Minkowski spacetime is the affine space ℝ 1 , 3 equipped with the Lorentzian metric

The null structure associated with this metric is fundamental to special relativity [1][2]. Standard treatments of Lorentzian and semi-Riemannian geometry provide the mathematical framework for null curves, causal cones, and their differential-geometric properties [3][4].

The purpose of this article is not to alter Equation (1). Instead, we study an auxiliary sequence of geometric maps beginning with a normalized cube and ending with a non-bijective representation on the standard Minkowski null cone. The intermediate Euclidean geometry is treated with standard surface theory [5], while the global map structure is formulated using the language of smooth maps, quotient spaces, and fibers [6][7].

The principal clarification developed here is that two terminal constructions play different roles. First, an independent map

parametrizes the complete double null cone. Second, a map

constructed from toroidal coordinates has bounded radial image and is therefore not surjective onto the full cone. Its significance lies instead in its non-bijective quotient structure.

Accordingly, the mathematically compatible structural chain studied below is

The full cone is recovered separately by P J .

Let

For q ∈ [ 0 , 1 ] , define

Its Jacobian is

Hence

Proposition 1.For every q ∈ [ 0 , 1 ] ,the image G q = κ J ( q ) ( G ) is compact,path connected,and contains Δ J .At q = 1 ,the map undergoes a controlled rank reduction from three dimensions to the z -axis.

Proof. The cube G is compact and path connected, and κ J ( q ) is continuous. Therefore its image is compact and path connected. Moreover, κ J ( q ) ( Δ J ) = Δ J . The rank statement follows directly from the Jacobian.

Let

Define Π J : G → D 2 by

where

Since 0 ≤ ρ < 1 , the image lies in D 2 . The apparent singularity at x = y = 0 is removable.

For visualization only, define the six-petal polar curve

Its enclosed area is

Remark1.The rosette is not used as a domain of the cylindrical immersion and is not required for the composition in Equation (2). It records a six-sector planar symmetry only. This separates the decorative symmetry representation from the mathematically necessary transition maps. The auxiliary six-petal symmetry visualization is shown inFigure 1.

Figure 1. The six-petal rosette is retained as an auxiliary visualization and is not an indispensable transition surface in the structural map chain.

The original schematic passage from a disk to cylindrical strip coordinates and then to angular torus coordinates requires explicit changes of variables. We now provide them.

Let D * 2 = D 2 \ { ( 0 , 0 ) } . Write a point of D * 2 in polar coordinates as

For fixed R > 0 and h > 0 , define the strip

The disk-to-strip transition map is

The angular coordinate is undefined at the disk center, so the center is excluded from this chart. This is a standard coordinate singularity rather than a singularity of the disk itself. A second local chart may be introduced around the center if a full atlas is required [6].

Next define

Thus the longitudinal strip coordinate becomes the major angular coordinate and the vertical strip coordinate becomes the minor angular coordinate. Endpoint identification in u produces the cylindrical closure, while endpoint identification in v completes the toroidal parameter domain.

Remark2.Equations (11) and (12) make the structural composition type-compatible. They are chosen transition maps; they are not claimed to be canonical.

Define

The tangent vectors are

Therefore

Proposition 2. C J is a local isometric immersion of the Euclidean strip:

This is the familiar developable geometry of a circular cylinder [5]. The statement is local; global closure requires the identification u ~ u + 2 π R .

Let R T > r T > 0 . Define the standard torus embedding

Its first fundamental form is

with area element

Consequently,

No global isometry between the Euclidean plane, cylinder, and torus is asserted. In particular, the cylinder is developable, whereas the torus has nonconstant Gaussian curvature.

The double null cone with vertex O = ( 0 , 0 ) is

Its causal interpretation is standard in Lorentzian geometry [8][9]. Let

For λ ≥ 0 and σ ∈ { − 1 , + 1 } define

Theorem 1 (Exact parametrization of the null cone).The map P J is surjective onto N and every point in its image satisfies the Minkowski null condition exactly.

Proof. Since ‖ n ‖ = 1 ,

Hence Im P J ⊆ N . Conversely, let ( c t , x ) ∈ N . At the vertex choose λ = 0 . Otherwise set λ = ‖ x ‖ = | c t | , σ = sgn ( t ) , n = x / ‖ x ‖ . Spherical angles represent n , so every null point is obtained.

For fixed ( χ , ϕ , σ ) and λ = c | t | , a generator obeys

and therefore Equation (1) gives d s 2 = 0 .

We now distinguish the independent surjective parametrization P J from the torus-induced map. Fix λ 0 > 0 and define

Define

Proposition 3. Q J is not surjective onto the full null cone.

Proof. For every ( ϕ , θ ) ∈ T 2 ,

Therefore no null point with radial parameter λ > λ 0 belongs to Im Q J . Moreover, χ and λ are coupled through the single parameter θ , so the accessible angular-radial combinations form only a restricted subset. Thus

This distinction removes any implication that the torus itself parametrizes the complete Minkowski null cone. The full cone is parametrized by P J ; Q J is a separate, restricted, non-bijective projection.

The topologically significant feature of Q J is not an equality of torus and cone topologies but the information loss generated by its fibers. Define the two toroidal circles

Because λ ( 0 ) = λ ( π ) = 0 ,

Thus entire one-dimensional subsets of T 2 are collapsed to the cone vertex.

