Fundamental photon orbits: black hole shadows and spacetime instabilities

TL;DR

This paper classifies fundamental photon orbits in black hole spacetimes, analyzing their stability and impact on shadows, revealing new phenomena like stable orbits and cuspy shadows in non-Kerr black holes with Proca hair.

Contribution

It introduces a classification of fundamental photon orbits in stationary, axi-symmetric spacetimes and explores their stability, extending the understanding of black hole shadows beyond Kerr solutions.

Findings

All FPOs are unstable in Kerr black holes, determining their shadows.

Stable FPOs can exist in non-Kerr spacetimes, leading to new instabilities.

Non-Kerr black holes with Proca hair can have cuspy shadows due to stable and unstable FPO interplay.

Abstract

The standard Black Holes (BHs) in General Relativity, as well as other ultra-compact objects (with or without an event horizon) admit planar circular photon orbits. These light rings (LRs) determine several spacetime properties. For instance, stable LRs trigger instabilities and, in spherical symmetry, (unstable) LRs completely determine BH shadows. In generic stationary, axi-symmetric spacetimes, non-planar bound photon orbits may also exist, regardless of the integrability properties of the photon motion. We suggest a classification of these fundamental photon orbits (FPOs) and, using Poincar\'e maps, determine a criterion for their stability. For the Kerr BH, all FPOs are unstable (similarly to its LRs) and completely determine the Kerr shadow. But in non-Kerr spacetimes, stable FPOs may also exist, even when all LRs are unstable, triggering new instabilities. We illustrate this for the case of Kerr BHs with Proca hair, wherein, moreover, qualitatively novel shadows with a cuspy edge exist, a feature that can be understood from the interplay between stable and unstable FPOs. FPOs are the natural generalisation of LRs beyond spherical symmetry and should generalise the LRs key role in different spacetime properties.

Equations

$$0=2\mathcal{H}=\underset{T⩾0}{\underbrace{g^{rr}{p_r}^2+g^{\theta \theta }{p_{\theta }}^2}}+\underset{V⩽0}{\underbrace{g^{tt}{E}^2-2g^{t\phi }E\Phi +g^{\phi \phi }{\Phi }^2}}.$$

Full text

Introduction. Light rings (LRs), $i.e.$ circular photon orbits, are an extreme form of light bending by ultracompact objects (UCOs). They have distinct phenomenological signatures in both the electromagnetic and gravitational wave channels. In the former, LRs are closely connected to the shadow of a black hole (BH) Bardeen (1973); Falcke et al. (2000). This is the absorption cross section of light at high frequencies, an observable that is being targeted by the Event Horizon Telescope Broderick and Loeb (2006); Doeleman et al. (2009). In the gravitational wave channel, LRs determine a perturbed BH’s early-time ringdown Goebel (1972), corresponding to the post-merger part of the recently detected gravitational wave transients by aLIGO Abbott et al. (2016a, b). The frequency and damping time of this early-time ringdown are set by the orbital frequency and instability time scale (Lyapunov exponent) of an (unstable) LR.

LRs also define other dynamical properties of UCOs. For horizonless UCOs, LRs often come in pairs, one being stable and the other unstable. The existence of a stable LR has been claimed to imply a spacetime instability Keir (2016); Cardoso et al. (2014). Finally, LRs impact on our Newtonian intuition for test particle motion: crossing (inwards) a LR swaps the perception of inwards/outwards, and reverses the centrifugal effect of angular motion Abramowicz (1993).

For spherical UCOs, LRs (which are always planar) are the only bound photon orbits. But for an axisymmetric (and stationary) spacetime more general photon orbits are possible, that neither escape to infinity, nor fall into a BH (if the UCO is a BH). In this letter, we analyse implications, and propose a classification, of this natural generalization of LRs, dubbed fundamental photon orbits (FPOs). In particular we argue they can trigger new spacetime instabilities and show they are paramount in understanding the detailed structure of BH shadows.

