The Pierre Auger Observatory in Argentina is the world largest detector for cosmic rays at the highest energies. It is realized as a hybrid detector, combining the detection of secondary particles at ground level with an array of 1660 water-Cherenkov detectors deployed over an area of about 3000 km² and the detection of emitted florescence light with 27 florescence telescopes at four sites around the array. For the Auger Engineering Radio Array, an extension of the Pierre Auger Observatory with antennas in the MHz range, it is necessary to monitor the local atmospheric conditions. These have a large influence on the radio emission of air showers. In particular, amplified signals up to an order of magnitude have been detected as an effect of thunderstorms. For a more detailed investigation and the detection of thunderstorms, a new lightning detection system has been developed and installed at the Pierre Auger Observatory in Argentina. Together with the data of an electric field mill that measures the electric field strength at ground level at the antenna array. The new lightning detection system allowed the development of a lightning-based trigger for the water-Cherenkov detectors to read individual stations when lightning strikes nearby. With these data a possible correlation between the formation of lightning and cosmic ray showers can be in future investigated even at low energies of about PeV. Within this work the development of the new lightning detection systems is described and the different data of the different systems are compared to each other.