$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \upsilon }$ Einstein’s General Relativity Equation for a manifold

Einstein’s geometric manifold explaining gravity related to energies and mass is expressed as a 2^nd order tensor equation. Replacing the Riemann manifold with a prism of thickness of “t” requires a 3^rd order tensor due to the extra dimension. The re-formulated Riemann curvature tensor now has a fourth spatial dimension with a boundary consisting of the prism thickness. The units of the tensor change from L⁻² to L⁻³. The corresponding energy mass tensor now is a gradient of energy and mass across the thickness of the prism. The advantage of adding an extra dimension is micro curvature of the prism predicts quantum properties, wave equations, and macro curvature reduces to the general relativity equation. The disadvantage is the complexity in the solutions. Einstein’s equation is a rank 2 symmetric tensor with 4 dimensions reducing to 10 simultaneous equations. The proposed equation will be a rank 3 symmetric tensor with 5 dimensions and a maximum of 125 components. Symmetry in uv reduces the components to 75. The proposed new equation for a prism uses a 3^rd order tensor.

$R_{\mu \nu \tau }-\frac{1}{2}R{g}_{\mu \nu \tau }=\frac{8\pi G}{c^4}{T}_{\mu \nu \tau }$ [7]

The cosmological component is omitted since energy can now be expressed in additional terms of the prism’s 3^rd order tensor. This formula may not be the final solution since the metric tensor $g_{\mu \nu }$ along with the Ricci tensor was introduced to conserve the order of tensors and to account for conservation of energy. The new form of the energy momentum equation is now third order and should be directly related to the Riemann tensor. The new form may be

$R_{\mu \nu \tau }=\frac{8\pi G}{c^4}{T}_{\mu \nu \tau }$

A third order tensor was previously suggested by Kaluza and Klein demonstrating the emergence of electromagnetic fields [13] [14] . Gravity is not related to the weak, strong, or the electromagnetic forces and is solely a function of curvature. The third order tensor formulation does not require a unified field theory that includes gravity. The weak, strong, and the electromagnetic are a fundamental property of the prism micro curvature. Gravity is a fundamental property of macro curvature. Geometry is the unifying metric between the micro and cosmic.

T₀₀₄ is energy density, driving mass generation via negentropy. T₁₁₄, T₂₂₄, T₃₃₄ represent surface tension, binding quarks geometrically, analogous to gluon cohesion. T_μν4, μ ≠ ν is zero for static proton, otherwise represents shear stresses (viscosity). T_4μ4, T_μ44 provides cross-dimensional momentum fluxes, potentially encoding weak, strong, or electromagnetic forces. T₄₄₄ is the shell’s self-energy, contributing to intrinsic prism properties (surface tension). These forces unify quantum (binding) and cosmic (gravity, rotation) phenomena via the prism’s curvature.

The following sections are not exact solutions to the proposed the higher order relativity equation. The solutions are approximate to demonstrate how curvature can determine micro and macro solutions to observe phenomena.

The geometry of particles shells and cores of these particles, determined by n, as shown in Figure 1 . The geometry is derived from the radius of the particle and thickness of the shell. The figures are scale adjusted. Shell role-Heavier particles (top, bottom, charm) have smaller cores and thicker shells indicating compact negentropy regions and stronger curvature. Lighter particles (up, down, electron have larger cores and thinner shells suggesting diffuse matter generation. The smaller the n the greater the deformation. A proton curvature is constructed from two up quarks and a down quark plus the cohesive energy from gluons.

Neutrino masses can be predicted with $n\approx 3\times {10}^6$ suggesting extremely diffuse curvature shells—consistent with neutrinos being barely tethered to spacetime. Flavor oscillations arise from wave interference within the prism shell. Mass differences reflect slight variations in shell geometry or curvature. Entanglement and propagation are governed by the shell’s thickness and refractive properties. The mass of particles predicted from this formula is graphically illustrated in Figure 2 .

