Static Optical Shortcuts in Quadratic Beyond-Horndeski Gravity A Numerical Reconstruction Study of Stability, Subluminality, and Their Obstructions Working preprint — September 2026 Abstract We investigate whether a localized, static, spherically symmetric geometry can produce a formal optical travel-time advance relative to a flat reference path while satisfying increasingly complete perturbative stability conditions in quadratic beyond-Horndeski gravity. The calculation is an inverse/reconstruction study, not a claim of a physical warp drive or a completed covariant solution. Starting from a smooth metric ansatz, we numerically construct a branch with formal travel-advantage factor [ \mathcal A_{\rm tr}\approx1.00978. ] Along this branch we impose an exact background relation used in the beyond-Horndeski stability formalism, obtain positive odd-sector kinetic coefficients, satisfy the odd-sector tachyon bound in the sampled domain, construct a positive even-sector no-ghost margin, choose the second radial characteristic to be subluminal, and solve the high-angular-momentum determinant condition as a boundary-value problem. The resulting high-(\ell) angular modes are real. However, evaluating the two even angular characteristic speeds shows that the larger mode becomes superluminal at multipole (\ell=20) for the reconstructed profile. Moreover, a direct implementation of the finite-(\ell) sufficient matrix conditions fails for the covariant splits tested. A local Taylor-jet embedding of the reconstructed combinations into (G_4(\pi,X)), (F_4(\pi,X)), and the scalar sector exists numerically, but requires large higher derivatives in the adopted normalization and does not yet satisfy all background field equations. We identify a useful tension: for the printed high-(\ell) characteristic equations, a fixed strictly positive angular determinant margin produces a product of squared angular speeds scaling as (\ell^4), so strict stability and all-multipole subluminality cannot coexist on such a fixed profile without a finite validity cutoff or further structural change. This result is model- and reconstruction-specific and is not presented as a general no-go theorem.
Introduction Localized spacetime shortcuts are strongly constrained in general relativity. Generic warp-field geometries have been shown to violate standard energy conditions, while gravitational time-delay theorems connect null focusing assumptions to the absence of localized fastest null paths. These results motivate asking a narrower modified-gravity question: Can the effective curvature sector provide the required null defocusing while the propagating degrees of freedom remain ghost-free, gradient-stable, tachyon-free, and subluminal? Quadratic beyond-Horndeski gravity provides a useful laboratory because complete linear stability conditions for arbitrary static, spherically symmetric backgrounds have recently been derived. Mironov and Volkova give sufficient conditions excluding ghosts, radial and angular gradient instabilities, tachyons, and superluminal modes in both parity sectors. Earlier reverse-engineering work demonstrated that the functional freedom of beyond-Horndeski theories can be used to construct backgrounds satisfying substantial subsets of these constraints. The present work follows a deliberately adversarial reconstruction strategy. At each stage we construct a candidate shortcut geometry, apply the next exact stability gate, and either reconstruct the remaining free functions or record the obstruction. Negative results are treated as part of the result. In particular, several apparently promising intermediate constructions fail when angular characteristics or finite-multipole conditions are imposed.
Scope and interpretation We study a static spherical line element [ ds2=-A®,dt2+\frac{dr2}{B®}+J®2d\Omega^2, ] with [ J®=r ] in the numerical reconstruction. A radial null ray obeys [ dt=\frac{dr}{\sqrt{A®B®}}. ] We define the formal optical travel-advantage factor over a finite interval ([r_0,r_1]) by [ \mathcal A_{\rm tr} \frac{r_1-r_0} {\displaystyle\int_{r_0}^{r_1}\frac{dr}{\sqrt{A®B®}}}. ] Thus [ \mathcal A_{\rm tr}>1 ] denotes a coordinate optical time smaller than that of the chosen flat reference interval. This quantity by itself does not demonstrate globally superluminal travel, a realizable propulsion device, or a causal shortcut between asymptotic observers. Those stronger statements require a complete global solution, appropriate boundary conditions, causal analysis, and a physically valid covariant theory.
