  We study the problem of heat conduction in general relativity by using
Carter's variational formulation. We write the creation rates of the entropy
and the particle as combinations of the vorticities of temperature and chemical
potential. We pay attention to the fact that there are two additional degrees
of freedom in choosing the relativistic analog of Cattaneo equation for the
parts binormal to the caloric and the number flows. Including the contributions
from the binormal parts, we find a $\textit{new}$ heat-flow equations and
discover their dynamical role in thermodynamic systems. The benefit of
introducing the binormal parts is that it allows room for a physical ansatz for
describing the whole evolution of the thermodynamic system. Taking advantage of
this platform, we propose a proper ansatz that deals with the binormal
contributions starting from the physical properties of thermal equilibrium
systems. We also consider the stability of a thermodynamic system in a flat
background. We find that $\textit{new}$ "Klein" modes exist in addition to the
known ones. We also find that the stability requirement is less stringent than
those in the literature.
