Абсолютная унифицированная математика квантовой гравитации

Geodesic deviation in curved spacetime is central to understanding gravitational interactions between two bodies within the framework of general relativity (GR). One of its most compelling predictions is the orbital precession of Mercury, approximately 0.43 arcseconds per Earth year, resulting from the influence of intense gravitational curvature near the Sun [1]. This subtle deviation exemplifies how spacetime curvature governs the motion of celestial bodies.

Building on Einstein’s 1915 formulation, the present study introduces a refined geometric model that visualizes an almond-shaped geodesic path, within the Sun–Mercury two-body system [2]. This path represents geodesic completeness and serves as a localized structure that captures transitions between spherical and hyperbolic curvature zones. These regimes are interpreted respectively as forward curvature R p + >  0 and reverse curvature R - < 0, with the perihelion precession attributed to a curvature flip from spherical to hyperbolic geometry.

This transition is represented geometrically by a closed parallelogram ‘ABCD’ constructed from the superposition of converging and diverging geodesics. In this configuration, sides AB and CD correspond to diverging geodesic paths, while sides BC and DA are shaped by converging geodesics. This structure encodes bidirectionality and leads to the formulation of a contrast-curvature field equation (10).

While GR provides a profound description of gravity through the Riemannian curvature tensor - R _ρ ^α _σλ , its inherent nonlinearity and the emergence of singularities introduce significant computational and conceptual challenges. To address these limitations, particularly in the context of singularity resolution and dark energy, the framework of Absolute Unified Mathematics of Quantum Gravitation (AUM) is proposed.

AUM reinterprets the conventional spacetime manifold M = Space ф Time as a contrast-driven geometry:

M = Space ⁽⁺⁾ ф Space ^(-) .

In this formulation, the negative space component associated with hyperbolic curvature symbolically reverses the effects of spherical curvature, and their superposition generates a bidirectional curvature field.

Notably, Space ^(-) offers a reinterpretation of gravitational time dilation: the experience of a free-falling object in a gravitational field is reframed as spatial expansion. Conversely, gravitational time contraction, encountered by a free-rising object, is reinterpreted as spatial compression governed by spherical curvature. The experience of gravitational time dilation or contraction depends on the object’s motion, free-fall or free-rise, within the bidirectional contrast curvature field. This field (10) acts as a diagnostic probe, identifying curvature dominance, transition zones, and equilibrium states (resolution of singularity). It encodes directional asymmetry and deformation, modeling gravity as a superposition of Ricci fields. This symbolic framework offers new tools for analyzing n-body dynamics and exploring extensions into the quantum gravitational regime.

• 1.    A generalized solution to Einstein’s field equation and its extension

We begin by recognizing that the Riemann zeta function, denoted as g(Z), is inherently non-linear [3]. Let us consider a complex variable Z = x ^e + ix ^v , where x ^e and x ^v span an N -dimensional space. The zeta function then takes the form:

N ∞

g (Z )= X X        e l^Og( a ^{n )} | =Л в л а .

e,v =0 n =1

Here, the term a ^—X acts as a scalar contraction, while e ^-i x ^{v lo}s( a n ) serves as a clockwise rotator in the complex plane. For Z > 1, the series converges, and the interplay between scalar shrinkage and rotational modulation becomes evident. To bridge this formulation with differential geometry, consider a vector A ^α undergoing transformation (2). First, it transform:

A

α

rotator/scalar

> ла А ^а .

Then, this transform vector is again transformed, as shown by equation (3):

Л _а А а       ■     Л в Л _а А а .

This transformation yields a new vector A ^a + SA ^a , where the differential change between (2) and (3) is:

DA ^a = dA ^a - 6A ^a = (1 - Л в )Л а А ^а .

Incorporating the Christoffel symbol, we write:

TL A ^ dx ^v = Л в Л а А ^а = g (ZW-

This expression (5) reveals that the non-linearity of the connection term is modulated by the zeta function, which rotates and contracts the vector A ^α , contributing to convergence. Conversely, for divergence (Z <  1), the scalar becomes expansive a x ^ , and the rotator reverses direction e ix ^log( a ^{n )} ,

leading to counter-clockwise rotation (as accounted by the known functional equation in analytical number theory) [3].

