warpax

Observer-robust energy condition verification for warp drive spacetimes.

warpax certifies the energy-condition structure of warp-drive spacetimes for every observer at once, from the eigenstructure of the mixed stress-energy tensor $T^a{}_b$, with exact curvature from JAX forward-mode autodiff. Because the Hawking--Ellis eigenvalue test never builds the Eulerian normal, it stays well-defined at all warp speeds, including superluminal $v_s \ge 1$ where single-frame (Eulerian) tools such as WarpFactory break down. The verdict is observer-independent: a Type-I point is decided exactly for every observer, while a Type-IV point has no rest frame and violates every condition unconditionally.

An Alcubierre warp bubble, straight out of warpax's autodiff curvature pipeline. The wireframe on top is the energy density an Eulerian observer measures — negative everywhere across the wall, ρ_Eul ≤ 0. The flat slab underneath is the NEC margin once you minimize over the whole null sphere; it never rises above zero, which is the exotic-matter problem drawn in a single frame. The sweep first sharpens the wall (σ: 1 → 16), then eases the velocity back down toward flat space. Geometric units, signature (−,+,+,+); and every frame is a frozen metric, a parameter sweep rather than a time evolution.

Highlights

• Frame-independent, all-observer energy-condition certification at every warp speed (including superluminal $v_s \ge 1$), from the eigenstructure of $T^a{}_b$ -- no Eulerian normal, no single-frame blind spots.
• Hawking--Ellis classification (Type I--IV) with explicit Type-IV detection, cross-checked by three eigensolvers and a 50-digit mpmath reference.
• Closed-form Type-I worst observer, with a multistart BFGS optimizer kept as a one-sided diagnostic at the residual non-Type-I points.
• Shift-vorticity control of the wall type: the vorticity of the ADM shift sets the Hawking--Ellis type, with universal $v_s$ scaling laws for the wall NEC deficit and curvature.
• Rigorous geodesic-integrated ANEC via a symplectic null integrator (with an on-cone witness), plus a Ford--Roman quantum-inequality diagnostic.
• Bondi four-momentum radiated-flux and Newman--Penrose peeling at null infinity (warpax.bondi).
• Exact curvature via forward-mode JAX autodiff -- no finite-difference stencils.
• Ten warp/shell metrics, constraint residuals, anisotropic TOV, ADM mass with falloff, Israel junctions, transport diagnostics, and source-first S-/T-shell construction with a five-criterion admissibility standard.

Two papers, one toolkit

warpax backs two separate papers with disjoint claims. If you cite a result, cite the paper it belongs to:

	Certification paper (arXiv:2602.18023)	Companion note (arXiv:2605.25417)
Question	Which observers see energy-condition violations, at which warp speeds?	Can source-first shells satisfy the energy conditions at all?
Results	Frame-free all-velocity certifier; velocity-resolved type map; shift-vorticity -> type control ($f=\kappa\omega$); closed-form worst observer; exoticity ranking + $v_s$ scaling laws	S-/T-shell constructions from the Einstein constraints; five-criterion admissibility standard; boundary-cost analysis
Modules	energy_conditions, geometry, averaged, quantum, analysis, geodesics, transport; metrics Alcubierre / Natário / Van den Broeck / Rodal / Lentz / WarpShell / Garattini	constraints (S-/T-shell solvers), tov, adm, junction, design, optimization; metrics/sshell.py, metrics/tshell.py
Examples	01–07	08–10

The S-/T-shells are constructed and certified in the companion note, not in the certification paper; neither paper's results depend on the other's.

Quick start

# Create environment and install
conda create -n warpax python=3.12 -y && conda activate warpax
pip install -e ".[dev,viz,design,solver]"

# Run a quick example
python examples/01_minkowski_sanity.py

See examples/README.md for a numbered learning path (01-10) and which optional extras each script needs.

For a 5--10 minute walkthrough from install to seeing an energy condition violation, see the Quickstart tutorial.

