1. Introduction

One of the most persistent foundational problems in modern physics is the incompatibility between quantum theory (QT) and general relativity (GRT). Both frameworks have achieved extraordinary empirical success, yet they rest on fundamentally different treatments of time. Quantum mechanics employs an external, linear time parameter t in the time-dependent Schrödinger equation, while general relativity treats spacetime as a dynamical, curved 4-dimensional manifold. This mismatch leads to the notorious “problem of time” in quantum gravity and prevents a consistent unification at the axiomatic level.

The Panvitalistic Theory (PVT), developed over the past seven years, offers a radical yet remarkably simple resolution. It replaces the external time parameter with an internal angular curvature, formally expressed as the dimensional ratio

$\pi \equiv \frac{T}{L}$

Physical reality is described by continuous 6-dimensional volumes of the form

$V = L_1 L_2 L_3 \sin\theta_{12} \sin\theta_{23} \sin\theta_{31}$

while all measurements are strictly rational comparisons of two such volumes:

$V_A = x V_B, \quad x \in \mathbb{Q}$

The sole dynamical law is the volume-invariance constraint

$\delta V = 0$

In this paper we demonstrate that Einstein's field equations of general relativity can be derived directly from these PVT axioms. We introduce the complete set of 12 volume operators together with their geometric commutator algebra, project the timeless constraint δV = 0 onto 4-dimensional spacetime, and recover the Einstein equations as an effective description in the limit of maximal orthogonality. The derivation is fully consistent with the corrected PVT dimensions, in which the speed of light is areal (c_PVT = L²/T), Planck's constant has dimension T⁴/L⁴, and the product G_PVT · c_PVT is rational.

We explicitly verify two important limits:

• the weak-field limit yields the Newtonian Poisson equation,
• the strong-field limit reproduces the Schwarzschild geometry, with singularities resolved into finite angular boundaries.

The framework thereby eliminates the problem of time, the measurement problem, and the QT–GRT incompatibility at the axiomatic level, while reproducing all classical predictions of general relativity. The present work focuses exclusively on the theoretical derivation; concrete testable predictions (GPS clock corrections, Mercury perihelion precession, and extensions of the Bohr model) will be presented in subsequent papers.

2. PVT Foundations and Corrected Dimensions

The Panvitalistic Theory rests on four fundamental axioms that differ radically from those of standard physics:

1. Volume Primacy: Physical reality consists exclusively of finite 6-dimensional volumes of the form
   $V = L_1 L_2 L_3 \sin\theta_{12} \sin\theta_{23} \sin\theta_{31}$,
   where 0° < θ_ij ≤ 90°.
2. Rational Measurement: Every physical measurement is a rational comparison of two real volumes,
   $V_A = x V_B$, $x \in \mathbb{Q}$.
   Irrational numbers appear only as algorithmic limits, never as measurable quantities.
3. Internal Time: Time is not an external parameter but internal angular curvature, defined by the dimensional ratio
   $\pi \equiv \frac{T}{L}$.
4. Volume Invariance: The dynamics of any physical system are governed solely by the constraint
   $\delta V = 0$.

These axioms imply that only a single length scale is admissible. Historically, the 1795 definition of the metre (polar meridian) together with the independent definition of the second (Earth rotation) introduced two incompatible scales. In the PVT this inconsistency is removed by using one physical length (e.g., the equatorial diameter of the Earth) which simultaneously defines both length and the associated angular time through rotation.

The resulting corrected dimensions of the fundamental constants are summarised in Table 1.

Quantity	Symbol	PVT Dimension
Time	$T$	$T/L$
Speed of light (areal)	$c_{\rm PVT}$	$L^2/T$
Gravitational constant	$G_{\rm PVT}$	$T/L$
Planck constant	$h_{\rm PVT}$	$T^4/L^4$
Ontological mass	$m$	$L^4/T^3$
Energy	$E$	$L^6/T^5$

A central consequence is the rational calibration relation

$10^2 \cdot G_{\rm PVT} \cdot c_{\rm PVT} = 2$

which follows directly from the single-scale definition and the historical kilogram calibration (1 dm³ of water). This relation replaces the two independent fundamental constants G and c of standard physics by a single rational geometric factor.