Beyond-Horndeski framework We use the quadratic beyond-Horndeski action and notation of Mironov and Volkova, with scalar [ \pi=\pi® ] and [ X=-\frac12B\pi’^2. ] Their perturbation analysis introduces background combinations [ \mathcal F,\quad\mathcal G,\quad\mathcal H ] and even-sector quantities including [ \mathcal P_1,\quad \xi,\quad \Xi,\quad \Gamma,\quad \Sigma,\quad \mathcal P_4,\quad \mathcal N, ] together with kinetic, gradient, and mass matrices. The odd-sector radial and angular squared characteristic speeds are [ c_{r,\rm odd}^2=\frac{\mathcal G}{\mathcal F}, ] and [ c_{a,\rm odd}^2=\frac{\mathcal G}{\mathcal H}. ] The complete stability formalism additionally supplies sufficient finite-(\ell) conditions for the even sector. For the principal numerical branch we adopt [ \mathcal H=\mathcal G=1 ] as a reconstruction gauge and determine (\mathcal F) from the exact background combination used in the reverse-engineering program. This is a restricted branch, not the most general theory.
Numerical reconstruction 4.1 Smooth shortcut direction and nonlinear continuation A linear program over smooth basis functions was first used to find localized perturbations [ A=1+2\epsilon a®, \qquad B=1+\epsilon b® ] that increase (\mathcal A_{\rm tr}) while maintaining a positive first-order margin in (\mathcal F). The survivor was then continued nonlinearly using [ A=e^{2\epsilon a}, \qquad B=e^{\epsilon b}. ] Introducing a first-order safety margin [ \delta\mathcal F\ge0.05 ] allowed a finite exact branch with [ \mathcal F\ge1. ] At [ \epsilon=0.019, ] the branch has [ \mathcal A_{\rm tr}=1.00977772 ] and [ 1.0000208 \lesssim\mathcal F \lesssim1.0217454. ] Consequently, [ c_{r,\rm odd}^2=\frac1{\mathcal F} ] lies approximately within [ 0.978717 \lesssim c_{r,\rm odd}^2 \lesssim 0.999979. ]
4.2 Even no-ghost and radial reconstruction We impose a positive even no-ghost margin [ 2\mathcal P_1-\mathcal F=\mu, ] with [ \mu=0.1. ] Since [ \mathcal P_1 \sqrt{\frac BA} \frac{d}{dr} \left[ \sqrt{\frac AB},\xi \right], ] this equation can be integrated directly to reconstruct (\xi®). On the branch [ \Xi=0,\qquad \mathcal H=\mathcal G=1,\qquad J=r,\qquad \pi’=1, ] the second radial characteristic can then be targeted explicitly. We choose [ c_{r,2}^2=0.8. ] The corresponding numerator (\mathcal N) and reconstruction quantity (\Sigma) are then determined from the published radial characteristic relation. Across the sampled interval, both radial characteristic gates pass by construction.
4.3 High-(\ell) angular determinant as a boundary-value problem The high-(\ell) angular determinant condition was initially difficult to satisfy using finite-dimensional trial profiles. The crucial change was to treat the published inequality as a first-order differential condition for (\mathcal P_4®). In our normalization, the flat GR-like endpoint value is [ \mathcal P_4=+2. ] We solve the differential equality with a strict positive stability margin (\eta), subject to [ \mathcal P_4(r_0)=2, ] and [ \mathcal P_4(r_1)=2. ] The boundary-matched solution gives [ \eta\simeq0.0223961. ] The reconstructed profile remains approximately within [ 1.994 \lesssim \mathcal P_4® \lesssim 2.141. ] The resulting high-(\ell) angular determinant margin remains positive throughout the sampled domain. Thus the determinant stability gate itself can be passed.
