Black Hole Shadow with Soft Hairs

Light bending around a heavy object such as the Sun has been a direct experimental confirmation of the theory of general relativity, and is coined as the name of “gravitational lensing". Moreover, a detailed study of the lensing pattern can reveal the nature of the astrophysical compact objects such as the mysterious black hole, which is thought of as the vacuum solution of Einstein gravity, or their surrounding environment such as the accretion disk. With the successful observation of the the first image of the supermassive black hole located at the center of the galaxy M87 via Event Horizon Telescope (EHT) Akiyama:2019cqa ; Akiyama:2019brx ; Akiyama:2019sww ; Akiyama:2019bqs ; Akiyama:2019fyp ; Akiyama:2019eap , a global radio observation network, electromagnetic investigation of black hole spacetime in strong gravity regime is now achievable. The light rays coming from uniformly distributed luminous objects in the universe reach the observer from all directions. If there is a black hole between the observer and the source, then the light ray will bend due to the gravity of the black hole. All the light rays which come under the effect of the black hole’s gravitational field can be classified into two classes. If we trace the light rays back from the point of observation, then the rays corresponding to the first class will end up at a light source, while those corresponding to the second class will hit the horizon Perlick:2021aok . The shadow of a black hole consists of a dark part in the observer’s sky, which corresponds to the light-rays of the second class. More specifically, from the observer’s image plane, the boundary of the black hole shadow is defined by the division of the observed null geodesics which neither flies to infinity nor goes to the horizon backward in time. The boundary of the shadow are contributed by photons which was tangential to unstable photon orbits with constant radius. For Schwarzschild black hole, the photon orbits has the same radius. The collection of all these possible photon orbits¹¹1The orbits may called ”light rings”, while a similar name ”photon ring” may refer to other meanings in different literature. See Perlick:2021aok and discussions therein. forms the "photon sphere". As solely determined by the backgroud metric, the shadow cast by the black hole provides a unique opportunity to test General Relativity(GR) in the regime of strong gravity Perlick:2021aok . For example, near-horizon properties of physical charges predicted by various black hole solutions within GR and beyond can be constrained, via the shadow sizes of different black hole solutions EventHorizonTelescope:2021dqv .

The properties of the shadow and its size represent valuable observable common to all metric theories of gravity and can be used to test them for their agreement with EHT measurements, see Cunha:2019hzj ; Li:2020drn ; Liu:2020ola ; Chen:2020aix ; Herdeiro:2021lwl ; kon16 ; you18 ; Konoplya:2021slg ; Li:2021mnx ; Kumar:2020owy ; Moura:2021eln ; Zhang:2021pvx ; Kasuya:2021cpk ; Addazi:2021pty ; Sarkar:2021djs for the recent progress on such investigations for various geometries of black hole or black hole alternative. Along this line of research, we consider an important class of black holes within the context of GR, which are black holes carrying the soft hairs associated with the asymptotic symmetries of spacetime. These asymptotic symmetries are the manifest of the infrared properties due to long-ranged features of the gravity force such as the loop infrared divergence and its cancellation by soft emission Weinberg:1965nx ; Chung:1965zza , and are supposed to yield the gravitational memory effect 1987Natur.327..123B ; Thorne:1992sdb ; Strominger:2014pwa . See Strominger:2017zoo for review of the subject from a modern perspective. As far as we know, the shadow of such black holes has not been intensively studied in the literature.