FPOs. In vacuum General Relativity (GR), the only regular (on and outside an event horizon) UCO is the Kerr solution Kerr (1963), wherein geodesic motion is Liouville integrable and separates in Boyer-Lindquist (BL) coordinates $(t,r_{BL},\theta,\varphi)$ Carter (1968). In this chart, FPOs with constant $r_{BL}$ and motion in $\theta$ exist, known as spherical orbits Teo (2003). The subset restricted to the equatorial plane are the two LRs, one for co-rotating and one for counter-rotating photons (with respect to the BH), both converging at $r_{BL}=3M$ in the Schwarzschild BH (mass $M$) limit Bardeen et al. (1972). Spherical orbits are related to the ringdown modes in BH perturbation theory Cardoso et al. (2009) and completely determine the Kerr BH shadow ($cf.$ Fig. 2). These are the most general FPOs in Kerr 111This follows from the inexistence of more than one radial turning point for photon motion on the Kerr solution., all of them unstable.

For generic stationary and axisymmetric spacetimes, we define FPOs as follows:

Definition: let $s(\lambda):\mathbb{R}\to\mathcal{M}$ be an affinely parameterised null geodesic, mapping the real line to the space-time manifold $\mathcal{M}$. $s(\lambda)$ is a FPO if it is restricted to a compact spatial region – it is a bound state – and if there is a value $T>0$ for which $s(\lambda)=s(\lambda+T),\forall\,\lambda\in\mathbb{R}$, up to isometries.

In coordinates $(t,r,\theta,\varphi)$ adapted to the stationarity and axi-symmetry vector fields, $\partial/\partial t$ and $\partial/\partial\varphi$ respectively, this definition requires periodicity only in ($r,\theta$). Generically, LRs can be determined via the $h_{\pm}(r,\theta)$ functions defined in Cunha et al. (2016). A LR is either a saddle point or an extremum of these functions, for fixed ($r,\theta$). The analogue of spherical orbits in non-separable spacetimes, however, is meaningless, since $r=const.$ is not preserved by mixing $r$ and $\theta$, and no key property, such as separability, singles out a particular coordinate chart.

Classification. The null geodesic flow on a spacetime $(\mathcal{M},g_{\mu\nu})$ is described by the Hamiltonian $\mathcal{H}=\frac{1}{2}g^{\mu\nu}p_{\mu}\,p_{\nu}=0$, where $p_{\mu}$ is the photon’s 4-momentum. Besides stationarity, axi-symmetry and asymptotic flatness, with the metric expressed in the aforementioned coordinates, we further assume a $\mathbb{Z}_{2}$ reflection symmetry on the equatorial plane $(\theta=\pi/2)$ and metric invariance under the simultaneous reflection $t\to-t$ and $\varphi\to-\varphi$ 222Gauge freedom is used to set $g_{r\theta}=0$..

In terms of the first integrals $p_{t}\equiv-E$ and $\Phi\equiv p_{\varphi}$, we define a potential $V(r,\theta)$ and a kinetic term $T\geqslant 0$ Cunha et al. (2016):

[TABLE]

$V>0$ defines a forbidden region in phase space. At its boundary, $V=0\Rightarrow p_{r}=0=p_{\theta}$. From Hamilton’s equations, $\dot{p}_{\mu}=-\frac{1}{2}\left(\partial_{\mu}g^{rr}p_{r}^{2}+\partial_{\mu}g^{\theta\theta}p_{\theta}^{2}+\partial_{\mu}V\right)$ 333$\dot{p}_{\mu}$ denotes derivative with respect to an affine parameter.. The limit $V\to 0$ leads to $\dot{p}_{\mu}\to-\frac{1}{2}\,\partial_{\mu}V$. Hence, photons can only hit the boundary of the allowed region $(V=0)$ perpendicularly. The null geodesic flow only depends on an impact parameter $\eta\equiv\Phi/E$; fixing $\eta$ determines the boundary of the forbidden region $V=0$.

