View source for Physics:Observer-Dependent G-Entropy and the Theory of Entropicity(ToE)

== Observer-Dependent Gravitational Entropy and the Theory of Entropicity (ToE) ==

=== Overview === 

The relationship between gravitational entropy and the observer's reference frame has generated significant debate within modern theoretical physics. A recent study published in the Journal of High Energy Physics (JHEP 07/2025) establishes that gravitational entropy is not an absolute quantity but instead depends fundamentally on the observer. This idea appears to challenge foundational assumptions of the [[Theory of Entropicity (ToE)]], which treats entropy as a physical, observer-independent scalar field that gives rise to gravitation through entropic gradients. This article explores the tension between these perspectives, reviews the claims and implications of the observer-dependent entropy paper, and proposes pathways for reconciling ToE with observer-relational entropy.

== Gravitational Entropy as an Observer-Dependent Quantity ==

The 2025 JHEP paper titled "Gravitational entropy is observer-dependent" introduces the idea that entropy in a gravitational system, specifically in a quantum gravitational context, varies depending on the observer's frame. This insight arises from the usage of Quantum Reference Frames (QRFs), which allow one to treat observers themselves as quantum systems. The entropy of a spacetime region is then calculated relative to a chosen QRF.

In this framework, the division of the total Hilbert space  into subsystems (e.g., system vs. environment, or inside vs. outside of a horizon) is not fixed. Instead, it varies with the observer. The von Neumann entropy associated with a region , for instance,

<math>
S_A = -\text{Tr}(\rho_A \log \rho_A),
</math>

can differ based on how entropy is defined from a particular frame. An inertial observer in Minkowski space sees the vacuum as pure, with zero entropy, while an accelerating observer perceives a thermal state with a non-zero entropy due to a Rindler horizon.

This observer-dependence implies that gravitational entropy, like energy in special relativity, has no absolute value but only a frame-dependent one. In the quantum gravity context, this is more than a coordinate artifact; it is a fundamental property of how physical systems are described relative to each other.

== Core Assumptions of the Theory of Entropicity (ToE) ==

ToE reinterprets entropy as a fundamental physical field  defined over spacetime. This field is not simply a thermodynamic state function, but a dynamical, causal field with the following properties:

'''Scalar Field''':  is postulated to be a Lorentz-invariant scalar at every spacetime point, similar to how temperature can be a local scalar field in relativistic thermodynamics.

'''Source of Gravity''': Gravity is not caused by spacetime curvature per se, but emerges from entropy gradients. A region with a higher  induces motion toward higher entropy states, mimicking gravitational attraction.

'''Master Action''': The dynamics of ToE are encoded in the Obidi Action:

<math>
\mathcal{A}_{\text{ToE}} = \int d^4x \sqrt{-g} \left( \mathcal{L}_{\text{matter}} + \mathcal{L}_{\text{entropy}} + \eta S(x) T^\mu_{\ \mu} \right),
</math>

where <math>\eta</math> is the trace of the energy-momentum tensor and  is a coupling constant.

This theory leads to the so-called entropic field equation:

<math>
\Box S(x) + \frac{\partial V(S)}{\partial S} = \eta T^\mu_{\ \mu},
</math>

thus establishing entropy as a source-coupled scalar field with independent dynamics.

== The Apparent Contradiction ==

The contradiction arises from the fact that if gravitational entropy is not an invariant scalar but varies depending on the observer, then  in ToE cannot be both a fundamental scalar and also represent gravitational entropy.

There are two immediate concerns:

1. Scalar Consistency: If  is truly a scalar field, then all observers should agree on its value at a point . However, if gravitational entropy is observer-relative,  must also vary with the observer. This undermines its scalar status.

2. Gradient Ambiguity: ToE posits that  drives motion via an entropic force. If different observers compute different , they may also compute different entropic gradients and thus different gravitational accelerations—an unacceptable result.

== Pathways for Reconciliation ==

Despite this tension, several strategies can be used to reconcile ToE with the notion of observer-dependent gravitational entropy.

=== Relational Entropy Interpretation === 

ToE can adapt to this challenge by treating  not as an absolute scalar field but as a relational field. In this view,  depends on both the spacetime point and the observer's quantum reference frame:

<math>
S(x; \text{QRF}) \rightarrow \text{a relational scalar field},
</math>

meaning that the entropy is a gauge-dependent section of a higher-dimensional bundle over both spacetime and observer Hilbert spaces.

This would then align ToE more closely with the way potentials and reference frames operate in general relativity and quantum theory.

===  Entropic Charge vs. Measured Entropy === 

Another interpretation is to distinguish between the "'''entropic charge'''" or '''intrinsic information content''' at a point , and the entropy an observer measures, which depends on coarse-graining.

