A Lorentzian Spacetime and Cosmological Structure from Informational Geometry
\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{hyperref} \usepackage{enumitem} \usepackage{microtype} \usepackage{booktabs}
\title{Emergence of Lorentzian Spacetime and Cosmological Structure from Informational Geometry} \author{C. L. Vaillant} \date{2024-2026 (Working Manuscript --- Not Peer Reviewed)}
\newtheorem{theorem}{Theorem} \newtheorem{proposition}{Proposition} \newtheorem{definition}{Definition} \newtheorem{assumption}{Assumption} \newtheorem{conjecture}{Conjecture} \newtheorem{remark}{Remark} \newtheorem{corollary}{Corollary}
\begin{document}
\maketitle
\begin{abstract} We investigate the extent to which Lorentzian spacetime structure can be reconstructed from informational geometry. Starting from the Fisher information metric on the probability simplex, we obtain a constant-curvature distinguishability geometry without presupposing space. A monotone informational flow defines an intrinsic ordering of states, from which a causal cone and Lorentzian metric emerge. We then extend the framework to quantum theory by identifying admissible updates with completely positive trace-preserving dynamics and replacing classical Fisher geometry with quantum monotone information metrics. At the quantum field level, local operator algebras and modular flow provide an intrinsic notion of temporal ordering, while entanglement structure organizes the emergence of effective geometry. Under symmetry assumptions, the resulting metric sector is shown to be compatible with FLRW Einstein cosmology. A conjectural relation between cosmological scale and informational capacity is proposed, and a holographic coarse-graining procedure reproduces the de Sitter entropy law within the homogeneous sector. The logical status of each step is stated explicitly. \end{abstract}
\tableofcontents
\section{Introduction}
General relativity describes spacetime geometry with extraordinary success, but it does not explain why spacetime should exist in the first place. Quantum theory, by contrast, is formulated most naturally in terms of states, operators, and transformations rather than in terms of a primitive geometric background. This motivates the question addressed in this paper:
\begin{quote} Can spacetime structure be reconstructed from informational structure? \end{quote}
The proposal explored here is that distinguishability between states is primary, and that geometry arises as an effective representation of that distinguishability together with the structure of admissible updates between states. In this sense, space corresponds to symmetric distinguishability, time to ordered informational change, and Lorentzian causality to the incompatibility between those two kinds of variation.
The framework is organized in four layers: \begin{enumerate}[label=(\roman*)] \item a classical statistical layer based on the probability simplex and the Fisher information metric, \item a quantum informational layer in which admissible updates are realized by completely positive trace-preserving maps and distinguishability is measured by a monotone quantum information metric, \item a quantum field layer in which local operator algebras, modular flow, and entanglement provide the natural language for intrinsic temporal ordering and geometric organization, \item a coarse-grained gravitational layer in which the informational metric sector is shown to be compatible with an FLRW Einstein cosmology. \end{enumerate}
The guiding chain is
\text{Information} \;\Longrightarrow\; \text{Causality} \;\Longrightarrow\; \text{Geometry} \;\Longrightarrow\; \text{Gravity}.
Throughout, the logical status of each step is made explicit. We distinguish: \begin{itemize} \item \textbf{Derived results}, which follow from the formal setup, \item \textbf{Assumptions}, which are introduced explicitly, \item \textbf{Conjectures}, which close the framework but are not derived here. \end{itemize}
\section{Classical Informational Geometry}
\subsection{The probability simplex}
\begin{definition}[Open probability simplex] For $N\ge 2$, define
\Delta_N^\circ
=
\left\{
p=(p_1,\dots,p_N)\in \mathbb{R}^N
\;\middle|\;
p_i>0,\ \sum_{i=1}^N p_i=1
\right\}.
A point in $\Delta_N^\circ$ is a probability distribution over $N$ distinguishable outcomes. The simplex therefore represents the space of all possible classical epistemic states of a finite outcome set.
\subsection{Fisher information metric}
\begin{definition}[Fisher metric] The Fisher information metric on $\Delta_N^\circ$ is
h
=
\sum_{i=1}^N \frac{dp_i^2}{p_i},
\qquad
\sum_{i=1}^N dp_i = 0.
