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 $x\rightarrow0$?