Spherically symmetric black holes and affine-null metric formulation of Einstein’s equations

The definition of well-behaved coordinate charts for black hole spacetimes can be tricky, as they can lead for example to either unphysical coordinate singularities in the metric (e.g., $r=2M$ in the Schwarzschild black hole) or to an implicit dependence of the chosen coordinates to physical relevant coordinates (e.g., the dependence of the null coordinates in the Kruskal metric). Here we discuss two approaches for coordinate choices in spherically symmetric spacetimes allowing us to explicitly discuss “solitary” and spherically symmetric black holes from a regular horizon to null infinity. The first approach relies on a construction of a regular null coordinate system (where regular is meant as being defined from the horizon to null infinity) given an explicit solution of the Einstein-matter equations. The second approach is based on an affine-null formulation of the Einstein equations and the respective characteristic initial value problem. In particular, we present a derivation of the Reissner-Nordström black holes expressed in terms of these regular coordinates.

Figure 1

In the past of an open set of future null infinity defined by those points for which their Bondi retarded time $u$ is in the range $u\in (u_0,\infty )$ we require the existence of a regular null function $w$ such that: $w=0$ at the horizon ${\mathcal{H}}^{+}$, and $w<0$ in the region of interest. In a vicinity of $i^{+}$, the null surfaces of constant retarded time $u$ are smooth all the way up to the event horizon for SBHs.

Figure 2

Penrose’s diagram for the collapsing null shell. For values $v<v_i$ the spacetime is Minkowski, $v=v_i$ is the null shell and $v>v_i$ is the conformal diagram of a Schwarzschild black hole.

Figure 3

A graphical representation of the geometry for the spherically symmetric characteristic initial value formulation. The boundary surface $\mathcal{B}$ where $y=0$ is given in green while the initial data surface ${\mathcal{N}}_0$ where $w=w_0$ is light blue. Every point in this diagram represents a two-dimensional sphere, in particular the red dots correspond to the two-dimensional common cross sections of the family of null hypersurfaces ${\mathcal{N}}_w$ with the boundary $\mathcal{B}$. In fact any other intersections of surfaces $y=\text{const}$ and $w=\text{const}$ are two-dimensional spaces. Note that in general, the surfaces $y=\text{const}$ are not null hypersurfaces, like in a double null formulation where the surfaces $y=\text{const}$ would be displayed parallel to one another.