[Paper Review] Rotating charged black hole in $4D$ Einstein-Gauss-Bonnet gravity: Photon motion and its shadow

Uma Papnoi, Farruh Atamurotov|arXiv (Cornell University)|Nov 30, 2021

Astrophysical Phenomena and Observations4 citations

TL;DR

This paper constructs a rotating charged black hole solution in 4D Einstein-Gauss-Bonnet (EGB) gravity using the Newman-Janis algorithm and investigates its photon trajectories, shadow morphology, horizon structure, effective potential, and energy emission. The Gauss-Bonnet coupling constant α and black hole charge Q significantly alter the shadow size, distortion, and energy emission rate, with higher α or Q reducing the shadow size and emission rate while increasing distortion.

ABSTRACT

We construct a charged rotating black hole in 4D Einstein-Gauss-Bonnet (EGB) gravity starting from charged black hole in 4D EGB gravity using complex coordinate transformations suggested by Newman-Janis. Further, we have studied the null geodesics to investigate the shape of the shadow cast by a rotating charged black hole in 4D EGB gravity. Also we have discussed their horizon properties and shadow cast. The phenomenon of black hole shadows alongwith the horizon structure and energy emission has been analysed to see the influence of Gauss-Bonnet term and black hole charge on horizon, shadow, effective potential and energy emission rate which are compared to their non rotating counterpart. It has been seen that the Gauss Bonnet parameter have an influence on the shape and size of the shadow as well as on the effective potential, horizon and energy emission rate.

Motivation & Objective

• To extend the static, charged black hole solution in 4D Einstein-Gauss-Bonnet gravity to a rotating, charged counterpart using the Newman-Janis algorithm.
• To analyze the influence of the Gauss-Bonnet coupling constant α and black hole charge Q on the horizon structure and photon geodesics.
• To investigate how α and Q affect the shadow size, shape distortion, and energy emission rate of the black hole.
• To compare the results with standard Kerr-Newman and Schwarzschild solutions in general relativity in the appropriate limits.
• To assess the implications for testing modified gravity theories using black hole shadow observations from missions like EHT and Black Hole Cam.

Proposed method

• Apply the Newman-Janis complex coordinate transformation to the static, charged 4D EGB black hole metric to generate a rotating solution.
• Derive the null geodesic equations from the metric to study photon trajectories and compute the photon sphere radius.
• Calculate the black hole shadow by tracing the boundary of photons that escape to infinity versus those captured by the horizon.
• Compute the effective potential for radial motion and analyze its peak location and height as a function of α, a (spin), and Q.
• Use the limiting absorption cross section σ_lim ≈ πR_sh² to derive the energy emission rate d²E/dωdt = 2π³R_sh²/(e^{ω/T}−1)ω³.
• Numerically plot and analyze the shadow shape, distortion parameter δ_s, and energy emission rate across varying α and Q.

Experimental results

Research questions

• RQ1 How does the Gauss-Bonnet coupling constant α affect the horizon structure of a rotating charged black hole in 4D EGB gravity?
• RQ2 What is the impact of α and black hole charge Q on the size and shape distortion of the black hole shadow?
• RQ3 How do α and Q influence the effective potential and the location of unstable circular photon orbits?
• RQ4 What is the dependence of the energy emission rate on frequency, α, and Q in 4D EGB black holes?
• RQ5 To what extent do the shadow and emission properties of 4D EGB black holes reduce to those in general relativity in the α→0 and Q→0 limits?

Key findings

• The horizon radius decreases with increasing α, a (spin), and Q, and there exists a critical α_ε beyond which no horizon exists for fixed a and Q.
• The effective potential peak increases and shifts leftward with higher α, a, and Q, indicating unstable photon orbits move closer to the black hole.
• The shadow size decreases with increasing α, a, and Q, and the distortion increases with higher α and a, while Q enhances distortion and reduces size.
• The distortion parameter δ_s increases monotonically with both α and Q, as confirmed by numerical plots in Fig. 7.
• The energy emission rate peaks at lower values of α and Q, and the rate decreases with increasing α or Q, indicating slower evaporation for larger α or Q.
• All results reduce to the Kerr-Newman, Kerr, and Schwarzschild solutions in the limit α→0 and Q→0, confirming consistency with general relativity.