Viscosity, surface tension, and other forces are determined by the stress energy density tensor subjected to curvature. The surface tension τ arises from the stress-energy tensor’s spatial components T₁₁₄, T₂₂₄, T₃₃₄ acting across the prism shell. The stress-energy tensor component T_μν4 is localized to the prism shell at y = ${\ell }_p$.

$T_{\mu \nu 4}=\left[\sigma h_{\mu \nu }+\eta \left({\nabla }_{\mu }{u}_{\nu }+{\nabla }_{\nu }{u}_{\mu }\right)\right]\delta \left(y-{\ell }_p\right)$

operating in the 5D equation $R_{\mu \nu \tau }-\frac{1}{2}R{g}_{\mu \nu \tau }=\frac{8\pi G}{c^4}{T}_{\mu \nu \tau }$

$h_{\mu \nu }$ is the projector tensor with

$h_{\mu \nu }=g_{\mu \nu }+u_{\mu }{u}_{\nu }$ representing the metric perpendicular to the four-velocity $u^u$.

η is the shear viscosity parameter, which can be associated with the calculated surface tension as a geometric analog for cohesive effects. $\eta \approx \tau$ the tensor’s stress (pressure-like, kg/m·s² in viscosity, but kg/s² in surface tension) due to surface tension localized to the prism’s surface via δ(y − ${\ell }_p$).

The energy density in the shell is: T₀₀₄ ≈ 2δ(y − ${\ell }_p$). Integrating over the extra dimension (y ≈ ${\ell }_p$) gives an effective 4D stress.

The surface tension is initially modeled as:

$\tau =\frac{\text{Energy Density}}{\text{Length Scale}}=\frac{\hslash c/{\ell }_p^4}{{\ell }_p}=\frac{\hslash c}{{\ell }_P^3}$

The term ${\ell }_P^3$ arises because the shell’s thickness confines the stress and the energy density projecting onto a 2D surface. The curvature of the surface further adjusts the density so a geometric term reflecting curvature, the negentropy term, must be included to relate radius to thickness of the prism. The other necessary component is mass which also effects curvature and energy density. This component is incorporated in the dimensionless component k with the ratio mass of the galaxy/reference galaxy mass. The k is empirically determined to match rotational velocities.

The negentropy −S adjusts this density based on the prism’s curvature (smaller n for higher curvature, e.g., protons; larger n for galaxies). The mass-dependent k ensures Tully-Fisher scaling ( $v\propto M^{0.25}$) [15] .

The final form

$\sigma \approx \frac{\hslash c}{{\ell }_P^3}{\left(-S\right)}^k$

$k=1.274-0.0125{\log }_{10}\left(M_{\text{total}}/M_{\text{ref}}\right)$ $M_{\text{ref}}={10}^{11}$ M_ref is the mass of medium size galaxy,

$S=-\left(\frac{1}{n^3}+\frac{3}{n^2}+\frac{3}{n}\right)$ for large R_d~3/n,

$n=R_d/{\ell }_p$ R_d is the size of the galaxy,

The observed galaxy velocities and the velocities calculated from scaled surface tension are listed below with rotational curves for various size galaxies illustrated in Figure 4 . Close to the core velocities begin to rise and flatten the farther from the core.

Small: ~106 km/s (matches dwarfs like IC 2574).

Medium: ~227 km/s (matches spirals like NGC 5055).

Large: ~261 km/s (matches giants like NGC 2903).

Milky Way: ~225 km/s (matches observed ~218 - 240 km/s).

