The Static Heavy Quark-Antiquark Potential within String Theory in Arbitrary Stationary Backgrounds

We consider a static open string in arbitrary stationary spacetime, which can represent a heavy quark-antiquark pair within the holographic framework or effective theory. Generally, the string profile is not symmetric with respect to the turning point, and the symmetry restores for a simple string configuration in backgrounds with certain constraints. We identify a wide family of metrics for which the symmetry is preserved, enabling a direct isolation of the linear-in-distance term in the static potential for simple symmetric string configurations, even in non-diagonal backgrounds. As a first example, we apply our formulas to the black brane dual to the $\mathcal{N}=4$ SYM plasma at finite temperature. We find that the separation distance between quarks, $L$, is given in terms of a hypergeometric function, while the potential, $V$, consists of two distinct contributions: a term linear in the separation and a term that involves its derivative by temperature. Analysis of the leading terms in the series expansion reveals that the temperature corrections of the separation distance leads to the "swallowtail" behavior. Further, applying our formulas to the Rindler-AdS spacetime dual to an accelerated $\mathcal{N}=4$ SYM plasma, we obtain analytic expressions for the distance and potential in terms of the elliptic integrals, which in the large Hawking temperature (large acceleration or small curvature) limit come to the conformal results for pure AdS. Then, we show that the distance between quarks decreases, the static potential between them increases, and the characteristic temperatures increase with an acceleration, $a_c$. However, we observe that an acceleration-scaled potential, $a_c V$ as a function of the acceleration-scaled distance, $a_c L$, does not depend on the certain value of the acceleration.