Constraints on Area Variables in Regge Calculus

We describe a general method of obtaining the constraints between area variables in one approach to area Regge calculus, and illustrate it with a simple example. The simplicial complex is the simplest tessellation of the 4-sphere. The number of independent constraints on the variations of the triangle areas is shown to equal the difference between the numbers of triangles and edges, and a general method of choosing independent constraints is described. The constraints chosen by using our method are shown to imply the Regge equations of motion in our example.Comment: Typographical errors correcte

Rosette Globulettes and Shells in the Infrared

Tiny, dense clumps of sub-solar mass called globulettes form in giant galactic HII regions. The young central clusters compress the surrounding molecular shells which break up into clumps, filaments, and elephant trunks that interact with UV light from the central OB stars. We study the nature of the infrared emission and extinction in the shell and globulettes in the Rosette Nebula (RN) and search for associated newborn stars. We imaged the northwestern quadrant of the RN in the near-infrared (NIR) through JHKs and narrow-band H2 1-0 S(1), Pbeta and continuum filters. NIR images were used to study the surface brightness of the globulettes and associated bright rims. NIR photometry was used to create an extinction map and to search for NIR excess objects. Archival images from Spitzer IRAC and MIPS 24 and Herschel PACS observations were used to further study the region and its stellar population and to examine the structure of the shell and trunks. The globulettes and elephant trunks have bright rims in the Ks band on the sides facing the central cluster. Analysis of 21 globulettes where surface brightness in the H2 1-0 S(1) line is detected shows that about a third of the surface brightness observed in Ks is due to this line: the observed average of the H2/Ks surface brightness is 0.26+-0.02 in the globulettes cores and 0.30+-0.01 in the rims. The estimated H2 1-0 S(1) surface brightness of the rims is 3-8*10^{-8} Wm^{-2}sr^{-1}um^{-1}. The H2/Ks surface brightness ratio supports fluorescence as the H2 excitation mechanism. The globulettes have number densities of n(H2)~10^{-4} cm^{-3} or higher. We confirm the results from previous optical and CO surveys that the larger globulettes contain very dense cores and dense envelopes, and that their masses are sub-solar. Two NIR protostellar objects were found in an elephant trunk and one in the most massive globulette in our study. (abridged)Comment: Accepted for publication in A&A. 24 pages, 27 figures. JHKs photometry will be available electronicall

Rosette nebula globules: Seahorse giving birth to a star

The Rosette Nebula is an HII region ionized mainly by the stellar cluster NGC 2244. Elephant trunks, globules, and globulettes are seen at the interface where the HII region and the surrounding molecular shell meet. We have observed a field in the northwestern part of the Rosette Nebula where we study the small globules protruding from the shell. Our aim is to measure their properties and study their star formation history in continuation of our earlier study of the features of the region. We imaged the region in broadband near-infrared (NIR) JsHKs filters and narrowband H2 1-0 S(1), P$\beta$, and continuum filters using the SOFI camera at the ESO/NTT. The imaging was used to study the stellar population and surface brightness, create visual extinction maps, and locate star formation. Mid-infrared (MIR) Spitzer IRAC and WISE and optical NOT images were used to further study the star formation and the structure of the globules. The NIR and MIR observations indicate an outflow, which is confirmed with CO observations made with APEX. The globules have mean number densities of ~$4.6\times10^4 \rm cm^{-3}$. P$\beta$ is seen in absorption in the cores of the globules where we measure visual extinctions of 11-16 mag. The shell and the globules have bright rims in the observed bands. In the Ks band 20 to 40% of the emission is due to fluorescent emission in the 2.12 $\mu$m H2 line similar to the tiny dense globulettes we studied earlier in a nearby region. We identify several stellar NIR excess candidates and four of them are also detected in the Spitzer IRAC 8.0 $\mu$m image and studied further. We find an outflow with a cavity wall bright in the 2.124 $\mu$m H2 line and at 8.0 $\mu$m in one of the globules. The outflow originates from a Class I young stellar object (YSO) embedded deep inside the globule. An H$\alpha$ image suggests the YSO drives a possible parsec-scale outflow. (abridged)Comment: 20 pages, 19 figures, accepted for publication in Astronomy and Astrophysics, figures reduced for astro-p

Mass and motion of globulettes in the Rosette Nebula

We have investigated tiny molecular clumps in the Rosette Nebula. Radio observations were made of molecular line emission from 16 globulettes identified in a previous optical survey. In addtion, we collected images in the NIR broad-band JHKs and narrow-band Paschen beta and H2. Ten objects, for which we collected information from several transitions in 12CO and 13CO were modelled using a spherically symmetric model. The best fit to observed line ratios and intensities was obtained by assuming a model composed of a cool and dense centre and warm and dense surface layer. The average masses derived range from about 50 to 500 Jupiter masses, which is similar to earlier estimates based on extinction measures. The globulettes selected are dense, with very thin layers of fluorescent H2 emission. The NIR data shows that several globulettes are very opaque and contain dense cores. Because of the high density encountered already at the surface, the rims become thin, as evidenced by our P beta images. We conclude that the entire complex of shells, elephant trunks, and globulettes in the northern part of the nebula is expanding with nearly the same velocity of ~22 km/s, and with a very small spread in velocity among the globulettes. Some globulettes are in the process of detaching from elephant trunks and shells, while other more isolated objects must have detached long ago and are lagging behind in the general expansion of the molecular shell. The suggestion that some globulettes might collapse to form planetary-mass objects or brown dwarfs is strengthened by our finding of dense cores in several objects.Comment: 15 pages, 15 figures Astronomy and Astrophysics 201

Microscopic Black Hole Pairs in Highly-Excited States

We consider the quantum mechanics of a system consisting of two identical, Planck-size Schwarzschild black holes revolving around their common center of mass. We find that even in a very highly-excited state such a system has very sharp, discrete energy eigenstates, and the system performs very rapid transitions from a one stationary state to another. For instance, when the system is in the 100th excited state, the life times of the energy eigenstates are of the order of $10^{-30}$ s, and the energies of gravitons released in transitions between nearby states are of the order of $10^{22}$ eV.Comment: 22 pages, 3 figures, uses RevTe

Spacetime Foam Model of the Schwarzschild Horizon

We consider a spacetime foam model of the Schwarzschild horizon, where the horizon consists of Planck size black holes. According to our model the entropy of the Schwarzschild black hole is proportional to the area of its event horizon. It is possible to express geometrical arguments to the effect that the constant of proportionality is, in natural units, equal to one quarter.Comment: 16 pages, 2 figures, improved and extended version with some significant changes. Accepted for publication in Phys.Rev.