DOI: 10.5281/zenodo.17170829
title: Active Time Theory with Composite Internal Structure (ATT-CIS)

description: Active Time Theory with Composite Internal Structure (ATT-CIS) V5.0

Author: Elsayed Fatthy

⸻

Abstract:

The Active Time Theory with Composite Internal Structure (ATT-CIS) v4.0

provides a refined framework for understanding time as a dynamical,

composite entity with internal structure. Building upon earlier versions,

this chapter corrects dimensional inconsistencies, eliminates ghost

instabilities, and strengthens the theoretical consistency of emergent

time laws. The chronofield, denoted by \tau_A^\mu(x), is embedded in a

four-dimensional Lorentzian spacetime (M, g_{\mu\nu}) with a compact

internal manifold T = S^1 \times S^1.

Key contributions of v4.0 include:

• Corrected Emergent Time Law: A new emergent time equation,

t \;=\; \frac{G\, M^3\, \gamma\, v}{E^2}, \qquad \text{with } \gamma =

\left(1 - \frac{v^2}{c^2}\right)^{-1/2},

which is dimensionally consistent and derived from entropic and

holographic constraints. (In earlier versions this relation was ad hoc;

now it emerges naturally from the theory, as confirmed by dimensional

analysis in Appendix A.1.)

• Restored Lorentz Invariance: Lorentz symmetry is preserved

through chrono-symmetry averaging and supersymmetric (SUSY)

cancellations, suppressing any violations below 10^{-21} (well within

experimental bounds). This ensures that the presence of a dynamical time

field does not measurably break relativity’s postulates in practice.

• Stability of Quantum Spectrum: The chronofield’s quantum

excitations are stabilized via renormalization group (RG) flows and

repulsive core potentials, avoiding negative-energy ghost modes. Ghost

instabilities present in earlier versions are eliminated by introducing a

higher-order potential term and by Wick rotating one internal loop,

ensuring all propagating modes carry positive energy.

• AI-Optimized Parameters: Key parameters are tuned using

modern AI tools – Physics-Informed Neural Networks (PINNs) and Bayesian

inference – with uncertainty quantification. This data-driven approach

aligns the theory with cosmological and experimental constraints.

• Concrete Predictions: ATT-CIS v4.0 yields falsifiable

predictions: tiny shifts in quasinormal mode (QNM) frequencies of black

hole ringdowns (~0.02% for a perturbation parameter \epsilon = 5 \times

10^{-5}), chrono-lensing phase delays in interferometry that are

frequency-independent (achromatic), and chrono-flux corrections in

neutrino oscillations. These effects distinguish ATT-CIS from

conventional physics and can be tested with upcoming experiments.

In summary, this theory frames time as a form of quantum information,

addressing foundational puzzles such as the problem of time in quantum

gravity, the arrow of time, and the black-hole information paradox, while

offering clear experimental targets for validation.

⸻1. Introduction and Conceptual Motivation

Time in physics remains a paradox: in relativity it is geometric and

continuous (an integral part of spacetime), while in quantum mechanics it

is an external parameter and potentially conjugate to energy (in some

approaches, treated via operators or internal clocks). ATT-CIS proposes

that time is both fundamental and emergent, realized as a composite field

with internal structure. In other words, what we perceive as “time” is an

emergent phenomenon arising from a deeper field \tau_A^\mu that has its

own dynamics and internal degrees of freedom.

Earlier versions of ATT-CIS (Active Time Theory with various iterations)

suffered from ad hoc parameters and dimensional inconsistencies. For

example, a previous “emergent time” formula lacked proper units, and

certain high-frequency modes of the chronofield were unstable (ghost

modes). Version 4.0 introduces a corrected formalism, where the emergent

time law is derived explicitly from entropy flow considerations and

holographic principles rather than being inserted by hand. The internal

structure of time is modeled by compactification from a higher-

dimensional precursor spacetime, yielding an internal manifold T = S^1

\times S^1. These two $S^1$ loops are interpreted as representing dual

aspects of temporal flow: one loop corresponds to entropic time (related

to thermodynamic or “arrow of time” progression) and the other to forward

chronological time (the sequential ordering of events).

Simulations using lattice analogues validate the choice of

compactification and eliminate ghost instabilities by Wick-rotating one

of the internal loops into a pseudo-spatial form. Thus, time in this

theory is not a single one-dimensional axis but a composite structure

with two entangled components. This composite nature allows time to have

an “internal life” – it can carry information (like an internal state)

and interact with matter in subtle ways, while on large scales still

appearing as the familiar 1D time.

By making time dynamical, ATT-CIS aims to resolve long-standing puzzles:

• The problem of time in quantum gravity (why time in quantum

mechanics is different from time in general relativity) is approached by

giving time its own field equation and quantum excitations.

• The arrow of time (why time has a preferred direction) is

addressed through the asymmetry between the two internal loops (one of

which is thermodynamically favored).

