  We introduce a class of relativistic fluid states satisfying the relativistic
local thermodynamical equilibrium postulate (abbreviated as relativistic (LTE)
postulate). States satisfying this postulate, are states "near equilibrium" (a
term defined precisely in the course of the paper) and permit us to attach a
fictitious "local thermodynamical equilibrium" state that fits event by event
the actual fluid state. They single out an admissible class of rest frames
relative to which thermodynamical variables like the energy density,
thermodynamical pressure, stresses, particle number density (or densities)
measured by observers at rest relative to these frames are becoming frame
independent provided second (or higher) order deviations from the fictitious
state of "local thermodynamical equilibrium" are ignored. We have verified this
property for a large class of theories of relativistic dissipation that include
the Hiscock-Lindblom class of first order theories, the Eckart and
Landau-Lifshitz theories, the Israel-Stewart transient thermodynamics, the
Liu-M\"uller-Ruggeri theory, fluids of divergence type and the latest developed
(BDNK) theory. Moreover, the phenomenological equations describing first order
deviations from the fictitious "local thermodynamical equilibrium" state
satisfy equations that remain form invariant under change of frame within the
class of admissible frames. We proved this property for the Hiscock-Lindblom
class of first order theories the Eckart and Landau-Lifshitz theories, the
Israel-Stewart transient thermodynamics and the Liu-M\"uller-Ruggeri theory of
relativistic dissipation the (BDNK) theory and we expect that the same property
to hold for the class of relativistic fluids of divergence type.
