Gamma-ray Burst Luminosity Relations: Two-dimensional versus
  Three-dimensional Correlations
  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. In this paper, we extend the two-dimensional (2D)
luminosity relations with $\tau_{\mathrm{lag}}$, $V$, $E_{\mathrm{peak}}$, and
$\tau_{\mathrm{RT}}$ as the luminosity indicators to three dimensions (3D)
using the same set of luminosity indicators to explore the possibility of
decreasing the intrinsic scatters. We find that, for the 3D luminosity
relations between the luminosity and an energy scale ($E_{\mathrm{peak}}$) and
a time scale ($\tau_{\mathrm{lag}}$ or $\tau_{\mathrm{RT}}$), their intrinsic
scatters are considerably smaller than those of corresponding 2D luminosity
relations. Enlightened by the result and the definition of the luminosity
(energy released in units of time), we discussed possible reasons behind, which
may give us helpful suggestions on seeking more precise luminosity relations
for GRBs in the future.
