#!/usr/bin/env python3 # SPDX-License-Identifier: MIT """Deterministic AdS3--Vaidya checks for the Volume 24 quench page. The calculation is intentionally bounded. It integrates the outgoing-null event horizon of a smooth planar AdS3--Vaidya shell, verifies regulated vacuum and BTZ geodesic lengths, and checks the analytic thin-shell crossing and causal-saturation conditions. It does not claim to solve a time-dependent extremal-surface boundary-value problem. """ from __future__ import annotations import argparse import json import math import platform import sys from pathlib import Path import numpy as np import scipy from scipy.integrate import solve_ivp from scipy.optimize import brentq REFERENCE_ENVIRONMENT = { "python": "3.11.4", "numpy": "2.0.2", "scipy": "1.13.1", "floating_point": "IEEE-754 binary64", } DEFAULT_OUTPUT = Path( "public/data/string-holographic-black-holes/" "holographic-quench-benchmark.json" ) def shell_mass(v: float | np.ndarray, width: float) -> float | np.ndarray: """Dimensionless final mass is one and the shell is centered at v=0.""" return 0.5 * (1.0 + np.tanh(np.asarray(v) / width)) def horizon_rhs(v: float, z: np.ndarray, width: float) -> np.ndarray: """Outgoing radial null curve for ds^2=z^-2[-f dv^2-2 dv dz+dx^2].""" f_value = 1.0 - float(shell_mass(v, width)) * z[0] ** 2 return np.array([-0.5 * f_value], dtype=float) def integrate_event_horizon( width: float, rtol: float, atol: float, max_step: float, terminal_z: float = 1.0, ) -> solve_ivp: """Integrate backward from the final planar-BTZ horizon z_h=1.""" solution = solve_ivp( lambda v, z: horizon_rhs(v, z, width), (4.0, -2.0), np.array([terminal_z], dtype=float), method="DOP853", rtol=rtol, atol=atol, max_step=max_step, dense_output=True, ) if not solution.success or solution.sol is None: raise RuntimeError(f"event-horizon integration failed: {solution.message}") return solution def max_integral_defect(solution: solve_ivp, width: float) -> float: """Trapezoidal integral defect on accepted ODE steps.""" v = solution.t z = solution.y[0] rhs = np.array([horizon_rhs(v_i, np.array([z_i]), width)[0] for v_i, z_i in zip(v, z)]) step = np.diff(v) defect = np.diff(z) - 0.5 * step * (rhs[:-1] + rhs[1:]) return float(np.max(np.abs(defect))) def regulated_geodesic_lengths(ell: float, cutoff: float) -> tuple[float, float, float]: """Vacuum and unit-temperature-scale BTZ lengths at finite z=cutoff.""" vacuum = 2.0 * math.asinh(ell / (2.0 * cutoff)) + 2.0 * math.log(cutoff) thermal = 2.0 * math.asinh(math.sinh(ell / 2.0) / cutoff) + 2.0 * math.log(cutoff) return vacuum, thermal, thermal - vacuum def environment_record() -> dict[str, object]: actual = { "python": platform.python_version(), "numpy": np.__version__, "scipy": scipy.__version__, "floating_point": "IEEE-754 binary64" if np.finfo(float).bits == 64 else "unexpected", } return { "reference": REFERENCE_ENVIRONMENT, "actual": actual, "matches_reference": actual == REFERENCE_ENVIRONMENT, } def compute() -> dict[str, object]: width = 0.15 ell = 1.2 exact_delta = 2.0 * math.log(2.0 * math.sinh(ell / 2.0) / ell) refinements = [ ("coarse", 1.0e-7, 1.0e-10, 0.04, 2.0e-2), ("medium", 1.0e-9, 1.0e-12, 0.02, 1.0e-2), ("fine", 1.0e-11, 1.0e-14, 0.01, 5.0e-3), ] reference = integrate_event_horizon(width, 5.0e-13, 5.0e-15, 0.002) reference_z0 = float(reference.sol(0.0)[0]) rows: list[dict[str, object]] = [] for label, rtol, atol, max_step, cutoff in refinements: solution = integrate_event_horizon(width, rtol, atol, max_step) vacuum, thermal, delta = regulated_geodesic_lengths(ell, cutoff) z0 = float(solution.sol(0.0)[0]) rows.append( { "label": label, "rtol": rtol, "atol": atol, "max_step": max_step, "radial_cutoff": cutoff, "event_horizon_z_at_shell_center": z0, "event_horizon_change_from_reference": abs(z0 - reference_z0), "max_step_integral_defect": max_integral_defect(solution, width), "vacuum_geodesic_length_renormalized": vacuum, "thermal_geodesic_length_renormalized": thermal, "thermal_minus_vacuum": delta, "geodesic_delta_error": abs(delta - exact_delta), } ) thin_widths = [0.15, 0.075, 0.0375] thin_shell_rows = [] for thin_width in thin_widths: solution = integrate_event_horizon(thin_width, 1.0e-11, 1.0e-14, 