Explicit solutions for relativistic acceleration and rotation

The Lorentz transformations are represented by Einstein velocity addition on the ball of relativistically admissible velocities. This representation is by projective maps. The Lie algebra of this representation defines the relativistic dynamic equation. If we introduce a new dynamic variable, called symmetric velocity, the above representation becomes a representation by conformal, instead of projective maps. In this variable, the relativistic dynamic equation for systems with an invariant plane, becomes a non-linear analytic equation in one complex variable. We obtain explicit solutions for the motion of a charge in uniform, mutually perpendicular electric and magnetic fields. By assuming the Clock Hypothesis and using these solutions, we are able to describe the space-time transformations between two uniformly accelerated and rotating systems.Comment: 15 pages 1 figur

Present and Future Prospects for GRB Standard Candles

Following our previous work, we conclude that a GRB standard candle constructed from the Ghirlanda et al. power-law relation between the geometry-corrected energy (E_gamma) and the peak of the rest-frame prompt burst spectrum (E_p) is not yet cosmographically useful, despite holding some potential advantages over SNe Ia. This is due largely to the small sample of \~20 GRBs with the required measured redshifts, jet-breaks, and peak energies, and to the strong sensitivity of the goodness-of-fit of the power-law to input assumptions. The most important such finding concerns the sensitivity to the generally unknown density (and density profile), of the circumburst medium. Although the E_p-E_gamma relation is a highly significant correlation over many cosmologies, until the sample expands to include many low-z events, it will be most sensitive to Omega_M but essentially insensitive to Omega_Lambda and w, with some hope of constraining dw/dt with high-z GRB data alone. The relation clearly represents a significant improvement in the search for an empirical GRB standard candle, but is further hindered by an unknown physical basis for the relation, the lack of a low-z training set to calibrate the relation in a cosmology-independent way, and several major potential systematic uncertainties and selection effects. Until these concerns are addressed, a larger sample is acquired, and attempts are made to marginalize or perform Monte Carlo simulations over the unknown density distribution, we urge caution concerning claims of the utility of GRBs for cosmography and especially the attempts to combine GRBs with SNe Ia.Comment: 5 pages, 2 figures, "Proceedings, Gamma-Ray Bursts in the Afterglow Era: 4th Workshop, Rome, Italy, Oct 18-22, 2004". Accepted to Il Nuovo Cimento. For more details, see astro-ph/0408413 (ApJ accepted), and other work from the cosmicbooms.net Team at http://www.cosmicbooms.net

The scalar complex potential and the Aharonov-Bohm effect

The Aharonov-Bohm effect is traditionally attributed to the effect of the electromagnetic 4-potential $A$, even in regions where both the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ are zero. The AB effect reveals that multiple-valued functions play a crucial role in the description of an electromagnetic field. We argue that the quantity measured by AB experiments is a difference in values of a multiple-valued complex function, which we call a complex potential or {pre-potential. We show that any electromagnetic field can be described by this pre-potential, and give an explicit expression for the electromagnetic field tensor through this potential. The pre-potential is a modification of the two scalar potential functions.Comment: 10 pages 2 figure