View source for Physics:Curved Spacetime Derived from Obidi's Theory of Entropicity(ToE)

== From Curved Spacetime in Einstein's Relativity to Curved Entropic Field in Obidi's Theory of Entropicity(ToE) ==
Conventional relativity asserts that '''matter curves spacetime''' and free bodies follow geodesics of that curved geometry. The [[Theory of Entropicity(ToE)]],<ref name="obidi2025reconcile">Obidi, John Onimisi. ''Einstein and Bohr Finally Reconciled on Quantum Theory: The Theory of Entropicity (ToE) as the Unifying Resolution to the Problem of Quantum Measurement and Wave Function Collapse''. Cambridge University.  (14 April 2025). https://doi.org/10.33774/coe-2025-vrfrx</ref> first formulated and developed by [[Biography:John Onimisi Obidi|John Onimisi Obidi]],<ref name="obidi_addendum">Obidi, John Onimisi. ''A Critical Review of the Theory of Entropicity (ToE) on Original Contributions, Conceptual Innovations, and Pathways towards Enhanced Mathematical Rigor: An Addendum to the Discovery of New Laws of Conservation and Uncertainty''. Cambridge University.(2025-06-30). https://doi.org/10.33774/coe-2025-hmk6n</ref><ref>Obidi, John Onimisi (2025). Master Equation of the Theory of Entropicity (ToE). Encyclopedia. https://encyclopedia.pub/entry/58596</ref><ref>Physics:Artificial Intelligence Formulated by the Theory of Entropicity(ToE). (2025, August 27). HandWiki, . Retrieved 03:59, August 27, 2025 from https://handwiki.org/wiki/index.php?title=Physics:Artificial_Intelligence_Formulated_by_the_Theory_of_Entropicity(ToE)&oldid=3742591</ref> proposes a complementary inversion: it is the '''entropic field''' that is fundamentally structured (curved), and what we call “spacetime curvature” is the effective bookkeeping of this deeper entropic structure.

=== Entropic Curvature Hypothesis (ECH) ===
* '''ECH:''' The primary curvature in nature is the curvature of the '''entropic field''' (entropy density and flux). The familiar spacetime curvature of General Relativity emerges as an effective description of this underlying entropic curvature.

=== Operational Reinterpretations ===
* '''Elapsed time ↔ entropic change:''' When we say “time has elapsed,” we register a nonzero entropic increment (a finite entropic event/tick).  
* '''Spatial separation ↔ entropic separation:''' When we perceive “distance,” we are registering the required entropic reconfiguration between states—an '''entropic separation'''—traced along an entropic geodesic (path of least entropic resistance).  
* '''Dynamics ↔ entropic geodesics:''' Motion follows trajectories that extremize an entropic action (least entropic resistance), which appear as geodesics in an emergent metric.

== Entropic Axiom of Reality (EAR) ==
ToE can be framed as a new axiom:

* '''EAR:''' Entropy is a fundamental field whose local density and flux dictate the organization of events. Time, distance, and geometry are emergent records of lawful, irreversible '''entropic flows'''.

=== Consequences ===
# '''No entropy, no time:''' Without finite entropic increments, no ticks occur and operational time is undefined.  
# '''No entropic gradients, no geometry:''' Without gradients, there is no structured organization of flows; the effective metric is trivial/degenerate.  
# '''With gradients, geometry emerges:''' Stable patterns of entropic flow induce an effective metric that we describe as spacetime curvature.

== Entropic Genesis of Metric ==
In ToE, the metric encodes how entropic flows organize:

* '''Metric as entropic bookkeeping:''' The effective metric summarizes how energy and information traverse entropic geodesics.  
* '''Curvature as entropic structure:''' What appears as curvature (tidal effects, lensing, time dilation) reflects variations in entropic density/flux and the constraints they impose.

(Heuristic schematic)
<math>g_{\mu\nu}^{\text{eff}} \;\sim\; \mathcal{F}\!\big[\partial_\mu \Lambda \, \partial_\nu \Lambda,\; J_\mu,\; \Pi_{\mu\nu}^{(\mathrm{irr})}\big]</math>  
where <math>\Lambda</math> is an entropy density functional, <math>J_\mu</math> an entropy flux, and <math>\Pi_{\mu\nu}^{(\mathrm{irr})}</math> captures irreversibility/production.

== Matter, Curvature, and Entropy: Who Sources What? ==
* '''GR view:''' Mass–energy sources curvature; curvature guides motion.  
* '''ToE view:''' Constraint asymmetry and irreversibility source the entropic field; the entropic field organizes flows; the observed metric is the effective summary of these flows.

{| class="wikitable"
! Framework !! Source term !! Governing law !! Physical path !! Observables
|-
| GR || Stress–energy <math>T_{\mu\nu}</math> || Einstein equations || Spacetime geodesics || Time dilation, lensing
|-
| ToE || Constraint asymmetry + irreversibility || Obidi Action / Vuli-Ndlela Integral (entropic field eqn) || Entropic geodesics (least entropic resistance) || Tick rates (ETL), entropic lensing, effective metric
|}

== Why ToE as a New Axiom ==
The following questions motivate ToE’s foundational status:
* “How can we say matter curves spacetime?” 

→ ToE's Approach: matter’s microstate structure and process irreversibility curve the '''entropic field'''; spacetime curvature is the emergent ledger.  
* “When time elapses, what truly changed?” 

→ ToE's Approach: a finite entropic increment occurred (an entropic event).  
* “When a distance separates two events, what created that space?” 

→ ToE's Approach: the entropic field structured a path of least entropic resistance between them, recorded as spatial separation.

=== Compact Statement ===
'''ToE elevates entropy from statistic to field:''' time is the count of entropic events, distance is entropic separation, and geometry is the emergent organization of irreversible entropic flows. Hence, it is more fundamental to say the '''entropic field is curved''', with spacetime as its effective manifestation.

== Programmatic Tests (for Future Work) ==
* '''Clock response to engineered entropic environments:''' Predictable rate shifts under controlled constraint asymmetries (ETL-linked).  
* '''Entropic lensing:''' Quantify deflection from calibrated entropy gradients and compare with GR lensing in matched regimes.  
* '''Geodesic selection under dissipation control:''' Track trajectory changes as irreversibility parameters are tuned.

== References ==

{{Sourceattribution|Curved Spacetime Derived from Obidi's Theory of Entropicity(ToE)|1}}