Calculated velocities from $$ \sigma $$≈ ℏc/ ${\ell }_p^3$ (−S)^k

Small R_d = 3.086 × 10¹⁹m n = 1.909 × 10⁵⁴ −S = 1.571 × 10⁻⁵⁴ $$ \sigma $$ = 1 × 10¹⁰ kg/s² v ≈ 100 km/s

Medium R_d = 9.258 × 10¹⁹m n = 5.727 × 10⁵⁴ −S = 5.239 × 10⁻⁵⁵ $$ \sigma $$ = 4.84 × 10¹⁰ kg/s² v ≈ 220 km/s

Large R_d = 1.543 × 10²⁰m n = 9.545 × 10⁵⁴ −S = 3.143 × 10⁻⁵⁵ $$ \sigma $$ = 6.2 × 10¹⁰ kg/s²v ≈ 248.9 km/s

Milky Way R_d = 1.08 × 10²⁰m n = 6.682 × 10⁵⁴ −S = 4.489 × 10⁻⁵⁵ $$ \sigma $$ = 5.071 × 10¹⁰ kg/s² v ≈ 225.2 km/s

Waves traveling in the shell must “fit” the discrete geometry, like standing waves on a closed loop. The model quantizes the circumference to integer multiples of a fundamental wavelength $\lambda \approx {\ell }_P$

$C=2\pi r=q\lambda$ with $q=1,2,3,\cdots$ $\lambda \approx {\ell }_p$ then $r=q{\ell }_p/2\pi$

This quantization imposes boundary conditions on the wave function, discretizing positions, momenta, and energies. The tiling spheres’ periodicity (lattice-like structure) introduces angular momentum quantization, as waves “twist” around the spheres, leading to the centrifugal term in the wave equation. The negentropy S ties to this via n

$n=\frac{r}{{\ell }_p}=q/2\pi$ for $q=1$ the lowest $n=0.16$

$S=-\left(\frac{1}{n^3}+\frac{3}{n^2}+\frac{3}{n}\right)$ quantizes the energy organization in the shell, with smaller

n (tighter curvature) for heavier particles.

The full wave function $\psi \left(r,\theta ,\phi \right)$ in spherical coordinates (fitting the spherical tiling) is separated as:

$\psi \left(r,\theta ,\varphi \right)=R\left(r\right)Y_q^{\ell }\left(\theta ,\varphi \right)$

with ℓ the angular quantum number (quantized due to the spheres’ symmetry) and q the magnetic quantum number ( $-\ell \le q\le \ell$).

The Schrödinger equation is:

$-\frac{{\hslash }^2}{2m}{\nabla }^2\psi +V\left(r\right)\psi =E\psi$

Substituting the separated form, the Laplacian ${\nabla }^2$ separates into radial and angular parts:

${\nabla }^2=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial }{\partial \theta }\right)+\frac{1}{r^2\sin \theta }\frac{{\partial }^2}{\partial {\Phi }^2}$

The angular part gives the eigenvalue $-\ell \left(\ell +1\right)$ for $Y_q^{\ell }$, leading to the radial equation for R(r):

$-\frac{{\hslash }^2}{2m}\left(\frac{{\text{d}}^2}{\text{d}{r}^2}+2\frac{\text{d}}{r\text{d}r}\right)R\left(r\right)+\frac{{\hslash }^2}{2m{r}^2}\ell \left(\ell +1\right)R\left(r\right)+V\left(r\right)R\left(r\right)=ER\left(r\right)$

The potential V(r) models the prism shell’s boundary, approximated as a delta

function at the Planck scale: $V\left(r\right)\approx \left(\frac{{\hslash }^2}{2m}\right)Q\delta \left(r-{\ell }_p\right)V\left(r\right)$, where Q is a constant

related to shell stiffness or curvature. Q is non dimensional scaling term consisting of negentropy term and exponent k relating to bound versus unbound mass.

Q is related to ${\left(-\frac{1}{{\left(r/{\ell }_p\right)}^3}+\frac{3}{{\left(r/{\ell }_p\right)}^2}+\frac{3}{r/{\ell }_p}\right)}^k$ where k = 1.274 − 0.0125

log₁₀(M_{unbound mass}/M_{bound mass}).

ħ²/2 m has units of energy × length², δ has 1/length, so V has energy units.

