In the late 1800s, Newton's gravitational theory faced significant challenges. While it accurately predicted the orbits of nearly all planets, Mercury, the closest planet to the Sun, exhibited an anomaly. First identified in 1845, this deviation was measured at approximately 42 arcseconds per Julian century. This seemingly minor discrepancy hinted at deeper issues within the framework of classical physics.

Various theories emerged to explain this anomaly. Some suggested the presence of an undiscovered inner planet exerting gravitational influence, while others recommended slight modifications to Newton's laws. Ultimately, however, no additional planets were found, and adjustments to Newton’s framework complicated the understanding of gravity rather than clarifying it.

Albert Einstein's work transformed this debate. By rethinking concepts of time, space, and matter, he developed a theory that addressed the inconsistencies while simplifying to Newton's equations under specific conditions, such as weak gravitational fields and lower velocities. The proximity of Mercury to the Sun resulted in a warping of space that Newton's theory could not adequately account for.

The concept of gravity faced further scrutiny in the 1930s with the observation of flat galactic rotation curves that defied predictions based on visible matter. Studies of distant galaxies, using light and radio emissions from hydrogen, revealed that these galaxies did not rotate like solar systems. Instead, their outer regions rotated at similar speeds to their inner sections, suggesting a rigid structure rather than the expected variation.

If solar systems seem to align closely with Newton’s principles, why would entire galaxies behave differently? Astrophysicists proposed the existence of dark matter—an unseen substance that interacts with gravity but not with light. This hypothesis suggests that dark matter must differ fundamentally from ordinary matter, as multiple candidates have been proposed, including axions, sterile neutrinos, and Weakly Interacting Massive Particles (WIMPs). However, none have emerged as a definitive explanation.

Some researchers argue that the search for dark matter may be misplaced and that the laws of gravity require a comprehensive revision, similar to the paradigm shift that Einstein's theory brought about in 1915. As galactic rotation curves exhibit characteristics not explained by existing models, a modification to gravitational theory that accounts for varying distances may be necessary.

One alternative is Modified Newtonian Dynamics (MOND), proposed by Israeli physicist Mordehai Milgrom in 1983. MOND modifies gravitational acceleration based on its strength. For high accelerations—like those within solar systems—the theory aligns with Newton's laws. However, for very low accelerations, it suggests a different relationship where force becomes proportional to the square of acceleration. This adjustment leads to flat rotation curves for distant objects, aligning well with the Tully-Fisher relation, which connects a galaxy's visible mass to the fourth power of its velocity.

While MOND effectively addresses some observed phenomena, dark matter theories based on density profiles like the Navarro-Frenk-White (NFW) struggle to explain the Tully-Fisher law without extensive fine-tuning. This becomes particularly evident in low surface brightness (LSB) galaxies, where the expected dominance of dark matter is not observed.

Cosmological simulations indicate a hierarchy of dark matter halos, with multiple smaller halos surrounding larger ones. However, this pattern isn't consistently observed in our universe, raising questions about the existence of these smaller halos.

Given the challenges facing both dark matter theories and MOND, researchers have begun exploring connections to a more robust theory of gravity. MOND excels in empirical predictions but falls short of providing a comprehensive framework for understanding multiple-body interactions and the curvature of space-time as described by Einstein.

A potential avenue involves AQUAL (A Quadratic Lagrangian), a theory that respects conservation laws. AQUAL can reconcile Newtonian dynamics within stars while explaining MOND-like behavior in galactic contexts. It also accounts for the external field effect, confirmed in 2020, which demonstrates that clusters subjected to high external accelerations exhibit near-Newtonian rather than MONDian dynamics.

Despite these advancements, AQUAL and similar theories encounter limitations, particularly in explaining gravitational lensing—an area where dark matter models thrive. The introduction of additional gravitational fields, like vector potentials, may help address this, but previous attempts have struggled to maintain relativistic consistency.

In 2004, Jacob Bekenstein presented the Tensor-Vector-Scalar (TeVeS) theory, which aimed to merge MOND with relativity. However, questions remain about its compatibility with solar system dynamics and its ability to replicate all observed phenomena.

Both MOND and TeVeS theories, along with dark matter explanations, share a common challenge: they are largely empirical and lack a unifying framework. The scientific community continues to seek a comprehensive theory that can elegantly resolve these outstanding issues—much like Einstein did over a century ago.

Update: In 2018, the detection of gravitational waves at LIGO, arriving simultaneously with light bursts, presented a significant challenge to dark matter emulators like TeVeS. These models predict differing behaviors for gravity and light waves, a disparity that the recent findings undermine. However, the implications for MOND and its generalizations may be less clear, leaving room for continued exploration in the quest for understanding the cosmos.