In our universe, space has three dimensions and time has one. Why aren't the numbers of spatial and temporal dimensions symmetric? In 1908, the mathematician Hermann Minkowski presented the concept of four-dimensional "spacetime," which unified space and time and gave later relativity theory an important mathematical foundation. But the fundamental question of why the metric, which defines distances and time intervals in spacetime, has a signature corresponding to three spatial dimensions and one time dimension still has no unified explanation.
Mihailo Djordjevic, Igor Prlina, and Marko Vojinović of the University of Belgrade in Serbia examined this question from the standpoint of discrete geometry. On October 22, 2026, they posted to arXiv the preprint "Minkowski metric signature as a statistical prediction of piecewise-flat quantum gravity". The team built a simplified quantum gravity model that divides four-dimensional spacetime into small geometric units, then numerically examined which metric signatures arise when the edge lengths of each unit are varied randomly. They found that under certain conditions, the same Minkowski-type signature as in real spacetime appears 85.34% of the time. However, changing the conditions on the edges makes other signatures dominant, so the results depend strongly on the model's assumptions.
データを表で見る
| 出現頻度 (%) | |
|---|---|
| ミンコフスキー型 (3,1,0) | 85.34 |
| 分割型 (2,2,0) | 8.3 |
| ユークリッド型 (4,0,0) | 6.36 |
The 85.34% figure does not represent the probability that the real universe has three spatial dimensions and one time dimension. What the study measured is how often hypothetical four-dimensional geometric blocks have a Minkowski-type local metric under specified conditions. The calculation also incorporates none of the dynamics that govern the real universe, such as gravity or particle interactions. To evaluate the result, it is necessary to distinguish between the structure of the model, the sampling conditions, and the physical elements that were deliberately omitted.
A statistical model that divides cosmic geometry into four-dimensional simplices
The idea of representing spacetime not as a smooth continuum but as a collection of small geometric elements goes back to "Regge calculus," proposed in 1961 by the Italian physicist Tullio Regge. In Einstein's general relativity, the curvature of spacetime is described by the metric tensor and curvature. In quantum gravity research, meanwhile, discretization methods have developed that approximate continuous spacetime as a combination of a finite number of flat simplices and study their geometric properties.
For example, a two-dimensional surface can be approximated by joining triangles, and three-dimensional space can be divided into tetrahedra. Extending the same idea to four dimensions gives the "4-simplex," a four-dimensional simplex with 5 vertices and 10 edges. In this model, the 4-simplex is treated as the basic building block of spacetime. The interior of each simplex is assumed to be flat, and its metric tensor and metric signature are determined by the combination of its 10 edge lengths.
A metric signature indicates how many of the metric tensor's eigenvalues are positive, negative, and zero. In real four-dimensional spacetime, the spatial and temporal directions have opposite signs, giving a structure generally called Minkowski-type. The team assigned random edge lengths to 4-simplices, computed the eigenvalues of the resulting metrics, and examined which signature each simplex fell into.
The simulations also tested the effect of changing the global shape of spacetime. They prepared three types of topology: a 4-ball with a boundary, a 4-sphere without a boundary, and a collection of roughly 100,000 mutually unconnected 4-simplices. For the 4-ball and the 4-sphere, they constructed 10 different triangulations each, and together with the disconnected simplex set they compared 21 geometric configurations. Each configuration contained about 100,000 4-simplices, and the calculations were repeated while varying the edge-length sampling conditions and cutoff parameters.
The computation was large in scale. With 30 edge-length conditions set for each of the 21 configurations, and 10 independent draws for each, the dataset totaled 6,300 sets. In addition, for each 4-simplex, the 5 tetrahedra and 10 triangles that compose it were also evaluated for metric signature using two independent methods. According to the paper, the total number of evaluations across 2-, 3-, and 4-dimensional simplices came to 20.16 billion. This does not mean that 20.16 billion independent four-dimensional spacetime blocks were generated; it is a sum of two kinds of evaluations across simplices of different dimensions.
To determine signatures, the team used two approaches: reconstructing the metric tensor from the edge lengths and counting the signs of its eigenvalues, and a geometric method using the Cayley-Menger determinant. The latter determines the signature from the sign relationships of determinants corresponding to a simplex and its sub-simplices, without having to compute eigenvalues directly. The team confirmed that the two results agree, validating the consistency of the computational methods. However, this agreement supports the correctness of the numerical calculation; it does not guarantee that the model accurately reproduces the real universe.