By carefully analyzing the asymptotic fall-off conditions of the metrics of the asymptotic flat spacetimes, Bondi, van der Burg, Metzner, and Sachs discovered the underlying asymptotic symmetries, now known as Bondi-Metzner-Sachs (BMS) symmetries Bondi:1962px ; Sachs:1962wk ; Sachs:1962zza . The BMS symmetries are not exact isometries of the spacetime, but are defined by choosing appropriate fall-off conditions for the metric components. From Noether’s theorem, the BMS symmetries give rise to conserved charges, which satisfy the so-called BMS algebra. Besides the charges associated with the asymptotic Poincaré symmetry, the so-called hard charges, these conserved charges also include an infinite number of “soft” charges, which can be thought of as the localized modes of momentum and angular momentum on the asymptotic celestial sphere, called by the names of supertranslations and superrotations, respectively. Therefore, these soft charges can be understood as the generators of conformal symmetries on the asymptotic celestial sphere Pasterski:2016qvg . Moreover, it was understood that the BMS symmetries, as the symmetries near the future and null infinities, act non-trivially on the asymptotic scattering states Strominger:2013jfa thus S-matrix via the associated Ward identities He:2014laa to yield the soft theorem discovered long ago by Weinberg Weinberg:1965nx , stating the factorization of the soft factor from the S-matrix.

Since there is no local observable in the general covariant theory such as GR, all BMS conserved charges are defined non-locally. Thus, these charges will also dress the other asymptotic boundaries, such as “hairs" on the horizons of black holes Pasterski:2020xvn . Just like the hard charges become the hard hairs of the black holes, the soft charges will then be dressed as the supertranslation/superrotation soft hairs and change the near horizon geometry. The black holes with linear soft hairs are in agreement with the leading terms of the BMS metrics Hawking:2016msc ; Hawking:2016sgy . However, it is nontrivial to construct black holes with soft hairs as the exact vacuum solutions of the Einstein equations. For 4D GR Compère and Long obtained exact black hole solutions with supertranslation hairs by novel coordinate transformations Compere:2016hzt . In Chakraborty:2017pmn ; Raju:2020smc , fundamental issues related to black hole physics like information paradox and its possible solutions are discussed.

Unlike the black holes, the usual stars have no event horizon to support the soft hairs. This is because that the soft hair generated by the singular coordinate transformation can, however, be moved freely outside the stars to yield no physical effect. Therefore, there is no nontrivial supertranslation state in the usual solar-system like regime to be observed by the light-bending tests. On the other hand, the supertranslation effect can be encoded in the gravitational memory in the usual gravitational wave observations of the mergers of binary black holes Thorne:1992sdb ; 1987Natur.327..123B . However, such memory effect is of higher post-Newtonian order, and it requires better resolution than the one of the current LIGO/Virgo/KAGRA interferometers to observe it unambiguously. The other way to observe the supertranslation hair as the near-horizon effect of a black hole is through black hole shadow observations, as we will discuss in this work.

Some properties of the above supertranslated black holes have been studied, e.g., the proposal of utilizing entanglement of the soft modes to resolve the information paradox Hawking:2016msc ; Bousso:2017dny ; Bousso:2017rsx ; Haco:2018ske , the Hawking radiation flux Chu_2018 ; Compere:2019rof ; Lin:2020gva and the flux-balance laws Compere:2019gft . In this work, we will study the black hole shadow of such supertranslated black holes. Naively, one may expect that any observable such as the size or shape of the shadow should be the same as the Schwarzschild counterpart since the supertranslated black holes are just obtained from the Schwarzschild black hole by large gauge transformations. However, due to the soft hairs, the polar symmetry is broken so that the null condition, i.e., the Hamilton-Jacobi equation of the null geodesic looks very complicated. This implies that the effect of the soft hair on the black hole shadow could be nontrivial. Moreover, the shadow is some kind of non-local observable, and it is not clear how it will be affected by the soft hair generated by the large gauge transformation. This is also an interesting issue we would like to examine in this work.

The rest of the paper is organized as follows. In section 2, we summarize the properties of supertranslated black holes. In section 3, as a precursor study we consider the analytical construction of the shadows for the black holes with linear supertranslation soft hairs. In section 4, we adopt the numerical ray-tracing method to construct the shadows for the supertranslated black holes with the full non-linear soft hair and compare their features with the Schwarzschild ones. Finally, we conclude our paper in section 5. Some technical details are given in the appendices. In Appendix A, we lay down the null geodesic equations in the background metric of the supertranslated black hole, and obtain the separated equations from Hamilton-Jacobi equation. In Appendix B, we sketch the analytical method of shadow construction adopted in section 3. In Appendix C we sketch an accretion model adopted in section 4 to construct the accretion pattern near the supertranslated black hole.

