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The first gravitational wave (GW) - gamma-ray burst (GRB) association,GW170817/GRB 170817A, had an offset in time, with the GRB trigger time delayedby $\sim$1.7 s with respect to the merger time of the GW signal. We generallydiscuss the astrophysical origin of the delay time, $\Delta t$, of GW-GRBassociations within the context of compact binary coalescence (CBC) -- shortGRB (sGRB) associations and GW burst -- long GRB (lGRB) associations. Ingeneral, the delay time should include three terms, the time to launch a clean(relativistic) jet, $\Delta t_{\rm jet}$; the time for the jet to break outfrom the surrounding medium, $\Delta t_{\rm bo}$; and the time for the jet toreach the energy dissipation and GRB emission site, $\Delta t_{\rm GRB}$. ForCBC-sGRB associations, $\Delta t_{\rm jet}$ and $\Delta t_{\rm bo}$ arecorrelated, and the final delay can be from 10 ms to a few seconds. ForGWB-lGRB associations, $\Delta t_{\rm jet}$ and $\Delta t_{\rm bo}$ areindependent. The latter is at least $\sim$10 s, so that $\Delta t$ of theseassociations is at least this long. For certain jet launching mechanisms oflGRBs, $\Delta t$ can be minutes or even hours long due to the extended enginewaiting time $\Delta t_{\rm wait}$ to launch a jet. We discuss the cases ofGW170817/GRB 170817A and GW150914/GW150914-GBM within this theoreticalframework and suggest that the delay times of future GW/GRB associations willshed light into the jet launching mechanisms of GRBs.

The Delay Time of Gravitational Wave -- Gamma-Ray Burst Associations