Diameter of Tiling Spheres: Given as ${\ell }_P$, the radius of each sphere is

$\frac{{\ell }_p}{2}\approx 8.081145\times {10}^{-16}$

Tiling Assumption: The prism is a thin shell (thickness t ≈ ${\ell }_P$tiled by these spheres without gaps, forming a lattice. In 3D Euclidean space, the densest packing of spheres is the face-centered cubic or hexagonal close-packed arrangement, achieving a packing fraction of

$\frac{\pi }{3\sqrt{2}}\approx 0.7405$. However, in the 5D prism, the extra dimension (index 4) may

allow a higher effective density due to compactification or discrete stacking along y.

Curvature Scale: The curvature is inversely related to the scale of the tiling. The smallest radius ( ${\ell }_P/2$) sets the fundamental length scale, suggesting curvature scales as $1/{\left({\ell }_P/2\right)}^2=4/{\ell }_P^2$

In general relativity, the Ricci scalar R (units m⁻²) represents the intrinsic curvature of spacetime. For a discrete lattice, the curvature is approximated by the discrete Laplacian or the geodesic deviation over the tiling scale. The maximal curvature occurs where the tiling spheres are most tightly packed, i.e., at the Planck scale, where the prism’s thickness and sphere diameter coincide.

The model uses a 5D framework, so the relevant curvature includes the third-order term $R_{\mu \nu \tau }$, which couples to $T_{\mu \nu \tau }$. The Ricci scalar in 5D (R⁽⁵⁾) is the trace of the 5D Ricci tensor, but the prism’s discrete nature suggests focusing on the local curvature induced by the tiling.

The negentropy $S=-\left(\frac{1}{n^3}+\frac{3}{n^2}+\frac{3}{n}\right)$, with $n=R/{\ell }_P$, quantifies order. For the maximal curvature, consider the smallest R, the radius of a single tiling sphere ( $R\approx \frac{{\ell }_p}{2}$):

$n=\frac{{\ell }_p}{2{\ell }_p}=\frac{1}{2}$ $S=-\left(\frac{1}{{\left(\frac{1}{2}\right)}^3}+\frac{3}{{\left(\frac{1}{2}\right)}^2}+\frac{3}{\left(\frac{1}{2}\right)}\right)$, $S=-\left(8+12+6\right)=-26$ $-S=26$

The surface tension $\tau \approx \frac{\hslash c}{{\ell }_p^3}{\left(-S\right)}^k$ with $k\approx 1.274$ for minimal systems) contributes to $T_{\mu \nu \tau }$

For $k=1.274$

$\tau \approx \left(7.488\times {10}^{78}\right)\times 1.2\times {10}^3\approx 9.0\times {10}^{81}\text{kg}/{\text{s}}^{\text{2}}$

This extreme tension reflects the maximal stress t the Planck scale, feeding into $T_{\mu \nu \tau }~\tau /{\ell }_p~{10}^{116}$ kg⋅m⁻²⋅s⁻².

The field equation $R_{\mu \nu \tau }=\kappa T_{\mu \nu \tau }$ relates curvature to the tensor. The maximal $T_{\mu \nu \tau }$ occurs where the tiling density is highest, i.e., $n=1/2$. The curvature $R_{\mu \nu \tau }$ has units m⁻³ and κ~8πG₅/c⁴ (units kg⁻¹⋅m³⋅s²).

To find the scalar curvature R⁽⁵⁾, trace over indices. Assuming isotropy at maximal density, R $R_{\mu \nu \tau }~R/{\ell }_p^3$ (projecting 5D curvature to 3D scale), the maximal $R~{\ell }_p^{-2}\cdot \left(-S\right)\cdot \kappa T~{10}^{-26}$ kg⁻¹⋅m³⋅s² (estimated from 5D compactification):

$R_{\max }\approx 2.2\times {10}^{211}{\text{m}}^{-2}$

This is an upper bound, as κ and tensor projection need precise 5D calibration. The standard Planck-scale curvature $1/{\ell }_p^2\approx 3.8\times {10}^{69}{\text{m}}^{-2}$ is exceeded due to negentropy amplification.

