Singularity-Free Gravitational Collapse: From Regular Black Holes to Horizonless Objects

Abstract

Penrose’s singularity theorem implies that if a trapped region forms in a gravitational collapse, then a singularity must form as well within such region. However, it is widely expected that singularities should be generically avoided by quantum gravitational effects. Here we shall explore both the minimum requirements to avoid singularities in a gravitational collapse as well as discuss, without relying on a specific quantum gravity model, the possible regular spacetimes associated to such regularization of the spacetime fabric. In particular, we shall expose the intimate and quite subtle relationship between regular black holes, black bounces and their corresponding horizonless object limits. In doing so, we shall devote specific attention to those critical (extremal) black hole configurations lying at the boundary between horizonful and horizonless geometries. While these studies are carried out in stationary configurations, the presence of generic instabilities strongly suggest the need for considering more realistic time-dependent dynamical spacetimes. Missing specific dynamical models, much less rigorous statements can be made for evolving geometries. We shall nonetheless summarize here their present understanding and discuss their implications for future phenomenological studies.

Appendices

Appendix 1: Extremal Horizons

A general feature implicit in the discussion above is that special things happen to the spacetime geometry at horizons, and that even more special things happen at extremal horizons. However different special things might happen for inner versus outer horizons. In this appendix we shall seek to present a coherent overview of this topic. (In a somewhat similar vein, it has been known for some time that special things happen at wormhole throats [76].)

We find it convenient to work with static spacetimes in area coordinates:

$$\begin{aligned} ds^2 = -e^{-2{\varPhi }(r)}\left( 1-\frac{2m(r)}{r}\right) dt^2 + \frac{dr^2}{1-\frac{2m(r)}{r}} + r^2\,d{\varOmega }^2_{2}. \end{aligned}$$

(9.31)

The horizons are located at solutions (if any) of the equation $r_H = 2 m(r_H)$. If we are dealing with a wormhole throat, we will need two coordinate systems of this type, carefully matched at the throat [114].

A purely geometrical result is that in a suitable orthonormal basis [114]

$$\begin{aligned} G_{\hat{t} \hat{t}} = {2m'(r)\over r^2}; \qquad \qquad G_{\hat{r} \hat{r}} = - {2m'(r)\over r^2}+ \left( 1-{2m(r)\over r} \right) {2{\varPhi }'(r)\over r}; \end{aligned}$$

(9.32)

Thence at any horizon (inner or outer, extremal or non-extremal) one has

$$\begin{aligned} G_{\hat{t} \hat{t}} + G_{\hat{r} \hat{r}} \quad \rightarrow \quad 0. \end{aligned}$$

(9.33)

This on-horizon “enhanced symmetry” for the Einstein (and Ricci) tensors is a recurring theme in near horizon physics [85]. Another useful and very general result is that the surface gravity is [113]:

$$\begin{aligned} \kappa _H = e^{-{\varPhi }(r_H)} \; {1-2m'(r_H) \over 2r_H }. \end{aligned}$$

(9.34)

An extremal horizon is characterized by the vanishing of the surface gravity which we see requires $2m'(r_H)=1$.

At any extremal horizon (outer or inner) the Einstein tensor (and Ricci tensor) become particularly simple. Specifically

$$\begin{aligned} G_{\hat{t} \hat{t}|r_E} = - G_{\hat{r} \hat{r}|r_E} = {1\over r_E^2}; \qquad \qquad G_{\hat{\theta }\hat{\theta }|r_E} = G_{\hat{\phi }\hat{\phi }|r_E} = -{m''(r_E) \over r_E}. \end{aligned}$$

(9.35)

In fact at any extremal horizon all orthonormal components of the Riemann tensor are proportional to either $1/r_E^2$, or $m''(r_E) /r_E$, or are zero. Thence at any extremal horizon all of the polynomial curvature invariants are simply multi-nomial functions $f(1/r_E^2,m''(r_E) /r_E)$ of these two quantities. Furthermore at any extremal horizon all nonzero orthonormal components of the Weyl tensor are simply miltiples the single quantity $(1+r_E m''(r_E))/r_E^2$. So the spacetime geometry simplifies quite drastically on any extremal horizon (either outer or inner).