0.005) thin_shell_rows.append( { "shell_width": thin_width, "event_horizon_z_at_shell_center": float(solution.sol(0.0)[0]), "difference_from_thin_shell_value_one": abs(float(solution.sol(0.0)[0]) - 1.0), } ) source_99_time = float(brentq(lambda v: float(shell_mass(v, width)) - 0.99, -1.0, 1.0, xtol=1.0e-14)) crossing_times = [0.2, 0.4, 0.59] crossings = [ { "boundary_time": time, "shell_crossing_depth": time, "positive_half_crossing_x": math.sqrt((ell / 2.0) ** 2 - time**2), } for time in crossing_times ] saturation_time = ell / 2.0 wrong_branch = integrate_event_horizon(width, 1.0e-9, 1.0e-12, 0.02, terminal_z=0.9) wrong_branch_error = abs(float(wrong_branch.sol(4.0)[0]) - 1.0) acceptance = { "fine_horizon_change_below_1e-10": rows[-1]["event_horizon_change_from_reference"] < 1.0e-10, "fine_geodesic_delta_error_below_1e-4": rows[-1]["geodesic_delta_error"] < 1.0e-4, "refinement_reduces_horizon_change": rows[0]["event_horizon_change_from_reference"] > rows[1]["event_horizon_change_from_reference"] > rows[2]["event_horizon_change_from_reference"], "refinement_reduces_cutoff_error": rows[0]["geodesic_delta_error"] > rows[1]["geodesic_delta_error"] > rows[2]["geodesic_delta_error"], "pre_quench_source_below_2e-6": float(shell_mass(-1.0, width)) < 2.0e-6, "thin_shell_crossing_saturates_causal_bound": math.isclose(saturation_time, ell / 2.0, rel_tol=0.0, abs_tol=1.0e-14), "wrong_terminal_branch_detected": wrong_branch_error > 1.0e-2, } return { "schema_version": 1, "model": { "metric": "ds^2=z^-2[-(1-m(v)z^2)dv^2-2dv dz+dx^2]", "shell_profile": "m(v)=(1+tanh(v/w))/2", "boundary_dimension": 2, "final_horizon_radius_z": 1.0, "shell_width_w": width, "boundary_interval_length_ell": ell, "initial_state": "Poincare AdS3 vacuum", "final_state": "unit-radius planar BTZ black brane", "integration_domain_v": [-2.0, 4.0], "subtraction": "subtract 2 log(1/z_cutoff) from each regulated geodesic length", }, "environment": environment_record(), "solvers": { "ode": "scipy.integrate.solve_ivp, DOP853, backward from z(4)=1", "root": "scipy.optimize.brentq, xtol=1e-14", "reference_ode": {"rtol": 5.0e-13, "atol": 5.0e-15, "max_step": 0.002}, }, "refinements": rows, "thin_shell_limit": { "rows": thin_shell_rows, "analytic_extremal_curve": "vacuum semicircle z^2+x^2=(ell/2)^2 with v=t-z", "shell_crossings": crossings, "saturation_time": saturation_time, }, "diagnostics": { "source_time_at_99_percent_final_energy": source_99_time, "source_mass_at_v_minus_one": float(shell_mass(-1.0, width)), "exact_late_thermal_minus_vacuum_length": exact_delta, "reference_event_horizon_z_at_shell_center": reference_z0, "wrong_terminal_branch_error": wrong_branch_error, "interpretive_boundary": "The event horizon is teleological; it is not a boundary equilibration time. The geodesic check fixes only static limits and the thin-shell crossing bound.", }, "acceptance": acceptance, "passed": all(acceptance.values()), } def main() -> int: parser = argparse.ArgumentParser(description=__doc__) parser.add_argument("--output", type=Path, default=DEFAULT_OUTPUT) parser.add_argument( "--allow-environment-mismatch", action="store_true", help="run for diagnostics while marking the serialized environment mismatch", ) args = parser.parse_args() record = compute() environment_matches = bool(record["environment"]["matches_reference"]) if not environment_matches and not args.allow_environment_mismatch: print("ERROR: runtime does not match the pinned reference environment", file=sys.stderr) print(json.dumps(record["environment"], indent=2, sort_keys=True), file=sys.stderr) return 2 args.output.parent.mkdir(parents=True, exist_ok=True) args.output.write_text(json.dumps(record, indent=2, sort_keys=True) + "\n", encoding="utf-8") print("label rtol cutoff z_EH(v=0) horizon change geodesic error") for row in record["refinements"]: print( f"{row['label']:<7} {row['rtol']:<9.1e} {row['radial_cutoff']:<8.3g} " f"{row['event_horizon_z_at_shell_center']:<15.12f} " f"{row['event_horizon_change_from_reference']:<17.3e} " f"{row['geodesic_delta_error']:.3e}" ) print(f"output: {args.output}") print(f"passed: {record['passed']}") return 0 if record["passed"] else 1 if __name__ == "__main__": raise SystemExit(main())