Is the time lag - luminosity relation of GRBs a consequence of the Amati relation ?

Context. The lag-luminosity relation (LLR) provides a way of estimating GRB luminosity by measuring the spectral lags between different energy bands. Aims. We want to understand the origin of the LLR and test its validity. This appears especially important if the LLR is to be used as a distance indicator. Methods. We perform a linear analysis of the lag between two spectral bands. The lag is obtained as the time interval between the maxima of a given pulse in the two bands. Results. We get a simple expression for the lag, which shows in a very simple way how it is related to the spectral evolution of the burst via the variation of the peak energy and spectral indices. When this expression is coupled to the Amati relation, it leads to a LLR that agrees with the observational results only if the burst's spectral evolution is limited to a decrease in peak energy during pulse decay. However, when the variation of the spectral indices is also taken into account, the predicted LLR differs from the observed one. Conclusions. We briefly discuss some ways to solve this problem, such as a possible correlation between pulse spikiness and burst luminosity.