The shadow of a collapsing dark star

The shadow of a black hole is usually calculated, either analytically or numerically, on the assumption that the black hole is eternal, i.e., that it has existed for all time. Here we ask the question of how this shadow comes about in the course of time when a black hole is formed by gravitational collapse. To that end we consider a star that is spherically symmetric, dark and non-transparent and we assume that it begins, at some instant of time, to collapse in free fall like a ball of dust. We analytically calculate the dependence on time of the angular radius of the shadow, first for a static observer who is watching the collapse from a certain distance and then for an observer who is falling towards the centre following the collapsing star.     

Self-Similar 2d Euler Solutions with Mixed-Sign Vorticity

We construct a class of self-similar 2d incompressible Euler solutions that have initial vorticity of mixed sign. The boundaries between regions of positive and negative vorticity form algebraic spirals, similar to the Kaden spiral and as opposed to Prandtl’s logarithmic vortex spirals. Also unlike the Prandtl case, spirals are not initially present.     

Spatiotemporal Studies of the One-Dimensional Coordination Polymer [Fe(ebtz)2(C2H5CN)2](BF4)2: Tug of War between the Nitrile Reorientation Versus Crystal Lattice as a Tool for Tuning the Spin Crossover Properties**

Reaction of 1,2-di(tetrazol-2-yl)ethane (ebtz) with Fe(BF₄)₂⋅6 H₂O in different nitriles yields one-dimensional coordination polymers [Fe(ebtz)₂(RCN)₂](BF₄)₂⋅nRCN (n=2 for R=CH₃ ( 1 ) and n=0 for R=C₂H₅ ( 2 ) C₃H₇ ( 3 ), C₃H₅ ( 4 ), CH₂Cl ( 5 )) exhibiting spin crossover (SCO). SCO in 1 and 3 – 5 is complete and occurs above 160 K. In 2 , it is shifted to lower temperatures and is accompanied by wide hysteresis (T_1/2^↓=78 K, T_1/2^↑=123 K) and proceeds extremely slowly. Isothermal (80 K) time-resolved single-crystal X-ray diffraction studies revealed a complex nature for the HS→LS transition in 2 . An initial, slow stage is associated with shrinkage of polymeric chains and with reduction of volume at 77 % (in relation to the difference between cell volumes V_HS−V_LS) whereas only 16 % of iron(II) ions change spin state. In the second stage, an abrupt SCO occurs, associated with breathing of the crystal lattice along the direction of the Fe–nitrile bonds, while the nitriles reorient. HS→LS switching triggered by light (808 nm) reveals the coupling of spin state and nitrile orientation. The importance of this coupling was confirmed by studies of [Fe(ebtz)₂(C₂H₅CN/C₃H₇CN)₂](BF₄)₂ mixed crystals ( 2 a , 2 b ), showing a shift of T_1/2 to higher values and narrowing of the hysteresis loop concomitant with an increase of the fraction of butyronitrile. This increase reduces the capability of nitrile molecules to reorient. Density functional theory (DFT) studies of models of 1 – 5 suggest a particular possibility of 2 to adopt a low (140–145°) value of its Fe-N-C(propionitrile) angle.     

Half-a-century of gamma-ray astrophysics at the Max-Planck Institute for Extraterrestrial Physics

The European Physical Journal H - This is a tutorial for the many-worlds theory by Everett, which includes some of my personal views. It has two main parts. The first main part shows the emergence...