0. Package Capabilities

This package implements the methods of the paper:

JC Schindler, A Aguirre. Algorithms for the explicit computation of Penrose diagrams. Class Quantum Grav 35 105019 (2018). doi:10.1088/1361-6382/aabce2. [arxiv:1802.02263.]

Allowed metrics

It allows one to plot causal diagrams for 3+1 dimensional spacetimes in two classes:

  • Any metric of the form

\[ds^2 = -f(r) \, dt^2 + f(r)^{-1} \, dr^2 + r^2 \, d\Omega^2.\]

  • Metrics that are piecewise-of the above form, glued along radial null shell junctions. Arbitrarily many shells and piecewise regions can be included to approximate smooth dynamical evolution.

Nature of the diagrams

The diagrams are constructed by exactly constructing coordinate transformations from a defining coordinate system (e.g. Schwarzchild \((t,r,\theta,\phi)\) or Eddington-Finklestein \((u,v,\theta,\phi)\) coordinates) into a system of “diagram” coordinates \((U,V,\theta,\phi)\) with the following properties:

  • The diagram is simultaneously a Penrose diagram and an exact coordinate diagram.

    • The coordinates \((U,V)\) cover the entire spacetime in a global compact double-null coordinate patch, so causal cones are bounded by lines of unit slope, without losing conformal information.

    • Thus internal features such as matter content, observers, and arbitrary spherically symmetric functions, can be unambiguously plotted.

  • Each point in the diagram corresponds to a spherical symmetry surface with a well defined radius.

    • Each point \((U_0,V_0)\) on the diagram corresponds to a spherical symmetry surface \((U_0,V_0,\theta,\phi)\), and has a physical radius.

    • The radius \(r=r(U_0,V_0)\) is defined by the area of symmetry spheres.

  • The diagram extends the metric smoothly across arbitrarily many horizons, without coordinate singularity.

    • Horizons occur wherever \(f(r)=0\).

  • All coordinates and parameters are unitless. Unitful values are restored by a choice of an overall length scale.

Custumizability

The package makes available functions for performing coordinate transformations, creating regions based off metrics, adding curves and contours to regions, and plotting diagrams. One can:

  • Quickly plot basic diagrams (by default depicting causal boundaries and lines of constant radius).

  • Create custom diagrams depicting arbitrary curves and coordinate systems on the diagram.

  • Calculate the spherical radius at each point in the diagram and plot contours of spherically symmetric functions.

  • Plot internal features such as densities, curvature scalars, vector fields, trapped surfaces, etc, using the above.

  • Specify custom metrics via the metric function \(f(r)\).

  • Glue together regions across properly matched null shell junctions.