In this paper, we would like to consider the shadow formed by the light geodesics passing by a black hole carrying soft hairs. There are two kinds of soft hairs, the supertranslation ones, and the superrotation ones Hawking:2016sgy . However, the explicit metric has only been constructed for the black hole carrying the supertranslation hairs but not the superrotation ones. For our purpose, we will also focus on the supertranslated black holes in this paper.

The supertranslated black holes are first proposed and studied by Hawking, Perry, and Strominger in Hawking:2016msc . They also showed that the soft hairs can be implanted by shock waves. However, their black hole carries the soft hair only up to linear order. For the first time, Compère and Long Compere:2016hzt constructed the full metric of the supertranslated black hole by applying a nontrivial coordinate transformation to the metric of Schwarzschild black hole in isotropic coordinate so that the resultant metric still solves the vacuum Einstein equation. This soft hair of the metric in Compere:2016hzt is encoded in an arbitrary function on the celestial sphere denoted as $C(\theta,\phi)$. For simplicity, we will assume the $C$-function is $\phi$ independent. In this case, Iofa showed that the metric can be put into a neat Compére-Long-Iofa (CLI) form Iofa:2018pnf ; Iofa:2019jjm :

where

with ${}^{\prime}:=\frac{d}{d\theta}$ and $M$ the mass of the black hole. This CLI metric is reduced to the Schwarzschild black hole metric for $C(\theta)=0$ as expected. If we expand the CLI metric up to the first order in $C$, this metric will be equivalent to the static case of the one in Hawking:2016msc .

The Killing horizon of the CLI metric is determined by $g_{tt}=0$, thus is located at $r=2M$ as its Schwarzschild counterpart. However, the metric is no longer spherically symmetric, so is the near horizon geometry, and there exists some new type of coordinate singularity or even the naked singularity. For example, to avoid the appearance of naked singularities, it requires Iofa:2018pnf ; Compere:2016hzt ,

Otherwise, it yields $g_{rr}<0$ for $r>2M$. On the other hand, a new coordinate singularity characterized by the collapse of the spatial 3-metric, i.e., $\det g_{ab}^{(3)}=0$ defines the so-called supertranslation horizon at $r=r_{SH}$. Explicitly, $r_{SH}$ is determined by the following condition

Note that the supertranslation horizon is nonspherical so that $r_{SH}(\theta)$ is angular dependent. This nonspherical feature due to the soft hairs seems to violate the no-hair theorem. However, the soft hairs will not change the hard hairs of the black hole and are the mere effect of the large gauge transformation. Especially, one can check that the Hawking temperature and the associated Hawking fluxes remain the same as those for the Schwarzschild black hole Chu_2018 ; Compere:2019rof ; Lin:2020gva . This then makes our study of shadow interesting, as it is not so clear if the nonlocal nature of the soft hairs can be detected by the black hole shadow, which is kind of semilocal observable. On the other hand, in reality, the black hole will be surrounded by the accreting matter, whose hydrodynamics will then be affected by the near horizon soft hairs, and yield nontrivial shadow. Therefore, the soft hairs will definitely give observable imprints on the shadow of the black hole.

In this work, we will consider the black hole without the naked singularity to avoid the violation of the weak cosmic censorship conjecture Penrose:1969pc , so that the soft hair function $C(\theta)$ should satisfy (5). Since $C$ can be quite generic, for concreteness we choose the typical spherical harmonics $Y_{\ell,m}$, i.e., we will choose the following $C$’s for the consideration of shadow in this work,

where $N_{\ell}$’s are the normalization constants so that the bound (5) is saturated for ${\mathcal{E}}=1$. In Fig. 1, we have shown the angular dependence of the position of the supertranslation horizon for the choice of $C(\theta)$ given in (7) for $\ell=2,3$. We see that the supertranslation horizons for these two cases are outside the Killing horizon at $r=2M$, but still inside the photon sphere of the Schwarzschild black hole at $r=3M$. We did not show the case of $\ell=1$ because its supertranslation horizon is below the Killing horizon and unobservable. This implies that the supertranslation horizon may not directly change the shape of the shadow. On the other hand, the supertranslation horizon can exceed the photon sphere at $r=3M$ if $\mathcal{E}>1$ to directly change the shape of the black hole shadow. However, in such cases, there appear naked singularities and the CLI metric no longer describes a black hole.

