Tests of general relativity in pseudo-Newtonian approach

Abstract

We investigate the extent to which pseudo-Newtonian gravitational potentials can reproduce classic tests of general relativity without resorting to full general relativistic formalisms. This is useful for the researchers seeking intuitive insight into relativistic gravity. Focusing on the perihelion precession of Mercury, gravitational redshift, and gravitational light bending, we derive analytical expressions for orbital precession and demonstrate that, with suitable physically acceptable parameters, pseudo-Newtonian approaches can accurately reproduce the observed perihelion advance and gravitational redshift. However, we confirm that no single potential consistently captures all relativistic effects. In particular, while certain parameters yield agreement with general relativity for planetary motion and redshift, they fail to reproduce gravitational lensing over a broad range of impact parameters. Our results highlight both the usefulness and limitations of pseudo-Newtonian methods in modeling gravitational phenomena. Although the pseudo-Newtonian approach cannot serve as universal substitutes for general relativity, especially in strong-field regimes, it provides valuable semi-analytical insight and pedagogical simplicity. Our results indicate the usefulness of the pseudo-Newtonian approach to uncover more complicated phenomena involved with strong field gravity in possibly modification to general relativity.

Figures (3)

• Figure 1: Variation of gravitational redshift as a function of radial coordinate for the ABN pseudo-Newtonian potential with different $\beta$. The exact solution from GR (black dashed) is also shown which matches with high accuracy for $\beta=0.5$ (solid magenta). All models converge at large distances, as expected, but diverge significantly in the strong-field regime near the black hole horizon.
• Figure 2: Variation of the deflection angle $\delta$ as a function of radial coordinate $R$ (distance of closest approach) for the ABN potential with different $\beta$. The exact solution from GR (black) is shown for comparison. While there are respective specific $R$ for each $\beta$ where ABN potential matches with GR, there is no generic trend. Also for standard values like $\beta=2,~3$, ABN based results diverge significantly from the GR predictions in the weak and strong field regimes.
• Figure 3: Variation of the deflection angle $\delta$ in equatorial plane as a function of radial coordinate $R$ (distance of closest approach) for the Mukhopadhyay potential with different spin parameters. Solid lines represent the exact Kerr metric predictions; dashed lines correspond to the Mukhopadhyay pseudo-Newtonian potential. The spin parameter $a$ varies from -0.75 to +0.75.