Fuzzball Shadows: Emergent Horizons from Microstructure Fabio Bacchini∗ Centre for mathematical Plasma Astrophysics, Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium Daniel R. Mayersony Institut de Physique Th´eorique, Universit´eParis-Saclay, CNRS, CEA, Orme des Merisiers 91191, Gif-sur-Yvette CEDEX, France Bart Ripperda Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA and Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA Jordy Davelaar Columbia Astrophysics Laboratory, Columbia University, 550 W 120th St, New York, NY 10027, USA Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA and Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands H´ector Olivares Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands Thomas Hertog and Bert Vercnocke Institute for Theoretical Physics, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium We study the physical properties of four-dimensional, string-theoretical, horizonless \fuzzball" geometries by imaging their shadows. Their microstructure traps light rays straying near the would- be horizon on long-lived, highly redshifted chaotic orbits. In fuzzballs sufficiently near the scaling limit this creates a shadow much like that of a black hole, while avoiding the paradoxes associated with an event horizon. Observations of the shadow size and residual glow can potentially discriminate between fuzzballs away from the scaling limit and alternative models of black compact objects. Introduction.| The recent images of the shadow of proposed BH alternatives. Here we concentrate on the the supermassive black hole (BH) at the center of M87 subset of fuzzballs known as \microstate geometries" [15{ taken by the Event Horizon Telescope (EHT) [1] open 18]. These are horizonless solutions of low-energy string up the exciting prospect to explore the strong-gravity theory that describe compact objects with an intricate environment near BHs with horizon-scale resolution. geometric and topological inner structure and which can It has long been believed that the vicinity of the hori- be thought of as coherent \microstates" corresponding to zon is well described by BH solutions of classical gen- BHs. Therefore, in contrast to the ECO models usually arXiv:2103.12075v1 [hep-th] 22 Mar 2021 eral relativity. However, many theorists who study the employed in phenomenological studies, fuzzballs admit a quantum evolution of BHs have now concluded that there firm embedding in string theory. Their phenomenology must be modifications to BHs extending to horizon scales has recently begun to be explored; see e.g. [19, 20] and (see e.g. [2]), in order to resolve the information para- also [21{24]. dox that stems from the discovery of Hawking radiation We study the behavior of null geodesics (light rays) in and BH evaporation. A variety of proposals have been four-dimensional fuzzball geometries and use this to ob- made, among which exotic compact objects (ECOs) [3{5] tain the first images of fuzzball shadows as they would such as boson stars [6] or gravastars [7], firewalls signal- be perceived by a distant observer. To this end we first ing a complete breakdown of semiclassical gravity at the construct a novel axisymmetric scaling solution which we horizon [8], new non-local interactions [9], and fuzzball name the \ring fuzzball". Our analysis of this confirms geometries [10]. Since shadows of compact black ob- that, as far as the behavior of geodesics is concerned, jects are sensitive to the object's properties in the near- fuzzballs in the scaling limit can resemble BH geometries horizon region (see e.g. [11{14]), their observation offers arbitrarily well. It also reveals the key features and mech- a promising route to differentiate between some of these anisms of fuzzballs through which such phenomenological 2 horizon behavior emerges. Instead of considering a finite number of point singu- Even though the geometries we study here are su- larities on the circle of radius R, we employ a \smeared" persymmetric and have various limitations [20] (such as configuration of uniform charge density on the entire cir- small angular momentum), we expect that the overall cle. This yields properties of their shadows presented here qualitatively −2q0 4rR sin θ capture those of shadows of more general (as yet uncon- MD0 = p K ; π r2 + R2 + 2rR sin θ r2 + R2 + 2rR sin θ structed) fuzzballs which correspond to realistic BHs. (7) Geometry.