Finally, what can we say about $m''(r_E)$? This will depend on how many non-extremal horizons merge to yield the extremal horizon of interest. If two non-extremal horizons merge then $m''(r_E)\ne 0$; if three (or more) non-extremal horizons merge then $m''(r_E)=0$. For instance in the extremal Reissner–Nordström geometry (where two horizons merge) we have $m(r) = m - {1\over 2}q^2/r$ and at extremality we obtain $m''(r_E)= -q^2/r_E^3 = -1/r_E < 0$. In contrast for the extremal inner horizons explored in Ref. [39] we have three merging horizons, m(r) is a rational quartic and it is easy to check that $m''(r_E)=0$.

In short, although it is perhaps not all that well appreciated, geometrically it is guaranteed that very special things happen at all extremal horizons; and these special properties will have a role to play in both phenomenology and in stability analyses for RBHs.

Appendix 2: Light Rings

From the discussion above we have seen that interesting things happen for light rings in extremal, near-extremal, and super-extremal objects. See also [26, 78, 120]. That something unusual happens with light rings in the extremal limit can already be deduced from the very simple and explicit example of Reissner–Nordström spacetime. Since this situation already captures the key features of the discussion with an absolute minimum of fuss, we present some brief pedagogical comments here, before looking at the general situation.

Reissner–Nordström Light-Rings

The Reissner–Nordström spacetime in area coordinates is

$$\begin{aligned} d s^2 = -(1-2m/r+q^2/r^2) d t^2 +{d r^2\over 1-2m/r+q^2/r^2} + r^2d {\varOmega }^2 \end{aligned}$$

(9.36)

It is a quite standard result that in Reissner–Nordström spacetime the light rings can be found by inspecting the effective potential

$$\begin{aligned} V(r) = \left( 1-{2m\over r}+{q^2\over r^2} \right) \left( {L^2\over r^2}\right) \end{aligned}$$

(9.37)

The circular photon orbits are located at $r_c$ such that $V'(r_c)=0$, and stability depends on the sign of $V''(r_c)$. If $V''(r_c)>0$ then the light ring is stable; if $V''(r_c)<0$ then the light ring is stable; if $V''(r_c)=0$ then the light ring exhibits neutral (marginal) stability.

Unfortunately, the potential V(r) is not unique, a circumstance which can sometimes cause confusion. Indeed, let F(x) be any monotone increasing function and define $\tilde{V}(r) = F(V(r))$. Then

$$\begin{aligned} \tilde{V}(r)' = F'(V(r)) \;V'(r); \qquad \tilde{V}(r)'' = F''(V(r))\; [V'(r)]^2 + F'(V(r)) \;V''(r). \end{aligned}$$

(9.38)

So the extrema $r_c$ of V(r) coincide with extrema of $\tilde{V}(r)$. Furthermore, at these extrema one has $sign\{\tilde{V}''(r_c)\} = sign\{V''(r_c)\}$.

To locate the light rings we note

$$\begin{aligned} V'(r) = {2L^2\over r^5}(3mr-2q^2-r^2), \end{aligned}$$

(9.39)

and

$$\begin{aligned} V''(r) = {2L^2\over r^6}( 3r^2+10q^2-12mr). \end{aligned}$$

(9.40)

The outer and inner horizons are located at

$$\begin{aligned} r_H = m \pm \sqrt{m^2-q^2}. \end{aligned}$$

(9.41)

The outer and inner light rings are located at

$$\begin{aligned} r_c = {3m\over 2} \pm {\sqrt{9m^2-8q^2}\over 2}. \end{aligned}$$

(9.42)

Distinct inner and outer light rings exist for $9m^2 > 8q^2$, and merge at $9m^2 = 8q^2$. that is, beyond extremality. The light rings merge at $r_c= {3m\over 2}$.

• At the outer light ring

  $$\begin{aligned} V''(r_c) = - {64 L^2 \sqrt{9 m^2-8 q^2} \over (3m + \sqrt{9 m^2-8 q^2})^5 } < 0, \end{aligned}$$

  (9.43)

  so the outer light ring is always unstable.

• At the inner light ring

  $$\begin{aligned} V''(r_c) = {64 L^2 \sqrt{9 m^2-8 q^2} \over (3m - \sqrt{9 m^2-8 q^2})^5 } > 0, \end{aligned}$$

  (9.44)

  so the inner light ring is always stable.