This metric measures local distinguishability between nearby probability distributions.
\begin{proposition}[Sphere representation] Under the coordinate transformation
x_i = 2\sqrt{p_i},
S^{N-1}(r=2)\subset \mathbb{R}^N.
\begin{proof} Since $p_i=x_i^2/4$, one has $dp_i=(x_i/2),dx_i$, so
\frac{dp_i^2}{p_i}
=
\frac{(x_i^2/4)\,dx_i^2}{x_i^2/4}
=
dx_i^2.
h=\sum_{i=1}^N dx_i^2.
\sum_i p_i = 1
\quad\Longrightarrow\quad
\sum_i x_i^2 = 4,
\end{proof}
\begin{corollary} The induced sectional curvature is
K=\frac{1}{4}.
\paragraph{Interpretation.} A constant-curvature spatial geometry appears directly from distinguishability structure, without any prior assumption of physical space.
\section{Informational Dynamics and Ordering}
\subsection{Clock functional}
\begin{definition}[Informational potential] Define
\Phi(p)=\sum_{i=1}^N p_i\log p_i.
This is the negative Shannon entropy. It is minimal at the maximally mixed state and approaches $0$ near extremal pure-outcome corners.
\subsection{Gradient flow}
\begin{definition}[Informational flow] Let
\dot p = \nabla_h \Phi
\end{definition}
\begin{proposition}[Monotonicity] Along this flow,
\frac{d\Phi}{dt}
=
h(\nabla_h\Phi,\nabla_h\Phi)\ge 0.
\begin{proof} By definition of the gradient,
\frac{d\Phi}{dt}
=
d\Phi(\dot p)
=
d\Phi(\nabla_h\Phi)
=
h(\nabla_h\Phi,\nabla_h\Phi)\ge 0.
\paragraph{Interpretation.} The flow defines an intrinsic ordering of informational states. At this stage, the parameter is not yet a physical time coordinate, but it already distinguishes reversible differences within a level set from ordered evolution across level sets.
\section{Classical Emergence of Lorentzian Structure}
\subsection{Admissible cone}
\begin{definition}[Classical admissible cone] At each $p\in\Delta_N^\circ$, define
C_p^+
=
\left\{
v\in T_p\Delta_N^\circ
\;\middle|\;
d\Phi(v)>0,\
h(v,v)\le \beta^2 d\Phi(v)^2
\right\}.
\begin{assumption}[Classical admissibility] Classically admissible infinitesimal updates lie within $C_p^+$. \end{assumption}
\subsection{Effective metric}
\begin{definition}[Effective classical quadratic form] Define
g = h - \beta^2 d\Phi^2.
\begin{theorem}[Lorentzian signature] If
\beta^2 \|\nabla_h\Phi\|_h^2 > 1,
\end{theorem}
\begin{proof} Let $n=\nabla_h\Phi/|\nabla_h\Phi|_h$. If $v$ is tangent to a level set of $\Phi$, then $d\Phi(v)=0$, so
g(v,v)=h(v,v)>0.
g(n,n)
=
1-\beta^2 d\Phi(n)^2
=
1-\beta^2 \|\nabla_h\Phi\|_h^2<0.
\end{proof}
\paragraph{Interpretation.} The Lorentzian sign is not inserted manually. It appears because symmetric distinguishability and irreversible informational ordering are geometrically inequivalent kinds of variation.
\section{Quantum Informational Completion}
The classical cone above is not taken as fundamental. It is interpreted as the commuting limit of a more general quantum informational structure.
\subsection{Quantum state space}
Let $\mathcal{D}(\mathcal{H})$ denote the space of density operators on a Hilbert space $\mathcal{H}$:
\mathcal{D}(\mathcal{H})
=
\{\rho\ge0,\ \mathrm{Tr}\rho=1\}.
Classical probability distributions are recovered as the commuting diagonal sector of $\mathcal{D}(\mathcal{H})$.
\subsection{Admissible quantum updates}
A physical quantum update is represented by a completely positive trace-preserving (CPTP) map
\mathcal{E}:\mathcal{D}(\mathcal{H})\to\mathcal{D}(\mathcal{H}).
