Imaging topological solitons: The microstructure behind the shadow

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Imaging topological solitons: The microstructure behind the shadow

Pierre Heidmann, Ibrahima Bah, and Emanuele Berti

Phys. Rev. D 107, 084042 – Published 25 April 2023

Abstract

We study photon geodesics in topological solitons that have the same asymptotic properties as Schwarzschild black holes. These are coherent states in string theory corresponding to pure deformations of spacetime through the dynamics of compact extra dimensions. We compare these solutions with Schwarzschild black holes by computing null geodesics, deriving Lyapunov exponents, and imaging their geometries as seen by a distant observer. We show that topological solitons are remarkably similar to black holes in apparent size and scattering properties, while being smooth and horizonless. Incoming photons experience very high redshift, inducing phenomenological horizonlike behaviors from the point of view of photon scattering. Thus, they provide a compelling case for real-world gravitational solitons and topological alternatives to black holes from string theory.

Received 26 December 2022
Accepted 7 March 2023

DOI:https://doi.org/10.1103/PhysRevD.107.084042

Published by the American Physical Society

Physics Subject Headings (PhySH)

Alternative gravity theories General relativity Quantum gravity Supergravity

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Pierre Heidmann ^*, Ibrahima Bah^†, and Emanuele Berti^‡

Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA

^*pheidma1@jhu.edu
^†iboubah@jhu.edu
^‡berti@jhu.edu

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Issue

Vol. 107, Iss. 8 — 15 April 2023

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Images

Figure 1
Schematic description of a topological star. The spacetime is smooth and terminates at $r = r_{B}$ where the $y_{1}$ circle degenerates. This induces a large topological bubble of charge $Q$ that can be considered as the “surface” of the star.
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Figure 2
Schematic description of a Schwarzschild topological soliton. It is a neutral bound state of three topological stars. The spacetime is smooth and terminates at $r = ℓ + 2 σ$ where the $y_{1}$ and $y_{2}$ circles degenerate alternatively. This induces a large topological bubble that can be considered as the surface of the soliton.
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Figure 3
Scattering properties of topological stars as a function of the charge-to-mass ratio $μ$ . Top: the Lyapunov and critical exponents associated with their outer unstable photon sphere. Middle: description of the two kinds of topological stars. Bottom: their apparent size with respect to the Schwarzschild shadow, as defined in Sec. 3a2.
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Figure 4
Illustration of the artificial background grids. The camera (gray point) is on a “celestial” sphere centered around the gravitational objects (represented as a white ball). The sphere has a large radius with respect to the object size: $R = 20 M$ for the three first backgrounds, and $R = 40 M$ for the last. For the first background, the sphere is covered by a quadricolor patch with a grid of meridians and latitudes of angle $π / 20$ . For the second and third backgrounds, the celestial sphere is covered by pictures of the Milky Way. For the last background, the sphere is totally black, and there is a bright “accretion disk.” The disk has a $π / 3$ inclination angle with respect to the camera-object plane and the disk radius is in the range $[(3 + 1 / 3) M, 5 M]$ . The celestial spheres have been artificially cut here to improve the readability of the figures.
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Figure 5
Gravitational lensing effects of the smooth horizonless solitons compared to flat space and a Schwarzschild black hole. From left to right: the five different backgrounds, flat space, the two kinds of topological stars, the Schwarzschild topological soliton, and the Schwarzschild black hole. From top to bottom: the quadricolor screen, the elapsed time, the maximum redshift experienced, the two artlike sky screens, and the accretion disk picture.
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Figure 6
Gravitational lensing effects of the Schwarzschild topological soliton as a function of the angle between the observer and the axis of symmetry. From left to right: the observer is on the equatorial plane of the soliton $θ = π / 2$ , on the north-hemisphere side at $θ = π / 3$ , and on the axis at $θ = 0$ . The rows follow the same conventions as in Fig. 5, and the scale for the elapsed time (second row) and maximum redshift experienced (third row) are also the same.
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