  The large scatters of luminosity relations of gamma-ray bursts (GRBs) have
been one of the most important reasons that prevent the extensive applications
of GRBs in cosmology. Many efforts have been made to seek tight luminosity
relations. With the latest sample of 116 GRBs with measured redshift and
spectral parameters, we investigate 6 two-dimensional (2D) correlations and 14
derived three-dimensional (3D) correlations of GRBs to explore the possibility
of decreasing the intrinsic scatters of the luminosity relations of GRBs. We
find the 3D correlation of $E_{\mathrm{peak}}$--$\tau_{\mathrm{RT}}$--$L$ to be
evidently tighter (at the $2 \sigma$ confidence level) than its corresponding
2D correlations, i.e., the $E_{\mathrm{peak}}$--$L$ and
$\tau_{\mathrm{RT}}$--$L$ correlations. In addition, the coefficients before
the logarithms of $E_{\mathrm{peak}}$ and $\tau_{\mathrm{RT}}$ in the
$E_{\mathrm{peak}}$--$\tau_{\mathrm{RT}}$--$L$ correlation are almost exact
opposites of each other. Inputting this situation as a prior reduces the
relation to $L \propto (E_{\mathrm{peak}}^{'} / \tau_{\mathrm{RT}}^{'})^{0.842
\pm 0.064}$, where $E_{\mathrm{peak}}^{'}$ and $\tau_{\mathrm{RT}}^{'}$ denote
the peak energy and minimum rise time in the GRB rest frame. We discuss how our
findings can be interpreted/understood in the framework of the definition of
the luminosity (energy released in units of time). Our argument about the
connection between the luminosity relations of GRBs and the definition of the
luminosity provides a clear direction for exploring tighter luminosity
relations of GRBs in the future.
