Distortion Gravity

Distortion Gravity (DG) is a metric‑affine theory of gravity proposed by Luca Eliseo Pavesi in 2026. It extends general relativity by promoting the affine connection Γ^ρ_μν to an independent dynamical field, whose deviation from the Levi‑Civita connection Γ̃^ρ_μν is quantified by the distortion tensor D^ρ_μν. The theory propagates a massless graviton and two massive vector fields – the trace vector V_μ and the axial torsion vector a_μ – and has been shown to be ghost‑free.

Distortion Gravity provides a four‑dimensional realisation of the ER = EPR conjecture: the distortion tensor that sustains a traversable wormhole (ER bridge) also governs quantum entanglement (EPR) through the quantised flux of the axial torsion.

The theory was first presented in a series of four articles published on ScienceOpen in 2026, which were subsequently reviewed and indexed by Sciety (eLife). The foundational article on ghost‑free unitarity received two recommendations on ResearchGate. An expanded monograph, Distortion Gravity: A New Theory of Gravitation: From Foundations to Complete Verification, was published as a book in June 2026 (ISBN 979‑8181642256).

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The distortion tensor and its vectors

In metric‑affine geometry the distortion tensor is defined as

D^ρ_μν = Γ^ρ_μν − Γ̃^ρ_μν, encoding both torsion T^ρ_μν = D^ρ_μν − D^ρ_νμ and non‑metricity Q_ρμν = ∇_ρg_μν.

Under the general linear group GL(4,ℝ), D^ρ_μν decomposes into irreducible parts. The dynamical sector consists of two vectors:

$$V_\mu = D^\alpha{}_{\mu\alpha}, \qquad a_\mu = \frac{1}{6}\varepsilon_{\mu\nu\rho\sigma} D^{\nu\rho\sigma}.$$

Action

The ghost‑free action for Distortion Gravity is

$$S_{\text{DG}} = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa} \tilde{R} + \mathcal{L}_V + \mathcal{L}_a + \alpha_1 I_1 + \alpha_2 I_2 + \alpha_3 I_3 \right],$$ where R̃ is the Riemann scalar of the Levi‑Civita connection, ℒ_V and ℒ_a are Proca Lagrangians for the trace and axial vectors, and I_1, 2, 3 are quadratic invariants of D^ρ_μν.

Field equations

The field equations are obtained by varying the action with respect to the metric and the vector fields.

Variation with respect to the metric

The Einstein–Hilbert term yields the Einstein tensor G̃_μν. The Proca kinetic and mass terms for a generic vector B_μ give

$$\frac{2}{\sqrt{-g}}\frac{\delta}{\delta g^{\mu\nu}}\left(-\frac{1}{4}\sqrt{-g}F_{\alpha\beta}F^{\alpha\beta}\right)=F_{\mu\alpha}F_\nu^{\ \alpha}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta},$$

$$\frac{2}{\sqrt{-g}}\frac{\delta}{\delta g^{\mu\nu}}\left(\frac{1}{2}\sqrt{-g}m^2B_\alpha B^\alpha\right)=m^2\left(B_\mu B_\nu-\frac{1}{2}g_{\mu\nu}B_\alpha B^\alpha\right).$$

Applying these to V_μ and a_μ, the metric field equation is

G̃_μν = κ(T_μν^matter+T_μν^(V)+T_μν^(a)+T_μν^(pot)), where the Proca stress–energy tensors are

$$T^{(V)}_{\mu\nu}=F^{(V)}_{\mu\alpha}F^{(V)\alpha}_{\ \ \ \nu}-\frac{1}{4}g_{\mu\nu}F^{(V)}_{\alpha\beta}F^{(V)\alpha\beta}+m_V^2\left(V_\mu V_\nu-\frac{1}{2}g_{\mu\nu}V_\alpha V^\alpha\right),$$ and similarly for T_μν^(a).

Variation with respect to the vector fields

Varying with respect to V_μ and a_μ yields the Proca equations in curved spacetime:

$$\tilde{\nabla}_\mu F^{(V)\mu\nu}+m_V^2V^\nu=0,\qquad \tilde{\nabla}_\mu f^{(a)\mu\nu}+m_a^2a^\nu=0.$$ Taking the divergence gives the Lorenz conditions $\tilde{\nabla}_\mu V^\mu=0$ and $\tilde{\nabla}_\mu a^\mu=0$.

Linearised equations

Expanding around Minkowski space (g_μν = η_μν + h_μν, D^ρ_μν = 0 + d^ρ_μν), the linearised field equations are

□V_μ + m_V²V_μ = 0,   □a_μ + m_a²a_μ = 0,   ∂_μV^μ = 0,   ∂_μa^μ = 0. In Fourier space this gives the dispersion relation ω² = k² + m², confirming 3 degrees of freedom for each massive vector.

Spontaneous symmetry breaking

For a homogeneous configuration $V_\mu = (V_0, \vec 0)$, the potential is

$$V(V_0) = \frac{1}{2} m_V^2 V_\mu V^\mu + \frac{\lambda_V}{4} (V_\mu V^\mu)^2 = -\frac{1}{2} m_V^2 V_0^2 + \frac{\lambda_V}{4} V_0^4,$$ using the metric signature (−,+,+,+). When m_V² < 0 the potential has a double‑well shape with minima at V₀ =  ± v, where $v = \sqrt{|m_V^2|/\lambda_V}$. The Levi‑Civita point V₀ = 0 becomes a saddle point.

