Do Galaxies Obey Different Laws of Gravity?

Written By: Hargun Cheema

Introduction

In the year 1939, Horace Babcock was studying galaxies. The galaxy his telescope was pointing at was Andromeda. While studying its rotation, he noticed that the outer regions of the galaxy appeared to contain far more mass than the luminosity that was coming off of it. Rather than questioning the missing mass in the galaxy, he made his own alternative explanations. Around the same time period, a similar conclusion was made by Jan Hendrik Oort when examining the galaxy NGC 3315. Astronomers believed that every system was governed by classical gravity. Similar to our Solar System, where most mass is concentrated at the center, and as we move away from the center, the orbital velocity of the planets decrease. Until the late 1960s and early 1970s, Vera Rubin working with Kent Ford, ran precise measurements of rotation curves of the outer edge of galaxies. And, what they found astonished everyone. The stars on the outer edge of the galaxies were orbiting as fast as the stars at the center. Rubin’s results implied that galaxies contained vastly more mass than what the light coming from the galaxies suggest. Her calculations suggested that 90% of the mass was invisible.

This work was further revived and confirmed the earlier insight of Fritz Zwicky, who five decades ago studied and found that in the Coma Cluster the galaxies were moving too rapidly as though being held by an invisible matter. And while applying the Virial Theorem in 1933, Zwicky coined the term “dark matter” which was used to describe the missing mass binding the cluster. Although initially, his conclusions were just ignored at that time, today it is a widely accepted concept of dark matter halos surrounding galaxies.

However, this is not the only hypothesis for this discrepancy. There are other much more radical questions like: what if gravity behaves differently at galactic scales? These questions motivated alternative approaches that don't include dark matter, such as Modification Newtonian Dynamics (MOND) and other modified gravitational theories.

Newtonian Dynamics

To understand these theories, first, we would need to understand Newtonian Dynamics. It is a mathematical model that explains the motions of the objects in the universe. It is based on the general principles explained in the “Philosophiæ Naturalis Principia Mathematica” referred to as Principia, written by Isaac Newton. The Newtonian Dynamics framework is a fundamental part of classical physics. It helps to determine the motion of a body by knowing its initial conditions.

It is applicable when Newton’s laws work, which is when objects move slowly compared to light, and allowing inertial frames to be related using Galilean transformation.

Galilean transformation is when: -

-        Speed simply add or subtract

-        Time is the same for everyone

-        Space and time are separate

(Works at only low speeds. At very high speeds you need Lorentz transformations from special relativity)

But, despite Newtonian Dynamics being historically significant for key figures like Galileo, Descartes, and Huygens contributing to Newton’s theories, it is recognized as an incomplete theory of the world, as it has its own limitations especially when the speed approaches the speed of light. Another notable area of its limitation is when explaining the motion of the galaxies without assuming the presence of a hidden mass. This led to the proposal of MOND.

Space-time and Gravitational Modifications 

Space-time and gravitational modifications refer to a category of theories and concepts that propose new understandings of the fundamental nature of space, time and gravity. These theories involve modification of Einstein’s General Relativity (GR). GR explains that instead of gravity being a force, it is a manifestation of the curvature of space-time which is caused by massive objects. This visualization helped us explain numerous predictions which are experimentally validated, such as the bending of light by massive objects also known as gravitational lensing and existence of gravitational waves.

Despite its success, current GR is incompatible with quantum mechanics, which suggests that our understanding is still incomplete. This further motivates physicists to modify the GR. There are different types of theories in Space-time and Gravitational modifications; they are ways to alter the standard model of gravity. One of these are Modified Newtonian Dynamics (MOND) and TeVeS (which is a relativistic generalization of MOND).

Some of these modifications involve extra dimensions or new fields, going beyond the standard four dimensions of space-time. For example, some theories consider seven-dimensional space-time, where models suggest that gravitational interactions propagate through all seven dimensions. Such models may also explain the accelerated expansion of the Universe without invoking dark energy. Whereas, in some modifications there are additional scalar and vector fields alongside the tensor field of General Relativity. This is seen in theories like Tensor–Vector–Scalar gravity (TeVeS).

Another interesting modification in advanced gravity is space-time with torsion. It is a modification of GR, where the space-time manifold is not only curved but also "twisted". This concept introduces a degree of freedom to the gravitational theory. While curvature is related to mass and energy, torsion is associated with the intrinsic spin of matter. Under ordinary conditions torsion produces successful predictions of GR, but under extreme environments, such as the early universe or very high-density regions, it may lead to new behaviors. And, since it interacts with the intrinsic spin of particles, torsion provides a way for both gravity and quantum theory to be unified.