The tiling spheres’ discreteness limits curvature: Beyond ${\ell }_P$/2, quantum gravity effects (e.g., string theory) or prism rupture (containment vessel) may cap it. The maximal physical curvature is thus tied to the densest packing, where R_max~10⁶⁹ - 10²¹¹ m⁻² as a practical limit (matching Planck curvature without excessive negen-tropy boost.

The maximal curvature of the spacetime prism, considering tiling spheres with a diameter of the Planck length ${\ell }_P$≈ 1.616229 × 10⁻³⁵, is approximately:

$R_{\max }~4{\ell }_p^2\cdot {\left(-S\right)}_{\max }\approx 1.58\times {10}^{71}{\text{m}}^{-2}$⋅

This assumes a conservative negentropy scaling (n = 1/2, −S = 26) without full 5D tensor amplification. The upper bound, including tensor and coupling effects, may reach 10²¹¹ m⁻², but the Planck-scale limit $1/{\ell }_p^2\approx 3.8\times {10}^{69}{\text{m}}^{-2}$ is likely the physical maximum due to discreteness constraints.

Standard Einstein-Hilbert Action as a Starting Point

In 4D general relativity, the action S_a is given by the Einstein-Hilbert Lagrangian:

$S_a={\displaystyle \int {ℒ}_{EH}\sqrt{-g}{\text{d}}^{4}x}$, ${ℒ}_{EH}=c^{4}R/16\Pi G$

where:

R is the Ricci scalar,

g is the determinant of the metric $g_{\mu \nu }$,

G is the gravitational constant,

c is the speed of light.

This action describes the geometry of spacetime coupled to a second-order stress-energy tensor $T_{\mu \nu }$ via the Einstein field equations

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

In the two-compartment model, the manifold is replaced by a 5D prism with a third-order tensor $T_{\mu \nu \tau }$, sourced in $R_{\mu \nu \tau }=\kappa T_{\mu \nu \tau }$ (or the modified form

$R_{\mu \nu \tau }-\frac{1}{2}R{g}_{\mu \nu \tau }=\kappa T_{\mu \nu \tau }$), where κ is a 5D coupling constant analogous to 8πG/c⁴.

The prism introduces a fifth dimension (index τ = 4) corresponding to its thickness (~ ${\ell }_P$), tiled by spheres with negentropy-driven energy. The action must generalize to 5D, integrating over $x_{\mu }$ (0 - 3) and y (4th spatial coordinate):

$S_a={\displaystyle \int ℒ\sqrt{-g^{\left(5\right)}}{\text{d}}^{\left(5\right)}x}$

where g⁽⁵⁾ is the 5D metric determinant, and the Lagrangian $ℒ$ includes contributions from the prism’s curvature and the third-order tensor.

The 5D Ricci tensor $R_{\mu \nu \tau }^{\left(5\right)}$ (from the 5D metric) and the third-order curvature $R_{\mu \nu \tau }$ (specific to the prism’s discrete structure) replace the 4D R. The model suggests $R_{\mu \nu \tau }$ encodes additional degrees of freedom (e.g., surface tension

$\tau \approx \frac{\hslash c}{{\ell }_p^3}{\left(-S\right)}^k$, viscosity $\eta \approx \tau$, so the Lagrangian should couple $R_{\mu \nu \tau }$ to $T_{\mu \nu \tau }$.

The action Lagrangian $ℒ$ should consist of:

Gravitational Term: A 5D curvature scalar or third-order curvature term, generalizing R.

Matter Term: The third-order stress-energy tensor T, representing energy-mass density from tiling spheres.

Negentropy Contribution: S as a geometric order parameter, modulating the tensor’s strength.

