Connecting the exterior gravitational field with the energy–momentum tensor of axially symmetric compact objects

A method to construct interior axially symmetric metrics that appropriately match with any vacuum solution of the Weyl family is developed in Hernández-Pastora et al. (Class Quantum Gravity 33:235005, 2016). It was shown, for the case of some vacuum solutions, that the simplest solution for the interior metric leads to sources with well-behaved energy conditions. Now, we integrate the field equations to obtain the interior metric functions in terms of the anisotropies and pressures of the source. As well, the compatible equations of state for these global models are calculated. The interior metric and the suitable energy–momentum tensor describing the source are constructed in terms of the exterior metric functions. At the boundary of the compact object, the behaviour of a pressure T m , defined from the energy–momentum tensor, is shown to be related with the exterior gravitational field. This fact allows us to explore the differences arising at the matter distribution when the spherical symmetry of the global metric is dropped. Finally, an equation derived from the matching conditions is obtained which allows us to calculate the Weyl coefficients of the exterior metric as source integrals. Hence the Relativistic Multipole Moments of the global model can be expressed in terms of the matter distribution of the source.