arXiv2013
While the fluorescence and the ground counter techniques for the detection of ultra-high energy cosmic rays (UHECR) were being developed for decades, the interest in the radio detection diminished after the initial experiments in the 1960s. As a result, the fluorescence and the surface array techniques are more mature today, providing more reliable measurements of the primary cosmic particle energy, chemical composition and the inelastic cross-section. The advantages of the radio technique are 100 percent duty cycle and lower deployment and operational costs. Thus, the radio technique can greatly complement the fluorescence and the ground array detection and can also work independently. With the ANITA balloon detector observing UHECRs and the success of LOPES, CODALEMA and other surface radio detectors, the radio technique received a significant boost in recent years. Reliable Monte Carlo (MC) simulations are needed in order to obtain the energy and other parameters of the primary cosmic ray particle from the radio observations. Several MC techniques, like ZHairesS and the Endpoint Formalism, were proposed in recent years. While they seem to reproduce some of the observed data quite well, there is a divergence between the different approaches under certain conditions. In this work we derive these approaches from Maxwells equations and prove their identity under certain conditions as well as discuss their applicability to the UHECR air showers and to a proposed experiment at SLAC.