Generalizing Optical Geometry

We show that by employing the standard projected curvature as a measure of spatial curvature, we can make a certain generalization of optical geometry (Abramowicz and Lasota 1997, Class. Quantum Grav. 14 (1997) A23). This generalization applies to any spacetime that admits a hypersurface orthogonal shearfree congruence of worldlines. This is a somewhat larger class of spacetimes than the conformally static spacetimes assumed in standard optical geometry. In the generalized optical geometry, which in the generic case is time dependent, photons move with unit speed along spatial geodesics and the sideways force experienced by a particle following a spatially straight line is independent of the velocity. Also gyroscopes moving along spatial geodesics do not precess (relative to the forward direction). Gyroscopes that follow a curved spatial trajectory precess according to a very simple law of three-rotation. We also present an inertial force formalism in coordinate representation for this generalization. Furthermore, we show that by employing a new sense of spatial curvature (Jonsson, Class. Quantum Grav. 23 (2006) 1) closely connected to Fermat's principle, we can make a more extensive generalization of optical geometry that applies to arbitrary spacetimes. In general this optical geometry will be time dependent, but still geodesic photons move with unit speed and follow lines that are spatially straight in the new sense. Also, the sideways experienced (comoving) force on a test particle following a line that is straight in the new sense will be independent of the velocity.Comment: 19 pages, 1 figure. A more general analysis is presented than in the former version. See also the companion papers arXiv:0708.2493, arXiv:0708.2533 and arXiv:0708.253

An intuitive approach to inertial forces and the centrifugal force paradox in general relativity

Abstract. As the velocity of a rocket in a circular orbit near a black hole increases, the outwardly directed rocket thrust must increase to keep the rocket in its orbit. This feature might appear paradoxical from a Newtonian viewpoint, but we show that it follows naturally from the equivalence principle together with special relativity and a few general features of black holes. We also derive a general relativistic formalism of inertial forces for reference frames with acceleration and rotation. The resulting equation relates the real experienced forces to the time derivative of the speed and the spatial curvature of the particle trajectory relative to the reference frame. We show that an observer who follows the path taken by a free (geodesic) photon will experience a force perpendicular to the direction of motion that is independent of the observers velocity. We apply our approach to resolve the submarine paradox, which regards whether a submerged submarine in a balanced state of rest will sink or float when given a horizontal velocity if we take relativistic effects into account. We extend earlier treatments of this topic to include spherical oceans and show that for the case of the Earth the submarine floats upward if we take the curvature of the ocean into account. I Introduction Consider a rocket in a circular orbit outside the event horizon of a black hole. 1 If the orbit lies within the photon radius, the radius where free photons can move on circular orbits, 2 a greater outward rocket thrust is required to keep the rocket in orbit the faster the rocket moves. However, outside of the photon radius the outward thrust decreases as the orbital speed increases just as it would for a similar scenario in Newtonian mechanics (the thrust will be inward directed for sufficiently high speeds, see Analogous to the situation in Newtonian mechanics we can introduce in general relativity a gravitational force that is velocity independent. This force is fictitious (unlike in Newtonian mechanics). We can also introduce a velocity dependent (fictitious) centrifugal force that together with the gravitational force balances the real force from the jet engine of the rocket. By this definition, the centrifugal force is directed inward inside of the photon radius and directed outward outside of the photon radius. This reversal of the direction of the fictitious centrifugal force is described by the formalism of optical geometry (see Appendix A) in which the phenomena has been discussed. We start by illustrating how the fact that the rocket thrust increases with increasing orbital speed (sufficiently close to the black hole) follows naturally from the equivalence principle (reviewed in Appendix B). We do so by first considering an idealized special relativistic scenario of a train moving relative to an (upward) accelerating platform. We then consider a more general but still effectively two-dimensional discussion of forces perpendicular to the direction of motion for motion relative to an accelerated reference frame in special relativity. In a static spacetime, the reference frame connected to the static observers behaves locally like an accelerating reference frame in special relativity and the formalism can therefore be applied also to this case. We then illustrate how to apply the formalism of this paper to the submarine paradox. Next we generalize the formalism of forces and curvature of spatial paths to include three-dimensional cases, forces parallel to the direction of motion, and rotating reference frames. The acceleration and rotation of the reference frame will be shown (as in Newtonian mechanics) to introduce terms that can be interpreted as iner-

