The Schwarzschild Solution of Einstein's General Relativity (GR) from the Obidi Field Equations (OFE) of the Theory of Entropicity (ToE)

In this example we show how the Schwarzschild metric – the unique, static, spherically symmetric, vacuum solution of general relativity – arises as a classical limit of the GEFE/OFE, starting from an entropic field configuration. The goal is not to “re‑derive” every GR step from scratch, but to demonstrate that, under appropriate assumptions, the ToE formalism reproduces the Schwarzschild geometry and identifies the mass parameter in terms of entropic data.

Assumptions:

1. Static, spherically symmetric entropic field  

   S = S(r), depending only on the radial coordinate r.

2. Minimal coupling regime  

   ξ → 0 (so that the entropic field does not directly source curvature beyond the standard effective stress–energy).

3. Classical, weak‑field, large‑scale limit  

   where the GEFE reduce to the Einstein field equations in vacuum.

We proceed in four steps.

Step 1: Radial entropic field equation (MEE)

In the minimal coupling limit and in vacuum (no external matter), the entropic field S satisfies the Minimal Entropic Equation (MEE) of the form

□ S − V′(S) = 0,    (K.1)

where □ is the d’Alembertian with respect to the emergent spacetime metric, and V(S) is the entropic potential.

In the weak‑field, static, spherically symmetric regime, we approximate the background as effectively flat and write the radial equation as

(1 / r²) d/dr ( r² dS/dr ) − m² S = 0,    (K.2)

where m is an effective entropic mass scale (coming from V(S) ≈ (1/2) m² S² near S = 0).

The general static, spherically symmetric solution of (K.2) is the Yukawa form

S(r) = S₀ + (S₁ / r) exp(−m r),    (K.3)

where S₀ and S₁ are integration constants.

For the gravitational sector in the long‑range, classical limit, we take the massless limit m → 0, which yields the Coulomb‑like entropic profile

S(r) = S₀ + S₁ / r.    (K.4)

This S(r) will serve as the prototype entropic configuration associated with a spherically symmetric mass distribution.

Step 2: Fisher–Rao information metric for S(r)

To connect the entropic field to geometry, we consider a one‑parameter family of probability distributions p(x | r) whose macroscopic parameter is controlled by S(r). For concreteness, take a simple Gaussian family where the variance σ²(r) is proportional to S(r):

σ²(r) ∝ S(r).    (K.5)

For a one‑parameter family p(x | θ), the Fisher–Rao metric is defined by

g_I(θ) = E[ (∂ ln p / ∂θ)² ],    (K.6)

and in the present case the relevant parameter is θ = r. A straightforward computation for a Gaussian with variance σ²(r) gives a Fisher–Rao metric component of the form

g_I,rr(r) = C · [ S′(r) ]² / [ S(r) ]²,    (K.7)

where C is a positive constant depending on the normalization of the statistical model, and S′(r) = dS/dr.

For the Coulomb profile (K.4),

S(r) = S₀ + S₁ / r,    (K.8)

we have

S′(r) = − S₁ / r²,    (K.9)

and therefore

g_I,rr(r) ∝ (S₁² / r⁴) / [ S₀ + S₁ / r ]².    (K.10)

The angular components of the information metric are taken to respect spherical symmetry, so that the information manifold metric in the radial–angular sector can be written as

dsI² = gI,rr(r) dr² + r² ( dθ² + sin²θ dφ² ).    (K.11)

The exact overall normalization of g_I,rr is not essential here; what matters is that the information manifold is spherically symmetric and that its radial component is determined by the entropic profile S(r).

Step 3: Emergence map and physical spacetime metric

The emergence map identifies the physical spacetime metric gS with the Fisher–Rao information metric gI up to a constant factor λ:

gS,μν = λ gI,μν,    (K.12)

with λ a constant in the classical regime.

For a static, spherically symmetric spacetime, we write the physical metric in standard Schwarzschild‑like coordinates as

ds_S² = − A(r) c² dt² + B(r) dr² + r² ( dθ² + sin²θ dφ² ).    (K.13)

The emergence map (K.12) implies that, in the radial–angular sector,

B(r) ∝ g_I,rr(r),    (K.14)

and the angular part is already in the standard form r² dΩ². The time component A(r) is determined dynamically by the GEFE/OFE in the vacuum.

In the classical, large‑scale limit, the GEFE reduce to the vacuum Einstein equations

R{μν} − (1/2) R g{μν} = 0.    (K.15)

For the static, spherically symmetric ansatz (K.13), these equations yield the standard system of ordinary differential equations for A(r) and B(r). Solving (K.15) under the requirement of asymptotic flatness (A(r) → 1, B(r) → 1 as r → ∞) gives the unique Schwarzschild solution

A(r) = 1 − 2 G M / (r c²),    (K.16)

B(r) = [ 1 − 2 G M / (r c²) ]⁻¹.    (K.17)

Thus, in the classical limit where the GEFE reproduce the vacuum Einstein equations, the emergent spacetime metric generated by a static, spherically symmetric entropic configuration is necessarily the Schwarzschild metric.