Given the prism’s discrete nature, the Lagrangian is proposed as: Two-Compartment Action Lagrangian

Given the prism’s discrete nature, the Lagrangian is proposed as:

$ℒ=\frac{c^4}{16\Pi G_5}{R}^{\left(5\right)}+\kappa T_{\mu \nu \tau }{R}^{\mu \nu \tau }+\lambda \left(-S\right)T_{\mu \nu \tau }{T}^{\mu \nu \tau }$

where:

$R^{\left(5\right)}$: 5D Ricci scalar, with G₅ the 5D gravitational constant (related to 4D G via compactification, G₅~G/ ${\ell }_P$.

$R^{\mu \nu \tau }$: Dual of the third-order curvature tensor, contracted with $T_{\mu \nu \tau }$ to enforce field equations.

κ: Coupling constant, $\kappa ~8\pi G_5/c^4$, adjusted for 5D.

λ: A dimensionless constant scaling negentropy’s effect on tensor self-interaction.

$-S=\left(\frac{1}{n^3}+\frac{3}{n^2}+\frac{3}{n}\right)$: Negentropy, with n = R/ ${\ell }_P$ (radius of the system, e.g.,

proton or galaxy, over Planck length).

Varying the action $S_a={\displaystyle \int ℒ\sqrt{-g^{\left(5\right)}}{\text{d}}^{\left(5\right)}x}$ with respect to the 5D metric $g_{\mu \nu }^{\left(5\right)}$

$\frac{\delta \left(ℒ\sqrt{-g^{\left(5\right)}}\right)}{\delta g_{\mu \nu }^{\left(5\right)}}=0$

Gravitational Term: R⁽⁵⁾ governs large-scale curvature (e.g., galaxy rotation).

Tensor Coupling: $T_{\mu \nu \tau }{R}^{\mu \nu \tau }$ links prism geometry to forces (e.g., surface tension replacing dark matter).

Negentropy Term: $\left(-S\right)T_{\mu \nu \tau }{T}^{\mu \nu \tau }$ quantifies order, driving particle masses

$m_n=\frac{\alpha \hslash c}{{\ell }_p}\left(-S\right)$ and energy transport (e.g., coronal heating).

The action Lagrangian for the energy-mass density tensor in the two-compartment model is:

$ℒ=\frac{c^4}{16\Pi G_5}{R}^{\left(5\right)}+\kappa T_{\mu \nu \tau }{R}^{\mu \nu \tau }+\lambda \left(-S\right)T_{\mu \nu \tau }{T}^{\mu \nu \tau }$

where:

$R^{\left(5\right)}$ The 5D Ricci scalar,

$T_{\mu \nu \tau }$ is the third-order stress-energy tensor (units kg⋅m⁻²⋅s⁻²),

$R^{\mu \nu \tau }$ is the dual third-order curvature,

$\kappa ~8\pi G_5/c^4$ (units kg⁻¹⋅m³⋅s²),

λ is a coupling constant (units kg⁻¹⋅m⁴⋅s²),

$S=-\left(1/n^3+3/n^2+3/n\right)$ is the negentropy (dimensionless), with n = R/ ${\ell }_P$.

Planck-Scale Dominance: Both models are capped near the Planck curvature ∼10⁶⁹ m⁻², where quantum effects prevent infinite curvature (singularities). In the two-compartment model, discreteness of tiling spheres eliminates singularities, like how strings “smear” spacetime at ${\ell }_s$.

Unification of Forces: The prism’s tensor unifies gravity (macro-curvature) with quantum forces (micro-curvature, e.g., replacing gluons), echoing string theory’s unification via vibrational modes producing gravitons, gauge bosons, and matter particles.

Dimensional Compactification: The prism’s extra dimension (thickness ${\ell }_P$) is compactified, akin to string theory’s 6 - 7 compact dimensions (Calabi-Yau manifolds) at the Planck scale, generating EM-like fields (as in Kaluza-Klein extensions in both).