Within this setup, we categorized FPOs as $X^{n_{r}\pm}_{n_{s}}$, where $X=\{O,C\}$, and $n_{r},n_{s}\in\mathbb{N}_{0}$:

i) they either reach the boundary [class O (open)], or they do not [class C (closed)], in which case they loop;

ii) they are either even (subclass+) or odd (subclass-) under the $\mathbb{Z}_{2}$ reflection symmetry. For odd states a distinct mirror orbit exists;

iii) they cross the equatorial plane ($\theta=\pi/2$) at $n_{r}$ distinct $r$ values (subclass${}^{n_{r}}$). Orbits on the equatorial plane, such as LRs, have $n_{r}=0$ (they never cross it);

iv) They have $n_{s}$ self-intersection points (subclass${}_{n_{s}}$).

Some illustrations of these orbits are given in Fig. 1. Typical LRs and more generic planar orbits are type $O^{0+}_{0}$ (left and right panels). Examples of the latter have been found, $e.g.$ in Cunha et al. (2016). $\mathbb{Z}_{2}$ odd orbits, such as $O^{0-}_{0}$, exist for instance in the $\mathbb{Z}_{2}$ Majumdar-Papapetrou dihole Shipley and Dolan (2016). The Kerr FPOs are all of class $O^{1+}_{0}$. We have verified class $O^{2+}_{1}$ and $C^{2+}_{0}$ exist for rotating Proca stars Brito et al. (2016).

Stability. The stability of FPOs can be analysed with Poincaré maps (see $e.g.$ José and Saletan (1998)). The relevant phase space is the 4-dimensional manifold $\mathbb{M}$, parameterized by $(r,\theta,\dot{r},\dot{\theta})$. Consider a null geodesic $s$ on $\mathbb{M}$ and let $\mathbb{P}$ be a Poincaré section, a submanifold of $\mathbb{M}$, which is assumed to intersect $s$ at multiple points. Usually the dimension of $\mathbb{P}$ is taken to be $\dim(\mathbb{M})-1=3$, but since there is an additional Hamiltonian constrain, we consider dim($\mathbb{P}$)=2. A Poincaré map $f:\mathbb{P}\,\to\,\mathbb{P}$, sends a given point of intersection with $s$ to the next intersection point. Parameterising $\mathbb{P}$ by ${\bf x}=\{x^{1},x^{2}\}$, the Poincaré map reads $f({\bf x}_{n})={\bf x}_{n+1}$. This defines a discrete sequence of the intersection points, indexed by $n$.

For a FPO, it is always possible to find $\mathbb{P}$ having fixed points $\tilde{{\bf x}}$ of this map, at which $f(\tilde{\bf x})=\tilde{\bf x}$. Its stability is determined by the behaviour of $f$ in the neighbourhood of $\tilde{\bf x}$. Taylor expanding to first order reads $f({\bf x}_{n})\simeq f(\tilde{\bf x})+\nabla f(\tilde{\bf x})\cdot{\bf y}_{n}$, where $\nabla f$ is a $2\times 2$ matrix $A_{kj}\equiv(\nabla f^{k})_{j}=\partial_{j}f^{k}$ and ${\bf y}_{n}\equiv{\bf x}_{n}-\tilde{\bf x}$ is the deviation variable. Neglecting the higher order terms, ${\bf y}_{n+1}\simeq\nabla f(\tilde{\bf x})\cdot{\bf y}_{n}$, such that the $N^{th}$ term of a sequence starting with a deviation ${\bf y}_{0}$ is ${\bf y}_{N}\simeq\left[\nabla f(\tilde{\bf x})\right]^{N}\cdot{\bf y}_{0}$. The value of ${\bf y}_{N}$ may diverge depending on the properties of (the matrix) $\nabla f(\tilde{\bf x})$, and in particular, of the modulus of its eigenvalues $\Lambda_{k}$: if $|\Lambda_{k}|\leqslant 1,\textrm{for all }k$, the orbit is stable; if $|\Lambda_{k}|>1,\textrm{for at least one }k$, the orbit is unstable.

