## Wednesday, October 31, 2012

### gravitationally bound systems

imagine you've assembled a self-gravitating bound equilibrated system from $n$ particles of mass $m$, such that the total mass is $M=nm$ and such that the system has a gravitational binding energy $E$. now you reassemble the system from $p$ times more particles of masses $m/p$ so that the total mass would be identical, $M = np \times m/p=nm$. would the gravitational binding still be $E$?

bonus question: why are the traces $\mathrm{tr}(A^n)$ of a square matrix $A$ taken to the power $n$ invariants? how are they related to the eigenvalues of $A$? would that be a viable way for computing the eigenvalues?

## Wednesday, October 24, 2012

### CMB and SNe

are cosmic microwave background measurements enough to claim spatial flatness and supernova measurements sufficient for claiming accelerated expansion of a FLRW-metric?

bonus question: if a set contains $n$ elements, how many subsets can you form? can you prove this relation with combinatorial arguments and with induction?

## Wednesday, October 17, 2012

### gravitational interaction of photons

why do gravitational interactions of (CMB) photons with the cosmic large-scale structure such as gravitational lensing or the integrated Sachs-Wolfe effect conserve the Planckian shape of the photon energy distribution?

bonus question: what's $\sum_{i=0}^n{n\choose i}$ with the binomial coefficient ${n\choose i}=n!/i!/(n-i)!$?

## Wednesday, October 10, 2012

### dark energy equation of state

why can the equation of state $w$ of dark energy in a scalar field model never be larger than $+1$ and never be more negative than $-1$? are there physical arguments why this would be weird? what would happen to distance measures and to structure growth if $w$ were outside this range?

bonus question: does the Higgs-mechanism in quantum field theory give rise to mass or to inertia?

second bonus question: why is dark matter called dark matter if it does not interact with light? if that was the reason, it should rather be called transparent matter, don't you agree?

## Wednesday, October 3, 2012

### gravity and dimensionality

would the description of gravity as a metric theory work in any number of dimensions?

bonus question: can you show without the rule of de l'Hospital or Taylor expansion that $\sin(x)/x=1$ for $x\rightarrow0$?