Definition 1.Define an equivalence relation ~ Q on T 2 by

Let

be the associated quotient space, with quotient projection π Q : T 2 → X Q .

Proposition 4 (Quotient factorization).There exists a unique continuous map

such that

Moreover, Q ˜ J is bijective by construction.

Proof. Q J is constant on each equivalence class of ~ Q . The universal property of the quotient therefore yields a unique continuous map Q ˜ J satisfying Equation (32). Since the equivalence classes are precisely the fibers of Q J , two quotient classes have the same image only if they coincide. Surjectivity onto N Q follows from the definition N Q = Im Q J . ☐

Remark3.The proposition establishes a quotient representation, not a diffeomorphism between the torus and the null cone. In particular, the vertex has a non-discrete preimage containing F 0 ∪ F π . The quotient therefore records an explicit topological loss of information. The singular behavior here is map-induced: it should not be confused with a physical spacetime singularity.

Away from F 0 ∪ F π , the radial parameter is positive. The two open annuli

map respectively to the future and past sheets selected by σ . The collapsed circles are boundary fibers between these sectors. This gives a simple stratified description: regular two-dimensional parameter regions are attached through fibers that collapse to the zero-dimensional cone vertex.

The causal and global properties of null boundaries are normally analyzed in a much richer Lorentzian setting [10]. The present construction makes no claim to reproduce that theory; the quotient analysis concerns only the topology of the auxiliary map Q J . The resulting type-compatible structural chain and the distinct roles of P J and Q J are summarized in Figure 2.

Figure 2. Type-compatible structural chain. The independent map P J parametrizes the full cone N , whereas the torus-induced map Q J reaches only N Q ⊊ N .

For a trajectory with spatial velocity v = d x / d t , define

Hence

This is exactly the standard dimensionless special-relativistic factor 1 − v 2 / c 2 , used here only as a diagnostic label. It introduces no new dynamics.

Table 1separates standard geometric facts from definitions specific to the structural construction. The causal geometry of the terminal cone remains entirely standard. No gravitational field, curvature of spacetime, or alteration of Lorentz symmetry is inferred from the Euclidean intermediate surfaces.

Table 1. Standard and construction-specific elements.

The revised formulation resolves three logically distinct issues.

First, the structural rank reduction in Equation (4) is a property of a Euclidean linear map. At q = 1 two directions collapse, but this does not imply a spacetime singularity.

Second, the intermediate maps are now domain-compatible. The explicit transitions A D C and A C T remove the need to interpret the composition only schematically. The disk center remains a coordinate-chart issue because polar angle is undefined there; it can be covered by an additional local chart if a global smooth atlas is desired.

Third, the two terminal maps must not be conflated. The surjectivity theorem applies to P J , whose independent variables ( λ , χ , ϕ , σ ) span the full double null cone. By contrast, Q J constrains λ and χ through the same toroidal coordinate θ . Its bounded radial image and collapsed fibers make it valuable as a quotient-geometric construction, not as a global parametrization of N .

The most mathematically distinctive feature is therefore the fiber structure of Q J . The circles F 0 and F π collapse to the cone vertex, while the complementary annuli map to nonzero future and past null sectors. A deeper treatment could classify all fibers, study the local topology of X Q , analyze whether alternative choices of λ ( θ ) and χ ( θ ) change the quotient type, and formulate analogous maps between more general compact parameter manifolds and Lorentzian null hypersurfaces.

No phenomenological Lagrangian is introduced in the present revision. A physical dynamics would require independently motivated measurable degrees of freedom, characteristic scales, and falsifiable observables. Keeping the geometric map separate from such a dynamics prevents an auxiliary representation from being mistaken for a new physical law.

We have reformulated the structural compactification construction as a mathematically type-compatible sequence of maps and separated the complete null-cone parametrization from the restricted torus-to-cone projection. The principal conclusions are:

1) κ J ( q ) has a computable Jacobian and undergoes a controlled rank reduction at q = 1 .

2) The planar rosette is an auxiliary symmetry visualization rather than a necessary step in the continuous map chain.

3) Explicit transition maps connect disk coordinates, strip coordinates, and toroidal angles.

4) The cylindrical stage is locally isometric to a Euclidean strip, while the toroidal stage carries the standard induced torus metric.

5) P J is surjective onto the complete double Minkowski null cone and satisfies d s 2 = 0 exactly.

6) Q J is not surjective onto the complete cone; its image is a proper subset N Q ⊊ N .

7) The circles F 0 and F π are singular fibers of Q J in the sense that each is collapsed to the cone vertex.

8) The map Q J factors naturally through the quotient T 2 / ~ Q , providing a precise mathematical description of the information lost under the non-bijective projection.

The resulting construction is best interpreted as an auxiliary geometric and quotient-topological representation compatible with standard special relativity. It does not modify the Minkowski metric and does not, by itself, define a physical dynamics.

The author thanks colleagues and reviewers whose comments motivated a sharper distinction between independent null-cone parametrization, restricted toroidal projection, coordinate compatibility, and quotient topology.

Oscar Mauricio Jeno Soto: Conceptualization, formal analysis, methodology, visualization, and writing (original draft and revision).

Data sharing is not applicable to this article because no new data were created or analyzed.