Results For the principal reconstructed survivor: Test Numerical result Status Formal optical advantage (\mathcal A_{\rm tr}\approx1.0097777) Pass Background (\mathcal F) gate (\mathcal F_{\min}\approx1.0000208) Pass Odd radial speed (0.9787\lesssim c^2<1) Pass Odd angular speed (c^2=1) Pass Odd tachyon bound minimum sampled margin (\approx4.23\times10^{-5}) Pass Even no-ghost (2\mathcal P_1-\mathcal F=0.1) Pass by reconstruction Second radial mode (c_{r,2}^2=0.8) Pass by reconstruction High-(\ell) angular determinant positive margin (\approx0.0214-0.0229) Pass Individual even angular speeds larger mode exceeds 1 at (\ell=20) Fail Finite-(\ell) sufficient block fails for tested reconstructions Fail Local covariant Taylor jet finite but large higher derivatives Partial Full on-shell covariant solution not constructed Open
5.1 Odd-sector completion Using the complete odd-sector potential, the sampled profile satisfies [ V®>-\frac6{r^2} ] everywhere. Numerically, [ -1.50\times10^{-4} \lesssim V® \lesssim 1.88\times10^{-4}, ] while [ V®+\frac6{r^2} ] has a minimum sampled margin of approximately [ 4.23\times10^{-5}. ] Thus the odd sector passes the sampled ghost, radial-gradient, angular-gradient, subluminality, and quoted tachyon conditions.
5.2 Angular characteristic obstruction Passing the high-(\ell) determinant does not guarantee that the two individual even angular characteristic speeds remain subluminal. Evaluating their published characteristic sum and product gives the following maximum values for the larger squared angular mode: [ \ell=10: \qquad c_{a,\max}^2\simeq0.2504, ] [ \ell=15: \qquad c_{a,\max}^2\simeq0.5634, ] [ \ell=19: \qquad c_{a,\max}^2\simeq0.9040, ] while [ \ell=20: \qquad c_{a,\max}^2\simeq1.00167. ] Thus the first sampled superluminal crossing occurs at [ \ell=20. ] At larger multipoles the effect rapidly increases: [ \ell=25: \qquad c_{a,\max}^2\simeq1.565, ] [ \ell=50: \qquad c_{a,\max}^2\simeq6.26, ] and [ \ell=100: \qquad c_{a,\max}^2\simeq25.0. ] The modes remain real; the failure is subluminality rather than loss of hyperbolicity in this particular test.
5.3 The multipole scaling The characteristic equations reveal the origin of this behavior. Let [ \Delta®>0 ] denote the strict high-(\ell) angular determinant margin. The product of the two even angular squared speeds can be written [ c_{a1}2c_{a2}2 \ell^4 \frac{ A\xi^2\Delta }{ J^6(2\mathcal P_1-\mathcal F) }. ] For a fixed background, everything multiplying (\ell^4) is independent of (\ell). If both squared speeds are required to satisfy [ 0<c_{a1}^2\le1, \qquad 0<c_{a2}^2\le1, ] then necessarily [ c_{a1}2c_{a2}2\le1. ] Consequently, [ \ell \le \left[ \frac{ J^6(2\mathcal P_1-\mathcal F) }{ A\xi^2\Delta } \right]^{1/4}. ] For the present reconstructed profile, the product alone yields a limiting scale near [ \ell\simeq41.6. ] The corresponding condition on the sum, [ c_{a1}2+c_{a2}2\le2, ] gives a stronger necessary ceiling around [ \ell\simeq27.5. ] The exact larger eigenvalue becomes superluminal earlier still: [ \ell=20. ] This produces a notable tension. For a fixed profile with [ \Delta>0, ] the characteristic product grows as [ \ell^4. ] Therefore strict angular determinant stability together with positive, subluminal angular modes for arbitrarily large (\ell) cannot persist on this fixed profile unless the coefficient of that scaling vanishes or the physical theory ceases to be applicable beyond a finite multipole/momentum scale. This is a derived property of the characteristic equations applied to the present reconstruction. It is not asserted as a general beyond-Horndeski no-go theorem. A finite effective-field-theory cutoff could materially alter its physical interpretation.
5.4 Finite-multipole sufficient conditions The complete even-sector analysis contains stronger finite-(\ell) conditions involving the matrices [ \mathcal G,\qquad \mathcal M, ] and the antisymmetric mixing matrix (Q). The sufficient stability conditions include [ \mathcal G_{11}>0, ] [ \det\mathcal G>0, ] [ \mathcal M_{22}>0, ] and [ \det\mathcal G\det\mathcal M
\frac{Q_{12}^2}{4} \mathcal M_{22}\mathcal G_{11}. ] We implemented the corresponding expressions for the reconstructed background. An initial explicit choice, [ \Gamma_1=\Gamma, \qquad \Gamma_2=0, ] fails for [ \ell=2,3,4,5,10,20. ] For example, at (\ell=2), [ \mathcal M_{22,\min} \simeq -1.89\times10^{-3}, ] and the full determinant inequality becomes negative. The same qualitative failure occurs at the other tested multipoles.