Step 4: Identification of the mass parameter in terms of entropic data

The remaining task is to relate the Schwarzschild mass parameter M to the entropic field parameters S₀ and S₁.

In the weak‑field, Newtonian limit, the Schwarzschild metric yields the gravitational potential

Φ(r) = − G M / r.    (K.18)

On the other hand, in the ToE framework, the entropic field S(r) = S₀ + S₁ / r generates an effective potential via the emergence map and the GEFE/OFE. To leading order in the weak‑field regime, we can write

Φ(r) ∝ S₁ / r,    (K.19)

so that matching (K.18) and (K.19) gives

G M ∝ S₁.    (K.20)

More explicitly, one can write

M = α S₁,    (K.21)

where α is a constant determined by the coupling constants of the theory (including λ, ξ, and numerical factors from the Fisher–Rao normalization). The precise value of α depends on the detailed normalization of the entropic field and the statistical model used to define g_I, but the key point is that the Schwarzschild mass M is not a free parameter: it is an emergent quantity proportional to the entropic “charge” S₁ of the underlying entropic field configuration.

Thus, the Schwarzschild solution appears in ToE as:

1. A static, spherically symmetric entropic configuration S(r) = S₀ + S₁ / r solving the MEE in the appropriate limit.

2. An emergent spacetime metric g_S obtained from the Fisher–Rao information metric via the emergence map.

3. A classical limit of the GEFE/OFE in which the vacuum equations reduce to the Einstein vacuum equations, yielding the Schwarzschild form for A(r) and B(r).

4. A mass parameter M identified in terms of the entropic field parameter S₁.

This worked example shows that, in the appropriate regime, the Theory of Entropicity reproduces the Schwarzschild solution of general relativity and interprets the mass of the black hole as an emergent entropic charge of the underlying information manifold.

Scholium 

Refinement of Step 4: explicit identification of M in terms of S₁

To make the relation

M = α S₁

more explicit, we introduce a simple and transparent weak–field identification between the entropic field and the Newtonian potential.

In the weak–field, static limit, the Schwarzschild metric gives

g_tt(r) ≈ − ( 1 + 2 Φ(r) / c² ),    (K.22)

with

Φ(r) = − G M / r.    (K.23)

On the ToE side, we assume that, to leading order, the entropic field S(r) = S₀ + S₁ / r induces an effective gravitational potential of the form

Φ(r) = − κ S₁ / r,    (K.24)

where κ is a positive constant determined by the coupling between the entropic field and the emergent spacetime metric (this κ encodes the effect of λ, ξ, and the normalization of the Fisher–Rao metric).

Matching (K.23) and (K.24) gives

G M = κ S₁,    (K.25)

so that

M = (κ / G) S₁.    (K.26)

Thus, the proportionality constant α in

M = α S₁    (K.21)

is explicitly

α = κ / G.    (K.27)

In a more detailed treatment, κ can be computed from the GEFE/OFE by expanding the emergence map and the Fisher–Rao metric in the weak–field limit and reading off the coefficient of the 1/r term in g_tt. For the purposes of this example, it is sufficient to note that:

1. The Schwarzschild mass M is proportional to the entropic “charge” S₁.

2. The proportionality constant α is fixed by the fundamental couplings of the theory (G, λ, ξ, and the statistical normalization entering g_I).

In other words, M is not an arbitrary parameter but an emergent quantity determined by the entropic configuration of the information manifold.

Remark: 

On the geometry–matter duality in this example [the geo-matter duality (GMD) of the Theory of Entropicity (ToE)]

This Schwarzschild example illustrates the geometry–matter duality [the geo-matter duality (GMD)] of the Theory of Entropicity (ToE) in a concrete way.

The starting point is purely entropic: a scalar entropic field S(r) on the information manifold. From this single object, two seemingly different structures emerge:

1. Geometry:  

The Fisher–Rao information metric gI, constructed from S(r), generates the physical spacetime metric gS via the emergence map gS = λ gI. The Schwarzschild geometry – encoded in A(r) and B(r) – is thus a manifestation of the amplitude structure of the entropic field.

2. Matter (mass):  

The parameter S₁ in the entropic profile S(r) = S₀ + S₁ / r is interpreted, through the weak–field potential Φ(r), as the mass M of the Schwarzschild solution. The mass is therefore an emergent dynamical attribute of the same entropic field, not an independent ontological input.

In this sense, the Schwarzschild solution shows explicitly how, in ToE, what general relativity treats as “geometry” (the metric) and “matter” (the mass parameter M) both arise from a single entropic structure. Geometry and matter are complementary manifestations of the entropic field, realizing the geometry–matter (geo-matter) duality at the level of a familiar classical solution.