Most of these theories are not designed to explain galaxy rotation directly, but they show that gravity itself may not be final. Now that we have seen alternative theories that do not involve dark matter, let us look at space-time modification theory that could potentially explain why galaxies rotate differently.

Modified Newtonian Dynamics (MOND)

MOND is a theoretical framework proposed to address these discrepancies between observed galactic dynamics and predictions of Newtonian gravity, serving as an alternative to the hypothesis of dark matter. It was first proposed by Mordehai Milgrom in 1983. This theory is a modification of Newton’s second law of motion. Newton’s law works great when gravity is strong, like earth revolving near the sun, which means high acceleration. But it breaks down when gravity is extremely weak, like the far-out edge of the galaxy where gravity is weak, which means low acceleration.

 MOND is a theoretical framework proposed to address these discrepancies between observed galactic dynamics and predictions of Newtonian gravity, serving as an alternative to the hypothesis of dark matter. It was first proposed by Mordehai Milgrom in 1983. This theory is a modification of Newton’s second law of motion. Newton’s law works great when gravity is strong, like earth revolving near the sun, which means high acceleration. But it breaks down when gravity is extremely weak, like the far-out edge of the galaxy where gravity is weak, which means low acceleration.

MOND suggests that below a certain acceleration, the force of gravity deviates from Newton’s law, which explains the unexpectedly high rotational speeds of stars in galaxies.

This means that objects feel more gravitational pull than Newton’s law predicted at very low acceleration.

Milgrom’s law is a very important aspect of MOND, as it introduces critical constant acceleration (a₀). The critical acceleration,( a₀), appears across independent galaxies, hinting at a universal scale. It states that when the acceleration falls below the value of approximately 10^-8cm/s² or 1.2 x 10^-10m/s² Its effective gravitational attraction departs from Newtonian predictions.

The equation Milgrom’s law is the modified version of F = ma, where force (F) is equal to mass (m) times acceleration (a). MOND’s modification is F = mμ(a/a₀)a, where μ(x) is interpolating function and x = a/a₀. This interpolating function is crucial in MOND, as it connects the general Newtonian regime at high accelerations where a >> a₀ (much greater than), μ(a/a₀) ≈ 1, and if μ(x) would be equal to 1, we would be left with F = ma, which means Newtonian dynamics work. For low accelerations where a << a₀ (much less than), μ(a/a₀) ≈ a/a₀, which becomes F = mμ(a/a₀)a.

This equation shows why in MOND, gravity does not weaken as quickly with distance. When gravitational acceleration becomes very small, gravity changes behavior and falls off more slowly. In MOND, at very large distances where accelerations are extremely small, gravity falls off closer to 1/r, rather than the 1/r² predicted by Newtonian gravity. This allows stars far from a galaxy’s center to remain in the fast and stable orbits, without the need of dark matter. Though initially, MOND was proposed to explain flat rotation curves of spiral galaxies without invoking dark matter, it has also been successful at explaining the Tully-Fisher Relation. An empirical relationship between a spiral galaxy’s luminosity (or baryonic mass) and its rotational velocity.

Despite these successes, MOND faces challenges at larger cosmological scales. It struggles to explain observation of galaxy clusters and the cosmic microwave background without invoking additional assumptions or extension. As a result, MOND is still an area of active research, with ongoing efforts to test whether it represents a fundamental modification of gravity.

Conclusion

The unexpected behavior of galaxies rotating fast presents one of the most persistent challenges in modern physics. Observations show that stars in the outer regions of galaxies move too quickly to be held together by the gravity of visible matter alone. To understand this, astronomers have largely embraced the idea of dark matter. This explanation has been highly successful across cosmology, yet it remains indirect. Modified Newtonian Dynamics offers fundamentally different perspectives. Rather than adding new matter, MOND proposes that gravity itself changes behavior at low accelerations. This simple adjustment produces a key feature of galactic rotation and reveals a consistent acceleration scale shared by many galaxies. But, MOND does not replace dark matter as a complete explanation of the universe, its successes in solving the discrepancy in galaxy dynamics is not random, but a repeatable pattern across many different galaxies. Whether the solution lies in unseen matter, MOND, or a deeper theory that unites it all is still yet to be known. What galaxy rotation curves have made clear is that our current formulas may not operate identically on all scales. It tests our understanding of nature’s most fundamental laws.