Fermat Principle in Finsler Spacetimes

It is shown that, on a manifold with a Finsler metric of Lorentzian signature, the lightlike geodesics satisfy the following variational principle. Among all lightlike curves from a point (emission event) to a timelike curve (worldline of receiver), the lightlike geodesics make the arrival time stationary. Here ``arrival time'' refers to a parametrization of the timelike curve. This variational principle can be applied (i) to the vacuum light rays in an alternative spacetime theory, based on Finsler geometry, and (ii) to light rays in an anisotropic non-dispersive medium with a general-relativistic spacetime as background.Comment: 18 pages, submitted to Gen. Rel. Gra

Strong lensing by fermionic dark matter in galaxies

It has been shown that a self-gravitating system of massive keV fermions in thermodynamic equilibrium correctly describes the dark matter (DM) distribution in galactic halos and predicts a denser quantum core towards the center of the configuration. Such a quantum core, for a fermion mass in the range of $50$ keV $\lesssim m c^2 \lesssim 345$ keV, can be an alternative interpretation of the central compact object in Sgr A*. We present in this work the gravitational lensing properties of this novel DM model in Milky Way-like spiral galaxies. We describe the lensing effects of the pure DM component both on halo scales, where we compare them to the effects of the Navarro-Frenk-White and the Non-Singular Isothermal Sphere DM models, and near the galaxy center, where we compare them with the effects of a Schwarzschild BH. For the particle mass leading to the most compact DM core, $m c^2\approx 10^{2}$ keV, we draw the following conclusions. At distances $r\gtrsim 20$ pc from the center of the lens the effect of the central object on the lensing properties is negligible. However, we show that measurements of the deflection angle produced by the DM distribution in the outer region at a few kpc, together with rotation curve data, could help to discriminate between different DM models. We show that at distances $\sim 10^{-4}$ pc strong lensing effects, such as multiple images and Einstein rings, may occur. Large differences in the deflection angle produced by a DM central core and a central BH appear at distances $r\lesssim 10^{-6}$ pc; in this regime the weak-field formalism is no longer applicable and the exact general-relativistic formula has to be used. We find that quantum DM cores do not show a photon sphere what implies that they do not cast a shadow. Similar conclusions apply to the other DM distributions for other fermion masses in the above specified range and for other galaxy types.Comment: 10 pages, 8 figures. v2: Version published in PR

Possible potentials responsible for stable circular relativistic orbits

Bertrand's theorem in classical mechanics of the central force fields attracts us because of its predictive power. It categorically proves that there can only be two types of forces which can produce stable, circular orbits. In the present article an attempt has been made to generalize Bertrand's theorem to the central force problem of relativistic systems. The stability criterion for potentials which can produce stable, circular orbits in the relativistic central force problem has been deduced and a general solution of it is presented in the article. It is seen that the inverse square law passes the relativistic test but the kind of force required for simple harmonic motion does not. Special relativistic effects do not allow stable, circular orbits in presence of a force which is proportional to the negative of the displacement of the particle from the potential center.Comment: 11 pages, Latex fil

Genauigkeit eines bildfreien Navigationssystemes für die Hüftpfannenimplantation – eine anatomische Studie

The position of the acetabular cup is of decisive importance for. the function of a total hip replacement (THR). Using the conventional surgical technique, correct placement of the cup often fails due to a lack of information about pelvic tilt. With CT-based and fluoroscopically-assisted navigation procedures the accuracy of implantation has been significantly improved. However, additional radiation exposure, high cost and the increased time requirement have hampered the acceptance of these techniques. The present anatomical study evaluates the accuracy of an alternative procedure-image-free navigation. This method requires little extra effort, does not substantially delay surgery, and needs no additional imaging. Press-fit cups were implanted in 10 human cadaveric hips with the help of the image-free navigation system, and the position of the cups was checked intraoperatively with a CT-based navigation system and postoperatively by computed tomography. All cups were implanted within the targeted safe zone with an average inclination of 44degrees (range 40degrees-48degrees, SABW 2.7degrees) and an average anteversion of 18degrees (range 12-24degrees, SABW 4.1degrees). Analysis of accuracy of the image-free navigation software revealed only a small, clinically tolerable deviation in cup anteversion and cup inclination in comparison with the CT-based navigation system and the post operative CT scans. The evaluated image-free navigation system appears to be a practicable and reliable alternative to the computer-assisted implantation of acetabular cups in total hip arthroplasty

Исследование термодинамических свойств галогенидов натрия (NaBr, NaCl, NaF, NaI) в области температур 293-673 К