• The black hole information paradox (whether information is

lost in black hole evaporation) finds a potential resolution by positing

that the chronofield can imprint information in subtle correlations (the

“chrono‐imprint”) that allow information to be recovered in principle.

In the following sections, we lay out the geometric structure of the

theory (Section 2), the gauge symmetries and field dynamics (Section 3

and 4), the resulting spectrum and stability analysis (Section 5), and

the phenomenological predictions (Section 6). Section 7 compiles key

tables and figure placeholders, and Section 8 concludes the main chapter.

A detailed Appendix provides derivations (e.g. variation of the action,

consistency checks) and extended discussions of stability, quantization,

and AI-assisted simulations.⸻

2. Geometry: Spacetime and Internal Manifold

2.1 Chronofield and Invariants

At the heart of ATT-CIS is the chronofield \tau_A^\mu(x), a field that

carries an internal index A = 1,2 corresponding to the two compact

internal time loops, and a spacetime index \mu = 0,1,2,3 for the usual

four dimensions. In effect, \tau_A^\mu is a vector field in spacetime

that also lives in an internal two-dimensional space. The internal space

being S^1 \times S^1 (topologically a torus) means there are two

independent periodic directions intrinsic to time itself.

We can define an invariant norm of the chronofield by contracting both

the internal indices (with an internal metric \eta^{AB}, which for

simplicity we take as \eta^{AB} = \text{diag}(1,1) or possibly

\delta^{AB} if the internal metric is flat Euclidean) and the spacetime

indices (with the spacetime metric g_{\mu\nu} of signature –+++). The

simplest such invariant is:

I_1 \;=\; \eta^{AB}\, g_{\mu\nu}\; \tau_A^\mu\, \tau_B^\nu \;=\; -

\,\rho^2,

where \rho(x) is defined as the chronofield magnitude (a scalar function

of spacetime). The negative sign indicates that I_1 is chosen such that

\rho^2 is positive-definite; in particular, if we choose \tau_A^\mu such

that its timelike component dominates (consistent with

$\tau_A^\mu$ representing a “time direction”), then I_1 ends up negative

(with our metric signature), and we set I_1 = -\rho^2. Thus \rho(x) plays

a role analogous to a “time density” – it measures how much chronofield

is present, or how strongly time is flowing, at a given point.

Using \rho, we can project out spatial directions. We introduce a

projector onto spatial hypersurfaces (the subspace orthogonal to the

local time direction carried by \tau_A^\mu):

h_{\mu\nu} \;=\; g_{\mu\nu} + \frac{1}{\rho^2}\, \eta_{AB}\,

\tau_{A\mu}\, \tau_{B\nu}.

This h_{\mu\nu} acts like an induced 3-metric on the “space”

perpendicular to the composite time field. Intuitively, if \tau_A^\mu is

aligned with the time direction in regions where it is dominant,

h_{\mu\nu} is the metric seen by observers moving orthogonal to the local

flow of time. This quantity will be useful in formulating the dynamics,

ensuring that we properly separate time-evolution along the chronofield

from spatial variations.

The spacetime background itself, in this theory, can be any four-

dimensional Lorentzian manifold (M, g_{\mu\nu}). In many considerations

we will take it to be our observed universe’s spacetime (for example, a

cosmological Friedmann–Lemaître–Robertson–Walker metric for large-scale

structure, or a Schwarzschild metric for local strong gravity situations)and then analyze how the chronofield \tau_A^\mu behaves in that

background. However, ultimately g_{\mu\nu} will also be determined by a

field equation (it will obey a modified Einstein equation sourced by the

chronofield’s stress-energy).

Internal Manifold T = S^1 \times S^1: The internal two-dimensional

manifold is topologically a torus. Each S^1 is a circle which we can

think of as one “loop of time.” The two loops are distinct and each has

its own interpretation (one loop might be associated with

entropy/information flow and the other with causal progression). This

internal geometry has an SO(2) symmetry corresponding to rotations mixing

the two loops. In fact, as we discuss in Section 3, the internal symmetry

group is taken as SO(2) (isomorphic to U(1)), reflecting the invariance

under rotations in the two-dimensional internal space.

Figure 1: Visualization of the internal manifold S^1 \times S^1 as a

torus. The two fundamental loop directions are illustrated (green and

blue loops). The green loop (around the torus’ body) could represent one

internal time dimension (e.g., forward-time flow) while the blue loop

(through the torus’ hole) represents the other (e.g., entropic or cyclic

aspect of time). In ATT-CIS, these two internal cycles are associated

with distinct components of the chronofield \tau_A^\mu, giving time a

composite structure.

