Asymptotic horizon formation, spacetime stretching, and causality

In this work we analyze asymptotically flat, spherically symmetric spacetimes in which an event horizon is present without any trapped surfaces. We identify two types of such spacetimes, each related to the asymptotic behavior (in time) of one of the two degrees of freedom of the metric. We study the causal structure of both types, showing that one almost always has a Cauchy horizon beyond which it is extendable, while the other is inextendable but has two separate future null infinity regions on either side of the horizon. We also study what energy conditions can be satisfied by the matter around the horizon. Some of these spacetimes were first introduced in an earlier work in which semiclassical effects near black hole horizons were analyzed. Here, we generalize this analysis to a larger family of geometries.

Figure 1

A qualitative representation of the relation between the radial coordinate $r$ and the proper length in the radial direction $l$ for a geometry in which $g$ diverges at some radius $r_{\mathrm{h}}$. This divergence corresponds to an infinite stretching of $l$, which completely severs the interior and exterior geometries.

Figure 2

Slice at constant time ($v=\text{const}$.) of the generalized redshift function $F(v,r)$ around $R_{\mathrm{h}}(v)$. There is a discontinuity in the first derivative of this function at $R_{\mathrm{h}}(v)$ if either $k_1\ne 0$ or ${\tilde{k}}_1\ne 0$, which through the Einstein equations can translate into either a thin shell of matter or into a discontinuity in pressure, depending on how the two individual degrees of freedom ($f$ and $g$) of the geometry, which comprise $F(v,r)$, behave.

Figure 3

Conformal diagram of the spacetime with $g\simeq 1$ and $f$ given by (7) satisfying the appropriate conditions for light-ray trapping. The dashed line is the event horizon, corresponding to the first trapped outgoing light ray. The dash-dotted line is the surface $r=r_{\mathrm{h}}$, which is described by a timelike curve and the Cauchy horizon. The curves to the left of $r_{\mathrm{s}}$ correspond to surfaces of $r=\text{const}<r_{\mathrm{h}}$, while to the right they are $r=\text{const}>r_{\mathrm{h}}$. The lines outside the conformal triangle indicate the need to extend the spacetime.

Figure 4

Relation between the radial coordinate $r$ and the proper length in the radial direction $l$ for a geometry in which $g$ tends to a divergence at $r_{\mathrm{h}}$. Outgoing light rays become trapped in this infinitely stretching region, while ingoing ones pass right through it.

Figure 5

Conformal diagram of the spacetime in which $g$ tends to a divergence. Left: the divergence is reached in infinite time at the surface $r=r_{\mathrm{h}}$ (the dash-dotted line), which becomes a separate part of future null infinity for outgoing light rays. This interior null infinity is denoted by ${\mathcal{I}}_{(1)}^{+}$, while the exterior one (for escaped light rays) is ${\mathcal{I}}_{(2)}^{+}$. Right: the divergence at $r=r_{\mathrm{h}}$ is reached at a finite moment $v=v_0$ (and remains thereafter), making $r=r_{\mathrm{h}}$ a future null infinity ${\mathcal{I}}_{(1)}^{+}$ for ingoing light rays with $v\ge v_0$. The symbols $i_{(1)}^{+}$ and $i_{(2)}^{+}$ indicate future timelike infinities for two different sets of observers (for the diagram on the right this is conditional; see discussion below). In general, this latter geometry is extendable past the surface $v=v_0$ for $r<r_{\mathrm{h}}$, marked as a Cauchy horizon. The diagram includes what the extension may look like, indicating that to be fixed it requires initial data from another surface, which is effectively another Cauchy horizon in the past of the region.