In the following, we will discuss the light-ray behavior around this CLI metric of the supertranslated black hole to examine the effect of soft hairs on the black hole shadow. As a precursor study, we first consider the CLI metric up to the linear order of soft hair function $C$ and construct the black hole shadow by the analytic method. Finally, we construct the black hole shadow for the full CLI metric by the numerical ray-tracing method.

To set up the framework of the black hole shadow construction as a precursor study, we first consider the case with linear supertranslation hair. We like to examine if the shadow problem can be analytically solved by proceeding with the linear perturbation around the known shadow of the Schwarzschild black hole. Although the CLI metric is diffeomorphic to the Schwarzschild metric, it breaks the polar symmetry even at the linear order of $C$ so that the integrability or separability of the null geodesic equations seems lost. However, the integrability could be hidden. Thus, we can obtain the CLI counterpart of the separable equations of the Schwarzschild metric by the coordinate transformation, although these separated equations are not decoupled. In this section and in the Appendix A and B, we explicitly obtain the separated geodesic equations and show that the black hole shadow can be constructed analytically with the help of the above separated geodesic equations. From the results, we will see that the black hole shadow will be affected by the linear soft hair. The effect of the full nonlinear soft hair on the black hole shadow will be studied by the numerical ray-tracing method in the next section.

Due to the strong gravity pull, a light-ray can be captured by the black hole if the impact parameter is not large enough. The shadow cast by the black hole on the celestial plane/sphere is schematically shown in Fig. 2. The procedure to construct the shadow is to trace the trajectories of the light-rays which are tangential to the photon sphere to the observer, and then mark down the positions of these light-rays when crossing the celestial image plane or sphere. When the geodesic equations are integrable, each light ray is uniquely determined by the conserved quantities, i.e., energy and angular momentum. Therefore, those light-rays coming from just outside the photon region will be characterized by specific energy and angular momentum. We can then match the tangent to this light-ray expressed in black hole coordinate representation with the same written in terms of local tetrads at the position of the observer to analytically determine the position of the shadow in the observer’s sky. This is the case for the Schwarzschild and Kerr black holes Grenzebach:2014fha . For the supertranslated black hole described by the CLI metric, the polar symmetry is broken so that the geodesic equations are not integrable. Despite that, in the following, we will solve the photon sphere up to the linear order of supertranslation hair and check if the shadow can be constructed analytically.

The light-ray geodesic equation can be obtained from the variation of the following Lagrangian:

where $\dot{}:=\frac{d}{d\tau}$ with $\tau$ the worldline time. Because the light-ray is null, so we should impose the null condition, which up to the first order of $C$ takes the form (Note in this section we will set $M=1$ for simplicity.):

Since the metric is static and independent of $\phi$, the energy $E$ and $z$-component of the angular momentum $L_{z}$ are conserved and are defined by

Furthermore, by varying ${\cal L}$ we can further obtain the geodesic equations. However, due to the complication of the CLI metric, the geodesic equations are quite involved and not very illuminating. In Appendix A we have written down the geodesic equations for the full CLI metric and explained the procedure of solving them for the null geodesics with given initial conditions. Besides, for this section we also expand the geodesic equations up to the linear order in $C$, which one can find in (37)-(40) of Appendix A.

Using the geodesic equations (37)-(40) and null condition (9), we can solve the conditions for the photon sphere, i.e., $\dot{r}=\ddot{r}=0$. The results are as follows:

where the $\theta_{0}$ and $\theta_{1}$ are the leading and first order solutions of the following two equations respectively,

The leading order result is the same as the ones for Schwarzschild black hole, it is interesting to see that the radius of the photon sphere remains the same at all orders.