PURPOSE: Computer-assisted knee surgery has become established in routine clinical practice. Still, there is no study investigating midterm clinical outcome after five to seven years postoperatively. We aimed to test the hypothesis that there is no difference either for subjective [Western Ontario and McMaster Universities (WOMAC) scores] or for objective (Knee Society Score, degree of flexion) criteria between computer-assisted total knee replacement (TKR) and freehand TKR after 5.6–7.3 years. METHODS: We performed a matched-pair analysis; 100 patients who received a primary TKR were investigated after a median follow-up of 6.25 years. Group A was operated on with the support of a computer system, while surgery on patients in group B was performed with the freehand technique. We determined WOMAC Score, Knee Society Score and degree of flexion. RESULTS: Overall we found similar results for WOMAC Score [group A: 42.98 (SD 13.80); group B: 41.54 (SD 15.01; p = 0.62)], Knee Society Score [group A: 168.20 (SD 21.94); group B: 166.60 (SD 21.44; p = 0.71)] and range of motion [group A: 106° (SD 9.19); group B 107° (SD 7.44; p = 0.62)]. CONCLUSIONS: No significant differences in midterm clinical outcome were found after TKR performed in the freehand vs computer-assisted technique

Classification of image distortions in terms of Petrov types

An observer surrounded by sufficiently small spherical light sources at a fixed distance will see a pattern of elliptical images distributed over the sky, owing to the distortion effect (shearing effect) of the spacetime geometry upon light bundles. In lowest non-trivial order with respect to the distance, this pattern is completely determined by the conformal curvature tensor (Weyl tensor) at the observation event. In this paper we derive formulas that allow to calculate these distortion patterns in terms of the Newman-Penrose formalism. Then we represent the distortion patterns graphically for all Petrov types, and we discuss their dependence on the velocity of the observer.Comment: 22 pages, 8 eps-figures; revised version, parts of Introduction and Conclusions rewritte

On the exact gravitational lens equation in spherically symmetric and static spacetimes

Lensing in a spherically symmetric and static spacetime is considered, based on the lightlike geodesic equation without approximations. After fixing two radius values r_O and r_S, lensing for an observation event somewhere at r_O and static light sources distributed at r_S is coded in a lens equation that is explicitly given in terms of integrals over the metric coefficients. The lens equation relates two angle variables and can be easily plotted if the metric coefficients have been specified; this allows to visualize in a convenient way all relevant lensing properties, giving image positions, apparent brightnesses, image distortions, etc. Two examples are treated: Lensing by a Barriola-Vilenkin monopole and lensing by an Ellis wormhole.Comment: REVTEX, 11 pages, 12 eps-figures, figures partly improved, minor revision

Spacetime Splitting, Admissible Coordinates and Causality

To confront relativity theory with observation, it is necessary to split spacetime into its temporal and spatial components. The (1+3) timelike threading approach involves restrictions on the gravitational potentials $(g_{\mu \nu})$, while the (3+1) spacelike slicing approach involves restrictions on $(g^{\mu \nu})$. These latter coordinate conditions protect chronology within any such coordinate patch. While the threading coordinate conditions can be naturally integrated into the structure of Lorentzian geometry and constitute the standard coordinate conditions in general relativity, this circumstance does not extend to the slicing coordinate conditions. We explore the influence of chronology violation on wave motion. In particular, we consider the propagation of radiation parallel to the rotation axis of stationary G\"odel-type universes characterized by parameters $\eta > 0$ and $\lambda > 0$ such that for $\eta 1$) chronology is protected (violated). We show that in the WKB approximation such waves can freely propagate only when chronology is protected.Comment: 25 pages, 3 figures; v2: minor typos corrected, accepted for publication in Phys. Rev.

Propagation of Light in the Field of Stationary and Radiative Gravitational Multipoles

Extremely high precision of near-future radio/optical interferometric observatories like SKA, Gaia, SIM and the unparalleled sensitivity of LIGO/LISA gravitational-wave detectors demands more deep theoretical treatment of relativistic effects in the propagation of electromagnetic signals through variable gravitational fields of the solar system, oscillating and precessing neutron stars, coalescing binary systems, exploding supernova, and colliding galaxies. Especially important for future gravitational-wave observatories is the problem of propagation of light rays in the field of multipolar gravitational waves emitted by a localized source of gravitational radiation. Present paper suggests physically-adequate and consistent mathematical solution of this problem in the first post-Minkowskian approximation of General Relativity which accounts for all time-dependent multipole moments of an isolated astronomical system.Comment: 36 pages, no figure

Quantum Energy Inequalities in Pre-Metric Electrodynamics

Pre-metric electrodynamics is a covariant framework for electromagnetism with a general constitutive law. Its lightcone structure can be more complicated than that of Maxwell theory as is shown by the phenomenon of birefringence. We study the energy density of quantized pre-metric electrodynamics theories with linear constitutive laws admitting a single hyperbolicity double-cone and show that averages of the energy density along the worldlines of suitable observers obey a Quantum Energy Inequality (QEI) in states that satisfy a microlocal spectrum condition. The worldlines must meet two conditions: (a) the classical weak energy condition must hold along them, and (b) their velocity vectors have positive contractions with all positive frequency null covectors (we call such trajectories `subluminal'). After stating our general results, we explicitly quantize the electromagnetic potential in a translationally invariant uniaxial birefringent crystal. Since the propagation of light in such a crystal is governed by two nested lightcones, the theory shows features absent in ordinary (quantized) Maxwell electrodynamics. We then compute a QEI bound for worldlines of inertial `subluminal' observers, which generalizes known results from the Maxwell theory. Finally, it is shown that the QEIs fail along trajectories that have velocity vectors which are timelike with respect to only one of the lightcones