5.5 Exploring reconstruction freedom The combination entering earlier reconstruction stages obeys [ \Gamma \Gamma_1+\frac{A’}A\Gamma_2. ] This leaves freedom to change (\Gamma_1) and (\Gamma_2) while keeping (\Gamma) fixed. We therefore parameterized [ \Gamma_1 \Gamma+\frac{A’}A u®, ] and [ \Gamma_2=-u®, ] where (u®) was expanded in six smooth modes. The coefficients were optimized simultaneously against the finite-(\ell) conditions for [ \ell=2,3,5,10. ] The optimization improved individual conditions over significant portions of the radial domain. For (\ell=2), for example, (\mathcal M_{22}) became positive over approximately (58%) of the interior. However, the complete sufficient determinant condition continued to fail over much of the interval. Thus the finite-(\ell) obstruction is not removed by this simple use of the (\Gamma_1/\Gamma_2) reconstruction freedom. This does not exclude more general covariant completions.
5.6 Local covariant reconstruction The preceding calculations manipulate combinations of covariant functions rather than directly specifying one global Lagrangian. To determine whether the reconstructed profiles can at least arise locally from common functions, we expand around the background trajectory [ X=X_b(\pi) ] and define [ y=X-X_b(\pi). ] We use local Taylor expansions [ G_4(\pi,X) g_0(\pi) + g_1(\pi)y + \frac12g_2(\pi)y^2 +\cdots, ] and [ F_4(\pi,X) h_0(\pi) + h_1(\pi)y +\cdots. ] The exact definitions of (\Gamma_1) and (\Gamma_2) can then be solved pointwise for the required values of [ G_4XX} ] and [ F_{4X}. ] The numerical residuals are approximately [ \simeq 1.9\times10^-12}, ] and [ \simeq 5.8\times10^{-11}. ] Thus the specified combinations admit a local numerical Taylor-jet embedding. However, in the adopted normalization the required derivatives become large: [ -1.07\times10^5 \lesssim G_{4XX} \lesssim 1.68\times10^4, ] [ -1.71\times10^4 \lesssim F_{4X} \lesssim 1.09\times10^5, ] and under one simple extension gauge the scalar-sector derivative is approximately [ -2.11\times10^5 \lesssim F_X \lesssim 3.46\times10^4. ] These numbers are normalization- and dimension-dependent and therefore should not be interpreted directly as physical coupling strengths. Nevertheless, their magnitude warns that strong coupling and effective-field-theory validity require explicit examination. Most importantly, a local Taylor jet is not an on-shell global covariant theory.
Discussion The reconstruction exhibits a recurring pattern. In general relativity, attempts to obtain localized optical advances encounter null-energy and null-convergence obstructions. In the present modified-gravity construction, substantial functional freedom allows several of those problems to be moved out of the radial sector. Both radial characteristics can remain stable and subluminal, the odd sector can satisfy its sampled complete conditions, and even the strict high-(\ell) angular determinant can be made positive. The obstruction then reappears in the individual angular characteristic cones. Once those modes are required to be not merely real but subluminal, the reconstructed branch fails at [ \ell=20. ] Separately, the stronger finite-(\ell) sufficient stability matrix remains problematic. This suggests that modified gravity does not trivially eliminate the cost associated with a localized optical advance. Instead, that cost may migrate among the background geometry, additional degrees of freedom, characteristic cones, strong coupling, and the regime of validity of the effective theory. The calculation therefore does not establish a physical warp-drive solution. The geometry is static and spherical. The optical advantage is defined over a finite coordinate interval. A single explicit global covariant Lagrangian satisfying all background equations has not been constructed. The global causal structure has not been established. The finite-(\ell) sufficient conditions fail for the tested reconstructions. And the even angular sector becomes superluminal above a finite multipole in the principal branch. The useful result is instead a sequence of reproducible gates culminating in a sharply identified obstruction.