Consider O${}^{1+}_{0}$ orbits and let $\mathbb{P}$ be the equator $\theta=\pi/2$. Using the Hamiltonian constraint, a local patch of $\mathbb{P}$ is parametrized by ${\bf x}=(r,\dot{r})$. At the fixed point, $\tilde{{\bf x}}=(\tilde{r},0)$, only two (symmetric) values of $\dot{\theta}$ are possible. For simplicity, restrict $\mathbb{P}$ to include only the fixed point with $\dot{\theta}\geqslant 0$ 444The O${}^{1+}_{0}$ actually intersects $\mathbb{P}$ with a symmetric $\dot{\theta}$ before returning to the initial point on $\mathbb{P}$. However, we could then redefine the map $f(x)\to(f\circ f)(x)$, so that $f(\tilde{x})=\tilde{x}$ without an intermediate point.. Defining $D=\det(A)$ and $T=\textrm{trace}(A)/2$, the eigenvalues are $\Lambda_{\pm}=T\pm\sqrt{T^{2}-D}$. For Hamiltonian systems $D=\pm 1$ José and Saletan (1998). The examples below have $D=1$ and fall into one of two cases. If $T^{2}>1$, one of the eigenvalues has modulus larger than unity, and the orbit is unstable. If $T^{2}\leqslant 1$, the eigenvalues $\Lambda_{\pm}=T\pm i\sqrt{1-T^{2}}$ have unit modulus, leading to a rotation of the Poincaré map around the fixed point, which is therefore stable 555This analysis provides a simple criterion for the stability of the FPO. $D=-1$ was not found within the cases analysed herein. In such case, the orbit is unstable for $T\neq 0$, and stable for $T=0$.

Kerr (and Kerr-like) FPOs. A generic Kerr solution has two LRs (see $e.g.$ Bardeen et al. (1972)), one for a negative impact parameter, $\eta_{-}^{LR}$, and the other for a positive one, $\eta_{+}^{LR}$ 666$\eta$ is always presented in units of the ADM mass $M$.. The specific value of $\eta_{\pm}^{LR}$ depends on the BH spin. A continuum of FPOs exists with $\eta_{-}^{LR}<\eta<\eta_{+}^{LR}$. Each of these is, in BL coordinates, a spherical orbit that crosses the equatorial plane at a given perimetral radius, $r_{\rm Peri}$ 777$r_{\rm Peri}$ is defined such that 2$\pi r_{\rm Peri}=\oint d\varphi\sqrt{g_{\varphi\varphi}}$, where the metric component $g_{\varphi\varphi}$ is taken at a spacelike slice and on the equatorial plane, and $\partial/\partial\varphi$ is the azimuthal Killing vector field., in between those of the two LRs, and attains a maximal/minimal angular coordinate $\theta_{\rm max}$. Observe that $\theta_{\rm max}=0,\pi$ for $\eta=0$, such that $\Delta\theta\equiv|\theta_{\rm max}-\pi/2|$ reaches $\pi/2$. The FPO with $\eta=0$ is actually the only complete spherical orbit; the remaining ones fail to reach high latitudes - Fig. 2 (left and middle panels).

All Kerr FPOs are unstable ($T^{2}>1$). Neighbouring orbits to FPOs either escape to infinity or fall into the BH. Hence, these unstable FPOs determine the edge of the BH shadow - Fig. 2 (right panel). Rotating BHs in modified gravity (or in GR with reasonable matter contents) have typically small deviations from Kerr, including in their shadows. Thus a similar picture for FPOs holds for many rotating BHs, leading, in particular, to (qualitatively) Kerr-like shadows. Examples exist both in GR and beyond GR de Vries (2000); Amarilla et al. (2010); Amarilla and Eiroa (2012, 2013); Grenzebach et al. (2014, 2015); Cunha et al. (2017); Abdujabbarov et al. (2016); Mureika and Varieschi (2016); Dastan et al. (2016).

