From massive gravity to modified general relativity II

We continue our investigation of massive gravity in the massless limit of vanishing graviton mass. From gauge invariance we derive the most general coupling between scalar matter and gravity. We get further couplings beside the standard coupling to the energy–momentum tensor. On the classical level this leads to a further modification of general relativity.     

Gravitational lensing and rotation curve

Based on the geodesic equation in a static spherically symmetric metric we discuss the rotation curve and gravitational lensing. The rotation curve determines one function in the metric without assuming Einstein’s equations. Then lensing is considered in the weak field approximation of general relativity. From the null geodesics we derive the lensing equation. The gravitational potential U(r) which determines the lensing is directly give by the rotation curve U(r) = −v ²(r). This allows to test general relativity on the scale of galaxies where dark matter is relevant.     

Subwavelength-size solid immersion lens

We report on the fabrication and characterization of nanoscale solid immersion lenses (nano-SILs) with sizes down to a subwavelength range. Submicrometer-scale cylinders fabricated by electron-beam lithography are thermally reflowed to form a spherical shape. Subsequent soft lithography leads to nano-SILs on transparent substrates for optical characterization. The optical characterization is performed using a high-resolution interference microscope with illumination at 642 nm wavelength. The focal spots produced by the nano-SILs show both spot-size reduction and enhanced optical intensity, which are consistent with the immersion effect.     

Classical and quantum chaos for a kicked top

We discuss a top undergoing constant precession around a magnetic field and suffering a periodic sequence of impulsive nonlinear kicks. The squared angular momentum being a constant of the motion the quantum dynamics takes place in a finite dimensional Hilbert space. We find a distinction between regular and irregular behavior for times exceeding the quantum mechanical quasiperiod at which classical behavior, whether chaotic or regular, has died out in quantum means. The degree of level repulsion depends on whether or not the top is endowed with a generalized time reversal invariance.     

Loudness reduction induced by a contralateral tone

The induced reduction in the loudness (ILR) of a weaker tone caused by a preceding stronger tone was measured with both tones in the same ear (ipsilateral ILR) and also in opposite ears (contralateral ILR). The two tones were always equal in duration and were presented repeatedly over several minutes. When the tone duration was 200 ms, for 24 listeners the loudness reduction averaged 11 dB under ipsilateral ILR and 6 dB under contralateral ILR. When the duration was 5 ms, ILR was 8 dB whether ipsilateral or contralateral. For each duration, ipsilateral and contralateral ILR were strongly correlated (r around 0.80).     

Experimental and theoretical characterization of the aromatization, epimerization, and fragmentation reactions of bi-2H-azirin-2-yls prepared from 1,4-diazidobuta-1,3-dienes

1,4‐Diazidobuta‐1,3‐dienes (Z,Z)‐ 10 , 17 , and 21 were photolyzed and thermolyzed to yield the pyridazines 13 , 20 , and 23 , respectively. To explain these aromatic final products, the generation of highly strained bi‐2H‐azirin‐2‐yls 12 , 19 , and 22 and their valence isomerization were postulated. In the case of meso‐ and rac‐ 22 , nearly quantitative formation from diazide 21 , isolation as stable solids, and complete characterization were possible. On the thermolysis of 22 , aromatization to 23 was only a side reaction, whereas equilibration of meso‐ and rac‐ 22 and fragmentation, which led to alkyne 24 and acetonitrile, dominated. Prolonged irradiation of 22 gave mainly the pyrimidine 25 . The change of the configuration at C‐2 of the 2H‐azirine unit was observed not only in the case of bi‐2H‐azirin‐2‐yls 22 but also for simple spirocyclic 2H‐azirines 29 at a relatively low temperature (75 °C). The fragmentation of rac‐ 22 to give alkyne 24 and two molecules of acetonitrile was also studied by high‐level quantum chemical calculations. For a related model system 30 (methyl instead of phenyl groups), two transition states TS‐ 30 – 31 of comparable energy with multiconfigurational electronic states could be localized on the energy hypersurface for this one‐step conversion. The symmetrical transition state complies with the definition of a coarctate mechanism.     

Synthesis of dichloroisoeverninic acid

3,5-Dichloro-4-hydroxy-2-methoxy-6-methyl-benzoic acid 1, the so-called dichloroisoeverninic acid, a food additive¹ resp. a constituent of the orthosomycin antibiotics avilamycin, curamycin, everninomycin, flambamycin and related derivatives,² is synthesized from methyl orsellinate 2 in five steps in 40% overall yield.     

Electron–positron pair production in the external electromagnetic field of colliding relativistic heavy ions

The results concerning the production in peripheral highly relativistic heavy-ion collisions presented in a recent paper by Baltz et al. are rederived in a very straightforward manner. It is shown that the solution of the Dirac equation directly leads to the multiplicity, i.e. to the total number of electron–positron pairs produced by the electromagnetic field of the ions, whereas the calculation of the single pair production probability is much more involved. A critical observation concerns the unsolved problem of seemingly absent Coulomb corrections (Bethe–Maximon corrections) in pair production cross sections. It is shown that neither the inclusion of the vacuum–vacuum amplitude nor the correct interpretation of the solution of the Dirac equation concerning the pair multiplicity is able the explain (from a fundamental point of view) the absence of Coulomb corrections. Therefore the contradiction has to be accounted to the treatment of the high energy limit. Received: 30 November 2001 / Published online: 22 February 2002