Euler observers for the perfect fluid without vorticity

There are many astrophysical problems that require the use of relativistic hydrodynamics, in particular ideal perfect fluids. Fundamentally, problems associated with intense gravitational fields. In this context, there are evolution problems that need to be addressed with techniques that are both as simple as possible and computer time inexpensive. A new technique using a new kind of tetrads was developed for the case where there is vorticity in order to locally and covariantly diagonalize the perfect fluid stress-energy tensor. The perfect fluid was already studied for the case where there is vorticity and several sources of simplification were already found. In this manuscript, we will analyze the case where there is no vorticity. We will show how to implement for this case the diagonalization algorithm that will differ from the previously developed for the case with vorticity. A novel technique to build tetrads using Killing vector fields will be introduced. We implement this new technique using only covariant and local manipulations of an algebraic nature, which will not add more substantial computational time and nonetheless bring about simplification in further applications like the construction of Euler observers for example. Precisely, as an application in spacetime dynamical evolution, a new algorithm will be formulated with the aim of finding Euler observers for this case without vorticity. It will be shown that the Einstein equations with the perfect fluid stress-energy tensor get substantially simplified through the use of these new tetrads. We will also show that the Frobenius theorem is covertly encoded in these new tetrads when Killing vector fields do exist.