Extending equation (4) to derive geodesic deviation, the absolute derivative of the separation vector χ ^α along a geodesic, parameterized by τ becomes:

Dx ^a = ЛА - (АУА Dτ dτ       dτ

Thus, the geodesic deviation equation reads:

- R a • _x , d   = D ( ЛА - (g ( z ))x ^a \

^ρσλ dτ dτ Dτ dτ        dτ ^.

Here, the Riemann curvature tensor (7) is further contracted to a Ricci tensor term:

RR a15 = (v^M^) - vRRAa5 ρλ          ρ λ        λ ρ representing positive curvature contributions (8), active when g(Z > 1), coverging in forward direction. Conversely, the negative curvature counterpart:

R P -) B ⁵ = v ■ v ■   - v ' v . ' в                              (9)

emerges when g (Z <  1), signifying geodesic divergence (indicated by reverse direction ^ ).

To formalize the duality outlined in equations (8) and (9), the manifold is defined in terms of contrast bidirectional Ricci curvatures, intrinsically asymmetric in nature, and this structure is encapsulated by the contrast-curvature tensor presented by equation (10):

Cp A = (v ' v, ■ A ⁵ + v'v.B ⁵ ) - (v, ■ v ' A ⁵ + v, v /A) .          (10)

• ^ρ   ρλ   ρλ    λρ   λρ

If C pX = 0, the bidirectional curvature field is out of equilibrium, indicating dominance of either spherical curvature R p R >  0 or hyperbolic curvature R p R <  0, such as the geodesic flipping observed near Mercury’s perihelion. Thus, the observed precession arises from a transition between regions of positive curvature and negative curvature within the manifold.

Conversely, if C _px = 0, the system achieves geometric equilibrium. In this state, positive and negative curvature effects cancel, forming a bidirectional field under equal tension. This balance may correspond to stationary configurations or critical curvature transitions, like perihelion, and may help resolve singularities, potentially explaining the critical strip of the Riemann zeta function [3].

This equilibrium resembles a spring suspended from the ceiling with a weight attached, stretched equally from both ends yet remaining motionless. It reflects the behavior of a balanced bidirectional curvature field, effectively emulating a flat Minkowski manifold. In such a regime, gravitational time dilation (interpreted as spatial expansion) is exactly countered by gravitational time contraction (interpreted as spatial compression). Space expands outward when neighboring geodesics diverge and contracts inward when they converge, maintaining dynamic yet stable geometric tension.

The formulation of the tensor (10) naturally implies the existence of a contrast Riemannian zeta function (11):

g(1 <  Z> 1) = g (Z> 1) - g (Z <  1).                          (11)

Building upon this framework, the formulation (7), (10) and (11) extends to a contrast Ricci curvatures flow equation governing the temporal evolution of the manifold volume:

Jv C ^pX dV =

This equation (12) captures the essence of bidirectional curvature dynamics and was formally introduced during the PIRT-2025 conference as a key outcome of the framework. In regimes where negative curvature R _p x dominates, the curvature flow tends outward, compressing the spherical curvature sector [4]. This results in densification of matter or dark matter within curvature wells, potentially contributing to high-density gravitational structures.

Conclusion

The framework concludes a contrast-curvature tensor C _ρλ , derived from the parallel transport of the forward and reverse vectors, as shown by equations (8) and (9), respectively. This tensor quantifies the bidirectionality between converging and diverging geodesics. Crucially, the Christoffel symbols embedded in these covariant derivatives are modulated by the Riemann zeta function g(Z)), where 1 >  Z > 1 spans the complexified manifold. Specifically, the Christoffel symbol equation (5) reveals that the zeta function acts as a non-linear modulator, rotating and scaling the vectors. For Z > 1, the zeta function converges, inducing clockwise rotation and scalar contraction, leading to geodesic convergence and shrinking of positive curvature. Conversely, for Z <  1, the zeta function diverges, producing counterclockwise rotation and scalar expansion, resulting in geodesic divergence and expansion of negative curvature. This dual behavior is encoded in the bidirectional Ricci curvature flow equation (12). Here, the volume evolution of the manifold is governed by the tension between positive and negative curvature flows. The presence of this non-linear function within the Christoffel symbol thus bridges analytical number theory with differential geometry. Thus, offering a new perspective of spacetime manifold.