Time is gravity with derived emergence of gravity where dS/dT>0

Stockholm 2026-01-15Author Mark Jacobson dS/dt>0 explains these conclusions HoLTZ gravity mechanism (as presented in the HoLTZ frameworkdiscussions on X) is a highly non-mainstream, speculative reformulation of gravity that completelyinverts the usual picture of general relativity.Instead of gravity being primarily a curvature of spacecaused by mass-energy (Einstein's geometric view), HoLTZ treats time as the fundamental arena andgravity as an emergent entropic effect driven by the irreversible flow of time.Core Ideas of the HoLTZGravity MechanismAbstractThe HoLTZ GL theory proposes a pre-geometric, timeless substrate consisting of directed relational networks, from which spacetime, matter, and gravity emerge solely through the fundamental principle of Directed Relational Entropy Growth (DREG): dS/dτ > 0, where S is a specially defined relational entropy and τ is an emergent causal ordering parameter. In this manuscript we derive the effective gravitational interaction as a consequence of entropy gradients in the substrate. Gravity manifests as an entropic "elastic" force that minimizes relational tension induced by local complexity gradients, leading in the continuum limit to an Einstein-like field equation. We present the key mathematical steps, from the definition of the substrate entropy to the recovery of the weak-field Newtonian limit and hints toward full general-relativistic curvature.1. IntroductionConventional approaches to quantum gravity typically assume spacetime as a pre-existing arena or quantize it directly. In contrast, HoLTZ GL adopts a radically minimalist ontology: reality begins as an abstract, pre-geometric directed acyclic graph (DAG) of proto-states connected by irreversible directed relations. No metric, no coordinates, and no fundamental time parameter exist a priori.The sole dynamical principle is DREG, which enforces strict monotonic increase of relational entropy along emergent causal chains. Gravity, in this picture, is not a fundamental interaction but an emergent response of the relational network to gradients in this entropy field, analogous to thermodynamic forces arising from entropy gradients in ordinary statistical mechanics.This derivation builds upon earlier thermodynamic routes to gravity (e.g., Jacobson 1995) but removes the assumption of a pre-existing spacetime manifold.2. The Relational Substrate and Entropy FunctionalThe fundamental structure is a directed relational graph Γ = (Σ, R), where Σ is the set of proto-states σ_i and R ⊆ Σ × Σ is the set of directed relations r_{ij}: σ_i → σ_j, with the graph remaining acyclic to encode irreversibility.Relational entropy on a local subgraph Γ_ℓ is defined asS(Γ_ℓ) = ∑_{maximal paths p in Γ_ℓ} log Ω_p ⋅ Θ(Δr_p),where Ω_p counts the multiplicity of equivalent irreversible branchings along path p, and Θ ensures only strictly irreversible steps contribute.In the large-system (continuum) limit, after suitable coarse-graining, S becomes a scalar field on an emergent manifold:S(x) ≈ ∫ ρ(x) log(ρ(x)/ρ₀) d³x + boundary corrections,with ρ(x) the local relational density.3. Directed Relational Entropy Growth (DREG) and Emergent Parameter τThe core axiom readsdS/dτ > 0 (strictly),where τ emerges as the extremal path length in the DAG:τ = sup_{chains C} length(C), with length measured in number of irreversible steps.Variationally, τ satisfies a self-consistent geodesic-like equation derived from the actionA[τ] = ∫ (∂S/∂τ)²/2 − V(S) dτ,with potential V(S) ≈ λ S²/2 (λ > 0) enforcing monotonicity.The resulting evolution equation is∂S/∂τ = √(2(V(S) + K)),where K collects fluctuation terms. Solutions with dS/dτ ≤ 0 are exponentially unstable, selecting the observed arrow.4. Entropy Gradients as Gravitational ForcesConsider a test subgraph Γ_test propagating through regions of varying S. The effective action along its worldline readsA = ∫ dτ [ m (∂S/∂τ) − ∫ ∇S ⋅ dx ],where m ∝ internal entropy of the subgraph (effective inertial mass).Stationarity (δA = 0) yields the equation of motionm d²x/dτ² = − ∇S.Thus the force law isF = − ∇S.This constitutes the Newtonian limit: regions of high relational complexity (high S, e.g. massive objects) attract low-complexity regions via steep entropy gradients.5. Relativistic Extension and Emergent CurvatureTo recover relativistic gravity, promote the description to a diffeomorphism-invariant effective field theory.The induced metric on the emergent manifold arises from averaging path lengths in the relational graph:g_{μν} ≈ ⟨ dx^μ dx^ν ⟩_{paths respecting DREG}.In the weak-field regime, perturbations h_{μν} around Minkowski η_{μν} satisfy∇² h_{00} ≈ − κ (∇S)² / S,with κ determined by matching to the Newtonian limit (κ → 8π in units G = c = 1).Full Einstein-like equations emerge by demanding DREG invariance under local coarse-graining diffeomorphisms, leading toR_{μν} − ½ R g_{μν} = 8π T_{μν}(S),where the effective stress-energy tensor derives from S variations:T_{μν}(S) ≈ (∂μ S ∂_ν S − ½ g{μν} ∂^λ S ∂_λ S) + higher-order terms.Black-hole horizons correspond to saturation points of local entropy growth, reproducing the Bekenstein-Hawking area law S_BH ≈ A/4 as maximal relational complexity on the causal boundary.6. Discussion and OutlookThe HoLTZ GL derivation achieves several desiderata for quantum gravity: complete elimination of fundamental spacetime and time, unification of the arrow of time with gravitational dynamics, natural incorporation of quantum fluctuations via √S noise in the entropy growth, emergent compatibility with the holographic principle. Open challenges include: precise renormalization of the coupling λ, derivation of the exact matter Lagrangian from topological invariants of the graph, quantitative predictions for Planck-scale phenomenology testable in analog systems or future quantum gravity experiments. Future work will focus on numerical simulations of finite relational graphs to verify stability of DREG solutions and emergence of curved geometries.AcknowledgmentsThis draft synthesizes concepts from the HoLTZ GL manuscript (Jacobson, 2026) and related thermodynamic derivations of gravity. Numerical explorations were conceptually supported by large-scale relational network simulations.References Jacobson, M. (2026). HoLTZ GL: A Relational Substrate Unification. [Zenodo record / arXiv equivalent]. Jacobson, T. (1995). Thermodynamics of Spacetime: The Einstein Equation of State. Phys. Rev. Lett. 75, 1260. [Additional references to entropic gravity, causal set theory, etc., would be expanded in a full manuscript.]Source:https://doi.org/10.5281/zenodo.18259128