With the photon sphere determined, the remaining work to construct the shadow is to find the light-rays connecting the observer and tangential to the photon sphere. Following the method of Grenzebach:2014fha ; Perlick:2021aok to analytically construct the black hole shadow, we need two linearly independent equations to determine the position of a light-ray on the celestial image plane. The detail of such construction is sketched in Appendix B. It turns out that it is necessary to separate the null condition, as in the cases of Schwarzschild or Kerr black holes, to obtain these two linearly independent equations. The separability relies on the integrability of the geodesic dynamics, which needs the additional conserved quantities such as the Carter constant. Even though the CLI metric seems breaking the polar symmetry, it is obtained from the Schwarzschild metric by the large diffeomorphsim so that the scalar quantities such as the Carter constant should be preserved. In Appendix A.1 we obtain the counterpart of two separated equations from the null condition of the Schwarzschild metric by performing the corresponding large diffeomorphsim. Although these two resultant equations are non-decoupled, they are helpful for obtaining the linearly independent equations in determining the position of a light-ray on the image plane.

Based on the above result, we can obtain the black hole shadow for the non-rotating black hole with linear supertranslation hair analytically without implementing the numerical ray-tracing method. The above construction and result are given in Appendix B, and a typical plot of black hole shadow is shown in Fig. 3 for the choice of $C(\theta)$ given in (23) for observer situated at $r_{O}=100M$ and $\theta_{O}=30^{\circ}$ and $90^{\circ}$.

In both the figures 3(a) and 3(b), the magenta dashed curve shows the boundary of the shadow of a Schwarzschild blac khole. In Fig. 3(a), we can see that the shadow of a supertranslated black hole is shifted vertically. In the next section, we will see that it is consistent with the shadow obtained by ray-tracing method. From the observational point of view, the shape deformation is the important factor to study. In this section and the next section, we calculate shape asymmetry $\mathcal{A}$ as defined in (26). The shape asymmetry for Orange and Cyan curve in Fig. 3(a) is $\mathcal{A}=0.0188$ and $0.0105$ respectively. On the other hand, in Fig. 3(b) we see that the shadows remain intact by the linear soft hair for $\theta_{O}=90^{\circ}$. In the next section, we will see that the shadows of the full CLI metric bear the similar features shown above for the linear CLI case.

In this section, we numerically calculate the shadow of the black hole with soft hair, which is described by the full CLI metric (1) for the choices of the soft hair functions specified in (7). After specifying the initial condition for rays at the observer’s image plane on the celestial sphere, we compute the null geodesics to trace each ray backward in time. The above ray-tracing procedure is schematically illustrated in Fig. 4.

In this paper, we studied the shadow of the black hole of soft hairs. Our work is motivated by the recent EHT observations of black hole shadows, and the proposal of black hole soft hairs motivated by the revival interest on the infrared structure of gravity. Our study can be seen as an attempt to explore the black hole information paradox and no-hair theorem from the observational point of view. Our results show that the correction to the shadow image due to soft hair, although tiny, is unique. Namely, within the numerical error, we find that the size and vertical position of the black hole shadow will be modified by the soft hairs, but not the horizontal position and its shape asymmetry deviating from the perfect circle of Schwarzschild case. Moreover, we also show that the accretion patterns of the matter surrounding the black hole will be also affected by the soft hairs.

Our study in this paper is quite preliminary in the observational sense, since we consider only Schwarzschild-like black holes, since the Kerr-like black holes could be more common in the universe. However, at this stage there is no known solution for the Kerr black hole with soft hairs having been constructed. Despite that, the same methodology can be applied once new black hole solutions with soft hairs are found, and the shadow features we have found in this paper can serve as the reference guide for the new constructions. Moreover, the more generic soft hair functions beyond what considered in this paper could introduce more features for the shadow morphology, which requires more intensive study for the future works.