The model includes physically significant simplifications. In ordinary quantum gravity theories, when handling different spacetime configurations with a path integral, weighting based on factors such as the gravitational action is taken into account. The team, however, set the classical action to and drew each edge length from a uniform distribution over a specified range. They therefore did not calculate how spacetime evolves under the Einstein equations; they examined which signatures appear statistically more often among the geometrically allowed configurations.
This simplification is intentional. Quantum gravity research includes various approaches that try to derive physical structure from discrete spacetime, such as spin foam models in loop quantum gravity and causal dynamical triangulations (CDT). In this study, the dynamical effects that matter in such theories were set aside for the time being, and the team examined whether a particular signature could be selected by the combination of edge lengths and the integration measure alone. If a geometric bias exists even before dynamics is introduced, it could offer a clue to the origin of the metric signature of real spacetime. Whether that bias would survive once actual physical laws are incorporated is a separate question.
Using only spacelike edges, the Minkowski type reaches 85.34%
The most notable result of the study is that the Minkowski-type signature, with three spatial dimensions and one time dimension, appeared with a frequency of . However, this value was obtained with a particular edge-length distribution and cutoff parameter, and it is not a property common to all four-dimensional geometries. To understand its meaning, we first need to look at the classification of metric signatures and the edge-length sampling conditions.
The paper writes the metric signature as $(p,q,r)$, where $p$ is the number of positive eigenvalues, $q$ the number of negative eigenvalues, and $r$ the number of zero eigenvalues. Under the sign convention adopted, the Minkowski type with three spatial dimensions and one time dimension corresponds to $(3,1,0)$, and the Euclidean type with only spatial directions corresponds to $(4,0,0)$. The signature $(2,2,0)$, with two spatial and two time dimensions, is called the "split signature," and sign-reversed forms such as $(1,3,0)$ and $(0,4,0)$ are also possible. Note that the number of time directions here is determined by the signs of the metric, and is a separate concept from the "arrow of time," the direction in which time runs from past to future.
The team set a condition in which all edges are spacelike and each edge length is drawn from a uniform distribution. In the calculation, edge lengths were made dimensionless, and a parameter $k$ representing the ultraviolet cutoff was varied from 0 to 9. This parameter is defined using the lower and upper edge-length bounds and as . If the minimum edge length is assumed to be the Planck length (about m), $k=2$ corresponds to drawing each edge from the range to $5l_p$.
The frequency of the Minkowski type peaked around this $k=2$. For the set of unconnected 4-simplices, averaging over 10 independent draws gave the following frequencies for each signature. The values listed as errors are sample standard deviations across repeated draws.
| モデルの条件 | 主なシグネチャ | 出現頻度 | 解釈上の注意点 |
|---|---|---|---|
| 空間的な辺のみ、、$k=2$ | ミンコフスキー型 $(3,1,0)$ | 力学的作用を除外した局所的な幾何学の出現頻度 | |
| 同条件 | 分割型 $(2,2,0)$ | 時間方向を2つ持つ計量も一定割合で出現 | |
| 同条件 | ユークリッド型 $(4,0,0)$ | 時間的な符号を持たない計量も出現 | |
| 空間的・時間的な辺を混合して抽出 | 分割型 $(2,2,0)$ | 約72～81% | 辺の条件を変えると時間2次元のシグネチャが優勢になる |
| 時間的な辺のみを抽出 | 符号反転ミンコフスキー型 $(1,3,0)$ | 約85%（$k=2$） | 空間的な辺のみの場合と符号を反転した関係にある |
Under these conditions, besides the Minkowski type, the split type appeared 8.30% of the time and the Euclidean type 6.36%. The sign-reversed Minkowski type $(1,3,0)$ and the Euclidean type $(0,4,0)$ were both 0% in the $k=2$ calculation. However, $(1,3,0)$ is not always impossible when using only spacelike edges; it was observed, at an extremely low frequency, in the $k=0$ calculation. The dominance of the Minkowski type must therefore be distinguished from the complete exclusion of other signatures.
