Distortion Gravity

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Distortion Gravity (DG) is a metric‑affine framework for gravity proposed in the work of Luca Eliseo Pavesi, in which the affine connection \(\Gamma^\rho{}_{\mu\nu}\) is promoted to an independent dynamical field. The deviation from the Levi‑Civita connection \(\tilde\Gamma^\rho{}_{\mu\nu}\) is encoded in the distortion tensor \(D^\rho{}_{\mu\nu}\). The theory propagates a massless graviton and two massive vector fields (the trace vector \(V_\mu\) and the axial torsion vector \(a_\mu\)) and has been shown to be ghost‑free.^[1][2]

Distortion Gravity provides a four‑dimensional realisation of the ER = EPR conjecture: the same distortion tensor that sustains a traversable wormhole (ER bridge) also controls the entanglement (EPR) via the quantised flux of the axial torsion.^[1][3]

Mathematical formulation

The distortion tensor and its vectors

In metric‑affine geometry, the distortion tensor is

\[D^\rho{}_{\mu\nu} = \Gamma^\rho{}_{\mu\nu} - \tilde\Gamma^\rho{}_{\mu\nu},\]

encoding both torsion \(T^\rho{}_{\mu\nu} = D^\rho{}_{\mu\nu} - D^\rho{}_{\nu\mu}\) and non‑metricity \(Q_{\rho\mu\nu} = \nabla_\rho g_{\mu\nu}\).

Under the general linear group \(GL(4,\mathbb{R})\), \(D^\rho{}_{\mu\nu}\) decomposes into irreducible parts.^[4] 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 \(\tilde{R}\) is the Riemann scalar of the Levi‑Civita connection, \(\mathcal{L}_V\) and \(\mathcal{L}_a\) are Proca Lagrangians for the trace and axial vectors, and \(I_{1,2,3}\) are quadratic invariants of \(D^\rho{}_{\mu\nu}\).^[1]

Field equations

The field equations are obtained by varying the action with respect to the metric and the vector fields.^[2][5][6]

Variation with respect to the metric

The Einstein–Hilbert term yields the Einstein tensor \(\tilde{G}_{\mu\nu}\). The Proca kinetic and mass terms for a generic vector \(B_\mu\) 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 \(V_\mu\) and \(a_\mu\), the metric field equation is

\[\tilde{G}_{\mu\nu}=\kappa\left(T^{\text{matter}}_{\mu\nu}+T^{(V)}_{\mu\nu}+T^{(a)}_{\mu\nu}+T^{\text{(pot)}}_{\mu\nu}\right),\]

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 \(T^{(a)}_{\mu\nu}\).

Variation with respect to the vector fields

Varying with respect to \(V_\mu\) and \(a_\mu\) 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 (\(g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\), \(D^\rho{}_{\mu\nu}=0+d^\rho{}_{\mu\nu}\)), the linearised field equations are

\[\Box V_\mu+m_V^2V_\mu=0,\qquad \Box a_\mu+m_a^2a_\mu=0,\qquad \partial_\mu V^\mu=0,\qquad \partial_\mu a^\mu=0.\]

In Fourier space this gives the dispersion relation \(\omega^2=\mathbf{k}^2+m^2\), 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 \(m_V^2 < 0\) the potential has a double‑well shape with minima at \(V_0 = \pm v\), where \(v = \sqrt{|m_V^2|/\lambda_V}\). The Levi‑Civita point \(V_0 = 0\) 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 \(\lambda_{\text{curv}} > 0\) and \(r_0\) the wormhole throat radius. Near the throat (\(r \sim r_0\)) the mass squared is negative (SSB phase), while far away (\(r \gg r_0\)) it is positive (symmetry restored). This localises the NEC violation at the throat and recovers GR asymptotically.^[1]

Traversable wormhole solution

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 \(\Phi(r)\) is the finite redshift function, \(b(r)\) the shape function with \(b(r_0) = r_0\), and \(H(r)\) a curvature‑dependent compression factor,

\[H(r) = 1 + \lambda(r)\bigl(e^{2\Phi(r)} - 1\bigr),\]

with \(\lambda(r)\) 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 \(\Phi(r) < 0\) (gravitational blueshift) near the throat, \(e^{2\Phi} < 1\) 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.^[1]

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.^[7]

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 \(\rho(r)\) 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 \(g^{-1}_{rr} = e^{2\Phi}(1 - b/r)/H(r)\).^[1]

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 \(r > r_0\)), traversability (proper crossing time \(\Delta\tau < \pi r_0\)), and angular stability. 57% of the configurations yielded fully valid traversable wormholes, with the best crossing time \(\Delta\tau = 0.1035\, r_0/c\), about three times shorter than the GR collapse timescale \(\pi r_0\). The NEC is violated at the throat and restored asymptotically.^[1]

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 \(a_\mu\) generates a quantised flux through the throat,

\[\Phi = \oint_{S^1} a_\mu dx^\mu = 2\pi a_\phi(r_0) = 2\pi n v_a r_0, \quad n \in \mathbb{Z}.\]

The entanglement entropy is proportional to this flux,

\[S_{\text{EE}} \propto \Phi.\]

Thus both the geometric connectivity (wormhole area) and the quantum correlations (entanglement) are controlled by the same integer \(n\).^[1]

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 (\(S_{\text{EE}} = 1\) bit) for all integer winding numbers \(n = 0,\dots,5\), 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”.^[8]

References

1. Pavesi, Luca Eliseo (2026). "Distortion Gravity: A Complete Proof of Ghost‑Free Unitarity with Full Analytical and Numerical Verification". SSRN (Elsevier). doi:10.2139/ssrn.6943658.
2. Pavesi, Luca Eliseo (2026). "Distortion Gravity: A Complete Proof of Ghost‑Free Unitarity with Full Analytical and Numerical Verification". Journal Article (ScienceOpen). doi:10.14293/PR2199.003864.v1.
3. Maldacena, Juan; Susskind, Leonard (2013). "Cool horizons for entangled black holes". Fortschritte der Physik. 61 (9): 781–811. doi:10.1002/prop.201300020.
4. Hehl, F. W.; McCrea, J. D.; Mielke, E. W.; Ne'eman, Y. (1995). "Metric‑affine gauge theory of gravity". Physics Reports. 258 (1–2): 1–171. doi:10.1016/0370-1573(94)00111-5.
5. Pavesi, Luca Eliseo (2026). "Experimental Signatures of Distortion Gravity: From Quantum Simulation to Astrophysical Predictions". Journal Article (ScienceOpen). doi:10.14293/PR2199.003893.v1.
6. Pavesi, Luca Eliseo (2026). "Experimental Verification of Distortion Gravity via Quantum Simulation with Realistic Noise". Journal Article (ScienceOpen). doi:10.14293/PR2199.003896.v1.
7. "Ponte di Einstein–Rosen – Spaziotempo ad anisotropia centrale di Pavesi". Wikipedia in italiano. Retrieved 18 July 2026. Link.
8. Clauser, J. F.; Horne, M. A.; Shimony, A.; Holt, R. A. (1969). "Proposed experiment to test local hidden‑variable theories". Physical Review Letters. 23 (15): 880–884. doi:10.1103/PhysRevLett.23.880.

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