[Paper Review] Effect of gravitational wave on shadow of a Schwarzschild black hole

Mingzhi Wang, Songbai Chen+1 more|arXiv (Cornell University)|Aug 13, 2019

Pulsars and Gravitational Waves Research49 references4 citations

TL;DR

This paper investigates the effects of a linearized gravitational wave on the shadow of a Schwarzschild black hole using a first-order perturbation of the spacetime metric. It shows that the shadow oscillates or deforms periodically depending on the Legendre polynomial order 𝑙: odd 𝑙 causes vertical oscillation of the shadow center, while even 𝑙 leads to periodic stretching and squeezing. The perturbation induces chaotic photon trajectories, resulting in self-similar fractal structures in the shadow boundary, with stronger influence on the vertical direction than horizontal.

ABSTRACT

We have studied the shadows of a Schwarzschild black hole under a special polar gravitational perturbation, which is a particular solution of Einstein equations expanded up to first order. It is shown that the black hole shadow changes periodically with time and the change of shadow depends on the Legendre polynomial order parameter l and the frequency σ of gravitational wave. For the odd order of Legendre polynomial, the center of shadow oscillates along the direction which is vertical to equatorial plane. For even l, the center of shadow does not move, but the shadow alternately stretches and squeezes with time along the vertical direction. Moreover, the presence of the gravitational wave leads to the self-similar fractal structures appearing in the boundary of the black hole shadow. We also find that this special gravitational wave has a greater influence on the vertical direction of black hole shadow.

Motivation & Objective

• To analyze how a specific gravitational wave perturbation alters the shadow of a Schwarzschild black hole.
• To investigate the influence of gravitational wave frequency 𝜎 and Legendre polynomial order 𝑙 on shadow dynamics.
• To explore the emergence of chaotic photon trajectories and their impact on shadow morphology.
• To quantify the time-dependent deformation of the shadow using deviation parameters 𝜀𝑜 and 𝜀𝑒.
• To examine the role of observer inclination angle 𝜃𝑜𝑏𝑠 in shaping observable shadow features.

Proposed method

• Adopts a first-order perturbation of the Schwarzschild metric using a special gravitational wave solution derived from Einstein's equations.
• Solves the null geodesic equations in the perturbed spacetime to trace photon trajectories.
• Numerically computes black hole shadows for observers at different inclination angles (𝜃𝑜𝑏𝑠 = 0° and 45°).
• Introduces deviation parameters 𝜀𝑜 and 𝜀𝑒 to quantify oscillation and stretching/squeezing of the shadow over time.
• Analyzes the emitted intensity of light rays to identify multiple images from the accretion disk (ring, thin image, peripheral region).
• Uses numerical simulations to map the time-evolving shadow and detect self-similar fractal structures in the boundary.

Experimental results

Research questions

• RQ1 How does a linearized gravitational wave affect the shape and position of the black hole shadow over time?
• RQ2 What is the dependence of shadow deformation on the Legendre polynomial order 𝑙 and wave frequency 𝜎?
• RQ3 Does the presence of a gravitational wave induce chaotic photon motion, leading to fractal structures in the shadow boundary?
• RQ4 How does the observer’s inclination angle 𝜃𝑜𝑏𝑠 influence the observed shadow dynamics?
• RQ5 What is the relative impact of the gravitational wave on the vertical versus horizontal dimensions of the shadow?

Key findings

• For odd 𝑙 (e.g., 𝑙=3), the center of the black hole shadow oscillates vertically over time, perpendicular to the equatorial plane.
• For even 𝑙 (e.g., 𝑙=4), the shadow center remains fixed but alternately stretches and squeezes along the vertical direction.
• The amplitude of shadow oscillation (𝜀𝑜) and deformation (𝜀𝑒) increases with increasing 𝑙 for a fixed frequency 𝜎.
• Gravitational wave perturbation induces non-integrable photon motion, resulting in self-similar fractal structures on the shadow boundary.
• The vertical dimension (height 𝑕) of the shadow varies more significantly than the horizontal width 𝑤, indicating stronger influence along the vertical axis.
• The average shadow width 𝑤 first increases then decreases with increasing observer radius 𝑟𝑜𝑏𝑠, while height 𝑕 and radius 𝑅 exhibit periodic variation with 𝑟𝑜𝑏𝑠.