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Numerical study of hypershadows in higher-dimensional black holes

Jianzhi Yang

TL;DR

This work develops a fully numerical framework in $D=5$ spacetime to compute the hypershadow—the 3D generalization of the black hole shadow—via backward ray tracing. By combining a ZAMO-based local observer basis, a robust definition of 3D impact parameters, and Hamiltonian geodesic integration, the authors reconstruct the full shadow volume for Schwarzschild--Tangherlini and Myers--Perry geometries and introduce visualization and symmetry-diagnostics such as reflection-difference maps. They quantify observer-position effects using distortion $\delta_s$ and displacement $\eta$, showing that rotation shrinks the shadow and, depending on the spin configuration, induces symmetry-breaking or rigid rotations; in the cohomogeneity-one case, a robust rotational symmetry is observed, while in the single-spin case the shadow deforms and translates with $\theta_{o}$ and spin $b$. The framework lays groundwork for numerically exploring more exotic geometries (e.g., black rings) and provides a concrete bridge between analytic and numerical treatments of higher-dimensional black-hole optics, with potential implications for understanding observable imprints of extra dimensions.

Abstract

We develop a fully numerical framework to compute and visualize the \emph{hypershadow}\cite{Novo:2024wyn}, the three-dimensional generalization of the black hole shadow in five-dimensional spacetimes. Our method is based on backward ray tracing and allows flexible control over observer position, enabling the reconstruction of the full shadow volume. For visualization, we combine discrete sampling with surface contouring and introduce reflection difference maps on central slices to quantify mirror symmetries. Applying this method to the Schwarzschild-Tangherlini and Myers-Perry geometries, we validate the former's spherical symmetry and systematically discuss the hypershadow's dependence on observer position and black hole spin parameters. We also provide compact quantitative measures for size reduction and global displacement, revealing clear monotonic trends. The framework is readily extendible to other metrics and opens the way to numerical studies of more exotic objects, such as black rings and their prospective toroidal hypershadows.

Numerical study of hypershadows in higher-dimensional black holes

TL;DR

This work develops a fully numerical framework in spacetime to compute the hypershadow—the 3D generalization of the black hole shadow—via backward ray tracing. By combining a ZAMO-based local observer basis, a robust definition of 3D impact parameters, and Hamiltonian geodesic integration, the authors reconstruct the full shadow volume for Schwarzschild--Tangherlini and Myers--Perry geometries and introduce visualization and symmetry-diagnostics such as reflection-difference maps. They quantify observer-position effects using distortion and displacement , showing that rotation shrinks the shadow and, depending on the spin configuration, induces symmetry-breaking or rigid rotations; in the cohomogeneity-one case, a robust rotational symmetry is observed, while in the single-spin case the shadow deforms and translates with and spin . The framework lays groundwork for numerically exploring more exotic geometries (e.g., black rings) and provides a concrete bridge between analytic and numerical treatments of higher-dimensional black-hole optics, with potential implications for understanding observable imprints of extra dimensions.

Abstract

We develop a fully numerical framework to compute and visualize the \emph{hypershadow}\cite{Novo:2024wyn}, the three-dimensional generalization of the black hole shadow in five-dimensional spacetimes. Our method is based on backward ray tracing and allows flexible control over observer position, enabling the reconstruction of the full shadow volume. For visualization, we combine discrete sampling with surface contouring and introduce reflection difference maps on central slices to quantify mirror symmetries. Applying this method to the Schwarzschild-Tangherlini and Myers-Perry geometries, we validate the former's spherical symmetry and systematically discuss the hypershadow's dependence on observer position and black hole spin parameters. We also provide compact quantitative measures for size reduction and global displacement, revealing clear monotonic trends. The framework is readily extendible to other metrics and opens the way to numerical studies of more exotic objects, such as black rings and their prospective toroidal hypershadows.

Paper Structure

This paper contains 15 sections, 36 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Backward Ray-tracing
  3. Local observer basis
  4. Impact parameters
  5. Geodesic Equation and Initial Condition
  6. 3D Hypershadow Visualization with 2D Shadow Comparison
  7. Hypershadow
  8. Schwarzschild--Tangherlini Black hole
  9. Myers-Perry Black Hole
  10. Cohomogeneity--one Solution: a=b
  11. Rotation in a single plane a=0
  12. Effects of Observer Position on the Black Hole Shadow
  13. Cohomogeneity--one case ($a=b$) vs. Schwarzschild--Tangherlini.
  14. Single--spin case ($a=0$) vs. Schwarzschild--Tangherlini.
  15. Conclusion

Figures (13)

  • Figure 1: Two--dimensional Kerr black hole shadows visualized with different pixel resolutions.
  • Figure 2: Hypershadows of the Schwarzschild--Tangherlini black hole rendered with different $N_{\rm pix}$. The observer position (the initial condition) is located at $\{65,\pi/2-1/20 \}$.
  • Figure 3: (a) Central slices in the $XY$, $XZ$, and $YZ$ planes. (b) H--$\Delta$: difference to the left--right mirror. (c) V--$\Delta$: difference to the up--down mirror. Resolution $N_{\rm pix}=512$.
  • Figure 4: Hypershadows of the Myers--Perry black hole ($a=b=0.45$) rendered with different $N_{\rm pix}$ and viewpoints. And the observer position is located at $\{65,\pi/2-1/20 \}$.
  • Figure 5: Top row: central slices of the hypershadow in the $XY$, $XZ$, and $YZ$ planes. Middle row: H--$\Delta$ maps. Bottom row: V--$\Delta$ maps. Parameters: $a=b=0.45$, observer at $\{65,\pi/2-1/20\}$, resolution $N_{\rm pix}=512$.
  • ...and 8 more figures