2026-08-29
From State to Time and Space: The Geometric Evolution of the Universe
Let's try to mentally approach the very beginning of the Universe, successively removing everything that could have arisen later. First, matter and free waves disappear, then spatial distances and directions. But is it possible to stop at empty time? If time is not a primary, ready-made coordinate, we must eliminate it too. Then we can no longer ask what happened in the first second: we must first understand how we could distinguish seconds in the first place.
In the article "The Temporal Field and Its Projections: The Origin of Space," we began our construction with the extremely simple state \(J(0,0)=1\) and proceeded from a directed sequence of changes to the temporal field, and then to its spatial projections. Now let's take another step toward the foundation of this chain. Do we have the right to immediately write \(J(\tau)\) if the parameter \(\tau\) has yet to emerge?
To avoid introducing time in a hidden way, we begin not with motion or a function of time, but with distinguishable states and the relationships between them. We will trace the conditions under which a simple order of states acquires a measure of duration, transforms into physical time, creates a temporal field, and finally allows for the emergence of space.
The Universe does not pass through its first states in external time. On the contrary, the stable order of these states subsequently manifests itself as physical time.
\[\tag{1} \boxed{ \text{states} \longrightarrow \text{order} \longrightarrow \text{time} \longrightarrow \text{temporal field} \longrightarrow \text{space} \longrightarrow \text{hot Universe} } \] 1. Why You Can't Start with a Function of Time
Ordinary physical evolution is written as \(Q(\tau)\). This notation presupposes that each state can already be associated with a moment in time, and neighboring states with a duration \(\Delta\tau\). For a formed Universe, this is natural. For the origin of time itself, this creates a logical circle: time is used to derive time.
Therefore, the initial level should be described without a time variable. Let us denote the possible states of the Universe as
\[\tag{2} \mathcal U_0,\quad \mathcal U_1,\quad \mathcal U_2,\quad\ldots \] The index \(n\) here is not time and does not measure duration. It is only a label that allows us to distinguish states in the mathematical description.
2. Initial Undifferentiated State
The initial state is associated with the global operator of Wave Electricity
\[\tag{3} \boxed{J_0=J(0,0)=1.} \] The formal notation \(a=b=0\) does not imply the object's rest in a pre-existing space. The parameter \(b\) describes the external motion of an already spatially manifested center, while at the initial level, external space does not yet exist. Therefore, \(b=0\) signifies the absence of an external projection, not a chosen frame of reference.
In the state \(\mathcal U_0\), distance, volume, velocity, temperature, and density are undefined. It also cannot be claimed that \(x=y=z=0\): such a notation would already presuppose the existence of space and its origin.
A single unchanging state by itself creates neither time nor a temporal field:
\[\tag{4} J_0=1,\qquad \Delta J=0. \] 3. The First Distinction of States
The next necessary level is the existence of at least two distinguishable states. Let the states \(\mathcal U_n\) be associated with the values \(J_n\). Then the primary change is determined by the finite difference.
\[\tag{5} \boxed{\Delta J_n=J_{n+1}-J_n.} \] This formula does not yet contain the rate of change. It does not divide by \(\Delta\tau\), since the duration of the transition is not yet determined. It expresses only the fact that the two states do not coincide.
Before the advent of time, the internal state can be written as
\[\tag{6} J_n=\j^{a_n}, \] where \(a_n\) is the state parameter. At this level, we cannot yet substitute \(a=\varpi\tau\), because the parameter \(\tau\) has not yet arisen.
What is primary is not the change per unit of time, but the difference between two states. Time only appears when such differences can be ordered and compared.
4. Order without Duration
Difference alone is also not enough. We must determine which states can be related and in what direction. Let's introduceStructural precedence relation:
\[\tag{7} \boxed{ \mathcal U_0\prec\mathcal U_1\prec\mathcal U_2\prec\ldots } \] The sign \(prec\) does not yet mean the usual "earlier in time." It indicates that the state on the right is a valid continuation of the state on the left. This is a primary ordering relation, not a movement through an existing temporal environment.
At this level, a sequence can be established, but it is impossible to determine whether two transitions are equally long. Therefore, a distinction must be made between chronological order and metric time:
\[\tag{8} \boxed{ \text{order answers the question "what follows what"} \qquad \text{time — "what is the interval between states"} } \] 5. Transition Does Not Occur in External Time
A natural question arises: how can one state transition to another if time does not yet exist? Here, the word "transition" should not be understood as a process occupying a certain duration. The entire structure of states, together with the relation \(prec\), is considered primary. It does not unfold against the background of additional external time.
After the emergence of an internal temporal measure, the relation \(prec\) will be observed as the "past-future" direction. But at the initial level, it is only a geometric rule for coordinating states.
Before the emergence of time, it is more correct to say not "the state has changed," but "two states are distinguishable and linked by a directional relationship."