Key results

Frame-independent type map across the luminal transition

On matched, wall-resolved grids, the Rodal irrotational geometry is globally Hawking--Ellis Type I at every speed from $v_s = 0.1$ to $2.5$, while the Alcubierre/Natario/Van den Broeck bubble walls are Type-IV dominated (no rest frame, no invariant energy density) at every speed. The split is controlled by a single geometric quantity, the vorticity of the ADM shift: the irrotational drive is the unique globally Type-I geometry, while a rotational shift drives the wall to Type IV. For Rodal's globally Type-I drive the Eulerian frame does not register ~72% of the wall weak-energy and ~73% of the wall dominant-energy violations seen by boosted observers, an exact eigenvalue statement rather than an optimizer artifact. A rigorous geodesic-integrated ANEC (symplectic integrator with an on-cone witness) and a Ford--Roman comparison preserve the ordering: every drive violates, and the irrotational Rodal geometry is the mildest by one to two orders of magnitude.

Invariant exoticity ranking and scaling laws

A boost-invariant ranking (NEC severity, Type-IV fraction, rigorous ANEC minimum) places the irrotational Rodal drive about a factor of seventy below the bubble-wall drives, driven by its vanishing Type-IV fraction and tiny averaged-null energy, not by a milder pointwise NEC. Two universal $v_s$ laws follow: the wall NEC deficit $\min(\rho+p_i) = -C,v_s^2$ makes the Santiago--Schuster--Visser no-go quantitative (measured, not asserted), and the wall curvature splits by the same vorticity that sets the type: vortical walls grow as $v_s^2$, the irrotational Rodal wall as $v_s^4$ ($R^2 \ge 0.996$).

Observer-robust vs Eulerian

Across six warp drives, 15--28% of DEC-violating grid points are invisible to the Eulerian observer. The Fuchs constant-velocity shell hides 92% of its shell-interior violations from an Eulerian-only check.

Custom metrics

Subclass ADMMetric and run the full pipeline. The figure below validates a Gaussian warp bubble on a 24x24x4 grid: SEC margins from the Eulerian observer (left), from the worst-case boosted observer found by BFGS (center), and 356 grid points the Eulerian frame reports as SEC-satisfied while the boosted observer sees them violated (right).

SEC comparison for a custom Gaussian warp bubble (v_s = 0.5). Red marks violations the Eulerian frame misses.

See examples/07_custom_warp_metric.py.

Shell admissibility

warpax ships a five-criterion admissibility standard for warp shells:

Criterion	Checks
A. Regularity	$C^2$ metric continuity (thick) or Israel conditions (thin)
B. Constraints	Hamiltonian + momentum residuals $\epsilon_{\mathcal{H}}$, $\epsilon_{\mathcal{M}}$
C. Matter model	Identifiable source (anisotropic fluid, elastic shell)
D. EC margins	Frame-free NEC/WEC/DEC from Hawking--Ellis eigenvalue slacks (exact, cap-free at Type-I; valid at all $v_s$)
E. Global	Positive ADM mass, asymptotic falloff, tidal forces, invariant transport

Fuchs constant-velocity shell: source-aware $\epsilon_{\mathcal{H}} \approx 3\times10^{-8}$; the bulk shell interior is Type-I and EC-compliant (0 of 13 probes violate), while the smoothing tail turns Type-IV. The source-first S-/ T-shells likewise pass criteria A--C and E with positive interior margins; the binding cost is a cap-free Type-I dominant-energy deficit at the inner shell edge ($\approx -4.4\times10^{-4}$), localized at the smooth source--vacuum transition, and the tilted T-shell's shift vorticity drives a Type-IV onset at its low-density edge. These shell results belong to the companion note; see The boundary cost of source consistency.

Examples

See examples/README.md for runtime estimates, install extras, and a suggested order for new users.

