Astronomical observations with a network of detectors of gravitational waves – I. Mathematical framework and solution of the five detector problem

Two different representations for the antenna pattern of Earth-based gravitational wave detectors (laser interferometers and bars) observing arbitrarily polarized waves are investigated in the long wavelength limit. We show that the response of a gravitational wave detector can be written either as a contraction between two symmetric trace-free (STF) tensors, one associated with the wave and the other with the detector, or as a linear combination of generalized spherical functions. We then apply this formalism to solve the simplest form of the 'inverse problem' for bursts: determining the direction of an incoming wave, the orientation of its polarization ellipse and the wave's two independent amplitudes, using only the response amplitudes of five wide-band detectors located in the same place. A monochromatic signal is assumed (e.g. coalescing binary).