The solution will be a retarded integral over the past light cone of the field point. The domain of the integration can be conveniently partitioned into a near-zone domain and a wave-zone domain. In the near zone, the field point distance from the source 





 is small compared with a characteristic wavelength 





, where tc is the characteristic timescale of the dynamics of the source and c is the speed of light. The boundary of the near and wave zones is situated roughly at an arbitrary radius 





. This radius is of the same order of magnitude as λc. The integrals can vanish at this boundary. In fact, by partitioning the domain of integration into these two zones, all integrations will be finite, and one can show that the contributions from the near-zone spatial integrals containing 





 (which diverge at 





) will be canceled by corresponding terms from the wave-zone integrals (Will & Wiseman 1996). As a result, this calculation does not depend on the arbitrary boundary radius 





. Keeping these facts in mind, one can find the PN metric as (Poisson & Will 2014)
1





where Ψ is defined as
2





Potentials that appear in the metric components are given by
3





where quantities in integrands are evaluated at time t and position 





. The matter variables in the PN approach are 





, where Σ* is the conserved surface mass density defined as
4





in which Σ is the proper surface density and g is the determinant of the metric tensor. Here u0 is the zeroth component of the velocity four-vector uα. It should be noted that in the Newtonian limit, there is no difference between Σ* and the proper surface density Σ. Moreover, p is the pressure, Π = /Σ is the internal energy per unit mass, and vj is the fluid’s velocity field defined with respect to the time coordinate t. Note that  is the proper surface density of internal energy and U is the gravitational potential for surface density Σ*. In the following subsection, we will introduce the Newtonian potential in terms of Σ. Using the PN metric and the conservation equation of the energy-momentum tensor, i.e., 





, one can find the PN equations of hydrodynamics (Poisson & Will 2014). In fact, in the 1pn approximation, the relativistic corrections appear as terms proportional to order c−2 in the equations. The continuity equation in the PN limit takes the following form:
5





In the cylindrical coordinate system (R, φ, z), where the velocity is 





, Equation (5) is given by
6





Moreover, the Euler equation in the PN approximation can be written as follows:
7





The terms inside the braces are PN corrections. In the 1pn approximation, we ignore terms proportional to order c−4 and higher. The left-hand side of Equation (7) can be written as 





. After some algebraic manipulations and using vector analysis in the cylindrical coordinate system, one can decompose Equation (7) to components R and φ. The R component of the Euler equation reads
8





Also, the φ component of the Euler equation can be written as
9





In order to have a complete set of differential equations, the abovementioned equations should be joined with the first law of thermodynamics and an equation of state (EOS). The first law of thermodynamics for a perfect fluid can be written as
10





On the other hand, the pressure and density of the fluid can be related to each other via an EOS. A simple case suitable for astrophysical aims is the barotropic equation, given as
11





where the conserved surface density is
12
Citation: Will & Wiseman 1996
Citation function: Uses
Citation: Poisson & Will 2014
Citation function: Uses; Uses
Citation: Poisson & Will 2014
Citation function: Uses