An off-shell conformally invariant Galilean Weyl tensor

Gravity is the force that keeps our feet on the ground and the planets in their orbits, but how we describe it depends entirely on the speed at which we are moving. For centuries, scientists have used two different rulebooks. The first, known as Newtonian gravity, works perfectly for slow-moving objects like cars, planets, and falling apples. It treats space as a fixed stage and time as a universal clock that ticks the same for everyone. The second rulebook, Einstein's general relativity, takes over when things move very fast or when gravity becomes incredibly strong. In this view, space and time are woven together into a flexible fabric that bends and stretches.

For a long time, these two descriptions seemed like separate worlds. However, modern physics has found that many phenomena, from the behavior of super-cooled atoms to the theoretical edges of black holes, can be understood by looking at the "slow-motion" version of Einstein's universe. This is the realm of Galilean geometry, a mathematical framework that describes gravity without the complications of high speeds. A central puzzle in this field has been how to define a specific geometric object called the Weyl tensor. In the fast-moving, relativistic world, this tensor is a crucial tool for understanding how gravity ripples through empty space and how the shape of the universe is distorted by mass. It is the part of gravity that remains even when there is no matter present. Physicists have long wanted to know: what does this object look like when we slow everything down to a standstill?

A team of researchers has now provided a clear, new answer to this question. They have constructed a definition of the Weyl tensor that works specifically within the slow-motion, Galilean framework, and they have done so in a way that does not rely on the specific equations that govern how matter moves. In physics, calculations that depend on the specific laws of motion are called "on-shell," while those that describe the geometry itself, regardless of the laws, are called "off-shell." By creating an off-shell definition, the researchers have isolated the pure geometric structure of Galilean gravity, separating it from the specific dynamics of any particular theory. This allows them to see the underlying shape of the universe in a way that was previously obscured.

The researchers approached this problem by taking the known definition of the Weyl tensor from Einstein's relativity and carefully shrinking the speed of light to infinity, a process known as the Galilean limit. This is not just a simple subtraction; it requires navigating a minefield of mathematical terms that behave differently as the speed limit of the universe is removed. They discovered that the resulting Galilean Weyl tensor splits into two distinct parts, which they call the electric and magnetic components. The electric part describes how space is stretched and squeezed, similar to how a rubber sheet deforms under a heavy weight. The magnetic part, however, is more subtle; it relates to how the flow of time and the rotation of space interact.

One of the most significant findings of the paper is that the magnetic part of this tensor does not automatically vanish, as it does in the standard, textbook version of Newtonian gravity. In classical Newtonian theory, the magnetic part is zero because the theory assumes a very specific, rigid structure of space and time. The new definition shows that if you relax those assumptions and look at the geometry more broadly, this magnetic part can be non-zero. This implies that the universe could have a "magnetic" gravitational character even in slow motion, provided the observers measuring it are moving in a certain way.

The team then asked a critical question: under what conditions does this magnetic part disappear, returning us to the familiar, simple Newtonian world? They found that it vanishes only if there exist specific observers who are "vorticity-free," meaning they are not rotating relative to the fabric of space, and whose expansion of space follows a very particular pattern. The authors propose that the existence of such observers should be a defining requirement for any theory to be called truly "Newtonian." In other words, a theory of slow-motion gravity is only Newtonian if it allows for observers who see no magnetic gravitational effects. This is a new, geometric way of distinguishing between a general Galilean theory and the specific Newtonian theory we use every day.

To reach these conclusions, the researchers also had to solve a long-standing problem regarding how to define a connection—a mathematical tool that tells you how to move vectors from one point to another in a curved space. In the relativistic world, there is a unique, natural way to do this. In the Galilean world, however, this uniqueness was lost unless extra, artificial structures were added to the theory. The team proved that there is, in fact, a unique way to define this connection using only the basic ingredients of Galilean geometry and a choice of observer, without needing to invent new fields or forces. This unique connection is robust and does not change depending on how fast the observer is moving, a property known as boost invariance.

The work also clarifies why previous attempts to define this tensor sometimes seemed to contradict each other. Some earlier studies found a tensor that was invariant under a different kind of transformation, which the authors explain is because those studies were looking at the geometry only when the equations of motion were already satisfied. By staying "off-shell," the new definition reveals a richer structure that includes spatial curvature, a feature that was often ignored or assumed to be zero in earlier work. This spatial curvature is essential for the new tensor to be conformally invariant, meaning it keeps its shape even if the entire universe is scaled up or down.

Ultimately, this paper provides a solid foundation for understanding gravity in the slow-motion limit. It shows that the geometry of a non-relativistic universe is more complex and flexible than previously thought, capable of supporting magnetic-like gravitational effects and spatial curvature. By defining exactly when a theory becomes "Newtonian," the authors have drawn a sharper line between general Galilean gravity and the specific laws of Newton. This clarity opens the door for future research into how gravitational waves might behave at low speeds, how to classify different types of non-relativistic universes, and how these geometric structures might appear in the study of exotic materials and black holes. The result is a cleaner, more complete picture of the geometry that underlies the world we see around us, where the speed of light is effectively infinite.