Reproducibility and numerical limitations All numerical results reported here come from one-dimensional radial discretizations and inverse/reconstruction calculations. Several important verification steps remain before the work should be treated as a formal physics result. First, all published stability equations should be independently transcribed and implemented by a second calculation. Second, the reported margins and characteristic speeds require systematic grid-refinement and preferably spectral-convergence studies. Third, dimensional conventions and field normalizations must be restored explicitly before interpreting reconstructed coupling magnitudes. Fourth, the complete background equations [ E_A=0, \qquad E_B=0, \qquad E_J=0, \qquad E_\pi=0 ] must be solved for one explicit covariant Lagrangian. Fifth, the perturbation analysis should then be rerun directly from that Lagrangian rather than from independently reconstructed background combinations. Finally, any interpretation as a physical shortcut requires global boundary conditions and causal analysis beyond the finite radial optical functional used here.
Conclusion We have constructed a static spherical inverse-geometry branch with a formal optical travel advantage of approximately [ 0.98%. ] The branch can be pushed through progressively stronger quadratic beyond-Horndeski stability tests. It passes, on the sampled domain: [ \text{the background }\mathcal F\text{ gate}, ] the complete tested odd sector, [ \text{the even no-ghost condition}, ] two radial subluminality conditions, and [ \text{a strict high-}\ell\text{ angular determinant condition}. ] Nevertheless, the larger even angular characteristic becomes superluminal at [ \ell=20 ] for the principal reconstructed profile. The complete finite-(\ell) sufficient matrix criterion also fails for the covariant splits tested. A local covariant Taylor jet can reproduce several of the required reconstruction combinations, but this does not constitute a complete on-shell theory. The principal technical observation is the scaling [ c_{a1}2c_{a2}2 \propto \ell^4\Delta, ] where (\Delta>0) measures the strict angular determinant margin. For a fixed reconstructed profile, this creates a direct tension between a nonzero stability margin and subluminality at arbitrarily high multipole. Whether this tension persists in fully on-shell solutions, alternative beyond-Horndeski branches, broader DHOST theories, or effective theories with explicit physical cutoffs is the natural next problem. The calculationtherefore does not provide a completed spacetime-shortcut theory. It does something more limited and testable: it identifies where a promising reconstructed shortcut survives, where it fails, and which mathematical degrees of freedom must be changed next. References S. Mironov and V. Volkova, Complete stability for spherically symmetric backgrounds in beyond Horndeski theory, arXiv:2404.06297 [gr-qc]. arXiv paper� S. Mironov, V. Rubakov, and V. Volkova, In hot pursuit of a stable wormhole in beyond Horndeski theory, Phys. Rev. D 107, 104061 (2023), arXiv:2212.05969. arXiv paper� J. Santiago, S. Schuster, and M. Visser, Generic warp drives violate the null energy condition, Phys. Rev. D 105, 064038 (2022), arXiv:2105.03079. arXiv paper� S. Gao and R. M. Wald, Theorems on gravitational time delay and related issues, Class. Quantum Grav. 17, 4999 (2000), arXiv:gr-qc/0007021. arXiv paper� Appendix A — Principal numerical survivor The numerical reconstruction discussed above uses and produces The principal diagnostics are and a high-� determinant margin of approximately The boundary-matched reconstruction satisfies with approximately The first sampled even-angular superluminal crossing occurs at where Appendix B — Computational artifacts The numerical work was developed through separate calculations for nonlinear continuation, radial reconstruction, angular boundary matching, the odd-sector tachyon gate, the even angular characteristic system, finite-� matrix tests, reconstruction-freedom searches, and local covariant Taylor reconstruction. The current research scripts are: Boundary-matched angular reconstruction Odd-sector full tachyon gate Even angular characteristic speeds Angular multipole ceiling Finite-� full matrix gate Finite-� reconstruction-freedom search Local covariant Taylor-jet reconstruction submitted by /u/-null_entry-
Originally posted by u/-null_entry- on r/ArtificialInteligence
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