Mixed modified-gravity--modified-inertia MOND theories

In the vast, dark silence between the stars, a quiet mystery has lingered for decades. When astronomers look at spinning galaxies, they see something that defies the rules of gravity as we know them. The stars on the outer edges of these cosmic islands are moving far too fast to be held in place by the visible matter of the galaxy alone. According to our standard understanding of physics, these stars should fly off into the void, yet they remain in orbit. To solve this, most scientists have proposed that invisible "dark matter" is providing the extra gravitational pull needed to keep the stars in line. However, a different school of thought suggests that the problem lies not with missing matter, but with our understanding of gravity itself. This alternative idea, known as Modified Newtonian Dynamics, or MOND, proposes that gravity behaves differently when the forces involved are extremely weak, such as those found in the deep reaches of a galaxy. Instead of a hidden substance, the universe simply follows a different set of rules in these low-acceleration environments.

For years, researchers have tried to build a complete theory of this modified gravity. Most attempts have focused on changing how gravity works, treating the force of attraction as the only thing that needs adjustment. Others have suggested that the resistance objects feel when they try to move—what physicists call inertia—might be the part that changes. But a new paper by Mordehai Milgrom at the Weizmann Institute of Science suggests that the truth might be more complex. He proposes that in a fundamental theory of this phenomenon, both gravity and inertia might need to be modified at the same time. He does not claim to have found the final answer, but rather maps out a vast landscape of possible theories where these two changes work together, showing how they could produce the same strange observations we see in the sky without requiring any dark matter.

Milgrom's work begins by looking at the mathematical "action" that describes how the universe operates. In physics, this is a way of calculating the path an object takes by balancing its energy of motion against the forces acting upon it. In the standard Newtonian view, there are three parts to this calculation: the energy of the gravitational field, the interaction between matter and that field, and the energy of the moving matter itself. Traditional MOND theories usually tweak just one of these parts. Some change the gravitational field, while others change the motion of the matter. Milgrom asks what happens if we change both the gravitational field and the motion of the matter simultaneously, while leaving the interaction between them untouched. He constructs families of theories that blend these two modifications, creating a spectrum of possibilities that ranges from pure gravity modification to pure inertia modification.

The key to his approach is a single number that acts like a dial, shifting the theory from one extreme to the other. At one end of the dial, the theory modifies only gravity, which is the approach most scientists have taken so far. At the other end, it modifies only inertia. In the middle, the theory modifies both. Milgrom shows that for any setting on this dial, the combined effect of these two changes can still reproduce the primary success of MOND: the ability to explain why stars in the outer regions of galaxies move at constant speeds regardless of their distance from the center. This is a crucial finding because it demonstrates that the phenomenon we observe does not depend on a single, specific way of modifying physics. Instead, it can emerge from a wide variety of underlying mechanisms, as long as the changes to gravity and inertia are balanced correctly.

One of the most interesting discoveries in this paper is that neither the modified gravity nor the modified inertia works on its own to create the MOND effect. If you take a theory that changes only gravity and pair it with standard Newtonian motion, you do not get the right results. Similarly, if you change only the motion of the particles but keep standard gravity, the theory fails. It is only the specific combination of the two that creates the observed behavior. This suggests that if a deeper, more fundamental theory of the universe exists, it might involve a single underlying principle that alters both how gravity acts and how objects resist changes in motion. It is a reminder that the universe might not be as simple as changing just one rule; it might require a coordinated shift in how space, time, and matter interact.

Milgrom also explores how these mixed theories behave in the deep, quiet regions of the universe where gravity is weakest. He finds that these theories naturally lead to a specific relationship between the mass of a galaxy and the speed at which it spins. This relationship, known as the mass-asymptotic-speed relation, is one of the most robust observations in astronomy. His work shows that this relationship holds true across his entire spectrum of mixed theories, regardless of how the dial is set. This reinforces the idea that the connection between mass and speed is a fundamental feature of the universe, one that appears whether we modify gravity, inertia, or both. However, he also notes that while these theories agree on the big picture, they might predict different outcomes for smaller, more specific details, such as how gravity behaves within our own solar system or how galaxies interact with their neighbors.

The paper serves as a heuristic tool, a way of exploring the boundaries of what is possible without claiming to have found the definitive theory. Milgrom admits that these models are somewhat artificial because they do not yet stem from a single, unifying idea that explains why both gravity and inertia would change together. He is not offering a finished product but rather a map of the territory. By showing that a wide range of mixed theories can work, he challenges the community to look beyond the current favorites and consider that the solution to the dark matter mystery might lie in a more complex interplay of forces. The work suggests that the path to understanding the universe may require us to rethink not just one aspect of physics, but the very nature of motion and attraction as a unified whole.

Ultimately, this research highlights the diversity of ways the universe could be constructed. It shows that the strange behavior of galaxies does not point to a single, unique solution, but rather to a family of possibilities. Whether the universe chooses to modify gravity, inertia, or a combination of both remains an open question. What is clear, however, is that the answer likely involves a delicate balance between how the cosmos pulls on objects and how those objects respond to that pull. Until we can observe the universe in a way that distinguishes between these different possibilities, the mystery of the missing mass—or the missing rules—will continue to drive the search for a deeper understanding of the laws that govern our existence.