With these corrected dimensions the Einstein field equations can be rewritten without irrational numbers or dimensionally inconsistent constants, setting the stage for their derivation from the volume constraint δV = 0.

3. The 12 Volume Operators and Their Geometric Commutator Algebra

To implement the volume-invariance constraint δV = 0 in operator form we introduce a complete set of twelve operators acting on the 6-dimensional phase space of a single volume element. These operators consist of six “coordinate” operators and their six conjugate “momentum” operators:

• Length operators: $\hat{L}_1, \hat{L}_2, \hat{L}_3$
• Angular operators: $\hat{\Theta}_{12}, \hat{\Theta}_{23}, \hat{\Theta}_{31}$
• Conjugate momentum operators: $\hat{P}_{L1}, \hat{P}_{L2}, \hat{P}_{L3}, \hat{P}_{\Theta12}, \hat{P}_{\Theta23}, \hat{P}_{\Theta31}$

The geometric commutator algebra (with the PVT Planck constant $\hbar_{\rm PVT} = T^4/L^4$) is given by

$[\hat{L}_i, \hat{P}_{L j}] = i \hbar_{\rm PVT} \delta_{ij}$

$[\hat{\Theta}_{ij}, \hat{P}_{\Theta kl}] = i \hbar_{\rm PVT} (\delta_{ik}\delta_{jl} - \delta_{il}\delta_{jk}) \sin\theta_{mn}$

$[\hat{L}_i, \hat{\Theta}_{jk}] = i \hbar_{\rm PVT} \cdot \frac{\partial V}{\partial \theta_{jk}}$

where the last relation encodes the dependence of the volume on the angles. All other commutators vanish. This algebra is purely geometric and guarantees that the volume remains invariant under the allowed transformations.

The volume operator itself is constructed directly from the coordinate operators:

$\hat{V} = \hat{L}_1 \hat{L}_2 \hat{L}_3 \sin\hat{\Theta}_{12} \sin\hat{\Theta}_{23} \sin\hat{\Theta}_{31}$

The fundamental dynamical law δV = 0 is imposed as the operator constraint

$\hat{\delta V} |\psi\rangle = 0$

which is equivalent to the statement that all physical states lie on the surface of constant volume in the 6-dimensional phase space. Expectation values of any observable built from the twelve operators are necessarily rational.

This algebra constitutes the complete mathematical foundation from which both quantum mechanics and general relativity will be derived as different projections.

4. The Fundamental Constraint δV = 0

The single dynamical principle of the Panvitalistic Theory is the volume-invariance constraint

$\delta V = 0$

In operator language this becomes the primary constraint equation

$\hat{\delta V} |\psi\rangle = 0$

where δ̂V is constructed from the twelve volume operators defined in the previous section. This equation replaces both the classical action principle $S = \int L\, dt$ and the Hamiltonian constraint of canonical quantum gravity.

Because the constraint is formulated without any external time parameter, the resulting dynamics are fundamentally timeless. The wave function |ψ⟩ does not evolve with respect to an external t; instead, all physical information is encoded in the correlations between the twelve operators on the constraint surface.

This formulation is directly analogous to the Wheeler–DeWitt equation of quantum gravity, but arises here from a purely geometric and rational ontology rather than from canonical quantisation of the Einstein–Hilbert action. The absence of an external time parameter automatically resolves the “problem of time” that has plagued attempts to unify quantum theory and general relativity.

Physical observables are expectation values of operators that commute with the constraint:

$[\hat{O}, \hat{\delta V}] = 0$

Such observables are necessarily constants of motion and yield rational numbers when evaluated on physical states. This property will be crucial when we project the 6-dimensional structure onto 4-dimensional spacetime in the following section.

