Godel's Solution to Einstien's GR

# Create a 3D animation (MP4) that visualizes Gödel's rotating universe features: # - Light-cone tipping along rings (as null-direction "wedges") # - The null ring at r = r_c # - A sample time-traveler worldline (helical path) that goes out, loops in the CTC region, and returns # - Camera rotation for spatial intuition # # If ffmpeg is not available, the code will fall back to GIF. # # Outputs: # /mnt/data/godel_3d.mp4 (preferred) # /mnt/data/godel_3d.gif (fallback if MP4 fails) # /mnt/data/godel_3d_still.png (key frame) import numpy as np import math import matplotlib.pyplot as plt from matplotlib import animation from mpl_toolkits.mplot3d import Axes3D # noqa: F401 import os # -------------------- Gödel-type helpers -------------------- def D_of_r(r, m): return np.sinh(m*r)/m def H_of_r(r, m, omega): return (4.0*omega/(m**2)) * (np.sinh(0.5*m*r)**2) def rc_analytic(omega): return float(np.arccosh(3.0) / (math.sqrt(2.0) * omega)) # Null slopes along a ring at radius r (dr = 0): dt/dphi = -H(r) ± D(r) def null_slopes(r, m, omega): H = H_of_r(r, m, omega) D = D_of_r(r, m) return -H + D, -H - D # -------------------- Scene construction -------------------- omega = 1.0 m = math.sqrt(2.0)*omega rc = rc_analytic(omega) # Time range for drawing (arbitrary units) t_min, t_max = -4.0, 4.0 # Radii to place "cone wedges" (inside, at, outside r_c) r_list = [0.5*rc, rc, 1.5*rc] # Build traveler worldline: # Segment A: move radially out (phi=0), t increases # Segment B: at r=r1>rc, loop phi from 0 -> 2π with timelike slope dt/dphi = -H(r1) (between null slopes) # Segment C: move radially back (phi=2π), small positive dt r0 = 0.3*rc r1 = 1.35*rc # Segment A Na = 120 ra = np.linspace(r0, r1, Na) phia = np.zeros(Na) ta = np.linspace(-1.0, 0.0, Na) # Segment B Nb = 260 phib = np.linspace(0.0, 2.0*np.pi, Nb) rb = np.full(Nb, r1) slope_mid = -H_of_r(r1, m, omega) # choose the mid between the two null slopes tb = np.linspace(ta[-1], ta[-1] + slope_mid*(phib[-1]-phib[0]), Nb) # Segment C Nc = 120 rc_seg = np.linspace(r1, r0, Nc) phic = np.full(Nc, 2.0*np.pi) tc = np.linspace(tb[-1], tb[-1]+0.5, Nc) r_path = np.concatenate([ra, rb, rc_seg]) phi_path = np.concatenate([phia, phib, phic]) t_path = np.concatenate([ta, tb, tc]) x_path = r_path * np.cos(phi_path) y_path = r_path * np.sin(phi_path) z_path = t_path # Precompute cylinder rings at r = r_c for a few t-levels phi_ring = np.linspace(0.0, 2.0*np.pi, 200) x_rc = rc * np.cos(phi_ring) y_rc = rc * np.sin(phi_ring) t_levels = np.linspace(t_min, t_max, 6) # Precompute null wedge lines at selected radii def wedge_lines_for_radius(r, t0=0.0, span=np.pi/8.0, samples=50): # two short lines representing the two null directions at phi0=0 over small |Δφ|≤span s_plus, s_minus = null_slopes(r, m, omega) dphi = np.linspace(-span, span, samples) # place wedges centered at phi0=0 and height t0 # First wedge (s_plus) phi1 = dphi t1 = t0 + s_plus * dphi r1 = np.full_like(dphi, r) x1 = r1 * np.cos(phi1) y1 = r1 * np.sin(phi1) # Second wedge (s_minus) phi2 = dphi t2 = t0 + s_minus * dphi r2 = np.full_like(dphi, r) x2 = r2 * np.cos(phi2) y2 = r2 * np.sin(phi2) return (x1, y1, t1), (x2, y2, t2) wedges = [] for idx, rr in enumerate(r_list): # stagger t0 so wedges don't overlap visually t0 = -2.0 + idx*2.0 wedges.append(wedge_lines_for_radius(rr, t0=t0)) # -------------------- Animation setup -------------------- fig = plt.figure(figsize=(7, 6)) ax = fig.add_subplot(111, projection='3d') # Set scene bounds Rmax = 1.8*rc ax.set_xlim(-Rmax, Rmax) ax.set_ylim(-Rmax, Rmax) ax.set_zlim(t_min, t_max) ax.set_xlabel("x") ax.set_ylabel("y") ax.set_zlabel("t") # Draw static elements # Critical cylinder rings for tz in t_levels: ax.plot(x_rc, y_rc, np.full_like(x_rc, tz), linewidth=1.0) # Null wedges wedge_artists = [] for (line1, line2) in wedges: x1, y1, t1 = line1 x2, y2, t2 = line2 h1, = ax.plot(x1, y1, t1, linewidth=2.0) h2, = ax.plot(x2, y2, t2, linewidth=2.0) wedge_artists.extend([h1, h2]) # Traveler worldline (animated) path_line, = ax.plot([], [], [], linewidth=2.5) # A null ring (closed photon curve) drawn at t=0 on r=rc ax.plot(x_rc, y_rc, np.zeros_like(x_rc), linewidth=2.0) # A horizontal slice (plane) hint via a thin ring grid (optional) for rgrid in np.linspace(0.3*rc, 1.6*rc, 5): ax.plot(rgrid*np.cos(phi_ring), rgrid*np.sin(phi_ring), np.zeros_like(phi_ring), linewidth=0.75) # Save a still still_path = "/mnt/data/godel_3d_still.png" plt.savefig(still_path, dpi=160, bbox_inches="tight") # Animation function total_frames = 360 def init(): path_line.set_data([], []) path_line.set_3d_properties([]) return [path_line] + wedge_artists def animate(i): # Reveal the path progressively k = max(2, int((i+1)/total_frames * len(x_path))) path_line.set_data(x_path[:k], y_path[:k]) path_line.set_3d_properties(z_path[:k]) # Rotate camera az = 30 + 360.0 * (i / total_frames) el = 22 + 5.0*np.sin(2*np.pi * i/total_frames) ax.view_init(elev=el, azim=az) return [path_line] + wedge_artists anim = animation.FuncAnimation(fig, animate, init_func=init, frames=total_frames, interval=30, blit=True) mp4_path = "/mnt/data/godel_3d.mp4" gif_path = "/mnt/data/godel_3d.gif" saved = {} try: Writer = animation.FFMpegWriter writer = Writer(fps=30, metadata=dict(artist="UDK"), bitrate=2400) anim.save(mp4_path, writer=writer, dpi=160) saved["mp4"] = mp4_path except Exception as e: # Fallback to GIF if ffmpeg is unavailable try: from matplotlib.animation import PillowWriter anim.save(gif_path, writer=PillowWriter(fps=20)) saved["gif"] = gif_path except Exception as e2: saved["error"] = f"Failed to save MP4 and GIF: {e}; {e2}" saved, still_path