No Arabic abstract
This Letter, i.e. for the first time, proves that a general invariant velocity is originated from the principle of special relativity, namely, discovers the origin of the general invariant velocity, and when the general invariant velocity is taken as the invariant light velocity in current theories, we get the corresponding special theory of relativity. Further, this Letter deduces triple special theories of relativity in cosmology, and cancels the invariant presumption of light velocity, it is proved that there exists a general constant velocity K determined by the experiments in cosmology, for K > 0, = 0 and < 0, they correspond to three kinds of possible relativistic theories in which the special theory of relativity is naturally contained for the special case of K > 0, and this Letter gives a prediction that, for K < 0, there is another likely case satisfying the principle of special relativity for some special physical systems in cosmology, in which the relativistic effects observed would be that the moving body would be lengthened, moving clock would be quickened. And the point of K = 0 is a bifurcation point, through which it gives out three types of possible universes in the cosmology (or multiverse). When a kind of matter with the maximally invariant velocity that may be superluminal or equal to light velocity is determined by experiments, then the invariant velocity can be taken as one of the general invariant velocity achieved in this Letter, then all results of current physical theories are consistent by utilizing this Letters theory.
We review the status of testing the principle of equivalence and Lorentz invariance from atmospheric and solar neutrino experiments.
The goal of this lecture is to introduce the student to the theory of Special Relativity. Not to overload the content with mathematics, the author will stick to the simplest cases; in particular only reference frames using Cartesian coordinates and translating along the common x-axis as in Fig. 1 will be used. The general expressions will be quoted or may be found in the cited literature.
It is shown how a Doubly-Special Relativity model can emerge from a quantum cellular automaton description of the evolution of countably many interacting quantum systems. We consider a one-dimensional automaton that spawns the Dirac evolution in the relativistic limit of small wave-vectors and masses (in Planck units). The assumption of invariance of dispersion relations for boosted observers leads to a non-linear representation of the Lorentz group on the $(omega,k)$ space, with an additional invariant given by the wave-vector $k=pi /2$. The space-time reconstructed from the $(omega,k)$ space is intrinsically quantum, and exhibits the phenomenon of relative locality.
The hodograph of a non-relativistic particle motion in Euclidean space is the curve described by its momentum vector. For a general central orbit problem the hodograph is the inverse of the pedal curve of the orbit, (i.e. its polar reciprocal), rotated through a right angle. Hamilton showed that for the Kepler/Coulomb problem, the hodograph is a circle whose centre is in the direction of a conserved eccentricity vector. The addition of an inverse cube law force induces the eccentricity vector to precess and with it the hodograph. The same effect is produced by a cosmic string. If one takes the relativistic momentum to define the hodograph, then for the Sommerfeld (i.e. the special relativistic Kepler/Coulomb problem) there is an effective inverse cube force which causes the hodograph to precess. If one uses Schwarzschild coordinates one may also define a a hodograph for timelike or null geodesics moving around a black hole. Iheir pedal equations are given. In special cases the hodograph may be found explicitly. For example the orbit of a photon which starts from the past singularity, grazes the horizon and returns to future singularity is a cardioid, its pedal equation is Cayleys sextic the inverse of which is Tschirhausens cubic. It is also shown that that provided one uses Beltrami coordinates, the hodograph for the non-relativistic Kepler problem on hyperbolic space is also a circle. An analogous result holds for the the round 3-sphere. In an appendix the hodograph of a particle freely moving on a group manifold equipped with a left-invariant metric is defined.
We show that depending on the direction of deformation of $kappa$-Poincare algebra (time-like, space-like, or light-like) the associated phase spaces of single particle in Doubly Special Relativity theories have the energy-momentum spaces of the form of de Sitter, anti-de Sitter, and flat space, respectively.