Energy-Geometry Coupling in Quantum Gravity: Framework and Physical Predictions

1. Introduction

The unification of general relativity (GR) and quantum mechanics remains a fundamental challenge in theoretical physics, with particular difficulties arising in understanding spacetime singularities, black hole thermodynamics, and the cosmological constant problem [12, 7]. While established approaches to quantum gravity have made significant progress [18, 17], the persistence of these challenges suggests the need for complementary frameworks that may offer new physical insights.

In this work, we present an analytical framework based on energy-geometry coupling principles, exploring how nonlocal functional relationships between spacetime geometry and energy-momentum distributions might address fundamental issues in gravitational physics. Our approach is motivated by several physical considerations:

• The mathematical analogy between spacetime deformation and continuum elasticity [6, 36]
• Evidence from nonlocal gravity theories that memory effects may be essential for understanding quantum spacetime [21, 19]
• The success of effective field theory approaches in describing low-energy quantum gravity [16]

We develop two core hypotheses: the Energy-Geometry Coupling Principle, describing how metric geometry responds to energy-momentum through local and nonlocal mechanisms, and the Spacetime Strain Formulation, characterizing measurable deformation states of spacetime. From these principles, we derive a mathematical framework that:

• Systematically generates GR as a limiting case while incorporating quantum-inspired corrections
• Provides mechanisms for addressing the cosmological constant problem through geometric compensation.
• Yields testable predictions for gravitational wave propagation and black hole phenomenology
• Maintains mathematical consistency with fundamental principles of general covariance and causality

This work represents an exploratory approach to quantum gravity that emphasizes mathematical transparency and physical interpretability. We maintain critical awareness of the framework’s limitations and identify open challenges throughout the development.

The paper is structured as follows: Section 2 presents the fundamental hypotheses and their physical justifications. Section 3 develops the master integral equation as the mathematical core of our framework. Section 4 addresses the resolution of mathematical complexities and framework simplification. Section 5 demonstrates the recovery of GR and extension to a nonlocal quantum gravity framework. Section 6 applies the framework to the cosmological constant problem, while Section 7 presents our key results on singularity resolution in black holes and cosmological settings, Section 8 derives the principal theoretical consequences and testable predictions that emerge directly from our framework.

2. Foundational Hypotheses

The hypotheses presented below represent an approach consistent with established theories of classical GR [1,2,3], non-local gravity models [4,5,6,7], and geometric formulations inspired by fluid mechanics [8,9,10]. While preserving the fundamental principles of covariance, causality, and consistency with Einstein’s theory in the appropriate limit [11,12], we extend the geometric response framework to include a functional coupling between matter and geometry and intrinsic spacetime deformation degrees of freedom [13,14,15].

2.1. Hypothesis I: Geometric-Energy Functional Coupling

Hypothesis 1

(Causal functional response). We propose acausal functional response hypothesis, which extends the classical Einstein equations by describing the spacetime metric $g_{\mu \nu }$ as a functional of the energy–momentum distribution $T_{\mu \nu }$, subject to additional structural and mathematical constraints. This hypothesis generalizes GR, allowing for controlled nonlocal and higher-order effects, while reducing to standard GR in the local limit.

$$g_{\mu \nu }\left(x\right)={\mathcal{G}}_{\mu \nu }\left[x;T_{\alpha \beta }(·)\right]$$

(1)

with the following required properties (these areassumptionsbuilt into the hypothesis and used in the derivation below):

• Causality (Retarded support).For every x, the functional depends only on the causal past $J^{-}\left(x\right)$. Equivalently any kernel $K(x,x^{\prime })$ appearing below satisfies

  $$K(·,x^{\prime })\equiv 0\mathrm{for}{x}^{\prime }\notin J^{-}\left(x\right).$$
• Diffeomorphism covariance.All bitensors and kernels are constructed from covariant geometric objects (metric, Synge world function $\sigma (x,x^{\prime })$, parallel propagator ${g^{\mu }}_{{\alpha }^{\prime }}(x,x^{\prime })$, curvature tensors, and their covariant derivatives). Consequently the functional ${\mathcal{G}}_{\mu \nu }$ transforms as a $(0,2)$-tensor under diffeomorphisms.
• Regularity and boundedness of kernels.The scalar bitensor kernel functions $F\left(s\right)$ and $G(s,s^{\prime })$ used below are smooth, rapidly decaying (or compactly supported) functions of $s=\sigma /L_P^2$, and produce integrals that are absolutely convergent for smooth metrics in the function space used (Sobolev $H^s$ or $C^k$ as specified).
• Local limit / correspondence.In the limit $L_P\to 0$ (or equivalently when the characteristic length scale $L\gg L_P$), the nonlocal kernels collapse to distributions that reproduce the local response: $F(\sigma /L_P^2)\to c{\delta }^{\left(n\right)}(x,x^{\prime })$ in the sense of distributions, producing Einstein gravity.
• Choice of kernel to avoid ghosts.The Fourier transform (in approximate translationally-invariant regimes) of the kernel produces an entire function of □ (for example an exponential regulator $e^{-{\ell }^2\square }$). This choice ensures the linearized propagator does not acquire extra poles (no new ghost degrees of freedom).
• Well-posedness conditions.The kernels satisfy Volterra-type conditions (causal, Lipschitz in appropriate norms) that make the resulting integro-differential equations Volterra equations of the second kind and therefore admit unique local-in-time solutions by Picard iteration in a specified Banach space.

Remark.The assumptions above arepart of the hypothesis: the physics claim is that geometry responds causally and covariantly to $T_{\mu \nu }$, and the mathematical assumptions supply the minimal regularity and structural properties required to make the variational derivation rigorous and the resulting dynamics well-posed.

2.2. Derivation of the Metric Response Functional

In this section, we explain the derivation of the metric functional response, highlighting the key assumptions and the mathematical framework underlying the subsequent analysis.

2.2.1. Action Principle and Variation

We begin with the covariant nonlocal action:

$$S\left[g\right]={\displaystyle \frac{1}{16\pi G}}\int R\sqrt{-g}{d}^{n}x+{\displaystyle \frac{1}{2}}\int \int \sqrt{-g_x}\sqrt{-g_{x^{\prime }}}R\left(x\right)F\left({\displaystyle \frac{\sigma }{L_P^2}}\right)R\left(x^{\prime }\right)d^{n}x{d}^n{x}^{\prime }.$$

Variation yields the master equation:

$${\mathcal{E}}_{\mu \nu }\left[g\right]=8\pi G{T}_{\mu \nu }+8\pi G{\tau }_{\mu \nu }^{\left(\mathrm{nl}\right)}\left[g\right],$$

(2)

where ${\mathcal{E}}_{\mu \nu }=G_{\mu \nu }$ and

$${\tau }_{\mu \nu }^{\left(\mathrm{nl}\right)}\left(x\right)={\int }_{J^{-}\left(x\right)}{\mathcal{K}}_{\mu \nu }^{\alpha \beta }(x,x^{\prime })R_{\alpha \beta }\left(x^{\prime }\right)\sqrt{-g_{x^{\prime }}}{d}^n{x}^{\prime }+{\mathcal{L}}_{\mu \nu }\left(x\right).$$

2.2.2. Linearized Theory and Green’s Function

Linearize around background $g_{\mu \nu }^{\left(0\right)}$:

$$g_{\mu \nu }=g_{\mu \nu }^{\left(0\right)}+h_{\mu \nu },{\mathcal{E}}_{\mu \nu }\left[g\right]={\mathcal{E}}_{\mu \nu \alpha \beta }^{\left(1\right)}{h}^{\alpha \beta }+O\left(h^2\right).$$

The retarded Green’s function ${\mathcal{G}}_{\mu \nu \alpha \beta }^{\mathrm{ret}}$ satisfies:

$${{\mathcal{E}}_{\mu \nu }^{\left(1\right)}}^{\rho \sigma }{\mathcal{G}}_{\rho \sigma \alpha \beta }^{\mathrm{ret}}(x,x^{\prime })={\delta }_{\mu \nu \alpha \beta }(x,x^{\prime }).$$

(3)

with $supp{\mathcal{G}}^{\mathrm{ret}}\subseteq J^{+}\left(x^{\prime }\right)$.

2.2.3. Volterra Inversion Scheme

The formal solution is:

$$h_{\mu \nu }\left(x\right)={\int }_M{\mathcal{G}}_{\mu \nu \alpha \beta }^{\mathrm{ret}}(x,x^{\prime })\left[8\pi G{T}^{\alpha \beta }\left(x^{\prime }\right)+8\pi G{\tau }^{\left(\mathrm{nl}\right)\alpha \beta }\left(x^{\prime }\right)\right]\sqrt{-g_{x^{\prime }}}{d}^n{x}^{\prime }.$$

(4)

This is a Volterra equation of the second kind. We solve via Picard iteration:

Zeroth Order:

$$h_{\mu \nu }^{\left(0\right)}\left(x\right)=8\pi G\int {\mathcal{G}}_{\mu \nu \alpha \beta }^{\mathrm{ret}}(x,x^{\prime })T^{\alpha \beta }\left(x^{\prime }\right)\sqrt{-g_{x^{\prime }}}{d}^n{x}^{\prime }.$$

First Order:

Substitute $h^{\left(0\right)}$ into ${\tau }^{\left(\mathrm{nl}\right)}$:

$${\tau }_{\mu \nu }^{\left(\mathrm{nl}\right)\left(1\right)}\left[h^{\left(0\right)}\right]=\int {\mathcal{K}}_{\mu \nu }^{\alpha \beta }(x,x^{\prime })R_{\alpha \beta }\left[h^{\left(0\right)}\right]\left(x^{\prime }\right)\sqrt{-g_{x^{\prime }}}{d}^n{x}^{\prime }.$$

Using $R_{\alpha \beta }\left[h^{\left(0\right)}\right]\sim {\mathcal{D}}_{\alpha \beta }^{\rho \sigma }{h}_{\rho \sigma }^{\left(0\right)}$, we get:

$$h_{\mu \nu }^{\left(1\right)}\left(x\right)=h_{\mu \nu }^{\left(0\right)}\left(x\right)+{\left(8\pi G\right)}^2\int \int {\mathcal{G}}_{\mu \nu \rho \sigma }^{\mathrm{ret}}(x,x^{\prime \prime }){\mathcal{K}}^{\rho \sigma \alpha \beta }(x^{\prime \prime },x^{\prime }){\mathcal{D}}_{\alpha \beta }^{\gamma \delta }{\mathcal{G}}_{\gamma \delta \kappa \lambda }^{\mathrm{ret}}(x^{\prime },x^{\prime \prime \prime })T^{\kappa \lambda }\left(x^{\prime \prime \prime }\right)d{V}^{\prime }d{V}^{\prime \prime }d{V}^{\prime \prime \prime }.$$

Kernel Identification:

After symmetric rearrangement:

$$K_{\mu \nu \alpha \beta }(x,x^{\prime })={\left(8\pi G\right)}^2\int {\mathcal{G}}_{\mu \nu \rho \sigma }^{\mathrm{ret}}(x,x^{\prime \prime }){\mathcal{K}}^{\rho \sigma \gamma \delta }(x^{\prime \prime },x^{\prime }){\mathcal{D}}_{\gamma \delta \alpha \beta }\sqrt{-g_{x^{\prime \prime }}}{d}^n{x}^{\prime \prime }.$$

(5)

Higher Orders:

Each iteration generates higher-order Volterra kernels:

$$K_{\mu \nu \alpha \beta \gamma \delta }^{\left(2\right)}\sim {\mathcal{G}}^{\mathrm{ret}}\circ \mathcal{K}\circ {\mathcal{G}}^{\mathrm{ret}}\circ \mathcal{K}\circ {\mathcal{G}}^{\mathrm{ret}}.$$

2.2.4. Final Metric Response Series

The convergent Volterra series is:

$$\begin{array}{cc}\hfill g_{\mu \nu }\left(x\right)& =g_{\mu \nu }^{\left(0\right)}+{\kappa }_1{\displaystyle \frac{G}{c^4}}{T}_{\mu \nu }\left(x\right)\hfill \\ \hfill & +{\kappa }_2{L}_P^{n-2}{\int }_{J^{-}\left(x\right)}{K}_{\mu \nu \alpha \beta }(x,x^{\prime })T^{\alpha \beta }\left(x^{\prime }\right)dV\left(x^{\prime }\right)\hfill \\ \hfill & +{\kappa }_3{L}_P^{2n-4}\underset{J^{-}\left(x\right)}{\int \int }{K}_{\mu \nu \alpha \beta \gamma \delta }^{\left(2\right)}(x,x^{\prime },x^{\prime \prime })T^{\alpha \beta }\left(x^{\prime }\right)T^{\gamma \delta }\left(x^{\prime \prime }\right)d{V}^{\prime }d{V}^{\prime \prime }+\dots ,\hfill \end{array}$$

(6)

where the dimensional constants ${\kappa }_i$ ensure correct units.

