Autor: A. Jaster | Datum: 2026/05/23 | DOI: 10.5281/zenodo.20354175
A causal-Machian model of time is formulated in which physical proper time is defined by the matter and energy in the causally relevant past of an event. The retarded horizon momentum of this matter selects a preferred rest frame UHμ(x). In that frame the maximum proper time is accumulated. A bounded scalar field Θ(x) ∈ [0,1] describes the degree to which temporal structure is realised: Θ=1 gives the General Relativity (GR) limit, 0 < Θ < 1 gives reduced temporal structure, and Θ=0 corresponds to a completely matter-free causal domain in which no physical proper time is defined. The physical clock metric modifies only the time direction selected by UHμ; it is not a conformal rescaling of space and time. Overdense regions therefore cannot make clocks run faster than the GR limit. All physical fields, including massless ones, propagate on the clock metric ĝμν; the resulting speed bound vH ≤ c√Θ is a prediction of the theory. The model provides a preferred frame for compact spatial topologies and thereby fixes the inertial twin comparison in such spaces. GR is recovered as a limiting case whenever Θ=1 and ∇Θ=0; in that sector the standard energy-momentum conservation law ∇μTμν=0 is recovered. The present paper establishes the kinematic and constraint structure of the theory; a complete variational closure and the regularisation of sharp causal boundaries are identified as open tasks.
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\fancyhead[L]{\small\textit{Emergent Time: A Causal-Machian Time Field
Coupled to Matter}}
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\begin{document}
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\begin{center}
{\LARGE\bfseries Emergent Time: A Causal-Machian Time Field\\[4pt]
Coupled to Matter}\\[1.2em]
{\large A.~Jaster}\\[0.4em]
{\normalsize \today}
\end{center}
\vspace{0.8em}
% ── Abstract ──────────────────────────────────────────────────────────────────
\begin{abstract}
A causal-Machian model of time is formulated in which physical proper
time is defined by the matter and energy in the causally relevant past
of an event. The retarded horizon momentum of this matter selects a
preferred rest frame $U_{H}^{\mu}(x)$. In that frame the maximum
proper time is accumulated. A bounded scalar field
$\Theta(x)\in[0,1]$ describes the degree to which temporal structure is
realised: $\Theta=1$ gives the General Relativity (GR) limit,
$0<\Theta<1$ gives reduced temporal structure, and $\Theta=0$
corresponds to a completely matter-free causal domain in which no
physical proper time is defined. The physical clock metric modifies
only the time direction selected by $U_{H}^{\mu}$; it is not a
conformal rescaling of space and time. Overdense regions therefore
cannot make clocks run faster than the GR limit. All physical fields,
including massless ones, propagate on the clock metric $\hat{g}_{\mu\nu}$;
the resulting speed bound $v_{H}\leq c\sqrt{\Theta}$ is a prediction
of the theory. The model provides a preferred frame for compact
spatial topologies and thereby fixes the inertial twin comparison in
such spaces. GR is recovered as a limiting case whenever $\Theta=1$
and $\nabla\Theta=0$; in that sector the standard energy-momentum
conservation law $\nabla^{\mu}T_{\mu\nu}=0$ is recovered.
The present paper establishes the kinematic and constraint structure
of the theory; a complete variational closure and the regularisation
of sharp causal boundaries are identified as open tasks.
\end{abstract}
\vspace{1em}
% =============================================================================
\section{Motivation}
\label{sec:motivation}
% =============================================================================
General Relativity relates proper time to spacetime geometry and
geometry to matter through the Einstein field equations
\citep{Einstein1915,Einstein1916}. It does not, however, require
temporal structure itself to be defined by matter: empty solutions
such as Minkowski or de~Sitter spacetime possess a well-defined time
direction even in the absence of material clocks or matter
distributions.
The model developed here adopts a Machian interpretation
\citep{Mach1883}. Inertial and temporal structures are meaningful
only relative to matter and energy. A domain with no matter or
energy in its causally relevant past should not contain a physically
realised proper time. Conversely, where matter is causally
available, that matter should determine both a preferred rest frame
and the strength of the temporal structure.
Additional motivation comes from canonical quantum gravity. The
Wheeler--DeWitt equation \citep{DeWitt1967}, $\hat{H}|\Psi\rangle=0$,
contains no explicit time parameter. Time arises through correlations
between physical subsystems \citep{Page1983}, in the spirit of a
relational picture of time closely aligned with the present approach.
The construction is conservative in the local limit. Ordinary GR is
recovered whenever the causal matter connection is complete. The
model nevertheless differs from GR in four conceptual regimes: fully
matter-free causal domains, compact spatial topology, early causal
build-up, and matter-poor regions where the causal time field is below
its saturated value.