The range of possible edge lengths is related to why the Minkowski type becomes more frequent. As $k$ increases, the range of edge lengths narrows and more 4-simplices have edges of nearly equal length. Such simplices can be embedded in Euclidean space, so the Euclidean type becomes more frequent once $k$ reaches 6 or higher. When $k$ is small, on the other hand, edge lengths vary more and the split type also becomes more frequent. The Minkowski type exceeds 85% as a result of choosing $k$ around 2 within this landscape, and this is not a universal proportion independent of that value.
The setting near the Planck length also calls for caution. The Planck length is a representative length scale at which quantum gravity effects are thought to become important, but it has not been experimentally confirmed that real spacetime has a discrete structure with the Planck length as its minimum unit. The model also assumes that edge lengths are drawn from a uniform distribution, and there is no evidence that the microscopic geometry of the real universe follows the same distribution. The results must be interpreted as geometric frequencies under these assumptions.
Furthermore, even if about 85% of simplices are of the Minkowski type, that alone does not mean the macroscopic universe as a whole has the same metric. The remaining roughly 15% includes the split type, with two time directions, and the Euclidean type, with no timelike sign. Under what boundary conditions simplices with different signatures can be joined when adjacent, and whether they can form continuous spacetime, must be verified separately. This simplex-by-simplex frequency calculation does not show that the consistent spacetime structure observed in the real universe would form naturally.
Allowing timelike edges makes a two-time-dimension structure dominant
What matters in this study is not only that the Minkowski type appeared with a high frequency of 85.34%. It also became clear that changing the edge-length sampling conditions greatly changes which signature dominates. In particular, the results when timelike edges are allowed, in addition to spacelike ones, show how important the model's premises are.
In a relativistic metric, spacelike and timelike intervals have opposite signs. In this calculation, spacelike edge lengths are represented as real numbers and timelike edge lengths as purely imaginary numbers, so that the squared edge lengths are positive or negative respectively. This is mathematical notation for handling the sign of the metric, and is not a claim that physical distance itself becomes imaginary in the ordinary sense. In addition to the condition where all edges are spacelike, the team examined conditions where all are timelike and where the two are mixed.
When spacelike and timelike edges were mixed, each edge was assigned to one or the other with equal probability, and its length was drawn uniformly from the specified range. Across the entire evaluated range of $k$, the dominance of the Minkowski type was then lost, and the split type $(2,2,0)$, with two spatial and two time dimensions, accounted for about 72–81%. For example, at $k=0$ the split type was , the Minkowski type , and the sign-reversed Minkowski type . Under this condition, the Euclidean and sign-reversed Euclidean types were extremely rare and almost never appeared.
When all edges were drawn as timelike, on the other hand, the result was the sign-reversed counterpart of the spacelike-only case. Around $k=2$, $(1,3,0)$ accounted for about 85%, a mirror-image relationship with the spacelike-only result in which $(3,1,0)$ dominated. According to the paper, if all edge lengths are multiplied by an imaginary unit and the sign of the metric tensor is reversed at the same time, the all-spacelike and all-timelike configurations correspond mathematically. The difference between the two therefore relates to the overall sign convention of the metric, and does not by itself mean that physically different universes are selected.
These results show that the bias toward the Minkowski type depends strongly on the condition of restricting all edges to be spacelike. No physical reason for excluding timelike edges has been established, and when spacelike and timelike edges are mixed, the split type, which differs from real spacetime, becomes overwhelmingly dominant. The 85.34% therefore cannot be treated as evidence that quantum gravity unconditionally selects Minkowski spacetime. What the team showed is that, when a particular domain of edge lengths is adopted, the Minkowski type becomes statistically dominant.
Relatively stable results were obtained, by contrast, for differences in topology and triangulation. Across the three types of configuration—the 4-ball with a boundary, the 4-sphere without one, and the set of unconnected 4-simplices—no large differences were seen in signature frequencies. Ten different triangulations were tried for each of the 4-ball and 4-sphere, and the results were nearly identical to those for the unconnected simplex set. However, there were small differences: the maximum sample standard deviation was 0.2% for the unconnected simplex set and 0.4% for the connected configurations.