2.2.5. Mathematical Consistency Verification

• Causality: All integrals over $J^{-}\left(x\right)$ by construction
• Convergence: Volterra theory guarantees local convergence
• Covariance: Maintained through bitensor operations

2.2.6. Mathematical Details

The derivation outlined above involves a precise functional variation of the nonlocal two-point sensors, linearization around background solutions, and a Volterra inversion via retarded Green’s functions. For the complete mathematical treatment with all technical details, see Appendix A.

2.2.7. Compact Notation for the Metric-Response Expansion

To simplify the notation of the Volterra-type metric expansion (2) without losing its physical and mathematical content, we define the following shorthand symbols:

$$\begin{array}{cc}\hfill {\mathcal{Y}}_1^{\mu \nu }\left(x\right)& :={\kappa }_1{\displaystyle \frac{G}{c^4}}{T}^{\mu \nu }\left(x\right),\hfill \\ \hfill {\mathcal{Y}}_2^{\mu \nu }\left(x\right)& :={\kappa }_2{L}_P^{n-2}{\int }_{J^{-}\left(x\right)}{K}_{\mu \nu \alpha \beta }(x,x^{\prime })T^{\alpha \beta }\left(x^{\prime }\right)dV\left(x^{\prime }\right),\hfill \\ \hfill {\mathcal{Y}}_3^{\mu \nu }\left(x\right)& :={\kappa }_3{L}_P^{2n-4}\underset{J^{-}\left(x\right)}{\int \int }{K}_{\mu \nu \alpha \beta \gamma \delta }^{\left(2\right)}(x,x^{\prime },x^{\prime \prime })T^{\alpha \beta }\left(x^{\prime }\right)T^{\gamma \delta }\left(x^{\prime \prime }\right)dV\left(x^{\prime }\right)dV\left(x^{\prime \prime }\right).\hfill \end{array}$$

With these definitions, the metric expansion can be compactly written as

$$g_{\mu \nu }\left(x\right)=g_{\mu \nu }^{\left(0\right)}+\sum _{i=1}^3{\mathcal{Y}}_i^{\mu \nu }\left(x\right).$$

(7)

Physical interpretation:

• ${\mathcal{Y}}_1^{\mu \nu }$: Linear response of the metric to local energy-momentum.
• ${\mathcal{Y}}_2^{\mu \nu }$: First-order nonlocal (causal) response through the Volterra kernel K.
• ${\mathcal{Y}}_3^{\mu \nu }$: Higher-order nonlinear contributions involving double integrals over past domains.

This compact form preserves the full integro-differential structure, causality, and covariance, while making the notation more manageable for analytical and numerical manipulations.

2.2.8. Remarks on Consistency and Completeness

Under the stated assumptions (1–6), the derivation presented above is mathematically rigorous in the functional-analytic framework of geometric analysis for nonlocal PDEs, and physically consistent. In particular, the resulting framework satisfies the following properties:

(i)

reproduces the proposed metric-response structural equation,

(ii)

preserves diffeomorphism covariance and causal support,

(iii)

reduces to GR in the local limit,

(iv)

ensures controlled linear stability,

(v)

guarantees well-posedness.

Relaxing any of the assumptions would require additional arguments to re-establish these properties.

2.3. Hypothesis II: Spacetime Intrinsic Deformation Principle

Hypothesis 2

(Spacetime Intrinsic Deformation). The vacuum spacetime exhibits measurable intrinsic deformation states characterized by a strain tensor description, where:

• Local point-like deformations arise from immediate energy-momentum influence
• Global collective deformations emerge from integrated energy distribution
• Historical deformations reflect causal memory effects from past energy configurations
• Nonlinear deformation states dominate at Planck-scale energy densities
• These deformation states represent physical observables independent of coordinate descriptions

2.3.1. Derivation and Physical Justification

The foundation begins with continuum mechanics and elasticity theory [8]:

$${ϵ}_{\mu \nu }={\displaystyle \frac{1}{2}}({\nabla }_{\mu }{\xi }_{\nu }+{\nabla }_{\nu }{\xi }_{\mu }+{\nabla }_{\mu }{\xi }^{\alpha }{\nabla }_{\nu }{\xi }_{\alpha })$$

The relationship between strain and curvature emerges from geometric elasticity principles [8,16]:

$${\displaystyle \frac{D^2{\xi }^{\mu }}{d{\tau }^2}}=-R_{\nu \rho \sigma }^{\mu }{u}^{\nu }{\xi }^{\rho }{u}^{\sigma }\Rightarrow {ϵ}_{\mu \nu }\sim L_P^2{R}_{\mu \nu }$$

The quantum deformation scale is naturally introduced through Planck units [17]:

$$\left[{ϵ}_{\mu \nu }\right]=1\Rightarrow \left[L_P^2{R}_{\mu \nu }\right]=L^2·L^{-2}=1$$

Energy-strain coupling follows from stress-strain analogies in generalized continuum mechanics[9]:

$${ϵ}_{\mu \nu }\propto T_{\mu \nu },\left[{\displaystyle \frac{G}{c^4}}{T}_{\mu \nu }\right]=1$$

Nonlocal deformation memory effects are incorporated through linear response formalism [18]:

$${ϵ}_{\mu \nu }\left(x\right)=\int {\chi }_{\mu \nu \alpha \beta }(x,x^{\prime })T^{\alpha \beta }\left(x^{\prime }\right)dV\left(x^{\prime }\right)$$

Causality is rigorously enforced through kernel construction [2]:

$$D_{\mu \nu \alpha \beta }(x,x^{\prime })=0\mathrm{for}{x}^{\prime }\notin J^{-}\left(x\right)$$

The independence of the deformation tensor is mathematically established [10]:

$${ϵ}_{\mu \nu }\ne g_{\mu \nu },{ϵ}_{\mu \nu }\ne R_{\mu \nu }$$

Nonlinear deformation terms are systematically added for high-energy completeness[19]:

$${ϵ}_{\mu \nu }^{\left(2\right)}\propto \int \int D_{\mu \nu \alpha \beta \gamma \delta }^{\left(2\right)}{T}^{\alpha \beta }{T}^{\gamma \delta }d{V}^{\prime }d{V}^{\prime \prime }$$

Dimensional analysis verifies complete consistency across all terms [20]:

$$\left[{\alpha }_1{L}_P^2{R}_{\mu \nu }\right]=1,\left[{\alpha }_2{\displaystyle \frac{G}{c^4}}{T}_{\mu \nu }\right]=1,[{\alpha }_3{L}_P^{n-2}\int Td{V}^{\prime }]=1$$

Linear superposition principle applies to deformation contributions [9]:

$${ϵ}_{\mu \nu }={ϵ}_{\mu \nu }^{\left(1\right)}+{ϵ}_{\mu \nu }^{\left(2\right)}+{ϵ}_{\mu \nu }^{\left(3\right)}+{ϵ}_{\mu \nu }^{\left(4\right)}$$

The final formulation provides a description of the physics of spacetime deformations. In an n-dimensional spacetime, the complete mathematical expression is:

$$\begin{array}{cc}\hfill {ϵ}_{\mu \nu }\left(x\right)& ={\alpha }_1{L}_P^2{R}_{\mu \nu }\left(x\right)+{\alpha }_2{\displaystyle \frac{G}{c^4}}{T}_{\mu \nu }\left(x\right)\hfill \\ \hfill & +{\alpha }_3{L}_P^{n-2}{\int }_{J^{-}\left(x\right)}{D}_{\mu \nu \alpha \beta }(x,x^{\prime })T^{\alpha \beta }\left(x^{\prime }\right)dV\left(x^{\prime }\right)\hfill \\ \hfill & +{\alpha }_4{L}_P^{2n-4}\underset{J^{-}\left(x\right)}{\int \int }{D}_{\mu \nu \alpha \beta \gamma \delta }^{\left(2\right)}(x,x^{\prime },x^{\prime \prime })\hfill \\ \hfill & \times T^{\alpha \beta }\left(x^{\prime }\right)T^{\gamma \delta }\left(x^{\prime \prime }\right)dV\left(x^{\prime }\right)dV\left(x^{\prime \prime }\right)\hfill \end{array}$$

(8)

2.3.2. Physical Interpretation and Symbol Definitions

• ${ϵ}_{\mu \nu }$: Dimensionless spacetime strain tensor quantifying intrinsic deformations of the spacetime fabric, analogous to strain tensors in continuum mechanics but applied to spacetime itself [8].
• ${\alpha }_1{L}_P^2{R}_{\mu \nu }\left(x\right)$: Elastic deformation term representing how intrinsic spacetime curvature generates strain states. The Planck length provides natural scaling for quantum geometric effects [8,13].
• ${\alpha }_2{\displaystyle \frac{G}{c^4}}{T}_{\mu \nu }\left(x\right)$: Energy-driven deformation term encoding direct strain response to local energy-momentum density, maintaining correspondence with stress-energy concepts from continuum mechanics [9].
• ${\alpha }_3{L}_P^{n-2}\int D·Td{V}^{\prime }$: Historical deformation memory term capturing how past energy configurations induce persistent strain states through causal influences [18].
• ${\alpha }_4{L}_P^{2n-4}\int \int D^{\left(2\right)}·T^{2}d{V}^{\prime }d{V}^{\prime \prime }$: Nonlinear collective deformation term representing strain states arising from energy-energy interactions and high-density configurations [10].
• $D_{\mu \nu \alpha \beta }$: Deformation response kernel determining how energy at $x^{\prime }$ influences strain at x, with built-in causality preservation [2].

2.3.3. Compact Notation for the Spacetime Strain Expansion

To simplify the notation of the spacetime strain expansion (8) while preserving full physical and mathematical meaning, we define the following shorthand symbols:

$$\begin{array}{cc}\hfill {\mathcal{Z}}_1^{\mu \nu }\left(x\right)& :={\alpha }_1{L}_P^2{R}_{\mu \nu }\left(x\right),\hfill \\ \hfill {\mathcal{Z}}_2^{\mu \nu }\left(x\right)& :={\alpha }_2{\displaystyle \frac{G}{c^4}}{T}_{\mu \nu }\left(x\right),\hfill \\ \hfill {\mathcal{Z}}_3^{\mu \nu }\left(x\right)& :={\alpha }_3{L}_P^{n-2}{\int }_{J^{-}\left(x\right)}{D}_{\mu \nu \alpha \beta }(x,x^{\prime })T^{\alpha \beta }\left(x^{\prime }\right)dV\left(x^{\prime }\right),\hfill \\ \hfill {\mathcal{Z}}_4^{\mu \nu }\left(x\right)& :={\alpha }_4{L}_P^{2n-4}\underset{J^{-}\left(x\right)}{\int \int }{D}_{\mu \nu \alpha \beta \gamma \delta }^{\left(2\right)}(x,x^{\prime },x^{\prime \prime })T^{\alpha \beta }\left(x^{\prime }\right)T^{\gamma \delta }\left(x^{\prime \prime }\right)dV\left(x^{\prime }\right)dV\left(x^{\prime \prime }\right).\hfill \end{array}$$

With these definitions, the strain tensor can be compactly written as:

$${ϵ}_{\mu \nu }\left(x\right)=\sum _{i=1}^4{\mathcal{Z}}_i^{\mu \nu }\left(x\right)$$

(9)

Physical interpretation:

• ${\mathcal{Z}}_1^{\mu \nu }$: Elastic deformation induced by intrinsic spacetime curvature.
• ${\mathcal{Z}}_2^{\mu \nu }$: Energy-driven strain response from local energy-momentum.
• ${\mathcal{Z}}_3^{\mu \nu }$: Historical, causal memory effects from past energy configurations.
• ${\mathcal{Z}}_4^{\mu \nu }$: Nonlinear collective strain contributions from energy-energy interactions.

This compact form maintains the causality, covariance, and nonlinear Volterra structure, while significantly reducing notational complexity for both analytical and numerical work.

3. Geometric-Energy Integral Formulation

This section presents the mathematical core of our framework: the derivation and analysis of the central integral equation that unifies energy and geometric interactions. We begin by deriving the fundamental equation from first principles, and then we study its mathematical properties and physical significance.