% =============================================================================
\section{Geometric and Causal Structure}
\label{sec:geometry}
% =============================================================================
\subsection{Reference Causal Geometry}
The theory employs a Lorentzian reference causal geometry $q_{\mu\nu}$
to define causal ordering, spacelike hypersurfaces, four-volume
elements, and the normalisation of the horizon momentum. This
separates the kinematic causal scaffolding from the physical clock
metric and avoids a circular definition of the preferred frame.
The reference geometry $q_{\mu\nu}$ is not a second physical metric
measured by rods and clocks. It is an auxiliary FLRW-normalised causal
geometry used to define the averaging domain, the foliation normal
$n_{q}^{\mu}$, and the reference density $\Omega_{\rm ref}$. In the
GR sector,
\begin{equation}
\Theta=1,\qquad \hat{g}_{\mu\nu}=q_{\mu\nu},
\label{eq:GRsector}
\end{equation}
so the auxiliary and physical metrics coincide. Away from this sector,
$q_{\mu\nu}$ remains the reference structure for the nonlocal
constraints, while $\hat{g}_{\mu\nu}$ determines physical proper time
and propagation. A full variational formulation in which the reference
structure is generated dynamically rather than prescribed is an open
task.
\subsection{Causal Horizon Domain}
For every event $x$ define the causally relevant past domain
\begin{equation}
\Dm(x)\subseteq J^{-}_{q}(x),
\label{eq:domain}
\end{equation}
where $J^{-}_{q}(x)$ denotes the past of $x$ according to
$q_{\mu\nu}$. The domain is not an arbitrary local sphere; it is the
causally accessible past of the event, bounded by the relevant
cosmological horizon, by the global topology, or by the causal
boundary of the solution.
The sharp boundary of $\Dm(x)$ is kept as an idealisation in this
paper. A smooth causal or Planck-scale boundary prescription is left
to a follow-up publication.
% =============================================================================
\section{Matter-Defined Horizon Frame}
\label{sec:frame}
% =============================================================================
\subsection{Causal Averaging Kernel}
For any scalar or vector density $X(x')$ averaged over $\Dm(x)$, use
the top-hat causal average
\begin{equation}
\langle X\rangle_{H}(x)
= \frac{1}{V_{H}(x)}\int_{\Dm(x)}X(x')\,\dint V_{q}(x'),
\label{eq:tophat}
\end{equation}
with $V_{H}(x)=\int_{\Dm(x)}\dint V_{q}(x')$. Equivalently,
\begin{equation}
W_{H}(x,x')
= \frac{1}{V_{H}(x)}\quad\text{for }x'\in\Dm(x),
\qquad
W_{H}(x,x')=0\quad\text{otherwise}.
\label{eq:kernel}
\end{equation}
The kernel is causal, non-negative, normalised with respect to
$\dint V_{q}$, and contains no free dimensional parameter. The
top-hat boundary is an idealisation; smoothing it changes the
regularity but not the definitions of the frame and scalar field.
\subsection{Retarded Horizon Momentum}
Let $T_{\mu\nu}$ denote the matter energy-momentum tensor and let
$n_{q}^{\nu}$ be the future-directed unit normal to the reference
time foliation with respect to $q_{\mu\nu}$. The retarded horizon
momentum density, averaged over the causal domain, is
\begin{equation}
\Pi_{H}^{\mu}(x)
= \frac{1}{V_{H}(x)}\int_{\Dm(x)}
T^{\mu}{}_{\nu}(x')\,n_{q}^{\nu}(x')\,\dint V_{q}(x').
\label{eq:PiH}
\end{equation}
Only the direction of $\Pi_{H}^{\mu}$ is needed for the preferred
frame. If $\Pi_{H}^{\mu}$ is timelike with respect to $q_{\mu\nu}$,
the horizon rest-frame vector is
\begin{equation}
\boxed{
U_{H}^{\mu}(x)
= c\,\frac{\Pi_{H}^{\mu}(x)}
{\sqrt{-q_{\alpha\beta}(x)\,\Pi_{H}^{\alpha}(x)\,\Pi_{H}^{\beta}(x)}},
}
\label{eq:UH}
\end{equation}
so that $q_{\mu\nu}U_{H}^{\mu}U_{H}^{\nu}=-c^{2}$. The normalisation
uses $q_{\mu\nu}$, not the clock metric $\hat{g}_{\mu\nu}$ that is
constructed later; there is no circularity.
If $\Pi_{H}^{\mu}=0$ or if $\Pi_{H}^{\mu}$ is null, no massive
horizon rest frame is defined. Such cases are boundary cases of the
theory. In particular, a domain filled only with perfectly directed
null radiation, with no matter component giving a timelike total
momentum, does not define a full proper-time frame by itself.