Continuous Markovian evolution is represented by a CPTP semigroup
\mathcal{E}_t=e^{t\mathcal{L}},
\begin{definition}[Quantum admissible tangent vector] A tangent vector $v\in T_\rho\mathcal{D}(\mathcal{H})$ is admissible if there exists a differentiable CPTP semigroup $\mathcal{E}_t$ such that
v=\left.\frac{d}{dt}\mathcal{E}_t(\rho)\right|_{t=0}.
\paragraph{Interpretation.} Admissibility is no longer a postulated cone condition. It is the infinitesimal envelope of physically realizable quantum channel dynamics.
\subsection{Quantum information geometry}
In the quantum setting, the classical Fisher metric is replaced by a monotone quantum information metric. We adopt the Bures metric $g_B$, equivalently the symmetric logarithmic derivative quantum Fisher metric, because it is contractive under CPTP maps and reduces to the classical Fisher metric on commuting states.
\subsection{Quantum speed limit and effective cone}
Quantum speed limits imply that physically realizable evolution cannot traverse arbitrarily large Bures distance in arbitrarily small proper time. This motivates the local condition
ds_B^2 \le d\tau^2,
Define
g_{\mathrm{eff}} = g_B - d\tau^2.
\begin{proposition} The null cone of $g_{\mathrm{eff}}$ coincides with the local envelope of admissible quantum evolutions saturating the quantum speed bound. \end{proposition}
\paragraph{Classical limit.} On the commuting diagonal sector,
g_B \longrightarrow h,
\section{Quantum Field Layer: Modular Flow and Entanglement}
The finite-dimensional quantum information picture extends naturally to quantum field theory by replacing global state spaces with local operator algebras.
\subsection{Local operator algebras}
To each spacetime region $\mathcal{O}$, algebraic quantum field theory associates a von Neumann algebra $\mathcal{A}(\mathcal{O})$. A state restricted to such an algebra determines a modular structure via Tomita--Takesaki theory.
\subsection{Modular flow}
Given a faithful state $\omega$ and a local algebra $\mathcal{A}$, modular theory defines an intrinsic one-parameter automorphism group
\sigma_t^\omega(\mathcal{A}).
\paragraph{Interpretation.} This is an algebraic notion of temporal ordering that does not require a prior external clock.
For the Minkowski vacuum restricted to a Rindler wedge, the Bisognano--Wichmann theorem identifies modular flow with Lorentz boosts. This is a strong indication that local time flow can arise from state-and-algebra structure alone.
\subsection{Entanglement and geometry}
At the quantum field and holographic levels, entanglement is not peripheral to geometry. Relative entropy, entanglement first laws, and entanglement-wedge relations all indicate that geometric structure is constrained by entanglement organization.
The present framework therefore admits the following interpretation: \begin{enumerate} \item primitive informational depth controls the availability of distinguishable bulk states, \item entanglement organizes these states into connected effective geometry, \item modular flow provides the local ordering structure that the classical model approximates via the entropy gradient flow. \end{enumerate}
\paragraph{Status.} This paper does not derive Einstein dynamics uniquely from entanglement. Rather, it aligns the informational construction with the known direction in which modular flow, relative entropy, and entanglement first-law arguments constrain gravitational geometry.
\section{Coarse-Grained Gravitational Sector}
\begin{assumption}[Homogeneity] The informational manifold has no preferred point beyond the structures introduced above. \end{assumption}
Under homogeneity, the scale parameter in the classical construction is constant:
\beta = \beta_0.
\begin{assumption}[Normalization] The limiting admissible update speed defines unit proper time. \end{assumption}
Thus
d\tau = \beta_0\, d\Phi.
\begin{assumption}[Isotropy] After coarse-graining, the level sets of $\Phi$ are isotropic. \end{assumption}
Then the emergent metric takes FLRW form:
ds^2 = -d\tau^2 + a(\tau)^2 \bar h_{ab}d\theta^a d\theta^b.
\begin{assumption}[Einstein vacuum compatibility] The coarse-grained metric sector is compatible with Einstein vacuum dynamics with positive cosmological constant. \end{assumption}
\paragraph{Result.} Under this assumption, the emergent metric sector reproduces the standard FLRW vacuum equations. In $3+1$ dimensions,
3\left(\frac{\dot a^2}{a^2}+\frac{K}{a^2}\right)=\Lambda,
\qquad
\frac{\ddot a}{a}=\frac{\Lambda}{3},
\paragraph{Important qualification.} The present framework does not derive Einstein dynamics uniquely. It shows that the emergent Lorentzian metric sector is compatible with Einstein vacuum cosmology under explicit assumptions.