Discrete vs. Continuous Structure: The two-compartment model’s prism is fundamentally discrete (tiling spheres), leading to a “quantized” curvature limit tied to sphere packing density (e.g., face-centered cubic packing fraction ~0.74 constrains maximal “bending”), potentially allowing slightly higher effective curvatures (~10⁷¹ m⁻²) via negentropy amplification. String theory is continuous at larger scales but introduces fuzziness via string world sheets, with curvature resolved at $\alpha \prime ={\ell }_s^2/4\pi$ $\alpha \prime$ (Regge slope parameter), typically capping at ~10⁷⁰m⁻² without discrete caps.

Curvature Amplification: The model’s negentropy −S and tensor coupling can amplify curvature beyond the base Planck value (up to 10²¹¹ m⁻² theoretically), driven by geometry and order/disorder. String theory curbs this via higher-order corrections (e.g., R⁴ terms in the effective action), preventing divergences but without explicit amplification factors like −S.

Dimensionality and Forces: The model uses 5D with a third-order tensor to emerge forces (e.g., surface tension replacing dark matter), while string theory requires 10D/11D, with branes and fluxes for force unification. String theory’s maximal curvature often relates to black brane horizons or AdS throats (~10⁶⁰ - 10⁷⁰ m^⁻² in warped throats), lower than the model’s amplified limit but similar in base scale. Comparison is in Table A1 .

The two-compartment model’s curvature is comparable in base scale to string theory’s but potentially higher due to negentropy-driven amplification, offering a discrete alternative to string fuzziness for quantum gravity resolution. Both avoid singularities at ~10⁷⁰ m⁻², but the model’s third-order tensor provides unique predictions (e.g., coronal heating via EM conduits).

In general relativity, the metric is perturbed as $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$ where $\left|h_{\mu \nu }\right|\ll 1$,

${\eta }_{\mu \nu }=\text{diagonal}\left(1,-1,-1,-1\right)$ is the Minkowski metric, and we use the harmonic gauge ${\partial }^{\mu }{h}_{\mu \nu }^{-}=0$, ${\partial }^{\mu }{h}_{\mu \nu }^{-}=h_{\mu \nu }-1/2{\eta }_{\mu \nu }h$,

The linearized Einstein equations are:

$\square h_{\mu \nu }^{-}=-\left(16\pi G/c^4\right)T_{\mu \nu }$

where $\square ={\partial }^{2}t/c^2-{\nabla }^2$ is the d’Alembertian. For static, low-velocity sources (non- relativistic limit, T₀₀ ≈ ρc², other components small):

${\nabla }^2{h}_{00}=\left(8\pi G/c^2\right)\rho$, $h_{00}=-2\Phi /c^2$

recovering Newton’s law ${\nabla }^2\Phi =4\pi G\rho$ with Φ the Newtonian potential.

The model is in 5D (indices A, B, C = 0 - 4), with metric $g_{AB}^{\left(5\right)}={\eta }_{AB}^{\left(5\right)}+h_{AB}^{}A{B}^{\left(5\right)}$,

${\eta }_{AB}^{\left(5\right)}=\text{diagonal}\left(1,-1,-1,-1,-1\right)$, and the extra dimension y (compactified to ~ ${\ell }_P$). The third-order curvature $R_{\mu \nu \tau }$ is non-standard (not the usual rank-4 Riemann) but interpreted as a contracted Ricci-like tensor with extra index for prism thickness. From Kaluza-Klein analogies we compactify y, reducing to effective 4D fields.

Linearization:

Perturb the 5D metric: $h_{AB}^{\left(5\right)}\ll 1$.

The 5D Ricci tensor linearizes as

$R_{AB}^{\left(5\right)}=-1/2{\square }^{\left(5\right)}{h}_{AB}^{\left(5\right)}+1/2{\partial }_A{\partial }_C{h}_{CB}^{\left(5\right)}+1/2{\partial }_B{\partial }_C{h}_{CA}^{\left(5\right)}-1/2{\partial }_A{\partial }_B{h}_{CC}^{\left(5\right)}$ (in gauge).

