Geodesic deviation in the q-metric

Abstract We consider the tidal forces between test particles falling along geodesics in the exterior spacetime generated by a static and axially symmetric compact matter source with a non-vanishing mass quadrupole. Specifically, we analyze the radial and angular geodesic deviation, compare it with that of the Schwarzschild spacetime, and investigate the impact of the deformation parameter q at different polar angles ( $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> ) with respect to the vertical symmetry axis. Furthermore, we examine the geodesic deviation for the case of non-constant $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> during the radial fall. We show that the presence of the deformation parameter affects the behavior of the geodesic deviation vectors, depending on its value. In particular, we observe that for arbitrary values of q and $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> , the behavior of the deviation vector differs as it approaches the singularity at $$r = 2m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> . Above all, we can witness either stretching or compression of the deviation vector for various combinations of q and $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> . These findings provide insight into the effects of quadrupole deformation on the motion of test particles in the vicinity of the central object.