DOI: 10.5281/zenodo.17094716
title: HTT Hyperdimensional Time Theory

description: HTT

Hyperdimensional Time Theory (HTT): Scale-Dependent Spatial Dimension in

Gravity

By Elsayed Fatthy

Abstract: We present an updated and enhanced version of Hyperdimensional

Time Theory (HTT), a covariant scalar–tensor framework in which the

effective spatial dimension $D(x)$ is promoted to a dynamical field that

modulates local clock rates and signal speeds. Proper time and light

speed vary as $\frac{d\tau}{dt} = N(D)$ and $c_{\rm eff}(D) = c/n(D)$,

with the constraint $N(D),n(D)=\Pi_0$ (set to 1 in macroscopic units)

enforcing a conserved “information capacity” across scales. This

invariant arises naturally from deformed symmetries in multifractional

spacetimes (a fractional analog of diffeomorphism invariance) and ensures

no net loss of causal flow. We construct a covariant HTT action coupling

$D(x)$ to gravity, and derive modified Einstein equations with scalar

stress-energy and a generalized Klein–Gordon equation for $D$. Choosing

power-law forms $N(D)=(D/3)^{-\alpha}$, $n(D)=(D/3)^{+\alpha}$ (ensuring

$Nn=1$) with $\alpha=1/2$ motivated by quantum-gravity critical

exponents, we find that for $D=4$ (our usual 3 spatial dimensions), local

clocks run about 13% slower and local light cones are narrower (maximal

signal speed $\sim13%$ lower) than in general relativity. The scalar

potential is taken as $V(D)=k(D-3)^2+\lambda(D-3)^4$, inspired by

thermodynamic arguments that entropy peaks near integer $D$ and by loop

corrections stabilizing $D=3$. Fractional calculus is employed for non-

integer dimensions: in particular, the d’Alembertian becomes $\Box_D =

\partial_t^2 - (-\Delta)^{D/3}$, which softens UV divergences and has

been shown to preserve unitarity at one-loop order . Signals follow an

“optical metric” that ensures all null rays propagate at $c_{\rm eff}$,

avoiding vacuum Cherenkov radiation. HTT offers a novel mechanism for

resolving classical singularities: as curvature increases,

$D(x)$ dynamically drops toward 2 and $N(D)\to0$, so that time

essentially “freezes” before a singularity can form (a black hole

interior or a collapsing universe can bounce instead). To reconcile this

with quantum-gravity expectations of dimensional reduction ($D\to2$ in

the UV), we allow a mild scale-dependence

$\Pi_0(\ell)=1+\gamma(\ell_P/\ell)^2$ (with $\gamma<0$) so that

microscopically $\Pi_0>1$ can yield effective $D<3$ while keeping $c_{\rm

eff}\le c$. The theory recovers general relativity in the limit

$D\to3$ (with $N=n=1$), consistent with solar-system tests that bound

$|D-3|\lesssim10^{-10}$ locally. Yet HTT predicts unique signatures in

strong fields: frequency-independent (achromatic) time delays for signals

skimming massive objects, directional anisotropies in a stochastic

gravitational-wave background, and slight mass-dependent shifts in black-

hole quasinormal mode spectra. We confront the model with current data —

notably the multimessenger observation of GW170817/GRB170817A

constraining $|c_g-c|/c<10^{-15}$ and thus $|D-3|\lesssim10^{-15}$ — and

find that HTT survives but is forced to mimic GR extremely closely on

cosmic scales (deviations $\lesssim10^{-20}$). Nevertheless, future

observations (e.g. precise pulsar/FRB timing, next-generation GW

observatories, high-accuracy black-hole spectroscopy) could detect the

tiny deviations HTT predicts. We also outline how techniques from AI(deep learning solvers for fractional differential equations) and quantum

gravity (fractional Wheeler–DeWitt equations, holographic duals) can

further develop HTT. Overall, HTT provides a unified setting that weaves

together scale-dependent dimensionality and relativistic symmetries,

yielding a rich phenomenology to probe the “dimensional fabric” of the

Universe.

Introduction

Modern quantum-gravity research often suggests that spacetime

dimensionality is scale-dependent, effectively reducing from 4 at large

scales to $\sim2$ near the Planck length. For example, asymptotically

safe gravity and causal dynamical triangulations (CDT) both predict a

spectral dimension dropping to around 2 in the ultraviolet . Numerical

lattice simulations (CDT) even measure $D_s \approx 1.5$ at short

distances , implying that the nature of spacetime changes with scale.

These “dimensional reduction” results hint that time and light might

behave differently in extremely dense or high-energy regimes, while still

preserving an ultimate limiting speed. HTT is a framework that realizes

this idea by promoting the local spatial dimension $D(x)$ to a dynamical

field influencing clocks and rulers. Intuitively, a higher effective

dimension densifies space, dilating local time flow and narrowing light

cones, whereas a lower $D$ does the opposite. This concept is depicted in

Figure 1, which shows a 3D fractal structure as an analogy for a geometry

with variable dimensional density. HTT builds on the insights of scalar–

tensor gravity (like Brans–Dicke theory) and incorporates elements from

multifractional spacetimes and Einstein-Æther models, but with a new

twist: the scalar field here is not an inflaton or quintessence, but a

“dimension dilaton” that governs how many spatial degrees of freedom are

locally available.

 

Figure 1: A three-dimensional fractal structure (Mandelbulb fractal

rendering) illustrating how space might have a varying “density” or

effective dimension at different scales. HTT identifies the local spatial

dimension $D(x)$ with such an intrinsic fractal property of spacetime,

which can change in different regions.

This theory is partly motivated by recent calls for more unified

frameworks in fundamental physics. Despite the success of general

relativity (GR) and the Standard Model, there remain deep puzzles they do

not resolve — for instance, the cosmological constant problem or the

matter–antimatter asymmetry in the Universe . As noted in a recent

review, standard approaches leave such questions open, indicating that

new ideas (like a scale-dependent spacetime fabric) may be needed for

progress. HTT aims to address some of these puzzles by linking the

quantum-gravity idea of dimensional flow (the change of effective

dimensions with scale) to observable relativity and cosmology. Crucially,

HTT is constructed to be falsifiable: it reduces to GR in everyday

conditions, but in extreme regimes it predicts clear deviations.

Throughout this paper, we derive HTT’s formulation (Sections 2–4),

examine its physical implications and limits (Sections 5–7), compare to

other theories (Section 8), confront observational constraints and

prospects (Section 9), and explore computational tools and extensions

(Section 10). The goal is to demonstrate that HTT provides a robust yettestable extension to GR that encodes the “fractal” nature of spacetime

in a covariant manner