The model’s third order $R_{\mu \nu \tau }$ may be $R_{\mu \nu \tau }={\partial }_{\tau }{R}_{\mu \nu }^{\left(4\right)}$ + prism terms, but since τ = 4 is distinguished, we approximate $R_{\mu \nu 4}\approx {\partial }_4{R}_{\mu \nu }^{\left(4\right)}\approx R_{\mu \nu }^{\left(4\right)}/{\ell }_P$ (gradient over thickness).

The field equation becomes $R_{\mu \nu 4}=\kappa T_{\mu \nu 4}$ with $T_{\mu \nu 4}=\left[h_{\mu \nu }+\eta \left({\partial }_{\mu }{u}_{\nu }+{\partial }_{\nu }{u}_{\mu }\right)\right]\delta \left(y-{\ell }_P\right)$, as given.

Compactification and Effective 4D Equation: Integrate over y (compact dimension): Effective 4D tensor $T_{\mu \nu }^{eff}={\displaystyle \int T_{\mu \nu 4}\text{d}y}/{\ell }_P\approx h_{\mu \nu }+\eta \left({\partial }_{\mu }{u}_{\nu }+{\partial }_{\nu }{u}_{\mu }\right)$, since ${\displaystyle \int \delta \left(y-{\ell }_P\right)\text{d}y}=1$.

The 5D linearized Ricci reduces (from Kaluza-Klein weak field) to 4D Einstein + Electromagnetic-like terms:

$R_{\mu \nu }^{\left(4\right)}\approx \kappa {\ell }_P{T}_{\mu \nu }^{eff}+\left(1/{\ell }_P^2\right)F_{\mu \sigma }{F}_{\nu }^{\sigma }-\left(1/4{\ell }_P^2\right)g_{\mu \nu }^{\left(4\right)}$

where $F_{\mu \nu }$ emerges from off-diagonal $h_{\mu 4}\approx A_{\mu }$ (Kaluza-Klein gauge field), scaled by $1/{\ell }_P^2$ for compactification.

Negentropy correction: $T_{\mu \nu }^{eff}\to \left(-S\right)T_{\mu \nu }^{eff}$, amplifying for small n (high curvature).

Similarities: In the limit ${\ell }_P$→ 0 (ignoring extra dimension) or S → 0 (no negen-tropy), the model reduces to general relativity linearized equation, recovering Newtonian gravity. The effective $T_{\mu \nu }^{eff}$ mimics general relativity $T_{\mu \nu }$ for macroscopic scales (e.g., galaxy rotation via surface tension as “dark matter” correction).

Differences:

Extra Fields: Model introduces Electromagnetic-like terms from compactification ( $1/{\ell }_P^2{F}^2$ ~10⁷⁰ m⁻² corrections, negligible in weak fields but amplifying binding at Planck scales).

Negentropy Amplification: (-S) scales ρ by ~10²–10²⁰ (depending on n), boosting effective mass density for particles (e.g., proton binding) but vanishing for cosmic n ~10⁵⁴ (general relativity-like).

Viscous/Shear Terms: $\eta \left({\partial }_{\mu }{u}_{\nu }+{\partial }_{\nu }{u}_{\mu }\right)$ adds fluid-like dissipation, absent in vacuum general relativity weak field, potentially explaining galactic rotation without dark matter (viscous drag ~τ v/r).

Dimensional Effects: general relativity is purely 4D; model has 5D corrections $~1/{\ell }_P^2$, diverging at Planck scales (quantum regime), while general relativity breaks down there without such terms.

In summary, the model’s weak field recovers general relativity’s Newtonian limit for large n but adds negentropy-boosted densities, electromagnetic fields, and viscous stresses for micro/macro deviations, aligning with its unification goals.