Understanding black hole evaporation using explicitly computed Penrose diagrams

Explicitly computed Penrose diagrams are plotted for a classical model of black hole formation and evaporation, in which black holes form by the accretion of infalling spherical shells of matter and subsequently evaporate by emitting spherical shells of Hawking radiation. This model is based on known semiclassical effects, but is not a full solution of semiclassical gravity. The method allows arbitrary interior metrics of the form $d{s}^2=-f(r)d{t}^2+f(r{)}^{-1}d{r}^2+r^{2}d{\mathrm{\Omega }}^2$, including singular and nonsingular models. Matter dynamics are visualized by explicitly plotting proper density in the diagrams, as well as by tracking the location of trapped surfaces and energy condition violations. The most illustrative model accurately approximates the standard time evolution for black hole thermal evaporation; its time dependence and causal structure are analyzed by inspection of the diagram. The resulting insights contradict some common intuitions and assumptions, and we point out some examples in the literature with assumptions that do not hold up in this more detailed model. Based on the new diagrams, we argue for an improved understanding of the Hawking radiation process, propose an alternate definition of “black hole” in the presence of evaporation, and suggest some implications regarding information preservation and unitarity.

Figure 1

Conventional Penrose diagram of an evaporating black hole. The hole forms via the infall of null dust (red arrow) and forms an event horizon (black dotted line). At some point the black hole evaporates, leaving Minkowski space. Such a diagram usually does not represent any spacetime—the mechanism of evaporation is left ambiguous, and the nature of point $B$ is totally unclear. In the obvious classical-spacetime interpretations of this diagram, however, point $B$ must be considered a naked singularity that creates a Cauchy horizon $H$ (red dotted line), raising questions about the applicability of this diagram for analyzing potentially unitary BH evaporation. Below, we attempt to eliminate these ambiguities by constructing Penrose diagrams for well-defined spacetime models of BH evolution.

Figure 2

Schematic illustration of Penrose diagrams for the shell model in simple cases. (a) The simplest singular case: a Schwarzschild black hole forms by collapse of a spherical null shell and evaporates by emitting a single burst of Hawking radiation, which nucleates at a radius $r_{\mathrm{ev}}=r_{\mathrm{hor}}+l_{\mathrm{ev}}$ just outside the apparent horizon at $r_{\mathrm{hor}}=2m$. (b) The simplest nonsingular case: a Hayward black hole (see below) forms and evaporates in the same way.

Figure 3

Matter content of the Hayward spacetime. Density and pressure are maximized at $r=0$, with maximum density ${\rho }_0=\frac{3}{8\pi l^2}=m/(\frac{4}{3}\pi r_{\mathrm{core}}^3)$. The FEC violation function ${\chi }_{\mathrm{FEC}}$ (see Appendix) shows non-negligible violations occur near the core surface; NEC and WEC are violated at negative-mass shells in the dynamic model, but not here in the bulk. The plot shows a small BH, but is mostly parameter independent: increasing $m$ pushes the horizons at $r_{-}\approx l$ (dashed) left and at $r_{+}\approx 2m$ (off scale) right, with no other effects. Spheres are trapped surfaces for all $r_{-}<r<r_{+}$, which we call the trapped spheres region.

Figure 4

Penrose diagrams for (a) singular and (b) nonsingular black holes that form by accreting a single shell of infalling matter and evaporate by emitting a single blast of Hawking radiation (see Fig. 2 for an illustrative schematic). Parameters are chosen to illustrate qualitative features, but the time evolution and relative length scales are not realistic (see Fig. 5 for an improved model). Positive-mass (accretion and outgoing Hawking radiation, gray dashed lines) and negative-mass (ingoing Hawking radiation, gray dotted lines) shells separate the spacetime into piecewise regions, with Hawking radiation nucleating at a tiny radial distance $l_{\mathrm{ev}}$ outside the horizon of the region to its past. In diagrams with many shells, shell masses are indicated by grayscale darkness (darkness proportional to $1+2\mathrm{\Delta }m/M$). The curvature cutoff length scale $l$ (which has physical significance only in Hayward regions) is held fixed across all regions, while the mass parameter $m$ (which in every region determines the gravitational mass measured by a distant observer) varies. The total mass $M$ is the maximum value of $m$ in any region, and locally $m$ is visualized by the linewidth of the conformal boundary at $r=\infty$ in each region (linewidth proportional to $1+2m/M$). Tick marks (gray) along $r=\infty$ mark off equal increments of proper time for an infinitely distant observer at constant radius (i.e., constant increments of $du$ and $dv$ along null infinity). The trapped spheres region (black dot-hatch fill), bounded by horizons (black) where $f(r)=0$, contains closed trapped spheres. Background coloring is determined by the local proper density $\rho$ (orange color scale) scaled by the maximum density ${\rho }_0=3/(8\pi l^2)$. The Hayward core is clearly visible as a dark orange region in the density plot, and the core surface almost exactly corresponds to the singularity location in the Schwarzschild case. Notably, distant observers near future null infinity begin to observe Hawking radiation at the same moment they see the infalling accretion shell fall through its own horizon. Lines of constant radius are shown at small ($dr=l/2$, teal) and large ($dr=2M/2$, magenta) length scales; even where they appear bundled or strongly kinked, they do in fact remain continuous. One strange-looking feature of this diagram is the appearance of a set of wiggly kinks and a few stray tick marks to the future (measured along future infinity) of the final evaporation shell, before the very stretched out area; this is an artifact of the unrealistic parameters, and in more realistic models these kinks and tick marks all coincide with the final shell. Coordinates $V$ and $U$ defining the axes are basically arbitrary null global coordinates, defined further in [5]. The same visualization scheme described here is used in all examples below.

