Curvature Corrections to the Yukawa Potential in Tolman Metrics

Abstract

This work investigates curvature-induced modifications to the Yukawa potential in static, spherically symmetric spacetimes described by Tolman metrics, focusing on their implications for compact stellar objects, with particular application to solutions IV and VI. Motivated by the interplay of quantum interactions and strong gravitational fields in systems like neutron stars, we derive explicit corrections to the Yukawa potential for these metrics. Contrary to previous findings suggesting that curvature corrections break the radial symmetry of the interacting potential near a highly charged black hole, we demonstrate that Tolman metric corrections preserve this symmetry in the local inertial frame. Numerical estimates for astrophysical objects reveal energy shifts of the order of $10^{-34}$ MeV for solution IV. The Tolman VI solution, while singular at the center, yields comparable corrections for most of the fluid sphere radius. A detailed analysis of the repulsive or attractive nature of these curvature corrections for a local observer is provided for each scenario. These results highlight the role of spacetime geometry in shaping quantum interactions and provide a foundation for future studies of nuclear interactions within the context of relativistic stars.

Figures (11)

• Figure 1: Leading order diagram for the process $\Phi+\Phi \rightarrow \Phi+\Phi$.
• Figure 2: Example of choice of coordinate system for two particles interacting via an interacting potential dependent on local position $\vec{r}$, in the interior of a spherically symmetric object of radius $r_\star$, around some point in the position $\vec{r'}$ relative to the metric origin.
• Figure 3: Diagram of vectors and directions given by a spherical metric at point $x'$, depicting our choice for the local unit vectors orientations and labeling.
• Figure 4: Values of $R_1$ and $R_2$ depend on $r_\star/r_s$, that is, the compactness. For this example, a model of $M=2\,M_\odot$ was considered.
• Figure 5: Values of the magnitude of the curvature corrections for the Yukawa potential for a range of values of $r'/r_\star$ for the object Pulsar J0740+6620 with mass $M=2.1 M_\odot$ and a radius $r_\star=12.32\ \mathrm{km}$.