6. Repeatability as the Basis of Clocks
An ordered chain creates a direction, but does not yet define equal intervals. For measurable time to emerge, a reproducible process is necessary, with which other changes can be compared. In the geometry of the operator \(J\), the periodic phase component becomes a natural standard.
Let some component of state \(C_n\) return to the same phase after \(N\) order steps:
\[\tag{9} C_{n+N}=C_n. \] This does not mean that the entire Universe returns to its previous state. Only a selected phase component repeats, while the overall order of states remains preserved. One complete cycle can be taken as a unit of duration \(T_*\). A separate step is then associated with an interval.
\[\tag{10} \Delta\tau=\frac{T_*}{N}, \qquad \tau_n=n\Delta\tau. \] This is how discrete order acquires a metric measure. The absolute scale \(T_*\) requires physical calibration, but the duration ratios are already determined by the number of stably reproducible phase steps.
\[\tag{11} \boxed{ \text{physical time} = \text{measure of ordered changes relative to a stable cycle} } \] 7. Transition from state \(a_n\) to function \(a(\tau)\)
Only after the emergence of a time measure can the discrete state parameter \(a_n\) be represented by a continuous function \(a(\tau)\). For uniform internal dynamics, the already known relations are introduced.
\[\tag{12} a(\tau)=\varpi\tau, \qquad \omega=\pi\varpi. \] The global operator takes a time representation
\[\tag{13} J(\tau)=\j^{a(\tau)} = \ep+\em e^{i\omega\tau}. \] Now the finite difference of states can be divided by a measurable interval and taken to the limit:
\[\tag{14} \boxed{ \mathcal T_J = \lim_{\Delta\tau\to0} \frac{J(\tau+\Delta\tau)-J(\tau)}{\Delta\tau} = \frac{dJ}{d\tau} } \] For uniform internal dynamics
\[\tag{15} \mathcal T_J = i\omega\em e^{i\omega\tau}. \] It is from this level that the temporal field, in the sense of the basic article, appears. Before it, there were states, differences, and order, but no derivative with respect to physical time.
8. Energy Appears After Time
Since the norm of the temporal field is determined by frequency,
\[\tag{16} |\mathcal T_J|=\omega, \] After the introduction of the quantum of action, the energy scale emerges
\[\tag{17} E=\hbar\omega. \] Consequently, energy in the model under consideration should not be attributed to the initial unchanging state \(J_0=1\). It appears as a measure of the stable change of state after the emergence of time and frequency.
The initial state cannot be characterized by temperature or energy density: these quantities require frequency, energy, volume, and an already formed temporal-spatial geometry.
9. Splitting of the Temporal Field
After the emergence of the temporal field, its idempotent splitting into mutually orthogonal channels becomes possible. In the simplest case,
\[\tag{18} \mathcal T_J = \prp\mathcal T_J+\prm\mathcal T_J, \qquad \prp\prm=0. \] Repeated splitting can create multiple independent final components. For the proposed four-dimensional implementation, we denote them by the projectors \(P_0,P_1,P_2,P_3\):
\[\tag{19} P_0+P_1+P_2+P_3=1, \qquad P_\mu P_\nu=\Delta_{\mu\nu}P_\mu. \] Then the complete temporal field is represented as
\[\tag{20} \boxed{ \mathcal T_J = \mathcal T_{\mathrm t} +\mathcal T_1+\mathcal T_2+\mathcal T_3 } \] Algebraic orthogonality does not yet make channels physical coordinates. For space to emerge, stability of projections, a rule for measuring distance, and a common metric that allows for comparison of the positions of different wave structures are necessary.
10. The Birth of Space
When the three orthogonal projections acquire a stable metric mapping, the primary temporal structure expands into spacetime:
\[\tag{21} \boxed{ \mathbb R_{\tau} \longrightarrow \mathbb R_{\tau}\oplus\mathbb R^3 } \] Before this transition, spatial scale is not zero—it is undefined. After the transition, distance, volume, propagation, and spatial energy density become meaningful.
Therefore, the origin of space cannot be imagined as a small ball located in an even larger empty space. It is the emergence of the very rule by which positions are distinguished and distances are measured.
11. Where does the Big Bang fit into this scheme?
The initial state \(\mathcal U_0\) cannot be directly identified with the hot Big Bang. It does not yet have time, energy, space, volume, or temperature. It is a deeper, pre-geometric boundary.
In the physical sense, the Big Bang corresponds not to the first state, but to a transition in which the already existing temporal field acquires a spatial metric and forms a high-energy wave medium:
\[\tag{22} \boxed{ \text{temporal field} \longrightarrow \text{spatial metric} \longrightarrow \text{hot wave universe} } \] Such a transition can be called the geometric Big Bang. It is not an explosion of matter from a point into the surrounding void, but the birth of space and the possibility of wave propagation.
