There is a unique variant of Absolute Parallelism, which is very simple as it has no free parameters: nothing (nor D=5) can be changed if to keep the theory safe from emerging singularities of solutions. On the contrary, eternal solutions of this t
heory, due to the linear instability of the trivial solution, should be of great complexity which can in some scenarios (with a set of slowly varying parameters of solutions) provide a few phenomenological models including a modified (better to say, new or another) gravity and an expanding-shell cosmology (the longitudinal polarization gives the anti-Milne model). The former looks (mostly) like a variant of tensor-Ricci-squared gravity on a brane of a huge scale L along the extra-dimension. The correction to Newtons law of gravity, which depends in this theory on two parameters (bi-Laplace equation) and behaves as 1/r on large scales, r>L (kpc>L>pc), can start from zero (the Rindler term vanishes) if a constraint is imposed on these parameters. On further consideration, one can conclude that generation of gravitational `short waves, lambda<L, is inhibited in this new gravity.
It is shown that modified gravity theories with a Lagrangian composed of the three quadratic invariants of the Riemann curvature tensor are not appropriate. The field equations are either incompatible and/or irregular [like f(R)-gravities], or, if co
mpatible, lead to the linear instability of polarizations relating to the Weyl tensor. A more relevant modification is the frame field theory, namely the best and unique variant of Absolute Parallelism; it has no free parameters (D=5 is a must) and no singularities arising in solutions. I sketch few remarkable features of this theory.
Quasi-achromat lattices (small dispersion is allowed in their straight sections, between their cells) are considered; in a cell, there are bending magnets of two kinds, of unequal magnetic field. Minimization of the effective emittance is carried out
by the following algorithm (which follows Teng, and partly Lee). (1) Every inner dipoles contribution to the natural emittance is minimized with respect to all optics parameters (relating to dispersion and beta-function), except for the shift parameter, s_0, which specifies the interval between the beta-function minimum and the center of a magnet; for a side bending magnet, its contribution to an integral relating to the effective emittance (the relation uses the fact that the arithmetic mean majorizes the geometric mean) is minimized. (2) The other parameters of dipoles (fields, lengths or angles ratios, shifts) are restricted with the boundary conditions: the equality of Courant-Snyder invariants on the exit from a magnet and on the entrance to the following one. (3) The minimum of effective emittance and the last free parameters (two or three) can be found by computation. The accuracy of this method falls with decreasing of the number of internal dipoles in a cell, and still the isomagnetic Tanaka-Ando minimum (for the modified DBA*-lattice which has no inner dipoles) is reproduced with accuracy better than half a percent. If the number of dipoles per cell does not exceed four, the smallest effective emittance (14% lower than TA-limit) is achieved for QBA**-lattice where all dipoles have nonzero shifts.
Supernovae red shifts are fitted in a simple 5D model: the galaxies are assumed to be enclosed in a giant S^3-spherical shell which expands (ultra) relativistically in a (1+4)D Minkowski space. This model, as compared with the kinematical (1+3)D mode
l of Prof Farley, goes in line with the Copernican principle: any galaxy observes the same isotropic distribution of distant supernovae, as well as the same Hubble plot of distance modulus mu vs red shift z. A good fit is obtained (no free parameters); it coincides with Farleys fit at low z, while shows some more luminosity at high z, leading to 1% decrease in the true distance modulus (and 50% increase in luminosity) at zsim 2. The model proposed can be also interpreted as a FLRW-like model with the scale factor a(t)=t/t_0; this could not be a solution of general relativity (5D GR is also unsuitable--it has no longitudinal polarization). However, there still exists the other theory (with D=5 and no singularities in solutions), the other game in the town, which seems to be able to do the job.
It is proposed a novel design of an X-ray monochromator which uses three crystals. The exit beam turns in horizontal plane on the angle $2alpha$ with respect to the incoming beam. So, this kind of monochromator is suitable for side beam-lines; severa
l tunable beam-lines can be built for a one fan beam of synchrotron radiation (from wiggler or bending magnet). In contrast to the double-crystal scheme of general position considered previously (which also suitable for side beam-lines), the new variant does not lead to a tilt of exit beam profile.
Galactic rotation curves and lack of direct observations of Dark Matter may indicate that General Relativity is not valid (on galactic scale) and should be replaced with another theory. There is the only variant of Absolute Parallelism which solution
s are free of arising singularities, if D=5 (there is no room for changes). This variant does not have a Lagrangian, nor match GR: an equation of `plain R^2-gravity (ie without R-term) is in sight instead. Arranging an expanding O_4-symmetrical solution as the basis of 5D cosmological model, and probing a universal_function of mass distribution (along very-very long the extra dimension) to place into bi-Laplace equation (R^2 gravity), one can derive the Law of Gravitation: 1/r^2 transforms to 1/r with distance (not with acceleration).
