  In this work, a connection has been indicated between the different existing
hydrodynamic formalisms, which in order to be causal and stable, (i) either
requires to promote the `non-fluid' variables as the new degrees of freedom
apart from the velocity and temperature, or (ii) needs to be in a general
hydrodynamic frame other than the ones given by Landau and Eckart. We have
rewritten the BDNK stress tensor (originally in the general frame) into the
Landau frame by redefining the fluid variables such as velocity ($u^{\mu}$) and
temperature ($T$) via a hydrodynamic frame transformation. This `fluid frame'
transformation includes shift variables $\delta u^{\mu}$ and $\delta T$, which
are small enough to be treated linearly but encompass all orders of gradient
corrections. The redefinition indicates that though the BDNK formalism has a
finite number of derivatives, in the Landau frame, it will either have an
infinite number of derivatives or one has to resum them by introducing new
`non-fluid' variables much like the MIS theory. The infinite derivative series
are summed in two different ways that lead to two different methods of
`integrating in' new `non-fluid' variables, showing the non-uniqueness of the
process of `integrating in' new variables. Finally, the dispersion relations
and the corresponding spectra of these different systems of MIS type equations
have been analyzed to check that the systems of equations presented here are
indeed equivalent to the BDNK formalism, at least in the hydrodynamic regime.