Script	Description
01_minkowski_sanity.py	Flat-space sanity check (all ECs satisfied)
02_schwarzschild_verification.py	Schwarzschild ground-truth validation
03_alcubierre_analysis.py	Alcubierre warp drive EC analysis (quickstart entry)
04_warp_drive_comparison.py	Multi-metric comparison (six warp drives)
05_grid_analysis.py	Grid-based EC verification + comparison figure
06_geodesic_through_warp_bubble.py	Geodesic integration with tidal forces
07_custom_warp_metric.py	Custom warp manifold + robust EC validation
08_metric_design.py	Shape-function metric design (B-spline reproduction)
09_admissibility_diagnostics.py	Admissibility diagnostics on the Fuchs warp shell
10_phase_diagram.py	Parameter-space sweep and EC-admissible transport phase diagram

python examples/01_minkowski_sanity.py
python examples/10_phase_diagram.py          # 8x6 demo (~5 min)
python examples/10_phase_diagram.py --full   # 20x15 sweep (~30 min GPU)

Architecture

metrics -> geometry -> energy_conditions -> analysis
              |              |
          geodesics    classification (Hawking--Ellis)
              |
         transport / tidal / blueshift

Package	Description
geometry	JAX autodiff pipeline: metric $\to$ Christoffel $\to$ Riemann $\to$ Ricci $\to$ Einstein $\to$ $T_{\mu\nu}$; ADM 3+1 split; $C^2$ regularity diagnostics
energy_conditions	NEC/WEC/SEC/DEC via Hawking--Ellis classification, eigenvalue algebra, multi-start BFGS observer optimization
metrics	Nine warp/shell metrics: Natario, Lentz, Rodal, Van den Broeck, WarpShell, Fuchs, S-shell, T-shell, Garattini--Zatrimaylov (Alcubierre, Minkowski, and Schwarzschild ship in benchmarks, making ten warp metrics total)
constraints	Hamiltonian + momentum constraint residuals; S-shell and T-shell constraint solvers (pure JAX)
tov	Anisotropic TOV equilibrium checker
adm	ADM mass with surface integral and asymptotic falloff verification
junction	Israel/Darmois junction conditions and surface stress-energy
transport	Invariant diagnostics: geodesic deviation, null coordinate-time asymmetry, blueshift hazard
optimization	Bernstein basis, multi-objective loss, EC soft/hard constraints, parameter sweep
geodesics	Timelike/null geodesic integration via Diffrax, tidal deviation, blueshift extraction
design	Differentiable shape-function parametrization with constrained BFGS optimizer
analysis	Eulerian vs. robust comparison, Richardson convergence, kinematic scalars
io	External metric loaders: WarpFactory (.mat), EinFields (checkpoint), Cactus (HDF5)
visualization	Matplotlib publication figures, Manim animations, phase diagram plots
classify	Bobrick--Martire subluminal/superluminal taxonomy
averaged	ANEC/AWEC null-ray and geodesic line integrals
quantum	Ford--Roman quantum inequality evaluator

All metrics implement a common MetricFunction interface: a callable (4,) -> (4,4) mapping coordinates $x^\mu$ to the covariant metric tensor $g_{\mu\nu}$.

Running tests

pytest                      # Full suite (1000+ tests across 33 modules)
pytest -m "not slow"        # Skip @slow grid tests (~50 s with -n auto)
pytest -m smoke             # Visualization import / render smoke tests
pytest -n auto              # Parallel execution

Reproducing results

To pin the exact Python environment used to produce the published results:

export PYTHON=$(uv run which python)
bash reproduce_all.sh

Stages can be run individually:

bash reproduce_all.sh --stage core      # Core computation
bash reproduce_all.sh --stage ablation  # Ablation studies
bash reproduce_all.sh --stage figures   # Figure generation

Use --keep-cache to skip cache deletion and only recompute missing results.

For the per-figure, per-claim mapping that backs the warp-shell admissibility paper (On the boundary cost of source-consistent warp shells), see the dedicated how-to guide: Reproducing the warp-shell admissibility paper.