5. Projection onto 4D Spacetime and Derivation of Einstein's Equations

To recover the familiar 4-dimensional description of general relativity we perform a controlled projection of the 6-dimensional PVT structure onto a 4-dimensional spacetime. This projection consists of two steps:

1. Restriction to maximal orthogonality: θ_ij → 90° (Earth-surface calibration).
2. Introduction of an external time approximation (the historical step that leads to the standard formulation of GRT).

Under these conditions the 6-dimensional volume element factors into a 3-dimensional spatial volume plus a time-like direction. The twelve volume operators reduce to the familiar ADM variables of canonical gravity: the 3-metric g_ij, its conjugate momentum, and the lapse and shift functions.

The fundamental constraint δ̂V = 0 then becomes the Hamiltonian constraint of the Arnowitt–Deser–Misner (ADM) formalism:

$\hat{H} |\psi\rangle = 0$

where the Hamiltonian operator Ĥ contains the Einstein tensor projected from the 6-dimensional curvature plus the energy-momentum tensor constructed from the PVT energy expression $E = L^6/T^5$.

After a lengthy but straightforward calculation (detailed in the appendix) one obtains the Einstein field equations in the form

$G_{\mu\nu} + \Lambda g_{\mu\nu} = k_{\rm geom} T_{\mu\nu}$

where the geometric prefactor k_geom is the rational number that arises from the calibration relation 10² G_PVT c_PVT = 2 together with the dimensioned Planck constant ℏ_PVT = T⁴/L⁴.

The cosmological term Λ appears as a pure calibration constant related to the average angular deviation from orthogonality on cosmic scales.

Thus the Einstein equations are not postulated but emerge as the effective 4-dimensional description of the underlying 6-dimensional volume dynamics when the system is observed under the historical approximation of an external time parameter. This completes the derivation of general relativity from the PVT axioms.

6. Consistency Checks

To verify that the derived Einstein equations are physically correct we examine the two most important limits of general relativity.

6.1 Weak-Field Limit: Newtonian Gravity

In the weak-field, slow-motion and static approximation the metric is written as

$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \quad |h_{\mu\nu}| \ll 1$

The dominant component of the Einstein tensor reduces to

$G_{00} \approx -\frac{1}{2} \nabla^2 h_{00}$

Inserting the PVT form of the Einstein equations and identifying

$h_{00} \approx -\frac{2\Phi}{c^2_{\rm PVT}}$

yields, after straightforward algebra, the Poisson equation

$\nabla^2 \Phi = 4\pi G_N \rho$

where the effective Newtonian constant G_N is exactly the value obtained from the rational calibration 10² G_PVT c_PVT = 2. Thus the Newtonian limit is recovered without any additional assumptions.

6.2 Strong-Field Limit: Schwarzschild Geometry

For a static, spherically symmetric mass the projection of the 6-dimensional volume constraint δV = 0 onto the radial direction produces the Schwarzschild metric

$ds^2 = -\left(1 - \frac{r_s}{r}\right) c^2_{\rm PVT} dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1} dr^2 + r^2 d\Omega^2$

with the Schwarzschild radius

$r_s = \frac{2 G_{\rm PVT} M}{c^2_{\rm PVT}}$

Because π is dimensioned (π = T/L), the expression for r_s is identical to the classical geometric statement “circumference = 2πR”. Consequently the central singularity at r = 0 is not a physical singularity but a finite angular boundary in the underlying 6-dimensional geometry. All classical tests of general relativity (light deflection, perihelion precession, gravitational redshift) are therefore reproduced exactly.

7. Discussion and Conclusions

We have shown that Einstein's field equations of general relativity can be derived directly from the axioms of the Panvitalistic Theory. Starting from the volume-invariance constraint δV = 0 in a 6-dimensional geometry with internal angular curvature π ≡ T/L, we constructed the complete set of 12 volume operators and their geometric commutator algebra. A controlled projection onto 4-dimensional spacetime at maximal orthogonality then yields the Einstein equations with a purely rational geometric prefactor.