• At $r_c= {3m\over 2}$, where the light rings merge, $9 m^2=8q^2$ so $V''(r_c) = 0$, and the merged light ring exhibits neutral stability.

At extremality ($m=|q|$) the light rings are formally located at

$$\begin{aligned} r_c= {3m\over 2} \pm {m\over 2} = \{ m, 2m\}. \end{aligned}$$

(9.45)

Here $r_c=2m$ corresponds to a true light ring, while $r_c=m$ represents the light sheet defining the extremal horizon. (There is now no angular motion, so this is not a “ring”.) At extremality ($m=|q|$) for the outer light ring

$$\begin{aligned} r_c = 2m = 2 r_H; \qquad V''(r_c) \rightarrow -{ L^2 \over 16 m^4 } < 0. \end{aligned}$$

(9.46)

At extremality ($m=|q|$) for the inner light sheet

$$\begin{aligned} r_c=m= r_H; \qquad V''(r_c) \rightarrow +{ 2L^2 \over m^4 } > 0. \end{aligned}$$

(9.47)

So the extremal horizon is a stable light sheet. Perhaps counter-intuitively, the fact that the light sheet is stable will destabilize the spacetime—since the light sheet is stable, massless particles cam pile up there; eventually back-reaction will become large, and the spacetime detabilizes. The situation is summarized in Fig. 9.11.

The key observation here is that the Reissner–Nordström spacetime is already subtle enough to exhibit a stable light ring at extremality, and multiple light rings in a small region beyond extremality. This does have implications for more general RBHs, since one can always cut off the core of the Reissner–Nordström spacetime at some $r_{core} < |q|$ and replace it with a Reissner–Nordström-inspired RBH that would then (by construction) exhibit exactly the same light rings as the Reissner–Nordström spacetime itself. In short, the existence of unstable light rings exterior to generic extremal black holes should not really come as a surprise.

Fig. 9.11

Reissner–Nordström inner and outer horizons, and inner and outer light rings

Generic Light-Rings

What can we say about light rings in the generic case? We already have rather good intuition based on what we saw happening for Reissner–Nordström spacetime.

For any geometry of the form given in Eq. (9.31) it is easy to check that the effective potential governing the light rings is

$$\begin{aligned} V(r) = e^{-2{\varPhi }(r)} \left( 1-{2m(r)\over r}\right) {L^2\over r^2}. \end{aligned}$$

(9.48)

It is then easy to check that

$$\begin{aligned} V'(r) = e^{-2{\varPhi }(r)}\, {2L^2\over r^4}\, \left( \{3m(r)-r m'(r)-r\}- {\varPhi }'(r) r^2(1-2m(r)/r) \right) . \end{aligned}$$

(9.49)

Purely geometrically this leads to

$$\begin{aligned} V'(r) = e^{-2{\varPhi }(r)}\, {2L^2\over r^4}\, \{3m(r) -r + r^3 G_{\hat{r}\hat{r}}(r) \}. \end{aligned}$$

(9.50)

This can also be written as

$$\begin{aligned} V'(r) = e^{-2{\varPhi }(r)}\, {2L^2\over r^4}\, \{3m(r) -r -rm'(r) + r^3 [G_{\hat{t}\hat{t}}(r) + G_{\hat{r}\hat{r}}(r)] \}. \end{aligned}$$

(9.51)

Furthermore one can easily verify that

$$\begin{aligned} V''(r) = e^{-2{\varPhi }(r)} {2L^2\over r^5}\{3r-12m(r)+6 r m'(r) -r^2 m''(r)\} -2{\varPhi }'(r) V'(r) -2{\varPhi }''(r) V(r). \end{aligned}$$

(9.52)

Thence at any light ring that might be present the condition $V'(r_c)=0$ implies

$$\begin{aligned} r_c = {3m(r_c) + r_c^3 [G_{\hat{t}\hat{t}}(r_c) + G_{\hat{r}\hat{r}}(r_c)] \over 1 - m'(r_c) } \end{aligned}$$

(9.53)