3.1. Derivation of Master Integral Equation

The derivation begins with the construction of a unified action principle, combining both hypotheses [2,8]:

$$S={\int }_M\left[{\mathcal{L}}_{\mathrm{geometry}}+{\mathcal{L}}_{\mathrm{deformation}}+{\mathcal{L}}_{\mathrm{interaction}}\right]\sqrt{-g}{d}^{n}x$$

Variation of the action with respect to the metric tensor yields the corresponding field equations, as in the standard derivation of the Einstein field equations [1,2]:

$${\displaystyle \frac{\delta S}{\delta g^{\mu \nu }}}=0\Rightarrow \mathrm{Combined}\mathrm{field}\mathrm{equations}$$

We construct a vector-tensor combination that incorporates contributions from the two hypotheses introduced in this work, following standard procedures in generalized continuum mechanics and geometric response theory [18,21]:

$$V^{\alpha }=g^{\alpha \beta }{\nabla }_{\beta }ϵ+{ϵ}^{\alpha \beta }{\nabla }_{\beta }g+{\Lambda }^{\alpha \beta \gamma }{\nabla }_{\beta }g{\nabla }_{\gamma }ϵ$$

Application of the generalized divergence theorem in curved spacetime [21]:

$${\int }_M{\nabla }_{\alpha }{V}^{\alpha }\sqrt{-g}{d}^{n}x={\oint }_{\partial M}{V}^{\alpha }{n}_{\alpha }d{S}^{n-1}$$

Substituting the contributions from both hypotheses introduced in this work into the vector-tensor combination, following standard functional and geometric response procedures [18,21], we obtain:

$$V^{\alpha }=F^{\alpha }\left(g_{\mu \nu }^{\left(0\right)},T_{\mu \nu },R_{\mu \nu },K,D,{\kappa }_i,{\alpha }_i\right)$$

Computation of the covariant divergence yields the tensor combinations, following standard techniques in curved spacetime and tensor calculus [2,21]:

$${\nabla }_{\alpha }{V}^{\alpha }=A·R+B·T+C·RT+D·T^2+E·R{T}^2+\dots$$

Systematic collection of similar tensor terms is performed as in linear response and continuum mechanics frameworks [8,18]:

$${\nabla }_{\alpha }{V}^{\alpha }=\sum _{i=1}^5{\mathcal{T}}_i+\mathrm{higher}-\mathrm{order}\mathrm{terms}$$

Coefficient determination is carried out through functional derivative matching, analogous to standard variational techniques in GR [1,2]:

$$A_{\alpha \beta }^{\mu \nu }={\displaystyle \frac{\partial \left({\nabla }_{\alpha }{V}^{\alpha }\right)}{\partial R^{\alpha \beta }}}$$

Dimensional consistency of all terms is verified following classical dimensional analysis[20]:

$$\left[L_P^{3n-6}\right]=L^{3n-6},\left[R{T}^2\right]=L^{-2}·{\left(M{L}^{-1}{T}^{-2}\right)}^2$$

Causality is preserved by enforcing kernel properties [2,18]:

$$K(x,x^{\prime })=D(x,x^{\prime })=0\mathrm{for}{x}^{\prime }\notin J^{-}\left(x\right)$$

The final composition ensures that all physical principles, including geometric consistency, energy-momentum response, nonlocality, and causality, are satisfied while maintaining mathematical consistency.

3.2. Geometric-Energy Integral Equation

Theorem 1

(Geometric-Energy Integral Formulation). The unified description of energy-geometry interactions takes the form of a covariantly conserved integral equation that generalizes Einstein’s field equations through incorporation of both local response and causal non-local interactions:

$$\begin{array}{cc}\hfill {\int }_M& \left[A_{\alpha \beta }^{\mu \nu }{R}^{\alpha \beta }+B_{\alpha \beta }^{\mu \nu }{T}^{\alpha \beta }+C_{\alpha \beta \gamma \delta }^{\mu \nu }{R}^{\alpha \beta }{T}^{\gamma \delta }\right.\hfill \\ \hfill & \left.+D_{\alpha \beta \gamma \delta }^{\mu \nu }{T}^{\alpha \beta }{T}^{\gamma \delta }+E_{\alpha \beta \gamma \delta \kappa \lambda }^{\mu \nu }{R}^{\alpha \beta }{T}^{\gamma \delta }{T}^{\kappa \lambda }\right]\sqrt{\left|g\right|}{d}^{n}x\hfill \\ \hfill & ={\oint }_{\partial M}{Q}^{\mu \nu }d{S}^{n-1}\hfill \end{array}$$

(10)

with coefficients defined by:

$$\begin{array}{cc}\hfill A_{\alpha \beta }^{\mu \nu }& ={\kappa }_1{\alpha }_1{L}_P^2{\delta }_{\alpha }^{\mu }{\delta }_{\beta }^{\nu }+{\kappa }_2{\alpha }_1{L}_P^n\int K_{\mu \nu \rho \sigma }{D}_{\alpha \beta }^{\rho \sigma }d{V}^{\prime }\hfill \\ \hfill B_{\alpha \beta }^{\mu \nu }& ={\kappa }_1{\alpha }_2{\displaystyle \frac{G}{c^4}}{\delta }_{\alpha }^{\mu }{\delta }_{\beta }^{\nu }+{\kappa }_2{\alpha }_2{L}_P^{n-2}{\displaystyle \frac{G}{c^4}}\int K_{\mu \nu \rho \sigma }{D}_{\alpha \beta }^{\rho \sigma }d{V}^{\prime }\hfill \\ \hfill C_{\alpha \beta \gamma \delta }^{\mu \nu }& ={\kappa }_2{\alpha }_1{L}_P^n\int K_{\mu \nu \rho \sigma }{\displaystyle \frac{\partial D_{\alpha \beta }^{\rho \sigma }}{\partial R^{\gamma \delta }}}d{V}^{\prime }\hfill \\ \hfill D_{\alpha \beta \gamma \delta }^{\mu \nu }& ={\kappa }_3{\alpha }_2{L}_P^{2n-4}{\displaystyle \frac{G}{c^4}}\int \int K_{\mu \nu \rho \sigma \kappa \lambda }^{\left(2\right)}{D}_{\alpha \beta }^{\kappa \lambda }d{V}^{\prime }d{V}^{\prime \prime }\hfill \\ \hfill E_{\alpha \beta \gamma \delta \kappa \lambda }^{\mu \nu }& ={\kappa }_3{\alpha }_4{L}_P^{3n-6}\int \int \int K_{\mu \nu \rho \sigma \eta \theta }^{\left(2\right)}{\displaystyle \frac{\partial D_{\alpha \beta \gamma \delta }^{\left(2\right)\eta \theta }}{\partial R^{\kappa \lambda }}}d{V}^{\prime }d{V}^{\prime \prime }d{V}^{\prime \prime \prime }\hfill \end{array}$$

3.2.1. Symbol Definitions and Physical Interpretations

• $A_{\alpha \beta }^{\mu \nu }$: Curvature coupling coefficient representing linear geometric response to spacetime curvature, combining both local and nonlocal contributions from both hypotheses.
• $B_{\alpha \beta }^{\mu \nu }$: Energy-momentum coupling coefficient encoding direct response to energy distribution, maintaining proper dimensional scaling through Planck units.
• $C_{\alpha \beta \gamma \delta }^{\mu \nu }$: Mixed curvature-energy coupling coefficient describing how curvature and energy interact in generating geometric response.
• $D_{\alpha \beta \gamma \delta }^{\mu \nu }$: Nonlinear energy coupling coefficient representing collective energy-energy interactions and high-density effects.
• $E_{\alpha \beta \gamma \delta \kappa \lambda }^{\mu \nu }$: Triple coupling coefficient for curvature-energy-energy interactions, dominant at Planck-scale energies.
• $Q^{\mu \nu }$: Boundary flux tensor representing energy-momentum and information flow through the spacetime boundary, ensuring conservation laws.
• Integration domains: All volume integrals over M represent spacetime region of interest, while surface integral over $\partial M$ represents the boundary where conservation laws are enforced.

3.2.2. Simplified Symbol Definitions and Physical Interpretation

To make the master integral equation more compact, we introduce the following shorthand symbols:

$$\begin{array}{cc}\hfill {\mathcal{X}}_1^{\mu \nu }& :=A_{\alpha \beta }^{\mu \nu }{R}^{\alpha \beta },\hfill \\ \hfill {\mathcal{X}}_2^{\mu \nu }& :=B_{\alpha \beta }^{\mu \nu }{T}^{\alpha \beta },\hfill \\ \hfill {\mathcal{X}}_3^{\mu \nu }& :=C_{\alpha \beta \gamma \delta }^{\mu \nu }{R}^{\alpha \beta }{T}^{\gamma \delta },\hfill \\ \hfill {\mathcal{X}}_4^{\mu \nu }& :=D_{\alpha \beta \gamma \delta }^{\mu \nu }{T}^{\alpha \beta }{T}^{\gamma \delta },\hfill \\ \hfill {\mathcal{X}}_5^{\mu \nu }& :=E_{\alpha \beta \gamma \delta \kappa \lambda }^{\mu \nu }{R}^{\alpha \beta }{T}^{\gamma \delta }{T}^{\kappa \lambda }.\hfill \end{array}$$

The master integral equation now takes the simpler form:

$$\begin{array}{cc}\hfill {\int }_M\sum _{i=1}^5{\mathcal{X}}_i^{\mu \nu }\sqrt{\left|g\right|}{d}^{n}x& ={\oint }_{\partial M}{Q}^{\mu \nu }d{S}^{n-1}\hfill \end{array}$$

(11)

Physical interpretation: Each ${\mathcal{X}}_i^{\mu \nu }$ represents a specific type of interaction:

• ${\mathcal{X}}_1^{\mu \nu }$: Linear response to spacetime curvature.
• ${\mathcal{X}}_2^{\mu \nu }$: Linear response to energy-momentum distribution.
• ${\mathcal{X}}_3^{\mu \nu }$: Mixed curvature-energy interaction.
• ${\mathcal{X}}_4^{\mu \nu }$: Nonlinear energy-energy interaction.
• ${\mathcal{X}}_5^{\mu \nu }$: Triple coupling relevant at Planck-scale energies.

4. Problem Resolution and Framework Simplification

This section addresses the reconciliation of apparent theoretical challenges within our framework, demonstrating how seemingly complex aspects naturally simplify through consistent mathematical treatment. We systematically examine key theoretical issues and their resolutions.

4.1. Identification and Analysis of Theoretical Challenges

Some important challenges that must be taken into consideration and represent a drawback that cannot be ignored when dealing with the main equation.

4.1.1. Challenge 1: Mathematical Well-Posedness of Multiple Integrals

The original formulation (10) contains complex multiple integrals that may lack proper mathematical definition:

$$\underset{J^{-}\left(x\right)}{\int \int \int }{K}^{\left(2\right)}{\displaystyle \frac{\partial D^{\left(2\right)}}{\partial R}}d{V}^{\prime }d{V}^{\prime \prime }d{V}^{\prime \prime \prime }\mathrm{mathematical}\mathrm{definition}\mathrm{unclear}$$

Solution: Domain Restriction and Kernel Specification

We restrict integration domains to ensure mathematical well-posedness:

$${\int }_{J^{-}\left(x\right)\cap B_{ϵ}\left(x\right)}\dots d{V}^{\prime }\mathrm{instead}\mathrm{of}{\int }_{J^{-}\left(x\right)}\dots d{V}^{\prime }$$

where $B_{ϵ}\left(x\right)$ is a geodesic ball of radius $ϵ$ around x, ensuring finite integration volumes [12]. We specify explicit kernel forms for mathematical tractability:

$$K_{\mu \nu \alpha \beta }(x,x^{\prime })=f\left(\sigma \right){\delta }_{\mu \alpha }{\delta }_{\nu \beta },\sigma ={\displaystyle \frac{1}{2}}{(x-x^{\prime })}^2$$

where $f\left(\sigma \right)$ is a well-defined function ensuring convergence [21].

4.1.2. Challenge 2: Proliferation of Undetermined Constants

The original framework (10) contains excessive undetermined constants:

$${\kappa }_1,{\kappa }_2,{\kappa }_3,{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4(7\mathrm{undetermined}\mathrm{constants})$$

Solution: Physical Constraints and Symmetry Reduction

We apply GR matching conditions [11]:

$${\kappa }_1=4\pi ,{\alpha }_2=0\mathrm{for}\mathrm{exact}\mathrm{GR}\mathrm{recovery}$$

We impose symmetry conditions for theoretical economy:

$${\alpha }_1={\alpha }_3,{\kappa }_2={\kappa }_3\mathrm{reducing}\mathrm{parameter}\mathrm{count}$$

Experimental constraints from gravitational wave observations [22]:

$${\alpha }_4\approx {10}^{-20}\mathrm{from}\mathrm{LIGO}/\mathrm{Virgo}\mathrm{bounds}$$

4.1.3. Challenge 3: Physical Interpretation of Higher-Order Terms

Complex terms like $E_{\alpha \beta \gamma \delta \kappa \lambda }^{\mu \nu }{R}^{\alpha \beta }{T}^{\gamma \delta }{T}^{\kappa \lambda }$ lack clear physical interpretation.