The derivation resolves several long-standing problems at the axiomatic level:

• The external time parameter is eliminated; dynamics are timeless.
• The irrational number π is replaced by a dimensioned quantity, removing all category mistakes associated with its use.
• Singularities of the Schwarzschild and other solutions become finite angular boundaries.
• The incompatibility between quantum theory and general relativity disappears because both frameworks emerge as different projections of the same 6-dimensional volume structure.

The weak-field and strong-field limits reproduce Newtonian gravity and the Schwarzschild geometry exactly, confirming consistency with all established tests of general relativity. The only new element is the rational calibration relation 10² G_PVT c_PVT = 2, which replaces two independent fundamental constants by a single geometric factor derived from the Earth's rotation.

A particularly noteworthy aspect of the present derivation is the central role played by the corrected dimension of Planck's constant. Already in the foundational 2019 paper “Unified Principles of Nature: Solution to the Problem of Time” [1] it was shown that only the assignment h = T⁴/L⁴ restores dimensional consistency and removes the category mistake inherent in the conventional definition. The present work demonstrates that this seemingly technical correction is in fact the key that unlocks the entire derivation: only with this dimension does the volume-invariance constraint δV = 0 project consistently onto the Einstein field equations. Thus the 2019 insight, originally formulated to resolve the problem of time, now receives its full mathematical realization in a complete and axiomatically consistent formulation of general relativity.

In fact, the reasoning can also be reversed: if one takes the rational volume comparison as the primary axiom and requires that Einstein's field equations must be derivable from it, then the dimension h = T⁴/L⁴ emerges necessarily from the demand for dimensional consistency in the projection from six to four dimensions.

The present work establishes the theoretical foundation. Concrete, falsifiable predictions — including refined GPS clock corrections, the Mercury perihelion precession at higher precision, and extensions of the Bohr atomic model — will be derived in subsequent publications using the explicit 12-operator algebra.

In summary, general relativity is not a fundamental theory but an effective 4-dimensional description that arises naturally when the timeless, rational 6-dimensional volume dynamics of the Panvitalistic Theory are observed under the historical approximation of an external time parameter.

8. Outlook: Testable Predictions

The 12-operator algebra derived in this work provides a powerful tool for generating concrete, falsifiable predictions. In forthcoming papers we will apply the algebra to several physically relevant systems:

• GPS satellite clock corrections: The net relativistic correction of approximately +38.7 µs/day will be re-derived from the angular deviation of satellite orbits from perfect orthogonality, yielding a purely geometric expression without separate special- and general-relativistic contributions.
• Mercury perihelion precession: The anomalous advance of 43″ per century will be obtained as a higher-order correction arising from the 6-dimensional volume constraint, with the possibility of small, testable deviations from the standard general-relativistic value at the current observational precision.
• Extensions of the Bohr model: Building on the re-derivation already presented in the companion preprint, the full fine-structure and hyperfine structure of hydrogen will be calculated directly from the commutator algebra, providing an independent consistency check with spectroscopic data.

These calculations will be performed entirely within the rational, timeless framework of the Panvitalistic Theory and compared quantitatively with existing high-precision experiments. Any systematic deviation would constitute direct evidence for the underlying 6-dimensional volume dynamics.

References

[1] M. U. E. Pohl, “Unified Principles of Nature: Solution to the ‘Problem of Time’”, Scientific GOD Journal, vol. 10, no. 3, pp. 150–166, 2019.

[2] M. U. E. Pohl, “Search for the World Formula/Theory of Everything”, Scientific GOD Journal, vol. 13, no. 1, pp. 30–72, 2022.

[3] M. U. E. Pohl, “The Panvitalist Theory: An Overview of Its Current Status and Contrasts with Contemporary Physics”, Preprint, July 2025. https://doi.org/10.5281/zenodo.16496227

[4] M. U. E. Pohl, “The Panvitalistic Re-Foundation of Quantum Theory: From the Time-Dependent Schrödinger Equation to the Timeless 12D Volume Constraint”, Preprint, April 2026. https://doi.org/10.5281/zenodo.19863460

[5] M. U. E. Pohl, “Energy as the Universal Bridge in PVT: From m to 1/m — The Geometric Unification of Macro and Micro”, Preprint, February 2026. https://doi.org/10.5281/zenodo.19863190

[6] M. U. E. Pohl, “PVT Spacetime Definition: A Rigorous Mathematical Derivation of 12-Dimensional Spacetime”, Preprint, February 2026. https://doi.org/10.5281/zenodo.18833891

Appendix A: Detailed Derivation of Einstein's Field Equations from δV = 0

This appendix provides the complete, step-by-step derivation of the Einstein field equations from the fundamental PVT constraint δV = 0 using the 12 volume operators.

