Kruskal-Szekeres and Penrose-Carter diagrams: the true shape of spacetime

Schwarzschild coordinates \((t, r)\) blow up at the horizon \(r=2M\) – a coordinate artifact, since nothing physical happens there (a free-falling observer notices nothing special crossing it). Kruskal and Szekeres found coordinates \((X, T)\) that are perfectly smooth across the horizon: in the exterior (\(r>2M\)),

\[X = \sqrt{r/2M - 1}\, e^{r/4M} \cosh(t/4M), \qquad T = \sqrt{r/2M - 1}\, e^{r/4M} \sinh(t/4M)\]

(with \(\sinh\) and \(\cosh\) swapping roles inside the horizon), so that light rays always travel at \(\pm 45^\circ\) and the horizon itself maps to the lines \(X=\pm T\). These coordinates reveal a startling fact: the maximally extended spacetime contains two separate asymptotically flat exterior regions, joined by a non-traversable “Einstein-Rosen bridge.” Compactifying further – applying \(\arctan\) to the null combinations \(u=T-X\), \(v=T+X\) – gives the Penrose-Carter diagram, bringing the entire infinite spacetime, including future and past null infinity, into one finite diagram while preserving the \(\pm 45^\circ\) light cones, making the whole causal structure visible at a glance.

import matplotlib.pyplot as plt
import numpy as np

from physicskit.relativity.visualizers.spacetime_diagrams import kruskal_coordinates, plot_kruskal_diagram, plot_penrose_diagram

Kruskal-Szekeres diagram

fig, ax = plt.subplots(figsize=(6, 6))
plot_kruskal_diagram(M=1.0, ax=ax)
plt.tight_layout()

Penrose-Carter conformal diagram

fig, ax = plt.subplots(figsize=(6, 6))
plot_penrose_diagram(M=1.0, ax=ax)
plt.tight_layout()
plt.show()

Check

Kruskal-Szekeres coordinates: T^2 - X^2 = (1 - r/2M) e^(r/2M), so the horizon r = 2M is the pair of lines T = +-X and the singularity r = 0 the hyperbola T^2 - X^2 = 1; Schwarzschild time slices are lines T/X = tanh(t/4M).

t_k, r_k = np.meshgrid(np.linspace(-10, 10, 21), np.concatenate([np.linspace(0.1, 1.9, 10), np.linspace(2.1, 20, 10)]))
X, T = kruskal_coordinates(t_k, r_k, 1.0)
np.testing.assert_allclose(T**2 - X**2, (1 - r_k / 2) * np.exp(r_k / 2), rtol=1e-10)
outside = r_k > 2
np.testing.assert_allclose((T / X)[outside], np.tanh(t_k[outside] / 4), rtol=1e-10)

/Users/cpoli/Code/physicskit/examples/relativity/spacetime_geometry/plot_kruskal_penrose_diagrams.py:59: RuntimeWarning: divide by zero encountered in divide
  np.testing.assert_allclose((T / X)[outside], np.tanh(t_k[outside] / 4), rtol=1e-10)