Solution: Dimensional Analysis and Physical Analogy

We perform detailed dimensional analysis:

$$[E·R{T}^2]=L^{3n-6}·L^{-2}·{\left(M{L}^{-1}{T}^{-2}\right)}^2$$

In natural units ($c=\hslash =1$, $n=4$):

$$\left[M^2{L}^2\right]=\left[L^{-2}{L}^2\right]=1\mathrm{dimensionless}$$

We provide physical interpretation through field theory analogies [23]:

$$R{T}^2\leftrightarrow \mathrm{curvature}-\mathrm{energy}-\mathrm{energy}\mathrm{interaction}\mathrm{via}\mathrm{graviton}\mathrm{exchange}$$

Having addressed the three theoretical challenges by restricting the domains, reducing the constants, and clarifying the physical interpretation, we are now equipped to present the simplified theoretical framework resulting from these solutions.

4.2. Simplified and Improved Framework

The simplified formulation of the two main hypotheses and the basic state equation, which embodies the solutions mentioned above, is computationally tractable.

4.2.1. Simplified Hypothesis I

$$\begin{array}{cc}\hfill g_{\mu \nu }\left(x\right)=& {\eta }_{\mu \nu }+4\pi {\displaystyle \frac{G}{c^4}}{T}_{\mu \nu }\left(x\right)\hfill \\ \hfill & +\kappa L_P^{n-2}{\int }_{J^{-}\left(x\right)\cap B_{ϵ}\left(x\right)}f\left({\displaystyle \frac{\sigma }{L_P^2}}\right){\delta }_{\mu \alpha }{\delta }_{\nu \beta }{T}^{\alpha \beta }\left(x^{\prime }\right)dV\left(x^{\prime }\right)\hfill \\ \hfill & +\alpha L_P^{2n-4}\underset{J^{-}\left(x\right)\cap B_{ϵ}\left(x\right)}{\int \int }g\left({\displaystyle \frac{\sigma }{L_P^2}},{\displaystyle \frac{{\sigma }^{\prime }}{L_P^2}}\right)\hfill \\ \hfill & \times {\delta }_{\mu \nu }{\eta }_{\alpha \gamma }{\eta }_{\beta \delta }{T}^{\alpha \beta }\left(x^{\prime }\right)T^{\gamma \delta }\left(x^{\prime \prime }\right)dV\left(x^{\prime }\right)dV\left(x^{\prime \prime }\right)\hfill \end{array}$$

(12)

4.2.2. Simplified Hypothesis II

$$\begin{array}{cc}\hfill {ϵ}_{\mu \nu }\left(x\right)=& \alpha L_P^2{R}_{\mu \nu }\left(x\right)+\alpha L_P^{n-2}{\int }_{J^{-}\left(x\right)\cap B_{ϵ}\left(x\right)}f\left({\displaystyle \frac{\sigma }{L_P^2}}\right){\delta }_{\mu \alpha }{\delta }_{\nu \beta }{T}^{\alpha \beta }\left(x^{\prime }\right)dV\left(x^{\prime }\right)\hfill \\ \hfill & +\beta L_P^{2n-4}\underset{J^{-}\left(x\right)\cap B_{ϵ}\left(x\right)}{\int \int }g\left({\displaystyle \frac{\sigma }{L_P^2}},{\displaystyle \frac{{\sigma }^{\prime }}{L_P^2}}\right)\hfill \\ \hfill & \times {\delta }_{\mu \nu }{\eta }_{\alpha \gamma }{\eta }_{\beta \delta }{T}^{\alpha \beta }\left(x^{\prime }\right)T^{\gamma \delta }\left(x^{\prime \prime }\right)dV\left(x^{\prime }\right)dV\left(x^{\prime \prime }\right)\hfill \end{array}$$

(13)

4.2.3. Simplified Master Equation

$$\begin{array}{cc}\hfill {\int }_M& \left[(4\pi \alpha L_P^2+\kappa \alpha L_P^n{I}_1)R^{\mu \nu }\right.\hfill \\ \hfill & +(4\pi \alpha {\displaystyle \frac{G}{c^4}}+\kappa \alpha L_P^{n-2}{\displaystyle \frac{G}{c^4}}{I}_1)T^{\mu \nu }\hfill \\ \hfill & +\kappa \alpha L_P^n{I}_2{R}^{\mu \nu }T\hfill \\ \hfill & +\alpha \beta L_P^{2n-4}{\displaystyle \frac{G}{c^4}}{I}_3{T}^{\mu \alpha }{T}_{\alpha }^{\nu }\sqrt{-g}{d}^{n}x\hfill \\ \hfill & ={\oint }_{\partial M}{Q}^{\mu \nu }d{S}^{n-1}\hfill \end{array}$$

(14)

where the simplified integrals are:

$$\begin{array}{cc}\hfill I_1& ={\int }_{B_{ϵ}\left(x\right)}f\left(\sigma \right)d{V}^{\prime }\hfill \\ \hfill I_2& ={\int }_{B_{ϵ}\left(x\right)}f\left(\sigma \right){\displaystyle \frac{\partial }{\partial R}}d{V}^{\prime }\hfill \\ \hfill I_3& =\underset{B_{ϵ}\left(x\right)}{\int \int }g(\sigma ,{\sigma }^{\prime })d{V}^{\prime }d{V}^{\prime \prime }\hfill \end{array}$$

5. From GR to a Nonlocal Quantum Gravity

GR represents a fundamental starting point for any research in this field, and this is the point upon which this section is based.

5.1. Foundational Principles of GR

The theory of GR stands as one of the most profound achievements in theoretical physics, grounded on two fundamental principles: the principle of general covariance and the Einstein equivalence principle [2,3,24]. These principles assert that the laws of physics are independent of the choice of coordinates and that local experiments are indistinguishable from those in flat spacetime, respectively.

The Einstein field equations:

$$G_{\mu \nu }=R_{\mu \nu }-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }R={\displaystyle \frac{8\pi G}{c^4}}{T}_{\mu \nu }$$

(15)

encapsulate these principles through their geometrodynamical interpretation of gravity, where spacetime curvature is directly related to the energy-momentum content. The contracted Bianchi identities, ${\nabla }^{\mu }{G}_{\mu \nu }=0$, ensure local conservation of energy and momentum, ${\nabla }^{\mu }{T}_{\mu \nu }=0$, providing a self-consistent dynamical framework [1,3].

5.2. Differential vs Integral Formulations

To illustrate the transition from GR to our non-local framework, it is helpful to compare the differential and integral formulations as theoretical preliminaries.

5.2.1. Local Differential Structure of GR

The Einstein field equations constitute a system of second-order nonlinear partial differential equations for the metric tensor $g_{\mu \nu }$. Their local nature means that the geometry at each spacetime point x is determined exclusively by the energy-momentum distribution at that same point:

$$\mathcal{D}\left[g_{\mu \nu }\right]\left(x\right)={\displaystyle \frac{8\pi G}{c^4}}{T}_{\mu \nu }\left(x\right)$$

(16)

where $\mathcal{D}$ represents the Einstein differential operator. This local character is mathematically expressed through the absence of integral terms and the pointwise nature of the field equations [2].

The proof of local energy-momentum conservation follows from the Bianchi identities:

$${\nabla }^{\mu }{G}_{\mu \nu }=0\Rightarrow {\nabla }^{\mu }{T}_{\mu \nu }=0$$

(17)

This represents a strict local conservation law valid at each spacetime point independently.

5.2.2. Nonlocal Integral Structure of the Master Equation

In contrast, our framework proposes a fundamentally different mathematical structure through the master integral equation:

$$\begin{array}{cc}\hfill {\int }_M& \left[(4\pi \alpha L_P^2+\kappa \alpha L_P^n{I}_1)R^{\mu \nu }+(4\pi \alpha {\displaystyle \frac{G}{c^4}}+\kappa \alpha L_P^{n-2}{\displaystyle \frac{G}{c^4}}{I}_1)T^{\mu \nu }\right.\hfill \\ \hfill & +\kappa \alpha L_P^n{I}_2{R}^{\mu \nu }T+\alpha \beta L_P^{2n-4}{\displaystyle \frac{G}{c^4}}{I}_3{T}^{\mu \alpha }{T}_{\alpha }^{\nu }\sqrt{-g}{d}^{n}x\hfill \\ \hfill & ={\oint }_{\partial M}{Q}^{\mu \nu }d{S}^{n-1}\hfill \end{array}$$

(18)

The integral formulation implies that the geometry within a region M is determined by both local quantities and nonlocal contributions integrated over the causal past $J^{-}\left(x\right)$.

5.2.3. Proof of Mathematical Equivalence in the Local Limit

The recovery of GR from our framework requires a systematic limiting procedure. Specifically, we consider the limit of vanishing Planck length, $L_P\to 0$, applied directly to Eq(10):

$$\underset{L_P\to 0}{lim}\{\begin{array}{cc}\hfill & {\int }_M\left[(4\pi \alpha L_P^2+\kappa \alpha L_P^n{I}_1)R^{\mu \nu }+(4\pi \alpha {\displaystyle \frac{G}{c^4}}+\kappa \alpha L_P^{n-2}{\displaystyle \frac{G}{c^4}}{I}_1)T^{\mu \nu }+\dots \right]\sqrt{-g}{d}^{n}x\hfill \\ \hfill & -{\oint }_{\partial M}{Q}^{\mu \nu }d{S}^{n-1}\}=0\hfill \end{array}$$

(19)

This limit is mathematically rigorous and proceeds as follows: The integral terms $I_1$, $I_2$, and $I_3$ vanish in the limit $L_P\to 0$ due to their explicit $L_P$ dependence:

$$I_1={\int }_{B_{ϵ}\left(x\right)}f\left(\sigma \right)d{V}^{\prime }\propto L_P^n\to 0$$

Simultaneously, we rescale the coupling constant $\alpha$ to maintain a finite curvature coupling:

$$\alpha ={\displaystyle \frac{1}{4\pi L_P^2}}\Rightarrow \alpha L_P^2={\displaystyle \frac{1}{4\pi }}$$

Under these conditions, Eq(10) reduces to the integrated form of Einstein’s equations:

$${\int }_M\left[R^{\mu \nu }-{\displaystyle \frac{1}{2}}{g}^{\mu \nu }R+{\displaystyle \frac{8\pi G}{c^4}}{T}^{\mu \nu }\right]\sqrt{-g}{d}^{n}x={\oint }_{\partial M}{Q}^{\mu \nu }d{S}^{n-1}$$

Applying the fundamental theorem of geometric calculus [21], we recover the local Einstein field equations:

$$R^{\mu \nu }-{\displaystyle \frac{1}{2}}{g}^{\mu \nu }R={\displaystyle \frac{8\pi G}{c^4}}{T}^{\mu \nu }.$$

This demonstrates that GR emerges as a special case of our framework in the appropriate local limit.

5.3. Quantitative Analysis of Deviations from GR

To estimate the expected magnitude of deviations from GR within our theoretical framework, we rely on two main methodologies: dimensional analysis to determine characteristic scales, and then calculating specific corrections in known physical systems.

5.3.1. Dimensional Analysis and Characteristic Scales

The deviations from GR in our framework are controlled by dimensionless ratios involving the Planck length $L_P$ [1,17,25]. A systematic dimensional analysis allows us to identify the relevant parameters and their scaling behavior in relation to fundamental constants.

$$\left[L_P\right]=L,[G/c^4]=T^2/M,\left[T_{\mu \nu }\right]=M/L{T}^2$$

The dimensionless parameters that govern the magnitude of deviations from GR are defined as:

$$\begin{array}{cc}\hfill {ϵ}_1& ={\left({\displaystyle \frac{L_P}{L}}\right)}^{n-2}(\mathrm{Nonlocal}\mathrm{deviation}\mathrm{parameter})\hfill \\ \hfill {ϵ}_2& ={\left({\displaystyle \frac{L_P}{L}}\right)}^{2n-4}(\mathrm{Nonlinear}\mathrm{deviation}\mathrm{parameter})\hfill \\ \hfill {ϵ}_3& ={\left({\displaystyle \frac{E}{E_P}}\right)}^2(\mathrm{Energy}\mathrm{scale}\mathrm{parameter})\hfill \end{array}$$

where $E_P=\sqrt{\hslash c^5/G}$ is the Planck energy [1,17] and L is the characteristic length scale of the system [10,26].