\section{Planck Lower Bound Without Exact Saturation}
The informational framework introduces a microscopic update timescale $\ell_I$. The earlier saturation claim $\ell_I=t_P$ is stronger than what is currently justified. What can be derived is a lower bound.
\subsection{Quantum action bound}
For an elementary informational update of energy $E_I$ and duration $\ell_I$, quantum resolvability requires
E_I\,\ell_I \gtrsim \hbar.
\subsection{Localization against collapse}
If the update is localized to a region of size $\ell_x\sim c\ell_I$, then gravitational non-collapse requires
c\ell_I \gtrsim \frac{2GE_I}{c^4}.
Combining the two inequalities yields, up to order-one factors,
\ell_I \gtrsim t_P,
\qquad
t_P = \sqrt{\frac{\hbar G}{c^5}}.
\begin{proposition}[Planck lower bound] Any physically distinguishable informational update compatible with quantum action and gravitational non-collapse satisfies
\ell_I \gtrsim t_P.
\subsection{Gap parameter}
We therefore write
\ell_I=\alpha_P\, t_P,
\qquad
\alpha_P\ge 1.
\paragraph{Interpretation.} $\alpha_P$ measures the separation between the actual elementary informational timescale and the Planck lower bound.
\section{Primitive Capacity Law}
\begin{conjecture}[Capacity law] The cosmological scale is related to primitive informational depth by
\Lambda = \frac{1}{\alpha_P^2 t_P^2 (\log N)^2}.
Equivalently,
\log N = \frac{1}{\alpha_P\, t_P\sqrt{\Lambda}}.
\paragraph{Status.} This relation is not derived here from a microscopic counting principle. It is a conjectural closure of the framework.
\paragraph{Interpretation.} The cosmological constant is interpreted as the inverse square of the primitive informational depth of the vacuum, weighted by the elementary update scale.
\section{Holographic Coarse-Graining}
\subsection{Primitive depth and boundary entropy}
Define the primitive bulk depth
u:=\alpha_P \log N.
Assume an isotropic boundary encoding with effective screen radius
R = \alpha\, u\, \ell_P.
Then
A = 4\pi R^2 = 4\pi \alpha^2 u^2 \ell_P^2.
Applying the area law,
S = \frac{A}{4\ell_P^2},
S = \pi \alpha^2 u^2.
Matching to de Sitter entropy fixes
\alpha=\sqrt{3}.
Therefore
\boxed{
S = 3\pi (\alpha_P\log N)^2
}
\boxed{
S = \frac{3\pi}{\Lambda t_P^2}.
}
\paragraph{Interpretation.} The primitive quantity $\alpha_P\log N$ is not itself a horizon entropy. It is a bulk depth variable. Observable horizon entropy is the quadratic boundary coarse-graining of that primitive depth.
\section{Observational Sector}
\subsection{Effective vacuum depth}
Define
D(a):=\alpha_P(a)\,\log N(a).
Then
\Lambda_{\mathrm{eff}}(a)=\frac{1}{t_P^2 D(a)^2}.
Since $\rho_{\mathrm{DE}}(a)\propto \Lambda_{\mathrm{eff}}(a)$, the effective equation of state becomes
w_{\mathrm{eff}}(a)
=
-1-\frac{1}{3}\frac{d\ln\rho_{\mathrm{DE}}}{d\ln a}
=
-1+\frac{2}{3}\frac{d\ln D}{d\ln a}.
Hence: \begin{enumerate} \item if $D(a)$ is constant, then $w_{\mathrm{eff}}=-1$, \item if $D(a)$ increases with expansion, then $w_{\mathrm{eff}}>-1$, \item if $D(a)$ decreases with expansion, then $w_{\mathrm{eff}}<-1$. \end{enumerate}
\subsection{Minimal phenomenological ansatz}
A simple one-parameter deformation is
D(a)=D_0 a^\nu.
