THE CORRESPONDENCE OF THE EREZ-ROSEN SOLUTION WITH THE HARTLE-THORNE SOLUTION IN THE LIMITING CASE OF AND

Аннотация

The link between exterior solutions to the Einstein gravitational field equations such as the exact
Erez-Rosen metric and approximate Hartle-Thorne metric is established here for the static case in the limit of linear
mass quadrupole moment  and second order terms in total mass. To this end, the Geroch-Hansen multipole
moments are calculated for the Erez-Rosen and Hartle-Thorne solutions in order to find the relationship among the
parameters of both metrics. The coordinate transformations are sought in a general form with two unknown functions
in the corresponding limit of and. By employing the perturbation theory, the approximate Erez-Rosen
metric is written in the same coordinates as the Hartle-Thorne metric. By equating the radial and azimuthal
components of the metric tensor of both solutions the sought functions are found in a straightforward way. It is
shown that the approximation and, which is used throughout the article, is physical and suitable for solving
most problems of celestial mechanics in post-Newtonian physics. This approximation does not require the use of the
Zipoy-Voorhees transformation, which is a necessary strict mathematical requirement in the approximation, i.e.
when no other approximations are made. This implies that the explicit form of the coordinate transformations
depends entirely on the approximation that is adopted in each particular case. The results obtained here are in
agreement with the previous results in the literature and can be applied to different astrophysical goals. The paper
pursues not only pure scientific, but also academic purposes and can be used as an auxiliary and additional material
to the special courses of general theory of relativity, celestial mechanics and relativistic astrophysics.