We now wish to self-consistently bound the location of possible solutions to this equation to determine whether a light ring exists for $r_c > r_H$. The purely geometrical null convergence condition (guaranteeing the convergence of null geodesics), when applied to the radial null geodesics, would imply

$$\begin{aligned}{}[G_{\hat{t}\hat{t}}(r) + G_{\hat{r}\hat{r}}(r)] \ge 0. \end{aligned}$$

(9.54)

If the Misner-Sharp quasi-local mass is non-decreasing outside the horizon then this would imply $m'(r) \ge 0$. Combining, if we assume the light ring exists, then its location is bounded below by:

$$\begin{aligned} r_c \ge 3m(r_c) \ge 3 m(r_H) = {3\over 2}\; r_H. \end{aligned}$$

(9.55)

To establish an upper bound we start from Eq. (9.50). Setting $V'(r_c)\rightarrow 0$ yields

$$\begin{aligned} r_c = 3m(r_c) + r_c^3 G_{\hat{r}\hat{r}}(r_c) . \end{aligned}$$

(9.56)

Then from the DCC (dominant convergence condition): $|G_{\hat{r}\hat{r}}| \le |G_{\hat{t}\hat{t}}| = {2m'/r^2}$ we see

$$\begin{aligned} r_c \le 3m(r_c) + 2 r_c m'(r_c). \end{aligned}$$

(9.57)

We need one more condition to get a useful bound: $(m(r)/r^3)' < 0$, implying $m'(r) < 3 m(r)/r$. (This condition corresponds to the volume-averaged density decreasing as one moves outwards, and is a very popular condition used in building relativistic and Newtonian models.) Then

$$\begin{aligned} r_c \le 9m(r_c) \le 9 m_\infty . \end{aligned}$$

(9.58)

Overall, under plausible structural conditions, and assuming existence of the light ring, we have

$$\begin{aligned} {3\over 2} r_H \le r_c \le 9 m_\infty . \end{aligned}$$

(9.59)

However, proving actual existence of the light rings is slightly more subtle, and requires slightly different arguments for outer-non-extremal and outer-extremal horizons.

Outer Non-extremal Horizons

For an outer non-extremal horizon in terms of the surface gravity we can calculate

$$\begin{aligned} V'(r_H) = {2 L^2 e^{-{\varPhi }(r_H)} \; \kappa _H\over r_H^2} >0. \end{aligned}$$

(9.60)

On the other hand for an asymptotically flat geometry, at large r we will have $m(r)=m_\infty +O(1/r)$ and ${\varPhi }(r)=O(1/r)$ whence asymptotically⁴

$$\begin{aligned} V'(r) = -{2 L^2 \over r^3} + O\left( 1\over r^4\right) < 0. \end{aligned}$$

(9.61)

The sign flip guarantees that there will be at least one light ring somewhere between the outer horizon and spatial infinity.

Outer Extremal Horizons

For any extremal horizon we can calculate

$$\begin{aligned} V'(r_H) = 0; \qquad V''(r_H) = -{2L^2 e^{-2{\varPhi }(r_H)} m''(r_H)\over r_H^3}. \end{aligned}$$

(9.62)

If this is to be an outer extremal horizon then we must have $m''(r_H)<0$ and so $V''(r_H)>0$. But then $V'(r)>0$ in the region immediately above the horizon. On the other hand, for any asymptotically flat geometry we still have

$$\begin{aligned} V'(r) = -{2 L^2 \over r^3} + O\left( 1\over r^4\right) < 0. \end{aligned}$$

(9.63)

The sign flip again guarantees that there will be at least one light ring somewhere between the outer horizon and spatial infinity.

Regular Horizonless Objects

For a regular horizonless object, (cf. a super-extremal Reissner–Nordstróm geometry with a regularized core at $r_\textrm{core}< |q|$), at short distances we would demand $m(r)=O(r^3)$ and ${\varPhi }(r) = O(r^2)$. Consequently

$$\begin{aligned} V'(r) = -{2 L^2 \over r^3} + O\left( 1\right) < 0, \end{aligned}$$

(9.64)

while at large distances

$$\begin{aligned} V'(r) = -{2 L^2 \over r^3} + O\left( 1\over r^4\right) < 0. \end{aligned}$$

(9.65)

There is now no sign flip and consequently there must be an even number (possibly zero) of light rings.