12. Evolution of the States of the Universe
The complete qualitative sequence can be represented by the following levels.
State \(\mathcal U_0\): indistinction. The global operator has the value \(J_0=1\). There is no frequency, time, or space.
State \(\mathcal U_1\): distinguishability. States with \(J_{n+1}
e J_n\) appear. The finite difference is determined, but not yet the rate of change.
State \(\mathcal U_2\): order. A directed relation \(\mathcal U_nprecmathcal U_{n+1}\) arises. It defines the sequence, but not the duration.
State \(\mathcal U_3\): repeatability. A stable phase cycle is formed, allowing the intervals between changes to be compared.
State \(\mathcal U_4\): time. Order acquires the metric measure \(\tau\), and the state parameter becomes a function \(a(\tau)\).
State \(\mathcal U_5\): temporal field. The derivative \(\mathcal T_J=dJ/d\tau\), frequency, and energy scale are determined.
State \(\mathcal U_6\): space. Stable Orthogonal projections acquire a common metric and form three spatial directions.
State \(\mathcal U_7\): free waves. The spatial dynamics of the temporal field propagate along the emerging directions. A hot wave medium is formed.
State \(\mathcal U_8\): particles. Some of the wave structures are spatially closed. Stable frequencies, mass, internal sheets, and charge projections emerge.
State \(\mathcal U_9\): Matter and structures. Interacting closed waves form atoms, matter, and large-scale systems.
\[\tag{23} \boxed{ \begin{aligned} J=1 &\longrightarrow \text{distinguishable states} \longrightarrow \text{order} \longrightarrow \text{time}\\ &\longrightarrow \text{temporal field} \longrightarrow \text{space} \longrightarrow \text{waves}\\ &\longrightarrow \text{particles} \longrightarrow \text{matter}. \end{aligned} } \] 13. Expansion is not an external motion
After the emergence of space, its further development requires a separate cosmological quantity. Let us denote the general scale of the spatial metric by \(S(\tau)\):
\[\tag{24} d\ell^2=S^2(\tau)\,\gamma_{ij}d\xi^i d\xi^j. \] The expansion rate is determined by the expression
\[\tag{25} H(\tau)=\frac{\dot S}{S}. \] The scale \(S\) cannot be identified with the internal parameter \(a\) or the external parameter \(b\). The parameter \(a\) describes the internal state, and \(b\) describes the motion of the center in an already formed space. Cosmological expansion characterizes the change in the most general spatial metric, and not the motion of the Universe relative to the external environment.
At this stage, the function \(S(\tau)\) has not yet been derived from the operator \(J\). Therefore, the proposed sequence provides a geometric basis for cosmology, but does not replace the quantitative expansion equations.
14. A New Understanding of the Initial Singularity
In conventional backward extrapolation, the spatial scale tends to zero, and the density to infinity. In the proposed construction, a different boundary is possible. Before the state \(\mathcal U_6\), the spatial scale and volume are not zero, but rather undefined.
\[\tag{26} \boxed{ \mathcal U_0:\qquad S\;\text{not zero, but not yet defined} } \] Consequently, the concept of infinite spatial density cannot be applied to the pre-geometric state. This isn't a definitive proof of the absence of a singularity, but it does show that a singularity can represent a limit to the applicability of spatial quantities, rather than a physical object with infinite parameters.
15. What needs to be achieved next
The sequence constructed is currently a geometric program. A quantitative cosmological model requires deriving:
— why a stable metric manifestation has precisely three spatial directions;
— the law of spatial scale variation \(S(\tau)\);
— the relationship between energy density and the norm and spectrum of the temporal field;
— the conditions for the emergence of free waves and their subsequent closure into particles;
— the laws of cooling, cosmological redshift, and matter formation;
— the observable consequence that distinguishes the state model from standard cosmological extrapolation.
Until the law \(S(\tau)\) is obtained, the model describes the origin of cosmological variables, but does not yet calculate the observed history of the expansion of the Universe.
Conclusion
To trace the origin of time itself, we begin with the distinguishable states of the Universe and the directional relationships between them. Their order does not yet contain duration and is not based on an external time coordinate—physical time will emerge later as a measure of transitions between states.
The stable recurrence of one of the phase components makes it possible to compare transitions and introduce physical time. After this, the state parameter \(a_n\) becomes a function \(a(\tau)\), a temporal field \(\mathcal T_J=dJ/d\tau\) emerges, and its stable orthogonal projections can acquire a spatial metric.
In this picture, the Big Bang is neither the first state nor an explosion of matter in a pre-existing space. This is a geometric transition from a temporal field to a spatially manifested high-energy wave medium. The initial state \(J=1\) lies deeper: it marks the boundary where neither time, nor space, nor physical energy are yet defined.
\[\tag{27} \boxed{ \text{The Universe does not begin in time:} \qquad \text{Time arises as a measure of the order of its states.} } \]