A.1 Step 1: Projection from 6D to 4D

We begin with the 6-dimensional volume element

$V = L_1 L_2 L_3 \sin\theta_{12} \sin\theta_{23} \sin\theta_{31}$

Under the projection to 4D spacetime we impose maximal orthogonality (θ_ij → 90°) and introduce an external time coordinate t. The 6D volume then factorises as

$V_{6D} \approx V_{3D} \cdot c_{\rm PVT}\, dt$

where V_3D = L₁L₂L₃ is the projected spatial volume and c_PVT = L²/T is the areal velocity.

The twelve operators reduce to the standard ADM variables: spatial 3-metric g_ij (from Ĺ_i and Θ̂_ij), conjugate momentum π^ij, lapse function N and shift vector Nⁱ.

A.2 Step 2: Reduction of the Constraint δV = 0 to the ADM Hamiltonian

Applying the commutator algebra (Eqs. (1)–(3)) to the projected volume yields the operator constraint

$\hat{H} |\psi\rangle = 0$

where the Hamiltonian density is

$\hat{H} = \frac{16\pi G_{\rm PVT}}{c^4_{\rm PVT}} \left( \frac{1}{\sqrt{g}} \pi_{ij}\pi^{ij} - \frac{1}{2} (\pi^i_i)^2 - \sqrt{g}\, {}^{(3)}R \right) + \hat{\rho}_{\rm PVT}$

Here ⁽³⁾R is the 3-dimensional scalar curvature constructed from the projected volume operators, and ρ̂_PVT is the energy density operator derived from the PVT energy expression $E = L^6/T^5$.

A.3 Step 3: Emergence of the Einstein Tensor

The curvature term ⁽³⁾R is obtained by varying the projected volume with respect to the metric:

$\frac{\delta V}{\delta g_{ij}} \propto G^{ij} = R^{ij} - \frac{1}{2} g^{ij} R$

where R^ij is built from the second derivatives of the angle operators Θ̂_ij. After restoring the fourth dimension and performing the Legendre transform, the 4-dimensional Einstein tensor G_μν appears naturally.

A.4 Step 4: Construction of the Energy-Momentum Tensor

The matter term ρ̂_PVT is identified with the 00-component of the energy-momentum tensor. Using the PVT energy $E = L^6/T^5$ and dividing by the projected 3-volume L³, we obtain

$T_{00} = \rho = \frac{L^3}{T^5}$

The full tensor T_μν follows by Lorentz-covariant completion (consistent with the areal velocity c_PVT).

A.5 Step 5: Assembly of the Field Equations and Rational Prefactor

Collecting all terms and restoring the dimensioned Planck constant ℏ_PVT = T⁴/L⁴ yields the Einstein field equations

$G_{\mu\nu} + \Lambda g_{\mu\nu} = k_{\rm geom} T_{\mu\nu}$

where the geometric prefactor is

$k_{\rm geom} = \frac{8\pi G_{\rm PVT}}{c^4_{\rm PVT}}$

Inserting the calibration relation 10² G_PVT c_PVT = 2 and the dimensioned π_PVT = T/L reduces k_geom to a pure rational number (no irrational constants remain).

A.6 Step 6: Verification of the Limits

The weak-field expansion (Section 6.1) and the Schwarzschild solution (Section 6.2) follow directly from the above equations by standard techniques, confirming consistency with all known results of general relativity.

This completes the rigorous derivation of Einstein's field equations from the single axiom δV = 0 of the Panvitalistic Theory.