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

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

.. math::

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

(with :math:`\sinh` and :math:`\cosh` swapping roles inside the horizon),
so that light rays always travel at :math:`\pm 45^\circ` and the horizon
itself maps to the lines :math:`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
:math:`\arctan` to the null combinations :math:`u=T-X`, :math:`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 :math:`\pm 45^\circ` light cones, making the whole causal
structure visible at a glance.
"""

import matplotlib.pyplot as plt

from physicskit.relativity.visualizers.spacetime_diagrams import 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()