In curved spacetime the effective mass becomes curvature‑dependent,

$$m_{V,\text{eff}}^2(r) = m_V^2 + \lambda_{\text{curv}} \frac{r_0^2}{r^2},$$ with λ_curv > 0 and r₀ the wormhole throat radius. Near the throat (r ∼ r₀) the mass squared is negative (SSB phase), while far away (r ≫ r₀) it is positive (symmetry restored). This localises the NEC violation at the throat and recovers GR asymptotically.

Modified ansatz

A static, spherically symmetric wormhole is described by the metric

$$ds^2 = - e^{2\Phi(r)} dt^2 + \frac{H(r)}{e^{2\Phi(r)}\left(1 - \frac{b(r)}{r}\right)} dr^2 + r^2(d\theta^2 + \sin^2\theta\, d\phi^2),$$ where Φ(r) is the finite redshift function, b(r) the shape function with b(r₀) = r₀, and H(r) a curvature‑dependent compression factor,

H(r) = 1 + λ(r)(e^2Φ(r) − 1), with λ(r) a double Gaussian localised at the throat:

$$\lambda(r) = A_1 \exp\!\left(-\frac{(r - r_0)^2}{2\sigma_1^2}\right) + A_2 \exp\!\left(-\frac{(r - r_1)^2}{2\sigma_2^2}\right).$$

When Φ(r) < 0 (gravitational blueshift) near the throat, e^2Φ < 1 and H(r) < 1, compressing the proper distance. The double Gaussian allows independent control of the throat and the region where b(r)/r approaches unity, preventing the wormhole from pinching off.

This metric has been discussed in the context of the Italian Wikipedia entry for the Einstein–Rosen bridge, where its anisotropic and entropic features are described in relation to the Orch‑OR paradigm.

Field equations

The Einstein equations with the Proca energy‑momentum tensors yield

$$b'(r) = \kappa r^2 \rho(r), \qquad \Phi'(r) = \frac{b(r)/r + \kappa r^2 p_r(r)}{2r(1 - b(r)/r)},$$ where ρ(r) and p_r(r) are the total energy density and radial pressure. The Proca equations for the vector fields in the wormhole background are

$$V_0'' + \left(\frac{2}{r} + \frac{H'}{2H}\right) V_0' - \frac{m_{V,\text{eff}}^2(r)}{g^{-1}_{rr}} V_0 + \frac{\lambda_V}{g^{-1}_{rr}} V_0^3 = 0,$$

$$a_\phi'' + \left(\frac{H'}{2H} + \frac{2}{r}\right) a_\phi' - \frac{m_{a,\text{eff}}^2(r)}{g^{-1}_{rr}} a_\phi + \frac{\lambda_a}{r^2 g^{-1}_{rr}} a_\phi^3 = 0,$$ with g_rr⁻¹ = e^2Φ(1−b/r)/H(r).

Numerical exploration

The coupled system was solved numerically using the SciPy `solve_ivp` routine (RK45). A total of 7,600 configurations were tested over a 13‑dimensional parameter space. A wormhole is considered valid if it satisfies: throat condition, flaring‑out, openness (b(r) < r for all r > r₀), traversability (proper crossing time Δτ < πr₀), and angular stability. 57% of the configurations yielded fully valid traversable wormholes, with the best crossing time Δτ = 0.1035 r₀/c, about three times shorter than the GR collapse timescale πr₀. The NEC is violated at the throat and restored asymptotically.

== ER = EPR realisation ==

Entanglement entropy and axial torsion flux

In Distortion Gravity, the effective Newton constant is modified by the background vector fields,

$$G_{\text{eff}} = \frac{G_N}{1 + \alpha V_\mu V^\mu + \beta a_\mu a^\mu}.$$ The entanglement entropy across the wormhole is given by the Ryu–Takayanagi formula

$$S_{\text{EE}} = \frac{A_{\text{throat}}}{4 G_{\text{eff}}} = \frac{\pi r_0^2}{G_{\text{eff}}}.$$

The axial torsion field a_μ generates a quantised flux through the throat,

Φ = ∮_S¹a_μdx^μ = 2πa_ϕ(r₀) = 2πnv_ar₀,  n ∈ ℤ.

The entanglement entropy is proportional to this flux,

S_EE ∝ Φ. Thus both the geometric connectivity (wormhole area) and the quantum correlations (entanglement) are controlled by the same integer n.

Quantum simulation

The ER = EPR mechanism was tested on the Qiskit platform using a two‑qubit circuit. A Bell state was prepared and subjected to Aharonov–Bohm phase shifts determined by the torsion flux. The von Neumann entropy remained maximal (S_EE = 1 bit) for all integer winding numbers n = 0, …, 5, confirming that torsion preserves quantum correlations. A CHSH Bell test showed a continuous modulation of Bell violations by the torsion gradient, without any violent “firewall”.

External links

• Distortion Gravity information page
• Personal website of Luca Eliseo Pavesi
• SSRN (Elsevier) author page
• ScienceOpen profile