Non-Kerr FPOs. Significant non-spherical deformation of the Schwarzschild BH can lead to exotic features in its optical images Abdolrahimi et al. (2015). For rotating BHs arising in a reasonable GR model with energy conditions abiding matter, non-Kerr-like shadows have been reported Cunha et al. (2015) for Kerr BHs with scalar hair Herdeiro and Radu (2014, 2015). Here, we illustrate non-Kerrness using a “cousin” model: Kerr BHs with Proca hair Herdeiro et al. (2016). In these hairy BHs, the null geodesic flow is non-integrable and chaos occurs for some (sufficiently) hairy BHs Cunha et al. (2016). Recent work suggests the dynamical formation of Kerr BHs with Proca hair East and Pretorius (2017), justifying a detailed analysis of the theoretical and phenomenological properties of this family of solutions.

Amongst these hairy BHs we have chosen a solution which is a sharp and illustrative example of (non-Kerr-like) FPOs, including stable ones. Its lensing produces the cuspy shadow – Fig. 3 888The corresponding Kerr BH with Proca hair has ADM [horizon] mass and angular momentum $(M,J)=(1.075,0.948)$ [$(M_{H},J_{H})=(0.045,0.012)$]. The Proca field oscillates with frequency $w=0.8$. All these quantities are in units of the Proca mass.. The solution’s (ADM) quantities, $M,J$, match those of the Kerr BH shown in Fig. 2. This is a (very) hairy BH with $\sim$ 96% of the mass and $\sim$ 99% of the spin stored in the “hair” (Proca field).

The salient feature of the cuspy shadow is its non-smooth edge. This feature, which occurs also for some Kerr BHs with scalar hair, is a consequence of the FPOs of this solution, as can be observed by analysing the $r_{\rm Peri}$ and $\Delta\theta$ for these FPOs, in terms of the impact parameter $\eta$ – Fig. 4 (left panel).

Fig. 4 (left panel) informs us that, as for Kerr, there are two LRs, for $\eta_{\pm}^{LR}=-4.75;0.97$. However, differently from Kerr, these LRs are connected by a continuum of FPOs that can be split into three branches: two unstable (with $T^{2}>1$, that connect to the LRs) and a stable one, with $T^{2}\leqslant 1$, in between. A careful analysis of the two unstable branches reveals that only a part of each (green thicker lines) contributes to the edge of the shadow. The remaining unstable FPOs, as well as the stable FPOs, do not. Since the edge of the shadow on the equatorial plane is determined by the LRs, the FPOs that determine this edge must jump between the two branches. The jump occurs at the FPOs C1 and A4, which have the same $\eta\simeq-1.71M$ and attain the same angular deviation $\Delta\theta$. But there is a discontinuity in the size of these orbits, $r_{\rm Peri}(C1)>r_{\rm Peri}(A4)$, inducing the cusp in the shadow, precisely at $\eta\simeq-1.71$ (Fig. 3, right panel, blue line).

The unstable FPOs that are not associated to the shadow edge can, however, impact on the lensing properties of the spacetime. This is manifest in the eye lashes depicted in Fig. 3 (right panel, pink lines) which are associated to FPOs between C1 and B3, and form a clear lensing pattern (inset): a ghost shadow edge from that branch of unstable FPOs. Finally, if any photon bound orbit induces a spacetime non-linear instability Keir (2016); Cardoso et al. (2014), such instabilities would be missed by analysing solely LRs. Indeed, this example illustrates that non-planar stable FPOs may exist without planar ones (LRs).

Remarks. FPOs are the generic counterpart of LRs in a stationary, axisymmetric spacetime (see Shipley and Dolan (2016); Yoshino et al. (2017) for other discussions on extension of LRs). The illustrations herein show that FPOs can have a richer structure than in Kerr, and are instrumental in understanding BH shadows, lensing properties and spacetime stability. Thus, general FPOs can yield spacetime information beyond the scope of LRs. An extension of this concept, for generic spacetimes without any isometries, such as dynamical BH binaries, would be of interest.

Acknowledgements. P.C. is supported by Grant No. PD/BD/114071/2015 under the FCT-IDPASC Portugal Ph.D. program. C. H. and E. R. acknowledge funding from the FCT-IF programme. This work was partially supported by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904, and by the CIDMA project UID/MAT/04106/2013. Computations were performed at the Blafis cluster, in Aveiro University.