% =============================================================================
\section{Scalar Time Field}
\label{sec:scalar}
% =============================================================================
\subsection{Energy Density on the Reference Foliation}
The energy density sourcing the scalar time field is measured with
respect to the canonical normal of the reference foliation. This
avoids path-dependent parallel transport in curved spacetime and
removes nested nonlocality. With $n_{q}^{\mu}(x')$ the
future-directed unit normal to the $q_{\mu\nu}$-time slice at $x'$,
define
\begin{equation}
\mathcal{E}_{H}(x')
= \frac{T_{\mu\nu}(x')\,n_{q}^{\mu}(x')\,n_{q}^{\nu}(x')}{c^{2}}
\geq 0.
\label{eq:EH}
\end{equation}
The quantity is local at $x'$, path-independent, and non-negative for
ordinary matter satisfying the weak energy condition. In homogeneous
FLRW, $n_{q}^{\mu}$ coincides with the comoving four-velocity and
$\mathcal{E}_{H}$ reduces to the standard energy density.
\subsection{Causal Matter Connection}
The causal matter connection at $x$ is the retarded average
\begin{equation}
\Omega_{H}(x)
= \frac{1}{V_{H}(x)}\int_{\Dm(x)}
\mathcal{E}_{H}(x')\,\dint V_{q}(x').
\label{eq:OmegaH}
\end{equation}
This is an average causal energy density, not the total mass in an
arbitrary volume; it does not grow without bound in overdense regions.
The reference value is the same functional evaluated in the
homogeneous FLRW reference geometry at the reference-cosmic time
$T_{q}(x)$:
\begin{equation}
\Omega_{\rm ref}(T_{q}) = \Omega_{H}^{\rm FLRW}(T_{q}).
\label{eq:Oref}
\end{equation}
Here $T_{q}$ is the cosmic time of the auxiliary geometry $q_{\mu\nu}$,
not the physical proper time generated by $\hat{g}_{\mu\nu}$. Hence
$\Omega_{\rm ref}$ is a reference normalisation, determined entirely
by $q_{\mu\nu}$ and the background matter distribution, without any
dependence on the unknown clock metric. The dimensionless
causal-connection ratio is
\begin{equation}
r(x) = \frac{\Omega_{H}(x)}{\Omega_{\rm ref}(T_{q}(x))}.
\label{eq:r}
\end{equation}
If the reference matter density itself vanishes, the temporal
reference is absent and the theory assigns $\Theta=0$.
\subsection{Bounded Scalar Time Field}
The scalar time field is
\begin{equation}
\Theta(x) = S\!\bigl(r(x)\bigr),
\label{eq:Theta}
\end{equation}
where $S:[0,\infty)\to[0,1]$ satisfies
\begin{equation}
S(0)=0,\qquad S(1)=1,\qquad S(r)=1\quad\text{for }r\geq1.
\label{eq:Sprops}
\end{equation}
A simple smooth interpolation satisfying these conditions is
\begin{equation}
S(r)=
\begin{cases}
3r^{2}-2r^{3}, & 0\leq r\leq1,\\
1, & r\geq1.
\end{cases}
\label{eq:Schoice}
\end{equation}
This choice is not unique. It should be read as a minimal model
ansatz rather than a derived law; any admissible $S$ must satisfy
\eqref{eq:Sprops}. The conditions \eqref{eq:Sprops} are the only
model-independent constraints; quantitative predictions in the sector
$0{\bfseries}p{4.0cm} p{3.8cm} p{5.0cm}}
\toprule
Property & GR & Causal-Machian model \\
\midrule
Temporal structure & Geometric proper time &
Matter-defined causal time \\
Preferred frame & None globally &
Horizon momentum frame $U_{H}^{\mu}$ \\
Scalar time field & Not present & $\Theta\in[0,1]$ \\
GR sector & Fundamental & $\Theta=1$, $T^{\mathrm{C}}_{\mu\nu}\to0$ \\
Energy-momentum conservation & $\nabla^{\mu}T_{\mu\nu}=0$ &
Recovered in GR sector \\
Matter-free domain & Vacuum spacetime allowed &
No proper-time structure \\
Metric modification & None & Time direction only (disformal) \\
Overdense regions & GR dynamics & $\Theta$ saturates at 1 \\
Signal speed & $v\leq c$ & $v_{H}\leq c\sqrt{\Theta}$ \\
Compact topology & Global frame must be chosen &
Frame fixed by $U_{H}^{\mu}$ \\
Main open task & -- & Constraint action $S_{\rm constr}$ \\
\bottomrule
\end{tabular}
\end{table}
% =============================================================================
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\end{document}