This tendency is consistent with the fact that the metric signature of each 4-simplex is determined mainly by the 10 edge lengths that compose it. Adjacent 4-simplices share a tetrahedron, with 4 vertices and 6 edges, as their boundary, so their geometric properties are not completely independent. For example, if the metric of the shared tetrahedron is Minkowski-type, constraints arise on an adjacent 4-simplex being Euclidean-type. Even so, in this calculation the effect of such local correlations on overall frequencies was small enough to be buried in the variation from drawing edge lengths at random.
The team interprets this result as indicating that the signature of the local metric depends more strongly on the edge-length distribution of each simplex than on the global topology. However, only three topologies and a limited set of triangulations were compared, and other topologies, such as the four-dimensional torus, were not covered. It has not been proven that the same frequencies hold for any spacetime structure, and it is not known whether the same property would persist under different edge-length distributions or connectivity conditions.
The fact that local signature frequencies depend little on topology is also a separate matter from whether the geometry of the whole universe is independent of topology. The model contains no dynamics describing the expansion of the universe or the formation of large-scale structure, and how correlations between adjacent simplices behave over long distances is not fully understood. Even if the Minkowski type appears frequently in individual simplices, whether they can form a macroscopically consistent spacetime cannot be judged from these results alone.
The limits of a model without dynamics, and challenges ahead
The biggest caveat in evaluating this study is that it excludes the dynamical action of gravity and matter from the calculation. In general relativity, the distribution of matter and energy shapes the curvature of spacetime, and that geometry in turn affects the motion of matter. In quantum gravity theories too, weighting based on the action is important when evaluating the various configurations of spacetime. The University of Belgrade model, by contrast, sets the classical action to zero and examines only the frequency of geometric configurations allowed under the specified measure.
As a result, it is not known how the frequency of the Minkowski type would change if the dynamics of real gravity were incorporated. Weighting by the action might strengthen the bias toward the Minkowski type, or it might weaken it. The authors themselves state that whether this tendency would be maintained, strengthened, or suppressed once dynamical effects are added is a subject for future research. At this stage, the 85.34% frequency cannot be treated as a value predicted by an actual quantum gravity theory.
Another issue is the validity of the probability distribution and measure used to draw edge lengths. In this study each edge length is chosen from a uniform distribution, but the results might change if a uniform distribution were used on the square of the edge length rather than the length itself, or if a different physical measure were adopted. The condition of choosing spacelike and timelike edges with equal probability when mixing them is also not derived from real physical laws. It will be necessary to examine what measure is appropriate for quantum gravity theory and how much the distribution of signatures changes as a result.
The edge-length scale poses a similar problem. The 85.34% result corresponds to the condition in which the minimum edge length is assumed to be the Planck length and the maximum is five times that. But it is unconfirmed whether the Planck length is the actual minimum length of spacetime, and there is no guarantee that the same trend would appear if simplex sizes were set over a wider range. In particular, to derive macroscopic spacetime it will be important to verify the continuum limit when many tiny simplices are combined, and the conditions for connecting regions with different signatures.
Future verification will require not only the changes that occur when a dynamical action is introduced, but also comparisons using different measures and edge-length distributions and a greater variety of topologies and triangulations. It will also be necessary to examine correlations between adjacent simplices in detail and confirm whether the local dominance of the Minkowski type leads to a consistent spacetime structure at larger scales. The paper is a preprint released on October 22, 2026, so peer-review evaluation and replication by independent research teams will also be important.
Why spacetime has a structure of three spatial dimensions and one time dimension is a long-standing unsolved problem in fundamental physics. It has been studied from various frameworks, including the compactification of extra dimensions in string theory and causal dynamical triangulations. This study adds the perspective of the statistical properties of geometric configurations, and its significance lies in numerically showing that a particular metric signature can already be dominant before dynamics is introduced.
However, this result does not explain "why the universe has only one dimension of time." What was shown is that, in a simplified model adopting a particular edge-length distribution and cutoff condition, a local metric with three spatial dimensions and one time dimension appears with high frequency. In fact, an important constraint also emerged: when timelike edges are allowed, the split type with two time dimensions becomes dominant. Whether the bias toward the Minkowski type is maintained in models that incorporate the dynamics of gravity and an appropriate measure will be a key test of whether this geometric explanation can be connected to an understanding of real spacetime.