The outer-edge ($r \ge R_2$) Type-IV gate (log-log slope $1.01 \pm 0.01$) and the ANEC impact-parameter scan are reproduced by scripts/run_tshell_typeIV_gate.py and scripts/run_anec_impact_scan.py.

Documentation

warpax ships full documentation in docs/, organized following the Diataxis framework:

Tutorials

• Quickstart -- 5--10 minutes from install to seeing an energy condition violation
• First curvature computation -- full curvature chain on Minkowski as a warm-up

How-to guides

• Define a custom warp metric -- subclass ADMMetric and run the verification pipeline
• Interpret EC results -- read margin signs, Hawking--Ellis types, and worst-case observers
• Load an external metric -- use WarpFactory, EinFields, or Cactus data
• Reproduce the warp-shell admissibility paper -- per-figure, per-claim mapping to scripts and outputs

Reference

• API reference -- autodoc of the public API
• Metric catalog -- all ten shipped metrics
• Benchmarks -- asv regression harness

Explanation

• Architecture -- package structure and design decisions
• Theory: ADM 3+1 and Hawking--Ellis types -- mathematical background
• Release notes -- version history and release summary

Manim visualizations

Every scene below comes straight from the same autodiff curvature and energy-condition code that backs the papers. Everything is in geometric units (G = c = 1, signature −,+,+,+) on the z = 0 slice, and each frame is a frozen metric - a parameter sweep, not a time evolution.

Eulerian kinematics. Expansion θ = −K - space stretches behind the ship (red) and squeezes in front of it (blue), the kinematics Alcubierre is famous for. Shear σ² rides along as iso-contours, with the f = 0.5 wall traced on top.	Kretschmann invariant. The curvature scalar K = RabcdRabcd - the same number for every observer. It is sign-indefinite in Lorentzian signature, so it dips negative, and it spikes hard on the wall.	Observer-robust NEC. Six axis-aligned Eulerian nulls on the left, the dense worst case over the whole null sphere on the right (k·nEul = −1). The gap between the panels is exactly what observer-robust verification buys you.

The full scene set spans 3D embedding diagrams and 2D heatmaps:

• WallAndVelocitySweep / VelocitySweep - dual-layer 3D: Eulerian energy density ρ_Eul embedding above a flat NEC-margin slab, sweeping wall steepness σ then velocity v_s.
• BoostRapiditySweep - the energy density a boosted observer measures vs rapidity ζ; it deepens as cosh²ζ and diverges to −∞ (the worst case over unbounded boosts is exactly the NEC).
• EulerianKinematics2D - expansion θ = −K, shear σ², vorticity ω² ≡ 0 (hypersurface-orthogonal).
• KretschmannInvariant2D - the observer-independent curvature invariant.
• NECMargin2D / EulerianVsWorstCaseNEC - observer-robust null-energy-condition margins.
• WorstCaseNullDirections / WorstCaseBoostDirections - worst-case null / timelike-boost direction fields.

# System dependencies (Ubuntu/Debian)
sudo apt install texlive-latex-extra texlive-fonts-recommended dvipng cm-super ffmpeg

# Python dependencies (Python <= 3.13 recommended for the renderer)
pip install -e ".[manim]"

# Render all scenes (2D via Cairo, 3D via the GPU OpenGL renderer)
python scripts/render_all_scenes.py

Rendered videos and images are written to media/ (not tracked by git). The 3D scenes render through manim's OpenGL renderer (EGL, headless).

Citation

If you found this work useful, please consider citing:

@article{le2026observer,
  title={Observer-robust energy condition verification for warp drive spacetimes},
  author={Le, An T},
  journal={arXiv preprint arXiv:2602.18023},
  year={2026}
}

@article{le2026boundary,
  title={On the boundary cost of source-consistent warp shells},
  author={Le, An T},
  journal={arXiv preprint arXiv:2605.25417},
  year={2026}
}