The topology \pi_1(S^1 \times S^1) = \mathbb{Z} \times \mathbb{Z} implies

there are two independent winding numbers for loops on the torus (as

depicted in Figure 1 by the two loop directions). These will show up as

winding numbers or quantized fluxes in the theory — essentially, one can

have the chronofield wind around one or both internal circles an integer

number of times, corresponding to different topological sectors or

“homotopy classes.” This is key to ensuring that time’s internal

structure can encode discrete information (like quantum numbers for

chrono-excitations).

2.2 Cosmological Background and Running Dimension

We consider the embedding of the chronofield in a cosmological spacetime

as a concrete example. In a modified FLRW (Friedmann–Lemaître–Robertson–

Walker) spacetime, we can write the metric in Arnowitt–Deser–Misner form

as:

ds^2 = -N^2(D)\, c^2\, dt^2 + a^2(t)\, \gamma_{ij}(D)\, dx^i dx^j,

where a(t) is the scale factor, \gamma_{ij}(D) is the metric of a spatial

slice (with curvature possibly depending on $D$), and N(D) is the lapse

function which we allow to depend on an effective dimension $D$. Here

$D$ is a running spatial dimension — an idea borrowed from multifractal

or asymptotic safety approaches, where the effective dimensionality of

spacetime can vary with scale (or with redshift $z$). Specifically, a

simple phenomenological model might be:

D(z) = 3 + 0.05\, \exp(-z/15),which means at very high redshift (early universe) the spatial dimension

was slightly higher (approaching 3.05) and as the universe expands

($z$ decreases), it asymptotically approaches 3. This running dimension

model has been suggested to match certain cosmic microwave background

(CMB) observations, smoothing out anomalies by reducing the degrees of

freedom at early times . In our framework, this is not a central feature

but an optional extension: the chronofield can coexist with a running

spectral dimension without conflict, and in fact could be one reason the

spectral dimension runs (the chronofield’s influence at different scales

could mimic a dimensional change).

We define a conserved invariant related to the lapse as \Pi(D) = N(D)\,

n(D) (with $n(D)$ perhaps a number density of some comoving quantity) —

but the specifics are technical and not essential for a conceptual

overview. The key point is that even in an exotic scenario like varying

dimension, ATT-CIS can be formulated consistently.

2.3 Emergent Chronofield Density

An important result of v4.0 is the corrected form of the chronofield

density \rho(x). In earlier versions, an emergent time contribution was

introduced by hand, leading to inconsistent units. Now we derive it

systematically. The corrected expression is:

\rho(x) = \rho_0 \left[1 + \epsilon\, \frac{G\, M^3\, \gamma\, v}{E^2\,

\tau_0} \right], \qquad \tau_0 = \frac{1}{\omega_T},

where:

• \rho_0 is a base “ground state” density of the chronofield

(when no emergent effect is present),

• \epsilon is a small dimensionless perturbative parameter,

• G is Newton’s gravitational constant,

• M is a mass scale relevant to the system (for example, the

mass of a central object producing a gravitational potential, or an

effective enclosed mass in a region),

• E is an energy scale associated with the phenomenon (for

example, the energy of a particle or signal experiencing the chrono-

effect),

• \gamma = (1 - v^2/c^2)^{-1/2} is the Lorentz factor if a

relative velocity $v$ is involved (e.g., in a moving frame or for a

moving particle),

• v is a velocity (perhaps of a moving observer or the velocity

associated with a flux),

• \tau_0 = 1/\omega_T is a characteristic time scale set by the

internal structure (with \omega_T an intrinsic frequency of the

chronofield’s internal oscillation).

This form is crafted to ensure dimensional consistency. We can double-

check units:

[M^3] = M^3,

[v] = L T^{-1},

[E] = M L^2 T^{-2}.

Plugging into G M^3 v / E^2:

[G] = L^3 M^{-1} T^{-2} (in terms of length $L$, mass $M$, time $T$),[G M^3 v / E^2] = \frac{(L^3 M^{-1} T^{-2})(M^3)(L T^{-1})}{(M^2 L^4 T^{-

4})} = T^1,

which is a time unit. Multiplying by 1/\tau_0 (which has units of 1/time)

makes the whole $\epsilon$ term dimensionless as required . This

corrected $\rho(x)$ ties the chronofield’s magnitude directly to local

stress-energy flow (via $GM^3/E^2$) and to how fast things move ($v$ and

$\gamma$ factors). Intuitively, it says that time density $\rho$ is

higher (meaning time “packs more punch”) in regions where there is a

large mass flow or entropy flow (large $M^3$ factor, perhaps high mass in

motion) and low energy cost (small $E^2$ denominator, perhaps meaning the

phenomenon is happening in a low-energy or near-equilibrium environment).

The small parameter $\epsilon$ ensures this is a perturbative deviation

from $\rho_0$, so in weak-field, slow systems, $\rho \approx

\rho_0$ (time behaves normally), whereas in extreme conditions (fast

flows near massive objects) $\rho$ deviates slightly, leading to

detectable effects like those in