dimensions - but what about a cosmological constant?

## Wednesday, June 3, 2015

### cosmology and chemistry

can you estimate how many stars there are in the observable universe and give your answer in units of Avogadro's number?

## Wednesday, May 27, 2015

### balance dark matter and dark energy

can you compute the scale factor $a$ at which a $\Lambda$CDM-cosmology has a total deceleration of zero? is there an equivalent more general result for dark energy components with a constant dark energy equation of state parameter $w$?

## Wednesday, May 20, 2015

### thermal vs. kinetic SZ-effect

imagine there's turbulent motion in the ICM inside a cluster of galaxies: would that generate a thermal or a kinetic SZ-signal?

## Wednesday, May 13, 2015

### mechanical similarity and the moon landing

the astronauts of Apollo 14 measured the surface gravity of the Moon with a pendulum. of course you could argue that all of that was faked, that the experiment was carried out on Earth and that the footage was shown in slow motion. by how much would everything need to be slowed down? would there be another experiment like the golf ball shot by the same Apollo 14 astronauts or the dropped hammer and feather from Apollo 15 that would convince you?

## Wednesday, May 6, 2015

### FLRW-fluids

ideal cosmological fluids are characterised by their equation of state. can you present an argument why the equation of state is the only free function present in the field equation under the symmetry assumptions of a FLRW-model?

## Wednesday, April 29, 2015

### neutrinos everywhere!

please estimate how many cosmic neutrinos are in the observable Universe. are there more or less photons than neutrinos?

## Wednesday, April 22, 2015

### Lenz-Runge-Laplace vector

the Lenz-Runge-Laplace vector is stationary for the Newton-potential $\Phi\propto 1/r$. is that potential the only one with that property?

## Wednesday, April 15, 2015

### cosmic acceleration maxed out

what equation of state parameter and what density would a cosmic fluid need to have for producing a total deceleration larger than 1 or more negative than -1?

## Wednesday, April 8, 2015

### boosted electric and magnetic fields

electric and magnetic fields seem to transform into each other under Lorentz-transformation: is it possible to boost into a frame in which one type of field disappears completely?

## Wednesday, April 1, 2015

### Darwin-term in the Kepler-problem

planets are not pointlike objects and the total force in a gravitational field would need to be integrated over the planet's volume, i.e. the acceleration of the centre of gravity would need to be corrected by tidal forces. can you estimate how large that correction would be?

## Wednesday, March 25, 2015

### constant radius of the Hubble sphere

can you set up a FLRW-model in which the Hubble sphere has a constant comoving radius? what would be needed for such a construction?

bonus question: for which $n$ is the equality $\mathrm{i}^{n^2}=\pm\mathrm{i}^n$ with $\mathrm{i^2}=-1$ fulfilled?

bonus question: for which $n$ is the equality $\mathrm{i}^{n^2}=\pm\mathrm{i}^n$ with $\mathrm{i^2}=-1$ fulfilled?

## Wednesday, March 18, 2015

### charge asymmetry

if the Earth was electrically charged its orbit would decay due to energy loss by electromagnetic radiation. can you construct an upper limit on the charge asymmetry, i.e. the ratio between electron and proton charge, from the observation that the solar system is very long lived? which planet would provide the best bound?

bonus question: can you show that $(-1)^{n^2}=(-1)^n$?

bonus question: can you show that $(-1)^{n^2}=(-1)^n$?

## Wednesday, March 11, 2015

### mass content of the Universe

can you estimate and quote an error on the estimate of the total mass inside the Hubble-volume? what assumptions do you make and which measurements are needed?

## Wednesday, March 4, 2015

### oscillations in the expansion history

the age of the Universe is estimated from the inverse Hubble constant, i.e. from the current expansion speed. imagine now that the Universe underwent a series of expansion and contraction phases such that we would underestimate the Universe's true age from the Hubble constant. would there still be ways of figuring out the true age?

## Wednesday, February 25, 2015

### recollapsing FLRW-universes

can you construct a FLRW-universe with dark matter and dark energy (with constant equation of state $w$) that recollapses? what's a suitable condition?

## Wednesday, February 18, 2015

## Wednesday, February 11, 2015

### FLRW-universes with no big bang

is it possible to set up a FLRW-model with dark matter and dark energy that has a finite size in the inifinite past, i.e. a universe without a big bang?

## Wednesday, February 4, 2015

### ambiguity in the Planck-units?

the Planck-units are defined using the great constants of Nature: $c$ from relativity, $\hbar$ from quantum mechanics, $k_B$ from thermodynamics and finally $G$ as the gravitational coupling constant. but gravity brings a long a second coupling constant, the cosmological constant $\Lambda$, which now makes the Planck system of units overdetermined: is there a way to resolve this?

## Wednesday, January 28, 2015

### cosmology in SI-units

can you formulate the speed and acceleration of a FLRW-universe in human-understandable units? what would be your favourite choice?

## Wednesday, January 21, 2015

### Einstein's grandfather clock

the Einstein clock is a set of parallel mirrors with a flash of light bouncing between the mirrors serving as a pendulum. it is a particularly nice way of illustrating time-dilation in special relativity. how would the path of the flash of light look like if the Einstein clock was inside a gravitational field (or was accelerated)?

## Wednesday, January 14, 2015

### fine structure for Keplerium

we wish all CQW-readers a happy New Year 2015:

Keplerium is a gravitationally bound atom with angular momentum quantisation identical to the Bohr-model for atoms. can you draw analogies between atomic fine structure and gravitomagnetic effects arising in weak-field relativity?

Keplerium is a gravitationally bound atom with angular momentum quantisation identical to the Bohr-model for atoms. can you draw analogies between atomic fine structure and gravitomagnetic effects arising in weak-field relativity?

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