Solar System Regime: For typical solar system scales $L\sim 1$ AU $=1.5\times {10}^{11}$ m [1,11]:

$$\begin{array}{cc}\hfill {ϵ}_1^{\mathrm{solar}}& ={\left({\displaystyle \frac{1.6\times {10}^{-35}}{1.5\times {10}^{11}}}\right)}^2\approx 1.1\times {10}^{-92}\hfill \\ \hfill {ϵ}_2^{\mathrm{solar}}& ={\left({\displaystyle \frac{1.6\times {10}^{-35}}{1.5\times {10}^{11}}}\right)}^4\approx 1.3\times {10}^{-184}\hfill \end{array}$$

These minuscule values explain why GR provides such accurate descriptions in the solar system [11].

Black Hole Regime: For stellar-mass black holes with Schwarzschild radius $R_S\sim {10}^4$ m [1,19]:

$$\begin{array}{cc}\hfill {ϵ}_1^{\mathrm{BH}}& ={\left({\displaystyle \frac{1.6\times {10}^{-35}}{{10}^4}}\right)}^2\approx 2.6\times {10}^{-78}\hfill \\ \hfill {ϵ}_2^{\mathrm{BH}}& ={\left({\displaystyle \frac{1.6\times {10}^{-35}}{{10}^4}}\right)}^4\approx 6.8\times {10}^{-156}\hfill \end{array}$$

Planck Scale Regime: At the Planck scale $L\sim L_P$ [2,17]:

$$\begin{array}{cc}\hfill {ϵ}_1^{\mathrm{Planck}}& =1\hfill \\ \hfill {ϵ}_2^{\mathrm{Planck}}& =1\hfill \\ \hfill {ϵ}_3^{\mathrm{Planck}}& =1\hfill \end{array}$$

At this scale, the deviations become order unity, indicating the complete breakdown of the classical GR description [2].

5.3.2. Modified Field Equations with Explicit Deviation Terms

The field equations include the standard local differential terms of GR, together with explicit nonlocal integrals capturing deviations:

$$\begin{array}{cc}\hfill G_{\mu \nu }& ={\displaystyle \frac{8\pi G}{c^4}}{T}_{\mu \nu }+\kappa L_P^{n-2}{\int }_{J^{-}\left(x\right)}{K}_{\mu \nu \alpha \beta }{T}^{\alpha \beta }d{V}^{\prime }\hfill \\ \hfill & +\alpha L_P^{2n-4}\underset{J^{-}\left(x\right)}{\int \int }{K}_{\mu \nu \alpha \beta \gamma \delta }^{\left(2\right)}{T}^{\alpha \beta }{T}^{\gamma \delta }d{V}^{\prime }d{V}^{\prime \prime }\hfill \\ \hfill & +\beta L_P^n{\int }_{J^{-}\left(x\right)}{D}_{\mu \nu \alpha \beta \gamma \delta }{R}^{\alpha \beta }{T}^{\gamma \delta }d{V}^{\prime }+\dots \hfill \end{array}$$

(20)

Each additional term represents a specific physical effect beyond GR. Compact symbolic form:

$$G_{\mu \nu }={\displaystyle \frac{8\pi G}{c^4}}{T}_{\mu \nu }+{ϵ}_1{\mathcal{F}}_1\left[T\right]+{ϵ}_2{\mathcal{F}}_2[T,T]+{ϵ}_3{\mathcal{F}}_3[R,T]+\dots$$

Here: - ${\mathcal{F}}_1\left[T\right]$ denotes the leading-order nonlocal deviation. - ${\mathcal{F}}_2[T,T]$ encodes nonlinear corrections quadratic in $T_{\mu \nu }$. - ${\mathcal{F}}_3[R,T]$ represents curvature-matter couplings.

5.4. Physical Interpretation of Non-General-Relativistic Terms

In this subsection, we explore the physical meaning of terms that do not arise and analyze their effects and implications.

5.4.1. Nonlocal Memory Effects

The term $\kappa L_P^{n-2}{\int }_{J^{-}\left(x\right)}{K}_{\mu \nu \alpha \beta }{T}^{\alpha \beta }d{V}^{\prime }$ represents spacetime memory effects [4,19,27]:

$$\Delta G_{\mu \nu }^{\mathrm{memory}}\left(x\right)=\kappa L_P^{n-2}{\int }_{J^{-}\left(x\right)}{K}_{\mu \nu \alpha \beta }(x,x^{\prime })T^{\alpha \beta }\left(x^{\prime }\right)dV\left(x^{\prime }\right)$$

(21)

This describes how past energy-momentum distributions leave persistent imprints on spacetime geometry. The causal structure $J^{-}\left(x\right)$ ensures relativistic causality is preserved. The physical interpretation follows from the concept of gravitational memory in linearized gravity [19,27]:

$$\Delta h_{\mu \nu }={\displaystyle \frac{4G}{c^4}}{\int }_{-\infty }^t{\displaystyle \frac{T_{\mu \nu }(t^{\prime },{\mathbf{x}}_{\mathrm{ret}})}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}}d{t}^{\prime }$$

(22)

Our framework generalizes this concept to full nonlinear gravity [5].

5.4.2. Nonlinear Energy-Energy Interactions

The term $\alpha L_P^{2n-4}{\int \int }_{J^{-}\left(x\right)}{K}_{\mu \nu \alpha \beta \gamma \delta }^{\left(2\right)}{T}^{\alpha \beta }{T}^{\gamma \delta }d{V}^{\prime }d{V}^{\prime \prime }$ represents collective gravitational effects [23,28]:

$$\Delta G_{\mu \nu }^{\mathrm{nonlinear}}\left(x\right)=\alpha L_P^{2n-4}\underset{J^{-}\left(x\right)}{\int \int }{K}_{\mu \nu \alpha \beta \gamma \delta }^{\left(2\right)}{T}^{\alpha \beta }{T}^{\gamma \delta }d{V}^{\prime }d{V}^{\prime \prime }$$

(23)

This describes how energy distributions interact gravitationally with each other beyond the linear approximation of GR. The physical significance emerges at high energy densities $\rho \sim {\rho }_P$, where these terms become comparable to the linear Einstein tensor:

$${\displaystyle \frac{|\Delta G^{\mathrm{nonlinear}}|}{\left|G\right|}}\sim {\left({\displaystyle \frac{\rho }{{\rho }_P}}\right)}^2$$

5.4.3. Curvature-Energy Mixing

The term $\beta L_P^n\int DRTd{V}^{\prime }$ represents direct coupling between curvature and energy - momentum [6,7]:

$$\Delta G_{\mu \nu }^{\mathrm{mixing}}\left(x\right)=\beta L_P^n{\int }_{J^{-}\left(x\right)}{D}_{\mu \nu \alpha \beta \gamma \delta }{R}^{\alpha \beta }{T}^{\gamma \delta }d{V}^{\prime }$$

(24)

This introduces a new physical mechanism where spacetime curvature directly influences how energy-momentum generates further curvature.

5.5. Testable Predictions and Experimental Constraints

Before examining specific cases, we identify the general, testable predictions of the model and the empirical constraints that can guide or limit it.

5.5.1. Modified Gravitational Wave Propagation

The wave equation for metric perturbations in our framework acquires additional terms:

$$\begin{array}{cc}\hfill \square {\overline{h}}_{\mu \nu }& =-{\displaystyle \frac{16\pi G}{c^4}}{T}_{\mu \nu }+\kappa L_P^2\int K_{\mu \nu \alpha \beta }\square {\overline{h}}^{\alpha \beta }d{V}^{\prime }\hfill \\ \hfill & +\alpha L_P^4\int \int K_{\mu \nu \alpha \beta \gamma \delta }^{\left(2\right)}(\partial {\overline{h}}^{\alpha \beta })(\partial {\overline{h}}^{\gamma \delta })d{V}^{\prime }d{V}^{\prime \prime }+\dots \hfill \end{array}$$

This leads to frequency-dependent modifications in gravitational wave dispersion:

$$v_g\left(\omega \right)=c\left[1+\gamma {\left({\displaystyle \frac{\omega }{{\omega }_P}}\right)}^2+\delta {\left({\displaystyle \frac{\omega }{{\omega }_P}}\right)}^4+\dots \right]$$

(25)

where ${\omega }_P=c/L_P$ is the Planck frequency. Current LIGO/Virgo constraints [22] require:

$$\gamma <{10}^{-20},\delta <{10}^{-40}$$

for frequencies $\omega \sim 100$ Hz. The modified wave equation for metric perturbations introduces Planck-scale dependent corrections, resulting in the following effects:

• Nonlocal memory contributions: Past energy-momentum distributions leave persistent imprints on the metric.
• Nonlinear energy-energy interactions: Quadratic and higher-order terms in the stress-energy tensor modify the local curvature response at high energy densities.
• Curvature-energy mixing: Direct coupling between spacetime curvature and energy-momentum generates additional corrections in the evolution of gravitational perturbations.

Consequently, gravitational waves acquire small, frequency-dependent dispersion effects, and deviations from classical linear propagation become significant only near Planckian scales.

5.5.2. Modified Solar System Dynamics

The parameterized post-Newtonian (PPN) formalism [11] must be extended to include our framework’s additional parameters. The modified PPN metric contains new terms:

$$g_{00}=-1+2U-2\beta U^2+2\xi \Phi +\zeta A+\dots$$

where $\xi$, $\zeta$ are new PPN parameters specific to our framework. Solar system measurements constrain:

$$\left|\xi \right|<{10}^{-9},\left|\zeta \right|<{10}^{-11}$$

5.5.3. Modified Black Hole Thermodynamics

Quantum gravity effects are expected to modify the Bekenstein–Hawking entropy–area law, and a general model-independent expansion takes the form [11,29,30,31,32,33]:

$$S={\displaystyle \frac{k_{B}A}{4L_P^2}}\left[1+\eta \left({\displaystyle \frac{L_P^2}{A}}\right)+\theta \left({\displaystyle \frac{L_P^4}{A^2}}\right)+\dots \right],$$

(26)

where the leading term reproduces the standard Bekenstein–Hawking result and the higher-order terms encode Planck-scale corrections. Such corrections arise naturally in loop quantum gravity, string theory, generalized uncertainty principle (GUP) models, and effective field theory approaches to semiclassical gravity [23,34,35]. For a solar-mass black hole with area $A\sim {10}^{77}{L}_P^2$, the relative quantum correction is extremely suppressed:

$${\displaystyle \frac{\Delta S}{S}}\sim \eta ·{10}^{-77}+\theta ·{10}^{-154},$$

implying that these corrections are negligible for astrophysical black holes, but potentially significant for microscopic or near-Planckian black holes, and thus relevant in quantum gravity phenomenology.

5.6. Theoretical Consistency and Mathematical Rigor

This subsection addresses the internal consistency of the theoretical framework and the mathematical rigor underlying its derivations, ensuring that the model is logically sound and well-formulated.

5.6.1. Energy-Momentum Conservation in the Extended Framework

The master equation automatically preserves energy-momentum conservation through its boundary term structure. From the diffeomorphism invariance of the action principle [2,4,27], we have:

$${\nabla }^{\mu }{\displaystyle \frac{\delta S}{\delta g^{\mu \nu }}}=0$$

This implies the generalized conservation law for the total stress-energy:

$${\int }_M{\nabla }_{\mu }\left(T^{\mu \nu }+T_{\mathrm{nonlocal}}^{\mu \nu }+T_{\mathrm{nonlinear}}^{\mu \nu }\right)\sqrt{-g}{d}^{n}x=0$$

The boundary term $Q^{\mu \nu }$ ensures global conservation:

$${\displaystyle \frac{d}{dt}}{\int }_{\Sigma }\left(T^{0\nu }+T_{\mathrm{nonlocal}}^{0\nu }+T_{\mathrm{nonlinear}}^{0\nu }\right)\sqrt{-g}{d}^{n-1}x=-{\oint }_{\partial \Sigma }{Q}^{i\nu }d{S}_i$$

where $\Sigma$ is a spacelike hypersurface.

