Why is the Universe the way we observe it? Why does space have three dimensions, while time has only one? And is there a mathematical way to prove that travel into the past is impossible, at least under certain conditions? Two independent physics papers published on the arXiv preprint server offer a way to approach these questions through the geometry of spacetime. One explores why the familiar structure of the Universe might emerge statistically, while the other offers a remarkably short proof that time loops are impossible for an entire class of geometries.

Where Three Dimensions of Space and One of Time Come From

In everyday life, we take the structure of the Universe for granted: we can move forward and backward, left and right, up and down, but we cannot choose the direction in which we move through time. In physics, this is described by four-dimensional spacetime, consisting of three spatial dimensions and one temporal dimension.

Yet a fundamental question remains: why this particular configuration? Must a theory of the Universe assume a single dimension of time from the outset, or could this structure emerge naturally from more fundamental geometric principles?

A group of physicists from Belgrade set out to investigate this question. In a study posted on arXiv, the researchers sought to determine which forms of spacetime arise most frequently when constructing geometric models.

To do this, they used a simplified model of quantum gravity in which spacetime is represented as a collection of four-dimensional simplices—geometric building blocks analogous to triangles in two dimensions or tetrahedra in three dimensions.

The researchers varied the edge lengths of these building blocks and examined the resulting types of geometry. In other words, they tested whether the familiar structure of spacetime might be statistically favored over many other possibilities.

The result was intriguing: under certain conditions, a geometry with three spatial dimensions and one temporal dimension emerged in approximately 85% of cases.

But there was an important caveat. This result occurred when the model allowed only spacelike edges and used a particular range of edge lengths. When both spacelike and timelike edges were allowed, the picture changed: a geometry with two spatial and two temporal dimensions emerged most frequently, accounting for 72–81% of cases.

Thus, the familiar structure of the Universe was favored only under certain mathematical assumptions. The study does not prove that our spacetime originated this way, but it shows that its structure may be linked to the geometric properties of a more fundamental theory.

Why Travel into the Past Conflicts with Geometry

The second paper addresses another mystery of spacetime: the possibility of traveling into the past.

General relativity has mathematical solutions that allow so-called closed timelike curves. These are hypothetical trajectories along which an observer could travel at a speed no greater than the speed of light and eventually return to the original event. In effect, this means the possibility of ending up in one's own past.

Physicist Hongsheng Zhang of China's University of Jinan proposed a mathematical criterion that rules out such trajectories for a particular class of spacetime geometries.

The most striking feature of his work is the brevity of the proof. Its central reasoning fits into just three lines.

Zhang used a mathematical tool known as the eikonal equation, which is used, among other things, to describe the propagation of light. With it, the physicist considered special functions that can be defined throughout the region of spacetime under study and have a nonzero null gradient.

If such a function exists and remains single-valued, its value changes strictly monotonically along any timelike trajectory.

This leads to a simple mathematical contradiction for travel into the past. If an observer returns to the original event, the function's value at the endpoint must match its value at the starting point. Yet it must have changed monotonically throughout the trajectory. Both conditions cannot be satisfied simultaneously.

Consequently, closed timelike curves cannot exist in such geometries.

Notably, this proof did not require specifying additional properties of matter or imposing restrictions on its energy characteristics.

A Prohibition, but Not for the Entire Universe

Despite the striking brevity of the proof, Zhang's work does not mean that physics has definitively settled the question of time travel.

The proposed criterion applies only under certain conditions: the required function must exist throughout the region of spacetime being considered, remain single-valued, and possess the necessary mathematical properties.

If these conditions are not met, the proof does not apply. It therefore provides not a universal ban on time loops, but a way to identify geometries in which they are impossible.

It is also important to note that both papers are currently posted on arXiv and have not undergone scientific peer review. Their findings should be regarded as preliminary research rather than definitively established properties of the Universe.

Geometry Rather Than Science-Fiction Scenarios

At first glance, these two papers address entirely different questions. One attempts to explain why our world has exactly three spatial dimensions and one temporal dimension, while the other investigates the mathematical conditions under which returning to the past is impossible.

But they share a common approach: the researchers seek to understand the fundamental properties of spacetime through its geometry, without relying on familiar notions of space and time as something given in advance.

For now, these studies do not provide definitive answers to questions about the origin of the Universe's structure or the nature of time. However, they show just how much can be learned by studying the mathematical properties of spacetime itself. And sometimes, ruling out a truly exotic possibility really does require nothing more than a proof that fits into three lines.