## ABSTRACT

This chapter develops the energy-momentum tensor of a perfect fluid—an idealized fluid in which energy dissipation due to viscosity or heat flow is ignored. Classical hydrodynamics is based on balance equation for mass, energy, and momentum, concepts that are distinct in nonrelativistic physics, but which are interrelated in relativity, mass-energy, energy-momentum. The chapter reviews the nonrelativistic theory. In a perfect fluid dissipation is ignored, implying that flows are isentropic. In hydrodynamics it is usually more convenient to work with specific, “per mass” quantities. In the relativistic theory of hydrodynamics both the energy and momentum currents involve transport of enthalpy. The Euler equation specifies the acceleration of a fluid; a covariant version would involve spacelike quantities because four-acceleration is spacelike and orthogonal to the four-velocity. Nonrelativistic hydrodynamics is based on some equations expressing conservation of mass, momentum, and energy. In relativity, energy is conserved, and energy is tied to mass.