An intuitive approach to inertial forces and the centrifugal force paradox in general relativity

As the velocity of a rocket in a circular orbit near a black hole increases, the outwardly directed rocket thrust must increase to keep the rocket in its orbit. This feature might appear paradoxical from a Newtonian viewpoint, but we show that it follows naturally from the equivalence principle together with special relativity and a few general features of black holes. We also derive a general relativistic formalism of inertial forces for reference frames with acceleration and rotation. The resulting equation relates the real experienced forces to the time derivative of the speed and the spatial curvature of the particle trajectory relative to the reference frame. We show that an observer who follows the path taken by a free (geodesic) photon will experience a force perpendicular to the direction of motion that is independent of the observers velocity. We apply our approach to resolve the submarine paradox, which regards whether a submerged submarine in a balanced state of rest will sink or float when given a horizontal velocity if we take relativistic effects into account. We extend earlier treatments of this topic to include spherical oceans and show that for the case of the Earth the submarine floats upward if we take the curvature of the ocean into account.Comment: 14 pages, 21 figure

Inertial forces and the foundations of optical geometry

Assuming a general timelike congruence of worldlines as a reference frame, we derive a covariant general formalism of inertial forces in General Relativity. Inspired by the works of Abramowicz et. al. (see e.g. Abramowicz and Lasota, Class. Quantum Grav. 14 (1997) A23), we also study conformal rescalings of spacetime and investigate how these affect the inertial force formalism. While many ways of describing spatial curvature of a trajectory has been discussed in papers prior to this, one particular prescription (which differs from the standard projected curvature when the reference is shearing) appears novel. For the particular case of a hypersurface-forming congruence, using a suitable rescaling of spacetime, we show that a geodesic photon is always following a line that is spatially straight with respect to the new curvature measure. This fact is intimately connected to Fermat's principle, and allows for a certain generalization of the optical geometry as will be further pursued in a companion paper (Jonsson and Westman, Class. Quantum Grav. 23 (2006) 61). For the particular case when the shear-tensor vanishes, we present the inertial force equation in three-dimensional form (using the bold face vector notation), and note how similar it is to its Newtonian counterpart. From the spatial curvature measures that we introduce, we derive corresponding covariant differentiations of a vector defined along a spacetime trajectory. This allows us to connect the formalism of this paper to that of Jantzen et. al. (see e.g. Bini et. al., Int. J. Mod. Phys. D 6 (1997) 143).Comment: 42 pages, 7 figure

Development of a Model System to Identify Differences in Spring and Winter Oat

Our long-term goal is to develop a Swedish winter oat (Avena sativa). To identify molecular differences that correlate with winter hardiness, a winter oat model comprising of both non-hardy spring lines and winter hardy lines is needed. To achieve this, we selected 294 oat breeding lines, originating from various Russian, German, and American winter oat breeding programs and tested them in the field in south- and western Sweden. By assaying for winter survival and agricultural properties during four consecutive seasons, we identified 14 breeding lines of different origins that not only survived the winter but also were agronomically better than the rest. Laboratory tests including electrolytic leakage, controlled crown freezing assay, expression analysis of the AsVrn1 gene and monitoring of flowering time suggested that the American lines had the highest freezing tolerance, although the German lines performed better in the field. Finally, six lines constituting the two most freezing tolerant lines, two intermediate lines and two spring cultivars were chosen to build a winter oat model system. Metabolic profiling of non-acclimated and cold acclimated leaf tissue samples isolated from the six selected lines revealed differential expression patterns of 245 metabolites including several sugars, amino acids, organic acids and 181 hitherto unknown metabolites. The expression patterns of 107 metabolites showed significant interactions with either a cultivar or a time-point. Further identification, characterisation and validation of these metabolites will lead to an increased understanding of the cold acclimation process in oats. Furthermore, by using the winter oat model system, differential sequencing of crown mRNA populations would lead to identification of various biomarkers to facilitate winter oat breeding