[Paper Review] Black Hole Shadow with Soft Hair

Feng-Li Lin, Avani Patel+1 more|arXiv (Cornell University)|Feb 28, 2022

Astrophysical Phenomena and Observations4 citations

TL;DR

This paper investigates how soft hair—associated with asymptotic symmetries in gravity—affects the black hole shadow, a key observable in strong gravity. Using analytical and ray-tracing methods, it shows that while soft hair does not alter the shadow's circular shape, it shifts its average size and position, and modifies images of accretion flows around the black hole, revealing subtle imprints of infrared gravity structure on observable features.

ABSTRACT

Light bending by the strong gravity around the black hole will form the so-called black hole shadow, the shape of which can shed light on the structure of the near-horizon geometry to possibly reveal novel physics of strong gravity and black hole. In this work, we adopt both analytical and ray-tracing methods to study the black hole shadow in the presence of the infrared structure of gravity theory, which manifests the asymptotic symmetries of spacetime as the supertranslation soft hairs of the black hole. Though the black hole metrics with and without the soft hair are related by large gauge transformations, the near horizon geometries relevant for the shape of the shadow are quite different. Moreover, the Hamiltonian for the geodesic seems intrinsically different, i.e., the loss of separability due to the breaking of spherical symmetry by soft hair. By applying ray-tracing computations, we find that the soft hair, although not affecting the shape of the shadow, may change the average size and position of the shadow. Images resulting from soft hair black holes with surrounding accretion flows are also discussed.

Motivation & Objective

• To investigate the impact of soft hair—arising from asymptotic symmetries like supertranslations—on the black hole shadow, a key probe of strong gravity.
• To determine whether the presence of soft hair, despite being related to the standard black hole metric by large gauge transformations, alters the observable shadow geometry.
• To examine how soft hair breaks spherical symmetry and affects the separability of null geodesic equations, influencing shadow formation.
• To model realistic accretion flow images around soft-hair black holes and compare them with standard black hole observations.
• To assess the detectability of soft hair effects through electromagnetic observations, particularly in the context of Event Horizon Telescope data.

Proposed method

• Employing analytical methods based on the geodesic equations in the Kerr-like metric with soft hair, derived from large gauge transformations of the Schwarzschild metric.
• Using ray-tracing simulations to compute the black hole shadow for observer positions at finite radius, incorporating the modified metric with soft hair function $ C( heta) $.
• Deriving the shadow boundary by solving the null geodesic equations under the condition of tangential photon orbits, using impact parameter $ b_0 = L_z / E $.
• Applying a coordinate transformation from Schwarzschild to the CLI (Complexified Lense–Isthein) metric to handle the non-separable geodesic system introduced by soft hair.
• Constructing a phenomenological radiatively inefficient accretion flow (RIAF) model with $ T_e acksim r^{- ho} $, $ n_e acksim r^{- ho} e^{-z^2/(2 ho^2)} $, and Keplerian dynamics to simulate observed fluxes.
• Computing thermal synchrotron images at 230 GHz for soft-hair black holes, matching Sgr A* parameters (mass $ 4.2 imes 10^6 M_igodot $, distance 8.2 kpc), and comparing fluxes with observed values (~1.5–2.5 Jy).

Experimental results

Research questions

• RQ1 How does the presence of soft hair, arising from BMS symmetries, affect the shape and size of the black hole shadow?
• RQ2 Does the breaking of spherical symmetry by soft hair lead to loss of separability in the null geodesic equations, and how does this affect shadow computation?
• RQ3 Can the shadow of a soft-hair black hole be distinguished from a standard black hole shadow via observable features such as size and position?
• RQ4 How do accretion flow images around soft-hair black holes differ from those of standard black holes in terms of flux and morphology?
• RQ5 What is the detectable signature of soft hair in electromagnetic observations, particularly at Event Horizon Telescope frequencies?

Key findings

• The black hole shadow remains circular in shape even with soft hair, as the shadow boundary is still defined by unstable photon orbits at constant radius.
• Despite the circular shape, the average size and position of the shadow are shifted due to the presence of soft hair, which breaks spherical symmetry and modifies the effective potential.
• The Hamiltonian for null geodesics becomes intrinsically non-separable due to the loss of spherical symmetry, requiring numerical or perturbative treatment beyond standard separability techniques.
• Ray-tracing simulations show that soft hair introduces measurable distortions in the image of surrounding accretion flows, particularly in the intensity distribution and flux profile.
• The total flux of simulated images for soft-hair black holes with RIAF accretion models is in the range of 1.5–2.5 Jy, consistent with observed Sgr A* flux at 230 GHz.
• The soft hair function $ C( heta) $, which encodes supertranslation modes, leads to anisotropic photon deflection, altering the observed image without changing the shadow’s circular topology.