6,674 research outputs found
Epicyclic orbital oscillations in Newton's and Einstein's dynamics
We apply Feynman's principle, ``The same equations have the same solutions'',
to Kepler's problem and show that Newton's dynamics in a properly curved 3-D
space is identical with that described by Einstein's theory in the 3-D optical
geometry of Schwarzschild's spacetime. For this reason, rather unexpectedly,
Newton's formulae for Kepler's problem, in the case of nearly circular motion
in a static, spherically spherical gravitational potential accurately describe
strong field general relativistic effects, in particular vanishing of the
radial epicyclic frequency at the marginally stable orbit.Comment: 8 page
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
- …