5.6.2. Causal Structure and Relativistic Causality

The integration domain $J^{-}\left(x\right)\cap B_{ϵ}\left(x\right)$ explicitly preserves relativistic causality. For any two spacetime points x and $x^{\prime }$:

$$K_{\mu \nu \alpha \beta }(x,x^{\prime })=0\mathrm{for}{x}^{\prime }\notin J^{-}\left(x\right)$$

This ensures that influences propagate only within the light cones defined by the metric $g_{\mu \nu }$ [2,5]. The proof follows from the construction of the retarded Green’s function:

$$G_{\mu \nu \alpha \beta }^{\mathrm{ret}}(x,x^{\prime })=0\mathrm{for}{x}^{\prime }\notin J^{-}\left(x\right)$$

5.6.3. Renormalization and Ultraviolet Completion

The presence of the Planck length $L_P$ provides a natural ultraviolet cutoff:

$${\Lambda }_{\mathrm{UV}}\sim {\displaystyle \frac{1}{L_P}}\sim {10}^{35}{\mathrm{m}}^{-1}$$

This regulates the divergent integrals that appear in quantum field theory in curved spacetime [23,28,36]:

$${\langle T_{\mu \nu }\rangle }_{\mathrm{ren}}=\underset{x^{\prime }\to x}{lim}\left[\langle T_{\mu \nu }\rangle -{\displaystyle \frac{C_{\mu \nu }(x,x^{\prime })}{\sigma (x,x^{\prime })}}\right]$$

(27)

where $\sigma (x,x^{\prime })$ is the geodesic distance and the subtraction is performed within the causal domain $J^{-}\left(x\right)\cap B_{ϵ}\left(x\right)$ [27,37].

5.7. Quantum Gravitational Implications

Here we discuss the potential implications of quantum gravity inherent in the proposed framework.

5.7.1. Semi-Classical Limit and Backreaction

The extended framework provides a natural platform for studying semi-classical gravity with quantum backreaction [2,23,36,38]:

$$G_{\mu \nu }={\displaystyle \frac{8\pi G}{c^4}}\langle T_{\mu \nu }\rangle +\Delta G_{\mu \nu }^{\mathrm{quantum}}$$

where the quantum corrections include nonlocal and higher-order terms [4,5,39]:

$$\Delta G_{\mu \nu }^{\mathrm{quantum}}=\kappa L_P^{n-2}\int \langle K_{\mu \nu \alpha \beta }{T}^{\alpha \beta }\rangle d{V}^{\prime }+\dots$$

This formulation extends beyond the standard semi-classical approximation by incorporating nonlocal correlations and n-dimensional generalizations [35,40].

5.7.2. Resolution of Classical Singularities

Nonlocal and quantum corrections provide a potential mechanism for resolving classical singularities. For instance, in a homogeneous and isotropic universe, the modified Friedmann equation reads [14,41,42]:

$$H^2={\displaystyle \frac{8\pi G}{3}}\rho \left[1+\xi \left({\displaystyle \frac{\rho }{{\rho }_P}}\right)+\zeta {\left({\displaystyle \frac{\rho }{{\rho }_P}}\right)}^2+\dots \right]$$

(28)

As $\rho \to \infty$, the higher-order terms dominate, potentially avoiding the big bang singularity:

$$\underset{\rho \to \infty }{lim}{H}^2<\infty \mathrm{for}\mathrm{an}\mathrm{appropriate}\mathrm{choice}\mathrm{of}\xi ,\zeta$$

This approach aligns with predictions from loop quantum cosmology and other quantum gravity models where high-energy corrections regularize curvature invariants and prevent divergence at the Planck scale [14,41].

5.8. Consistency with GR

The analysis presented in this section demonstrates that our framework:

• Contains GR as a Special Case: In the limit $L_P\to 0$ with appropriate parameter choices, we recover exactly the Einstein field equations.
• Provides Controlled Extensions: The deviations from GR are parametrized by well-defined dimensionless quantities ${ϵ}_1,{ϵ}_2,{ϵ}_3$ that are minuscule in most physical situations.
• Preserves Fundamental Principles:General covariance, energy-momentum conservation, and relativistic causality are maintained throughout the extension.
• Offers Testable Predictions: Specific, quantitatively precise deviations from GR predictions are derived, amenable to experimental testing.
• Addresses Theoretical Challenges: The framework provides mechanisms for ultraviolet regularization, singularity resolution, and consistent semi-classical treatment.
• Maintains Mathematical Consistency: All extensions are derived from a well-defined action principle and maintain proper tensor transformation properties.

The master equation represents not a rejection of GR, but rather its completion through the incorporation of quantum gravitational effects in a mathematically consistent and physically well-motivated framework. The extremely small values of the deviation parameters in most astrophysical contexts explain the remarkable success of GR while simultaneously pointing toward potential discoveries in extreme gravitational regimes.

6. Emergence of the Cosmological Constant

In this section, we explore the application of the extended nonlocal framework to the cosmological constant problem. By considering deformation contributions and nonlocal corrections, we illustrate a possible dynamical mechanism that could lead to a small effective vacuum energy, offering a way to alleviate the fine-tuning issue.

6.1. Dynamical Emergent Cosmological Constant

To address this problem within our framework, we begin with Eq(10) in vacuum conditions ($T_{\mu \nu }=0$) [43,44]:

$${\int }_M\left[A_{\alpha \beta }^{\mu \nu }{R}^{\alpha \beta }+C_{\alpha \beta \gamma \delta }^{\mu \nu }{R}^{\alpha \beta }{T}^{\gamma \delta }+E_{\alpha \beta \gamma \delta \kappa \lambda }^{\mu \nu }{R}^{\alpha \beta }{T}^{\gamma \delta }{T}^{\kappa \lambda }\right]\sqrt{-g}{d}^{n}x={\oint }_{\partial M}{Q}^{\mu \nu }d{S}^{n-1}$$

(29)

Under vacuum conditions, the energy-momentum tensor vanishes, but the deformation terms from Hypothesis II persist, leading to a non-trivial vacuum structure [6,45].

6.1.1. Vacuum Energy-Momentum Tensor under Spacetime Deformations

Even in the absence of matter sources ($T_{\mu \nu }=0$), the deformation tensor ${ϵ}_{\mu \nu }$ contributes to the effective energy-momentum through the interaction terms [4]:

$$T_{\mu \nu }^{\left(\mathrm{eff}\right)}=T_{\mu \nu }^{\left(\mathrm{matter}\right)}+T_{\mu \nu }^{\left(\mathrm{deformation}\right)}$$

The deformation contribution is given by:

$$T_{\mu \nu }^{\left(\mathrm{deformation}\right)}={\alpha }_1{L}_P^2\left(R_{\mu \nu }-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }R\right)+\mathrm{nonlocal}\mathrm{terms}$$

This represents the energy-momentum associated with the intrinsic deformation states of spacetime itself, which can dynamically affect the effective vacuum energy [46].

6.1.2. Modified Vacuum Field Equations

Substituting the effective energy-momentum tensor into equation (10) and setting $T_{\mu \nu }^{\left(\mathrm{matter}\right)}=0$, we obtain the vacuum field equations [44]:

$${\int }_M\left[(1+{\alpha }_1{L}_P^2)R^{\mu \nu }-{\displaystyle \frac{1}{2}}{g}^{\mu \nu }R+{\Lambda }_{\mathrm{eff}}{g}^{\mu \nu }\right]\sqrt{-g}{d}^{n}x={\oint }_{\partial M}{Q}^{\mu \nu }d{S}^{n-1}$$

where the effective cosmological constant emerges from the deformation terms:

$${\Lambda }_{\mathrm{eff}}={\Lambda }_{\mathrm{bare}}+{\Lambda }_{\mathrm{deformation}}+{\Lambda }_{\mathrm{nonlocal}}$$

6.1.3. Explicit Form of the Deformation Contribution

The deformation contribution to the cosmological constant can be derived from the strain tensor formulation [44,45]:

$${\Lambda }_{\mathrm{deformation}}={\displaystyle \frac{1}{2}}{\alpha }_1{L}_P^{2}R-{\displaystyle \frac{1}{4}}{\alpha }_3{L}_P^{n-2}{\int }_M{D}_{\alpha \beta }^{\alpha \beta }{\langle T^{\gamma \delta }\rangle }_{\mathrm{vac}}d{V}^{\prime }$$

(30)

where ${\langle T^{\gamma \delta }\rangle }_{\mathrm{vac}}$ represents the vacuum expectation value of the energy-momentum tensor.

6.1.4. Quantum Vacuum Expectation Value

In quantum field theory in curved spacetime, the vacuum expectation value of the energy-momentum tensor is given by [36,38]:

$${\langle T_{\mu \nu }\rangle }_{\mathrm{vac}}={\displaystyle \frac{\hslash c}{L_P^4}}{g}_{\mu \nu }+\mathrm{curvature}-\mathrm{dependent}\mathrm{terms}$$

(31)

The leading term represents the zero-point energy contribution:

$${\langle T_{\mu \nu }\rangle }_{\mathrm{vac}}^{\left(0\right)}={\rho }_{\mathrm{vac}}{g}_{\mu \nu }={\displaystyle \frac{\hslash c}{L_P^4}}{g}_{\mu \nu }$$

(32)

6.1.5. Nonlocal Compensation Mechanism

The key insight from our framework is the presence of nonlocal compensation terms in the deformation energy-momentum [4,6]:

$$T_{\mu \nu }^{\left(\mathrm{deformation}\right)}=-{\displaystyle \frac{\hslash c}{L_P^4}}{g}_{\mu \nu }+\Delta T_{\mu \nu }^{\left(\mathrm{nonlocal}\right)}$$

(33)

The nonlocal correction term is:

$$\Delta T_{\mu \nu }^{\left(\mathrm{nonlocal}\right)}={\kappa }_2{L}_P^{n-2}{\int }_M{K}_{\mu \nu \alpha \beta }{\langle T^{\alpha \beta }\rangle }_{\mathrm{vac}}d{V}^{\prime }$$

(34)

This term provides a dynamical compensation mechanism for the vacuum energy, effectively reducing the large QFT vacuum contribution.

6.1.6. Self-Consistency Condition

The effective cosmological constant must satisfy a self-consistency condition derived from the master equation [44]:

$${\Lambda }_{\mathrm{eff}}{g}_{\mu \nu }=8\pi G\left({\langle T_{\mu \nu }\rangle }_{\mathrm{vac}}+T_{\mu \nu }^{\left(\mathrm{deformation}\right)}\right)$$

(35)

Substituting the expressions from Eq(31 . 33), we obtain:

$${\Lambda }_{\mathrm{eff}}=8\pi G\left({\displaystyle \frac{\hslash c}{L_P^4}}-{\displaystyle \frac{\hslash c}{L_P^4}}+\Delta {\rho }_{\mathrm{nonlocal}}\right)$$

(36)

The leading terms cancel exactly, leaving only the small nonlocal correction.

6.1.7. Calculation of the Nonlocal Correction

The nonlocal correction term can be evaluated using the explicit form of the kernel $K_{\mu \nu \alpha \beta }$ [6,45]:

$$\Delta {\rho }_{\mathrm{nonlocal}}={\kappa }_2{L}_P^{n-2}{\int }_M{K}_{\alpha \beta }^{\alpha \beta }(x,x^{\prime }){\displaystyle \frac{\hslash c}{L_P^4}}\sqrt{-g\left(x^{\prime }\right)}{d}^n{x}^{\prime }$$

(37)

For a maximally symmetric spacetime (de Sitter space), the integral can be computed exactly:

$$\Delta {\rho }_{\mathrm{nonlocal}}={\kappa }_2{\displaystyle \frac{\hslash c}{L_P^4}}{\left({\displaystyle \frac{L_P}{L_H}}\right)}^{n-2}$$

(38)

where $L_H=\sqrt{3/\Lambda }$ is the Hubble length scale [47].

6.1.8. Final Expression for the Effective Cosmological Constant

Combining all contributions, we obtain the final expression for the effective cosmological constant:

$${\Lambda }_{\mathrm{eff}}=8\pi G\Delta {\rho }_{\mathrm{nonlocal}}=8\pi G{\kappa }_2{\displaystyle \frac{\hslash c}{L_P^4}}{\left({\displaystyle \frac{L_P}{L_H}}\right)}^{n-2}$$

(39)

In four dimensions ($n=4$), this simplifies to:

$${\Lambda }_{\mathrm{eff}}=8\pi G{\kappa }_2{\displaystyle \frac{\hslash c}{L_P^4}}{\left({\displaystyle \frac{L_P}{L_H}}\right)}^2$$

(40)

6.1.9. Numerical Evaluation and Comparison with Observations

Using the observed value of the Hubble constant $H_0\approx 70$ km/s/Mpc [48], we have $L_H\approx 1.3\times {10}^{26}$ m. The Planck length is $L_P\approx 1.6\times {10}^{-35}$ m. Substituting these values:

$${\left({\displaystyle \frac{L_P}{L_H}}\right)}^2\approx {\left({\displaystyle \frac{1.6\times {10}^{-35}}{1.3\times {10}^{26}}}\right)}^2\approx 1.5\times {10}^{-122}$$

The Planck energy density is:

$${\displaystyle \frac{\hslash c}{L_P^4}}\approx {\displaystyle \frac{(1.05\times {10}^{-34})(3\times {10}^8)}{{(1.6\times {10}^{-35})}^4}}\approx 4.6\times {10}^{113}{\mathrm{J}/\mathrm{m}}^3$$

Therefore, the effective cosmological constant is:

$${\Lambda }_{\mathrm{eff}}=8\pi G{\kappa }_2(4.6\times {10}^{113})(1.5\times {10}^{-122})\approx {\kappa }_2\times 1.7\times {10}^{-8}{\mathrm{J}/\mathrm{m}}^3$$

The observed cosmological constant corresponds to ${\Lambda }_{\mathrm{obs}}\approx {10}^{-9}{\mathrm{J}/\mathrm{m}}^3$, which requires ${\kappa }_2\sim 0.06$ [47].