Figure 5

Penrose diagrams for a nonsingular Hayward black hole that forms gradually and then evaporates by slowly emitting thermal radiation with a standard time dependence [see (8)–(10)]. The visualization scheme and associated legend are the same as in Fig. 4. As discussed in Sec. 4, these diagrams accurately capture both global and local (interior) structure, since they are constructed by directly finding a compact global double-null coordinate system for the spacetime. Although the three diagrams depicted in (a), (b), and (c) appear vastly different, they all are derived from exactly the same spacetime—they have strictly the same causal structure and are related by conformal transformations in the $UV$ plane. (These “conformal” transformations are induced by coordinate transformations, so no geometric information is lost; see Sec. 4.) While the conformal freedom in Penrose diagrams is well known, it is not widely recognized how drastically different these transformations can make the spacetime appear, or that no single diagram will clearly depict the various widely different timescales of BH evolution (formation, evaporation, Planck scale) simultaneously. Inevitably, in any diagram, some features will be squished beyond recognition; hence the need to compare multiple diagrams to gain a full understanding of the spacetime. Here, different transformations are used to highlight features (a) during accretion, (b) during evaporation, and (c) near the end of evaporation. In comparing the causal structure between them, note that some lines appearing null are only nearly null, and that some features are hidden by being very squished. For example, in (a), the entire evaporation process [which includes important timelike features visible in (b) and (c)] is squished into a tiny, seemingly null, line. Similarly, parts of the high density region (orange) in (b), and all of the high density region in (c), are squished and hidden behind the future part of the nearly null segment of $r=0$. The important qualitative features of these diagrams are summarized in Sec. 8. Parameters were chosen to be as realistic as numerically allowed, providing a strong hierarchy of the formation, evaporation, and interior length/time scales (an overall scale factor is irrelevant). Since the BH length scales are so small compared to the evaporation rate, lines $r=\text{const}$ at $2M$ (magenta) and smaller length scales are very close together and each appears as a single bundle; an additional set of $r=\text{const}$ lines has been added with spacing $dr={\tau }_{\mathrm{ev}}/20$ (faint purple) to display larger scales. Although some lines of constant radius may seem discontinuous, they are, in fact, just strongly kinked; their continuity can be confirmed by gradually adjusting the parameters from less extreme values. For example, faint purple lines approaching the evaporating horizon in (a) do not disappear, but closely hug the horizon in a bundle until reappearing in the far future region. The opacity of the tick mark near the moment of evaporation in (a) shows that many tick marks have piled up there; this is the phase pileup usually associated with Hawking radiation in particle creation calculations. The same phase pileup can be observed in (b) and (c); in all cases one expects interesting results wherever the $dv$ ticks are very mismatched with traced-back $du$ ticks. Panels (d) and (e) confirm that the numerically generated dynamics closely approximate the desired behavior.

Figure 6

Detail view of horizon and core dynamics during formation and the early stages of evaporation, for a spacetime similar to Fig. 5 but with somewhat less realistic parameters (in the sense that the mass and lifetime are not mutually consistent). Although the parameters are less realistic, the features depicted are qualitatively accurate. Details such as these are not visible in more quantitatively accurate diagrams, where the extreme hierarchy of length and time scales prevents accretion and evaporation features from being depicted simultaneously. Compare to the region where accretion shells meet the core in Fig. 5. Note that the apparently teal region is, in fact, a bundle of closely spaced lines of constant radius just outside the core.

Figure 7

Penrose diagrams for single-burst shell models with (a) negative and (b) positive background curvature, with metrics (11) corresponding to the Hay-AdS and Hay-dS BH spacetimes. Parameters were chosen for maximum visibility of qualitative features. Density color scale here depicts $\rho -{\rho }_{\mathrm{bg}}$, with constant background density ${\rho }_{\mathrm{bg}}=\pm 3/(8\pi L^2)$ (positive for dS). In (b), an additional set of lines $r=\text{const}$ (faint purple) have been added with spacing $dr=L/2$ to depict larger length scales. Note that, in addition to BH trapping, the dS case includes cosmological trapped regions having nothing to do with the BH. Tick marks no longer correspond to proper time, but still represent equal increments of $du$ and $dv$. Notably, inspecting the tick marks shows that the phase pileup typically associated with Hawking radiation is still present in the $u$ and $v$ coordinates, even without the assumption of asymptotic flatness.

Figure 8

Two spherical null shells colliding in a spherically symmetric spacetime. Gray lettering indicates the standard labeling of each region in DTR calculations (see Appendix). Black lettering describes an example: Two shells, each of mass $\mathrm{\Delta }m$, collide in an initial Schwarzschild spacetime of mass $m_B=m$. Assuming $A$, $B$, $C$, $D$ are all Schwarzschild spacetimes, the masses $m_C$ and $m_D$ are fixed by shell junction conditions, with signs determined by whether each shell is radially ingoing ($+$) or outgoing ($-$) toward the future. The final mass $m_A=M$ is determined by the DTR relation. Assuming all shells (both initial and final states) have a positive mass, mass inflation can occur only if both shells are radially ingoing—for instance, inside the trapped region of a BH. Otherwise, restricting to positive-mass shells ensures that $|M-m|\le \mathrm{\Delta }m$.