#temporal coordinates

Abstract

We propose a theoretical framework for a “Temporal Ceiling” arising naturally in discrete models of spacetime. In contrast to continuous manifold descriptions based on the FLRW metric, we adopt a fundamentally discrete structure inspired by Causal set theory. In this model, spacetime is not pre-existing but is incrementally generated through the ordered addition of events. We show that such growth dynamics impose a strict upper bound—the Temporal Ceiling—on forward temporal displacement. This boundary is not a physical barrier but a consequence of the non-existence of future events beyond the current growth step. Implications for cosmology, causality, and branching structures are discussed.

I. Introduction: From Continuous Manifolds to Discrete Growth

Standard cosmology models spacetime as a smooth 4-dimensional manifold evolving under Einstein’s equations. Within this framework, time is treated as a continuous coordinate, and future states are mathematically well-defined.

However, several approaches to quantum gravity suggest that spacetime may be fundamentally discrete. In particular, models such as Causal set theory replace the continuum with a partially ordered set of spacetime events.

In such models:

Spacetime is composed of discrete elements (events)

Causal relations define temporal ordering

The universe grows via the sequential addition of new elements

This shift motivates a re-examination of temporal structure and the existence of the future.

II. Discrete Spacetime Structure

We define spacetime as a set:

[

\mathcal{U}_N = {E_1, E_2, \dots, E_N}

]

where:

Each (E_i) is a spacetime event

A partial order ( \prec ) encodes causality

[

E_i \prec E_j \quad \text{if } E_i \text{ causally precedes } E_j

]

Time is not a continuous parameter but is instead identified with the cardinality of the set:

[

T \sim N

]

Each increment:

[

\mathcal{U}N \rightarrow \mathcal{U}{N+1}

]

represents a fundamental “tick” of the universe, analogous in scale to the Planck time but interpreted here as an ontological update rather than a measurement limit.

III. Emergent Expansion

In continuous cosmology, expansion is described by a scale factor (a(t)). In the discrete framework, expansion emerges from the growth in the number of spacetime elements.

Define:

[

V(N) \propto |\mathcal{U}_N|

]

where:

(V(N)) represents effective spatial volume

(|\mathcal{U}_N|) is the number of events

Thus:

Cosmological expansion corresponds to the increasing cardinality of the causal set.

This reproduces large-scale expansion behavior without invoking a continuous metric.

IV. The Temporal Ceiling

We now define the central concept:

Definition (Temporal Ceiling):

At growth step (N), the Temporal Ceiling is the maximal index of realized spacetime events. No event (E_{N+k}) with (k > 0) exists.

This implies:

The future is not merely unknown—it is ontologically absent

Physical processes cannot reference or evolve into non-existent events

V. Kinematics Near the Ceiling

A worldline is a chain of causally ordered events:

[

\gamma = (E_1 \prec E_2 \prec \dots \prec E_N)

]

Forward evolution requires:

[

E_N \rightarrow E_{N+1}

]

However:

(E_{N+1}) does not exist until the next growth step

Thus:

Result 1: Forward Incompleteness

No trajectory can extend beyond the current maximal event.

Result 2: State Stagnation

At the Temporal Ceiling, all dynamical evolution halts until new events are generated.

This replaces the need for:

Metric singularities

Divergent energy densities

with a purely combinatorial constraint.

VI. Branching Structures

We model branching as divergence within the growth process:

A branch originates at event (E_k), (k < N)

It evolves via an independent extension:

[

\mathcal{U}_k \rightarrow \mathcal{U}_k'

]

Each branch has its own growth parameter (N'), but obeys:

[

N' \leq N_{\text{max}}'

]

Result 3: Local Temporal Ceilings

Each branch possesses its own Temporal Ceiling, determined by its growth history.

No branch can:

Access events beyond its own generated structure

“Outrun” its own event production

VII. Causality and Ontology of the Future

In this framework:

The past: fixed set of realized events

The present: current maximal layer of growth

The future: nonexistent set

This sharply contrasts with the block universe interpretation implicit in the FLRW metric, where all times are equally real.

Instead, we obtain:

A dynamically generated spacetime with an evolving boundary in the temporal direction.

VIII. Discussion

The Temporal Ceiling emerges as a natural consequence of discrete spacetime growth:

It is not a physical barrier

It does not require exotic energy conditions

It arises from the absence of future elements

This framework:

Preserves causality

Eliminates paradoxes associated with accessing undefined future states

Provides a natural interpretation of time as process rather than dimension

IX. Conclusion

We have reformulated the concept of a Temporal Ceiling within a discrete spacetime model. By identifying time with the growth of a causal structure, we show that the future is fundamentally inaccessible because it is not yet constructed.

The universe does not evolve within time—it generates time.

All observers, regardless of trajectory or branching history, are constrained to exist at the maximal realized layer of spacetime.

Keywords