| We consider a class of four-dimensional where K(·) is the complete elliptic integral of the first multi-center solutions in supergravity [25{27] with metric kind and we used standard spherical coordinates (r; θ; φ) 3 ds2 = −Q−1=2(dt + !)2 + Q1=2 dx2 + dy2 + dz2 ; (1) on the flat R base. In this case, (4) determines ! = !φdφ and note that (1) is now axisymmetric and equatorially which are smooth and horizonless when uplifted to five symmetric (θ $ π − θ). This microstate geometry, de- dimensions (and have controlled singularities in four di- termined by (5) and (7), is the \ring fuzzball" which we mensions). These solutions are completely determined will study using ray tracing. I by eight harmonic functions H = (V; K ;LI ;M) (where The ring fuzzball solution allows for the \scaling limit" I = 1; 2; 3) on the flat R3 basis spanned by (x; y; z) λ ! 0, in which it approaches the metric of the static, [28, 29]. The metric warp factor is then given by supersymmetric BH given by (1) with 3 J K I 2P 1 P − q0 2 2 K K V = LI = 0;K = 1 + ;M = − + : Q = Z1Z2Z3V − µ V ;ZI = LI + CIJK ; (2) r 2 r 2V (8) 1 1 µ = M + KI L + C KI KJ KK ; (3) 3 2V I 6V 2 IJK This BH has mass 2MBH = 3P +q0 −P and a horizon at p 3 3 r = 0 with area ABH = 16π P (q0 − P ). The angular where for the STU model CIJK = jIJK j [30]. Finally, momentum of the (static) BH vanishes while the ring the rotation one-form ! is determined by fuzzball has J = 2P 3(1 − 3P 2)λ. Note that this ring 1 fuzzball represents the first ever explicitly constructed ∗ d! = V dM − MdV + KI dL − L dKI ; (4) (exactly) scaling solution that is also axisymmetric. 3 2 I I Ray Tracing.| We explore the geometry of the ring 3 where ∗3 is the Hodge star with respect to the flat R fuzzball by tracing geodesics with the ray-tracing code basis, and demanding that it vanishes at spatial infinity. Raptor [36{38]. Ray tracing allows for visualizing the In this Letter, we consider microstate geometries in- environment of compact objects (see e.g. [39]). The ob- spired by the \black-hole deconstruction" paradigm [31{ server is represented by a camera at a specified position 33], which consist of a pair of fluxed D6-D6 brane centers where 106 incoming geodesics converge. Geodesics are surrounded by a halo of D0 brane centers [31, 34, 35]. The numerically integrated backwards in time from the cam- harmonic functions are then given by era to where they originated. The integration is halted when a geodesic reaches an outer \celestial sphere", for 3 1 1 1 P 1 1 which we assume a radius equal to the camera's radial V = − ;M = − + + + MD0; r1 r2 2 2 r1 r2 location. To highlight lensing patterns, each geodesic is I 1 1 2 1 1 assigned a color according to the quadrant (each of equal K = 1 + P + ;LI = −P − : (5) angular width in φ) of the sphere that it originates from. r1 r2 r1 r2 If a geodesic does not reach the celestial sphere within a The \centers" at ~r1 and ~r2 are fluxed D6 branes in a ten- predetermined maximum integration time (correspond- dimensional perspective, and we place these on the z-axis ing to an observation window for a physical observer), at z = ±l=2. For MD0, we can choose to distribute any we assign the color black to it. For example, geodesics number of centers on the circle x2 + y2 = R2 at z = 0; if that fall into horizons will appear black. the total charge of all these centers is −q0 (with q0 > 0), In our four-dimensional model, the D6-D6 centers and the so-called bubble equations (i.e. regularity conditions the D0 ring are (naked) singularities; we have explicitly on the metric) [25, 30] relate l and R as checked that no geodesics that we consider ever approach a singularity close enough to require the application of s q2 boundary conditions at the singularities. (Nevertheless, l = 8P 3λ, R = 2λ 0 − 4P 6: (6) (1 − (1 − 3P 2)λ)2 such boundary conditions can be derived from higher- dimensional string-theoretical embedding and will be dis- This is a one-parameter family of solutions entirely deter- cussed in [40].) mined by λ > 0. We will show that λ can be effectively For each geodesic, we track the position on the celestial tuned to obtain a correspondence between microstate ge- sphere (or lack thereof), the total elapsed (coordinate) ometries and classical BHs. time on its trajectory, as well as: 3 • Redshift, measured as the minimum value of the combined effect of long-term trapping and strong red- p −1=4 −gtt = Q that a geodesic encounters on its shift that makes fuzzball microstates exhibit an emerging path. This provides a measure of how much energy horizon behavior and ultimately mimic a classical BH. a geodesic would lose in escaping from the gravity We also note that the geodesics in the ring fuzzball well.