6.1.10. Physical Interpretation and Mechanism

The resolution of the cosmological constant problem in our framework arises from a dynamical compensation mechanism:

$$\underset{\sim {10}^{112}}{\underbrace{{\rho }_{\mathrm{vac}}^{\mathrm{QFT}}}}+\underset{\sim -{10}^{112}}{\underbrace{{\rho }_{\mathrm{deformation}}}}+\underset{\sim {10}^{-8}}{\underbrace{\Delta {\rho }_{\mathrm{nonlocal}}}}=\underset{\sim {10}^{-8}}{\underbrace{{\rho }_{\mathrm{eff}}}}{\mathrm{erg}/\mathrm{cm}}^3$$

(41)

The large quantum vacuum energy is exactly compensated by the deformation energy of spacetime, leaving only a small residual nonlocal correction that matches the observed cosmological constant [44,45].

6.2. Comparison with GR and Other Approaches

Based on the conceptual framework presented earlier, a set of comparative results can be obtained.

6.2.1. Contrast with Standard Quantum Field Theory in Curved Spacetime

In standard QFT in curved spacetime [36], the vacuum energy contributes directly to the cosmological constant without any compensation mechanism:

$${\Lambda }_{\mathrm{standard}}={\Lambda }_{\mathrm{bare}}+8\pi G{\rho }_{\mathrm{vac}}^{\mathrm{QFT}}$$

(42)

This requires extreme fine-tuning of the bare cosmological constant ${\Lambda }_{\mathrm{bare}}$ to cancel 120 decimal places.

6.2.2. Relation to Other Proposed Solutions

Our mechanism shares conceptual similarities with several other approaches while differing in implementation:

Unimodular Gravity: In unimodular gravity [49], the cosmological constant appears as an integration constant rather than a fundamental parameter. Our approach provides a dynamical origin for this integration constant.

Supersymmetry: Supersymmetry can cancel vacuum energy contributions between bosons and fermions, but supersymmetry is broken at low energies. Our mechanism operates at all energy scales.

Anthropic Principle: The anthropic principle [49] selects the observed value from a multiverse of possibilities, while our approach predicts the value dynamically.

6.2.3. Testable Predictions and Experimental Verification

The framework makes several testable predictions:

$${\displaystyle \frac{\Delta \Lambda }{\Lambda }}\sim {\left({\displaystyle \frac{L_P}{L_H}}\right)}^2\sim {10}^{-122}$$

This extremely small variation is consistent with current observational constraints but may be detectable in future precision cosmology experiments. Additionally, the framework predicts modifications to the equation of state of dark energy:

$$w\left(z\right)=-1+\delta w\left(z\right),\delta w\left(z\right)\sim {\left({\displaystyle \frac{L_P}{L_H}}\right)}^{2}f\left(z\right)$$

where $f\left(z\right)$ is a redshift-dependent function determined by the nonlocal kernel.

7. Singularity Resolution in the Proposed Framework

The resolution of spacetime singularities represents a fundamental challenge in theoretical physics. In GR, the singularity theorems of Penrose and Hawking establish that under certain physically reasonable conditions, singularities are inevitable [12,50]. These theorems show that gravitational collapse or the cosmological evolution of the universe can lead to regions where the classical theory breaks down. The central question addressed in this section is whether the proposed framework of geometric-energy coupling and spacetime deformation provides a mathematically consistent mechanism for singularity avoidance [45,51].

We define a spacetime singularity as a point where:

• Curvature invariants diverge:

  $$\underset{x\to x_s}{lim}{R}_{\alpha \beta \gamma \delta }{R}^{\alpha \beta \gamma \delta }\to \infty$$
• Geodesic incompleteness: Some geodesics cannot be extended beyond finite affine parameter [2,12]
• Breakdown of deterministic evolution: The field equations lose predictive power, signaling a failure of classical dynamics [1,21]

7.1. Regulating Singularities via Nonlocal Fields

Here, we study the application of the non-local field-theoretic framework to a specific problem related to spacetime singularities. By applying the modified field equations and imposing quantization conditions on the spacetime curvature, we derive upper bounds on the curvature that prevent the formation of singularities within a finite region.

7.1.1. Modified Field Equations and Nonlocal Structure

Beginning with Eq(10), applying the fundamental theorem of geometric calculus [21], we obtain the local differential form:

$$\begin{array}{cc}\hfill {\nabla }_{\mu }& \left[A_{\alpha \beta }^{\mu \nu }{R}^{\alpha \beta }+B_{\alpha \beta }^{\mu \nu }{T}^{\alpha \beta }+C_{\alpha \beta \gamma \delta }^{\mu \nu }{R}^{\alpha \beta }{T}^{\gamma \delta }\right.\hfill \\ \hfill & \left.+D_{\alpha \beta \gamma \delta }^{\mu \nu }{T}^{\alpha \beta }{T}^{\gamma \delta }+E_{\alpha \beta \gamma \delta \kappa \lambda }^{\mu \nu }{R}^{\alpha \beta }{T}^{\gamma \delta }{T}^{\kappa \lambda }\right]={\nabla }_{\mu }{Q}^{\mu \nu }\hfill \end{array}$$

This represents a system of integro-differential equations due to the nonlocal nature of the coefficients. The coefficient $A_{\alpha \beta }^{\mu \nu }$ has the structure:

$$A_{\alpha \beta }^{\mu \nu }\left(x\right)={\kappa }_1{\alpha }_1{L}_P^2{\delta }_{\alpha }^{\mu }{\delta }_{\beta }^{\nu }+{\kappa }_2{\alpha }_1{L}_P^n{\int }_{J^{-}\left(x\right)}{K}_{\mu \nu \rho \sigma }(x,x^{\prime })D_{\alpha \beta }^{\rho \sigma }(x,x^{\prime })dV\left(x^{\prime }\right)$$

Similar expressions hold for other coefficients. The causal structure is explicitly enforced through the integration domain $J^{-}\left(x\right)$, ensuring relativistic causality is preserved.

7.1.2. Strain Quantization and Maximum Curvature Bound

The spacetime strain tensor obeys quantization conditions inspired by approaches in loop quantum gravity and nonlocal gravity [41,45,52]:

$${\oint }_{\partial V}{ϵ}_{\mu \nu }d{S}^{\mu \nu }=n·{ϵ}_0,n\in \mathbb{Z}$$

where the fundamental deformation quantum is:

$${ϵ}_0=\alpha L_P^2{R}_0={\displaystyle \frac{\alpha }{L_P^2}}$$

This implies a maximum allowable curvature within any finite region. For a small geodesic ball of radius r, the integrated deformation satisfies:

$${\int }_V{ϵ}_{\mu \nu }d{V}^{\mu \nu }\le N{ϵ}_0$$

yielding the curvature bound:

$$R\le {\displaystyle \frac{N}{\alpha L_P^2}}{\left({\displaystyle \frac{L_P}{r}}\right)}^n$$

As $r\to 0$, this inequality prevents curvature divergence, ensuring singularities are avoided [4,6].

7.2. Black Hole Singularity Resolution

We attempt to explore the implications of the proposed framework when dealing with the singularities of black holes.

7.2.1. Modified Geometry and Regularized Sources

For a spherically symmetric mass distribution, we make the ansatz:

$$d{s}^2=-f\left(r\right)c^{2}d{t}^2+g\left(r\right)d{r}^2+r^{2}d{\Omega }^2$$

where $f\left(r\right)$ and $g\left(r\right)$ contain nonlocal contributions. The classical energy-momentum tensor for a point mass M:

$$T_{\mu \nu }={\displaystyle \frac{M}{4\pi r^2}}\delta \left(r\right)\mathrm{diag}(1,0,0,0)$$

is regularized through nonlocal smearing:

$$T_{\mu \nu }^{\left(\mathrm{eff}\right)}\left(x\right)=T_{\mu \nu }^{\left(\mathrm{matter}\right)}\left(x\right)+\kappa L_P^{n-2}{\int }_{J^{-}\left(x\right)}{K}_{\mu \nu \alpha \beta }(x,x^{\prime })T_{\left(\mathrm{matter}\right)}^{\alpha \beta }\left(x^{\prime }\right)dV\left(x^{\prime }\right)$$

For the kernel choice $K_{\mu \nu \alpha \beta }(x,x^{\prime })=f\left(\sigma \right){\eta }_{\mu \alpha }{\eta }_{\nu \beta }$ with $f\left(\sigma \right)=exp(-\sigma /L_P^2)$:

$$T_{00}^{\left(\mathrm{eff}\right)}\left(r\right)={\displaystyle \frac{M}{4\pi r^2}}\left[\delta \left(r\right)+{\displaystyle \frac{\kappa }{L_P^2}}{e}^{-r^2/L_P^2}\right]$$

7.2.2. Field Equations Solution and Regularity Analysis

Solving the modified field equations yields:

$$g\left(r\right)={\left[1-{\displaystyle \frac{2GM}{c^{2}r}}\left(1+\kappa \mathrm{erf}\left({\displaystyle \frac{r}{L_P}}\right)\right)\right]}^{-1}$$

The Kretschmann scalar becomes:

$$K=R_{\alpha \beta \gamma \delta }{R}^{\alpha \beta \gamma \delta }={\displaystyle \frac{48G^2{M}^2}{c^4{r}^6}}\left[1+\mathcal{F}(r/L_P)\right]$$

with modification function:

$$\mathcal{F}\left(x\right)=2\kappa \mathrm{erf}\left(x\right)+{\kappa }^2{\mathrm{erf}}^2\left(x\right)-{\displaystyle \frac{4\kappa }{\sqrt{\pi }}}x{e}^{-x^2}+\mathcal{O}\left({\kappa }^2{x}^4{e}^{-2x^2}\right)$$

As $r\to 0$:

$$\underset{r\to 0}{lim}K={\displaystyle \frac{48G^2{M}^2}{c^4{L}_P^6}}\left[1+2\kappa +{\kappa }^2\right]<\infty$$

Christoffel symbols remain finite, with ${\Gamma }_{tt}^r\sim {\displaystyle \frac{GM}{c^2{L}_P^2}}$ as $r\to 0$, ensuring geodesic completeness.

7.3. Cosmological Singularity Resolution

We aim to explore the contribution of this framework to addressing cosmological singularities.

7.3.1. Modified Cosmological Dynamics

For a homogeneous, isotropic universe, the modified Friedmann equations become:

$$H^2={\displaystyle \frac{8\pi G}{3c^2}}\rho \left[1+\xi \left({\displaystyle \frac{\rho }{{\rho }_P}}\right)+\zeta {\left({\displaystyle \frac{\rho }{{\rho }_P}}\right)}^2+\dots \right]$$

where ${\rho }_P=c^5/(\hslash G^2)$ is the Planck density, and:

$$\xi ={\displaystyle \frac{\kappa }{2\pi }},\zeta ={\displaystyle \frac{\alpha }{8{\pi }^2}}$$

The modified Raychaudhuri equation includes nonlocal corrections:

$${\displaystyle \frac{\ddot{a}}{a}}=-{\displaystyle \frac{4\pi G}{3c^2}}(\rho +3p)+{\Delta }_{\mathrm{nonlocal}}+{\Delta }_{\mathrm{deformation}}$$

with:

$${\Delta }_{\mathrm{nonlocal}}=\kappa L_P^2{\int }_{-\infty }^{t}K(t,t^{\prime })H\left(t^{\prime }\right)\rho \left(t^{\prime }\right)d{t}^{\prime }$$

$${\Delta }_{\mathrm{deformation}}=\alpha L_P^2\left(R+{\displaystyle \frac{1}{2}}{ϵ}_{\mu \nu }{ϵ}^{\mu \nu }\right)$$

7.3.2. Big Bang Avoidance and Bouncing Scenarios

The framework predicts a maximum density:

$${\rho }_{\max }={\rho }_P\left[{\displaystyle \frac{-\xi +\sqrt{{\xi }^2-4\zeta (1-\frac{8\pi G}{3c^2{H}^2}{\rho }_P)}}{2\zeta }}\right]$$

For $\xi >0$, $\zeta >0$, ${\rho }_{\max }$ is finite and positive, implying $a_{\min }>0$. The correction terms can violate the strong energy condition, enabling $\ddot{a}>0$ when $\rho \sim {\rho }_P$, leading to cosmological bounce scenarios.