Then
\Lambda_{\mathrm{eff}}(a)=\Lambda_0 a^{-2\nu},
\qquad
w_{\mathrm{eff}}=-1+\frac{2}{3}\nu.
Thus deviations from $\Lambda$CDM correspond directly to flow of primitive vacuum depth.
\subsection{Benchmark normalization}
Using the benchmark flat-$\Lambda$CDM values
H_0=67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}},
\qquad
\Omega_m=0.315,
\qquad
\Omega_\Lambda\approx 0.685,
\Lambda_{\mathrm{obs}} \approx 1.09\times 10^{-52}\,\mathrm{m}^{-2}.
With
\ell_P = 1.616255\times 10^{-35}\,\mathrm{m},
\alpha_P \log N
=
\frac{1}{\ell_P\sqrt{\Lambda_{\mathrm{obs}}}}
\approx 5.93\times 10^{60}.
Equivalently,
\alpha_P \log_2 N \approx 8.55\times 10^{60}\ \text{bits}.
\subsection{Current constraint on depth flow}
If one uses the benchmark constant-$w$ value
w_0=-0.978^{+0.024}_{-0.031},
\nu = \frac{3}{2}(w_0+1)
\approx 0.033^{+0.036}_{-0.047}.
Thus current benchmark cosmology remains compatible with
\nu=0,
\paragraph{Interpretation.} Within this framework, the observational question of evolving dark energy becomes the informational question of whether primitive vacuum depth evolves with cosmic expansion.
\section{Logical Status}
\subsection*{Derived} \begin{itemize} \item Fisher geometry on the simplex, \item monotone informational ordering from the entropy gradient flow, \item Lorentzian signature from the split between reversible distinguishability and irreversible ordering, \item quantum origin of admissible updates via CPTP/GKLS dynamics, \item Planck lower bound on the elementary informational timescale, \item compatibility of the primitive depth law with de Sitter entropy under holographic coarse-graining. \end{itemize}
\subsection*{Assumed} \begin{itemize} \item homogeneity of the informational manifold, \item isotropy of the coarse-grained sector, \item Einstein vacuum compatibility of the emergent metric, \item existence of a homogeneous isotropic boundary encoding. \end{itemize}
\subsection*{Conjectured} \begin{itemize} \item the primitive capacity law
\Lambda = \frac{1}{\alpha_P^2 t_P^2 (\log N)^2},
\item the full dynamical law governing $D(a)$,
\item the derivation of Einstein dynamics from the entanglement structure rather than its explicit assumption.
\end{itemize}
\section{Conclusion}
We have developed a layered informational framework in which Lorentzian spacetime structure emerges from distinguishability geometry and admissible state evolution. At the classical level, the probability simplex with the Fisher metric yields a constant-curvature geometry of distinguishability. A monotone informational flow defines an intrinsic ordering of states, and from the split between reversible and irreversible directions a Lorentzian metric emerges. At the quantum level, admissible updates are identified with CPTP/GKLS dynamics, and classical Fisher geometry is recognized as the commuting limit of quantum monotone information geometry. At the quantum field level, modular flow and entanglement provide the natural structural interpretation of emergent time and geometry. Under symmetry assumptions, the resulting metric sector is compatible with FLRW Einstein cosmology and standard horizon thermodynamics.
Within this interpretation, the classical simplex construction is not an isolated ansatz. It is the finite-dimensional commuting reduction of a broader quantum-entanglement-geometric picture. The cosmological constant is conjecturally tied to primitive informational depth, while observable horizon entropy is its holographically dressed boundary image.
The principal open problems are now sharply defined: \begin{enumerate} \item derive the primitive capacity law from a microscopic counting principle, \item determine the gap parameter $\alpha_P$ from deeper quantum-gravitational structure, \item derive the emergent metric dynamics rather than assuming Einstein compatibility, \item determine the dynamical law governing $D(a)$ and confront it directly with cosmological data. \end{enumerate}
Under its stated assumptions, the framework provides a coherent route from informational structure to Lorentzian spacetime, cosmological geometry, and horizon entropy.
\end{document}