Thus, entropy is invariant as a field, but its interpretation as thermodynamic entropy becomes observer-relative:

: fine-grained entropy density (field-theoretic).

: observer's coarse-grained entropy of a region , derived from partial access to the field.

===  Horizon-Dependent Entropy === 

Many instances of observer-dependent entropy arise near causal horizons. For example:

Rindler horizon, which possesses Unruh entropy,

Black hole horizon, which possesses Bekenstein-Hawking entropy.

ToE could therefore explain these phenomena by asserting that entropy accumulates or is integrated over inaccessible boundary regions, that is to say:

<math>
S_{\text{horizon}} = \int_{\mathcal{H}} S(x) dA,
</math>

and that the observer-dependence arises from differences in the accessible regions of entropy.

=== Coarse-Graining Rules Embedded in ToE === 

ToE might be revised to include a rule for coarse-graining based on observer type:

Static observer: coarse-grains beyond static horizon,

Infalling observer: uses full access, no coarse-graining.

Hence, the entropy measured depends on the degree of access to the global  field, but the field itself remains invariant.

== Formalizing Observer Dependence in ToE ==

To reconcile these views rigorously, ToE could introduce an entropic frame bundle , where:

1) Each observer corresponds to a section 

2) The field  lives on the bundle rather than directly on the (observer) frame.

Then entropy becomes a function of both location and observer-frame:

<math>
S: (x, e) \mapsto \mathbb{R},
</math>

but where physical predictions must be frame-invariant, such that:

<math>
\delta \mathcal{A}_{\text{ToE}} = 0 
</math>

under entropic frame transformation.

This approach echoes the shift from '''Galilean relativity''' to '''Lorentz invariance''' and might be formalized using '''fiber bundles''' and '''groupoids'''.

== Implications for Gravity from Entropic Gradients ==

ToE posits that gravity arises from gradients, and objects follow entropic geodesics. In a relativistic observer-dependent setting, this becomes:

<math>
\nabla^{(e)} S(x) = f_{\text{obs}}(x, e),
</math>

where the function on the right hand side is derived from the observer's coarse-grained effective entropy.

Therefore, the entropic force is not universal but perceived differently:

Inertial observer: flat geodesic, 

Accelerated observer: curved geodesic, 

'''
This elegantly matches how gravity appears to inertial and accelerated observers in general relativity.'''

== Conclusion on Observer-Dependence of Gravitational Entropy ==

While the observer-dependence of gravitational entropy seems at first to contradict the Theory of Entropicity, a deeper examination reveals that ToE can accommodate this result through relatively straightforward modifications:

Treating entropy as relational rather than absolute

Distinguishing entropic charge from coarse-grained entropy

Incorporating causal horizons into entropy field calculations

Embedding observer dependence directly into the ToE action and field structure


In this enriched form, ToE not only remains viable but gains additional explanatory power. It can now provide a comprehensive framework for understanding how gravity, thermodynamics, and information interact across reference frames.


Thus far, this paper, “Observer-Dependent Gravitational Entropy and the Theory of Entropicity (ToE),” has been written to investigate whether there is any compatibility between the ideas of the Theory of Entropicity (ToE) and the notion of Observer-dependence of Gravitational Entropy propunded by De Vuyst, J. et al. We have systematically:

# Analyzed the core claims from the JHEP (2025) paper on observer-dependence of gravitational entropy,
# Identified possible contradictions with ToE’s formulation of entropy as a scalar field,
# Proposed four reconciliation pathways (relational entropy, entropic charge vs. measured entropy, horizon-based structure, and coarse-graining rules),
# Introduced a refined structure via an entropic frame bundle to incorporate observer dependence formally,
# Concluded with implications and extensions of ToE’s power in light of the observer-relative paradigm.

==See Also==

[[Entropic gravity]]

[[Black hole thermodynamics]]

[[Quantum reference frame]]

[[Information geometry]]

[[Theory of Entropicity (ToE)]]

==References==

De Vuyst, J. et al. (2025). "Gravitational entropy is observer-dependent." JHEP, 07(2025), 146.

Obidi, J. O. (2024). "Foundations of the Theory of Entropicity." Encyclopedia of Entropicity, Cambridge Open Engage.

Obidi, J. O. (2025). "Unified Action in ToE and Entropic Field Equations." HandWiki Physics, Section: Theory of Entropicity.

Verlinde, E. (2011). "On the origin of gravity and the laws of Newton." Journal of High Energy Physics, 2011(4), 29.

Padmanabhan, T. (2010). "Thermodynamical aspects of gravity: New insights." Reports on Progress in Physics, 73(4), 046901.


[[Category:Gravitational entropy]] [[Category:Entropic gravity]] [[Category:Quantum gravity]] [[Category:Theory of Entropicity]] [[Category:Observer-dependent physics]]

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