7.4. Well-Posedness and Computational Validation of Nonlocal Field Equations

We assess the mathematical consistency of nonlocal field equations by proving existence and well-posedness theorems for their solutions. Subsequently, we illustrate the practical application of these equations through specific computational examples, demonstrating how the theoretical framework yields finite curvature constants in relevant physical scenarios.

7.4.1. Well-Posedness and Existence Theorems

The master equation represents a system of nonlinear integro-differential equations. Local existence and uniqueness are established using fixed-point methods [53,54,55]. The iterative scheme:

$$g_{\mu \nu }^{(k+1)}=F[g_{\mu \nu }^{\left(k\right)},T_{\mu \nu }]$$

satisfies Lipschitz continuity due to causal structure and finite integration domains [4,5].

The initial value formulation admits well-posed evolution with modified constraint equations:

$$\begin{array}{cc}\hfill D^i{K}_{ij}-D_{j}K& =8\pi G{T}_{0j}+{\Delta }_{0j}^{\mathrm{nonlocal}}\hfill \\ \hfill R^{\Sigma }+K^2-K_{ij}{K}^{ij}& =16\pi G{T}_{00}+{\Delta }_{00}^{\mathrm{nonlocal}}\hfill \end{array}$$

7.4.2. Specific Computational Examples

Regular Black Hole Interior: For $\kappa =1$, $\alpha =0.1$:

$$d{s}^2=-\left(1-{\displaystyle \frac{2GM}{c^{2}r}}{e}^{-r^2/L_P^2}\right)c^{2}d{t}^2+{\left(1-{\displaystyle \frac{2GM}{c^{2}r}}{e}^{-r^2/L_P^2}\right)}^{-1}d{r}^2+r^{2}d{\Omega }^2$$

Curvature invariants remain finite:

$$\underset{r\to 0}{lim}R={\displaystyle \frac{24GM}{c^2{L}_P^3}},\underset{r\to 0}{lim}K={\displaystyle \frac{80G^2{M}^2}{c^4{L}_P^6}}$$

Bouncing Cosmology Model: For $\xi =1$, $\zeta =0.1$:

$$a_{\min }={\left({\displaystyle \frac{{\rho }_0}{{\rho }_P}}{\displaystyle \frac{2\zeta }{1+\xi }}\right)}^{1/3(1+w)}$$

For radiation-dominated universe ($w=1/3$) with ${\rho }_0\sim {10}^{-123}{\rho }_P$, $a_{\min }\sim L_P$.

7.5. Potential Challenges and Experimental Constraints

We aim to explore the potential theoretical challenges and experimental limitations that may restrict or guide the applicability of the proposed framework.

7.5.1. Mathematical and Interpretational Challenges

• Multiple Integral Definitions: Triple integrals require Hadamard finite part regularization:

  $$\underset{\mathcal{D}}{\int \int \int }f(x,x^{\prime },x^{\prime \prime })dVd{V}^{\prime }d{V}^{\prime \prime }=\underset{ϵ\to 0}{lim}\left[\underset{{\mathcal{D}}_{ϵ}}{\int \int \int }fdVd{V}^{\prime }d{V}^{\prime \prime }-C\left(ϵ\right)\right]$$
• Convergence and Global Existence: Iterative schemes may converge slowly; global existence remains open [55,56].
• Physical Interpretation: Deformation tensor measurement:

  $${ϵ}_{\mu \nu }={\displaystyle \frac{1}{2}}\underset{\Delta x\to 0}{lim}{\displaystyle \frac{\delta L_{\mu \nu }}{L_{\mu \nu }}}$$
• Energy Conditions and Causality: Nonlocal terms can violate classical energy conditions; causal structure requires careful analysis [4,5].

7.5.2. Experimental and Observational Constraints

• Solar System Tests:$\left|\kappa \right|<{10}^{-5}$ from planetary ephemerides
• Gravitational Wave Constraints:$\left|\alpha \right|<{10}^{-20}$ from dispersion measurements
• Black Hole Astrophysics: Consistency with Hawking radiation and entropy [30,32,33]

8. Theoretical Implications and Predictions

The integral Equation (10) provides a foundation for predictions beyond GR [1,2,25]. These emerge from nonlocal terms [5,6,7], deformation contributions [8,10], and higher-order couplings [23,39].

8.1. Nonlocal Gravitational Memory with Frequency-Dependent Amplification

Nonlocal Memory Effect. The nonlocal term:

$${\mathcal{N}}_{\mu \nu }\left(x\right)=\kappa L_P^{n-2}{\int }_{J^{-}\left(x\right)\cap B_{ϵ}\left(x\right)}f\left({\displaystyle \frac{\sigma }{L_P^2}}\right){\delta }_{\mu \alpha }{\delta }_{\nu \beta }{T}^{\alpha \beta }\left(x^{\prime }\right)dV\left(x^{\prime }\right)$$

yields frequency-dependent memory:

$$\Delta h_{\mu \nu }^{\mathrm{total}}=\Delta h_{\mu \nu }^{\mathrm{GR}}\left[1+\gamma \left({\displaystyle \frac{\omega }{{\omega }_P}}\right)+\delta {\left({\displaystyle \frac{\omega }{{\omega }_P}}\right)}^2+\dots \right]$$

This represents new physics beyond GR’s frequency-independent memory [11,19]. □

8.2. Spacetime Strain Quantization at Planck Scale

Strain Quantization. Deformation flux quantization:

$${\oint }_{\partial V}{ϵ}_{\mu \nu }d{S}^{\mu \nu }=n·{ϵ}_0,n\in \mathbb{Z}$$

with fundamental quantum:

$${ϵ}_0={\displaystyle \frac{\hslash G}{c^3}}\sqrt{{\Lambda }_{\mathrm{eff}}}\approx {10}^{-61}$$

provides a novel approach to quantum gravity with testable distinctions from existing models. □

8.3. Spacetime Elasticity and Modified Tidal Forces

Modified Tidal Forces. The deformation tensor introduces elastic behavior:

$${\displaystyle \frac{D^2{\xi }^{\mu }}{d{\tau }^2}}=-R_{\nu \rho \sigma }^{\mu }{u}^{\nu }{\xi }^{\rho }{u}^{\sigma }-{\nabla }_{\nu }{ϵ}_{\sigma }^{\mu }{\xi }^{\sigma }{u}^{\nu }{u}^{\rho }$$

Modified tidal acceleration:

$$a^r=-{\displaystyle \frac{2GM}{r^3}}{\xi }^r\left[1+\alpha {\left({\displaystyle \frac{L_P}{r}}\right)}^2+\beta {\left({\displaystyle \frac{L_P}{r}}\right)}^4+\dots \right]$$

affects binary pulsars and gravitational waveforms. □

8.4. Quantum Gravitational Corrections to Newton’s Law

Modified Newtonian Potential. Modified Poisson equation yields:

$$\Phi \left(r\right)=-{\displaystyle \frac{GM}{r}}\left[1+\alpha e^{-r/L_P}+\beta {\left({\displaystyle \frac{L_P}{r}}\right)}^2+\gamma {\left({\displaystyle \frac{L_P}{r}}\right)}^3+\dots \right]$$

with experimental constraints [57]:

$$\left|\alpha \right|<{10}^{-5},\left|\beta \right|<{10}^{10},\left|\gamma \right|<{10}^{15}$$

for $r\gtrsim {10}^{-4}$ m. □

9. Conclusions

This work has produced an analytical framework for energy-geometry coupling in quantum gravity, synthesizing concepts from GR, continuum mechanics, and non-local field theory. The two fundamental hypotheses—energy-geometry functional coupling and spacetime strain formulation—provide a mathematically consistent foundation that generalizes Einstein’s field equations while preserving essential physical principles.

The master integral equation derived from these principles offers a unified description of local and non-local gravitational interactions, naturally incorporating quantum-inspired corrections through the Planck-scale parameterization. The framework’s versatility is demonstrated through its systematic recovery of GR in appropriate limits and its capacity to generate modified gravity theories through parameter variations.

Key achievements include:

• A mathematically rigorous formulation of energy-geometry coupling with dimensional consistency
• Mechanisms for addressing the cosmological constant problem through geometric compensation
• Regularization approaches for spacetime singularities via non-local effects
• Testable predictions for gravitational wave dispersion and black hole phenomenology

Significant challenges remain, particularly in the domains of operator non-commutativity, rigorous renormalization procedures, and complete background-independent quantization. The mathematical consistency of the integral equations requires further investigation, including existence theorems and global solution analysis.

Future research directions should focus on developing exact solutions in symmetric spacetimes, numerical implementation of the integral equations, and detailed comparison with observational data from gravitational wave detectors. The relationship between this framework and other quantum gravity approaches presents fertile ground for further theoretical exploration.

This work contributes to the ongoing quest for quantum gravity by providing a transparent, mathematically grounded framework that bridges classical and quantum descriptions while maintaining empirical testability and theoretical consistency.

Sketch Proof (Full Details are Standard but Lengthy)

Write

$$S_{\mathrm{nonlocal}}={\displaystyle \frac{1}{2}}\int \int \sqrt{-g\left(x\right)}\sqrt{-g\left(x^{\prime }\right)}R\left(x\right)F\left({\displaystyle \frac{\sigma }{L_P^2}}\right)R\left(x^{\prime }\right).$$

Varying, collect terms proportional to $\delta g^{\mu \nu }\left(x\right)$ and $\delta g^{\mu \nu }\left(x^{\prime }\right)$, integrate by parts the ${\nabla }_{\alpha }{\Theta }^{\alpha }$ terms and combine symmetric contributions (note: exchange of integration variables over causal domain produces symmetric kernel). The variation of $\sigma$ contributes terms involving $\delta g$ contracted with derivatives of F, which then can be expressed as integrals against $R_{\alpha \beta }\left(x^{\prime }\right)$ with a bitensor kernel. Boundedness and smoothness of F justify interchange of variation and integrals (dominated convergence). Boundary terms are isolated and cancelled by $S_{\mathrm{bnd}}$.

First-Order (Linear) Response

Linearize around a background $g_{\mu \nu }^{\left(0\right)}$ (e.g. Minkowski or another solution). Denote the linearized Einstein operator ${\mathcal{E}}^{\left(1\right)}$ and its retarded Green’s operator ${\mathcal{G}}^{\mathrm{ret}}$ (well-defined under standard hyperbolicity assumptions). Then the linear solution reads

$$h_{\mu \nu }\left(x\right)={{\mathcal{G}}_{\mu \nu }^{\mathrm{ret}}}^{\alpha \beta }*\left(8\pi G_n{T}_{\alpha \beta }+8\pi G_n{\tau }_{\alpha \beta }^{\left(\mathrm{nl}\right)}\right),$$

where * denotes causal convolution/integration over past domain. Expanding ${\tau }^{\left(\mathrm{nl}\right)}$ to first order in curvature/metric perturbation yields an expression of the form

$$h_{\mu \nu }\left(x\right)={\kappa }_1{\displaystyle \frac{G_n}{c^4}}{T}_{\mu \nu }\left(x\right)+{\kappa }_2{L}_P^{n-2}{\int }_{J^{-}\left(x\right)}{K}_{\mu \nu \alpha \beta }(x,x^{\prime })T^{\alpha \beta }\left(x^{\prime }\right)dV\left(x^{\prime }\right)+\mathcal{O}\left(T^2\right),$$

with K built from ${\mathcal{G}}^{\mathrm{ret}}$ and F. This reproduces the structural form of equation (1) in the weak-field limit.

Higher-Order (Nonlinear) Response

Iterating the Volterra expansion (Picard iteration) yields higher-order Volterra kernels and the double integral structure

$$g_{\mu \nu }\left(x\right)=g_{\mu \nu }^{\left(0\right)}+{\kappa }_1{\displaystyle \frac{G}{c^4}}{T}_{\mu \nu }+{\kappa }_3{L}_P^{2n-4}\underset{J^{-}\left(x\right)}{\int \int }{K}_{\mu \nu \alpha \beta \gamma \delta }^{\left(2\right)}(x,x^{\prime },x^{\prime \prime })T^{\alpha \beta }\left(x^{\prime }\right)T^{\gamma \delta }\left(x^{\prime \prime }\right)d{V}^{\prime }d{V}^{\prime \prime }+\dots ,$$

thus obtaining the full Volterra series (the same structural equation as (1)). Convergence of the series is guaranteed locally in time under the Lipschitz/Volterra conditions (assumption 6), by standard fixed-point theorems in an appropriate Banach space.