⚛️ general relativity

Analytical approach for calculating shadow of dynamical black hole

This paper develops a compact analytical framework to describe how time-varying mass in spherically symmetric spacetimes (such as Vaidya spacetimes) shifts the photon sphere and modifies the observable black hole shadow through a force-decomposed radial equation and a gauge-invariant energy-flux relation.

Original authors: Vitalii Vertogradov, Ali Övgün

Published 2026-02-10

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Vitalii Vertogradov, Ali Övgün

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Growing Shadow: A Simple Guide to Black Hole Dynamics

Imagine you are standing on a pier at night, looking out at a massive, dark whirlpool in the middle of the ocean. You can’t see the water itself because it’s pitch black, but you can see the way it bends the light from the distant stars behind it. That dark circle in the middle of the stars is the "shadow" of the whirlpool.

In space, black holes do something similar. They are so heavy and powerful that they bend light into circles, creating a dark "shadow" that tells us how big and heavy the black hole is.

Until now, most scientists have studied black holes as if they were frozen statues—unchanging and still. But in the real universe, black holes are "hungry." They are constantly swallowing gas and stars (accretion) or losing energy through radiation. They are dynamic, meaning they are constantly changing size and strength.

This paper, written by Vitalii Vertogradov and Ali Övgün, provides a new mathematical "map" to understand how a black hole's shadow changes as the black hole grows or shrinks.

The Three "Forces" of Light

The authors looked at how light travels near a black hole and realized that the light isn't just being pulled by gravity; it’s caught in a tug-of-war between three different "forces":

The Centrifugal Force (The "Spin" Factor): Think of a person on a merry-go-round. The faster they spin, the harder they are pushed outward. Light orbiting a black hole has this same outward "push."
The General Relativity Correction (The "Curvature" Factor): This is the extreme gravity of the black hole. It acts like a powerful inward vacuum, trying to suck the light into the center.
The Induced Term (The "Changing Weight" Factor): This is the paper's big breakthrough. Imagine you are trying to balance a spinning top on your finger, but someone keeps adding weight to the top while it's spinning. It becomes much harder to keep it steady! This "induced term" represents how the changing mass of the black hole pushes or pulls on the light.

The "Expanding Shadow" Metaphor

The authors used a specific model called the Vaidya spacetime to test their math. Think of it like this:

The Growing Black Hole (Accretion): Imagine a snowball rolling down a snowy hill, picking up more and more snow. As it gets bigger, its "shadow" on the ground gets larger. The paper proves mathematically that as a black hole eats more matter, its "photon sphere" (the ring where light orbits) expands, and its shadow gets bigger and wider.
The Shrinking Black Hole (Evaporation): Now imagine an ice cube melting in the sun. As it loses mass, it gets smaller. The paper shows that if a black hole loses mass, its shadow will shrink.

Why does this matter?

For years, we have been taking pictures of black holes (like the famous images of M87*). But those images are just "snapshots" in time.

If we want to truly understand how black holes live and die, we can't just look at one photo; we need to see how they change. This paper gives astronomers a mathematical ruler. It allows them to look at a changing shadow and work backward to figure out exactly how much "food" the black hole is eating or how fast it is losing weight.

In short: This paper provides the toolkit to watch the "breathing" of a black hole—seeing it grow and shrink through the changing shape of its shadow.

This paper, titled "Analytical approach for calculating shadow of dynamical black hole" by Vertogradov and Övgün, presents a theoretical framework for understanding how the shadow of a black hole evolves when the spacetime is not static but undergoing mass changes (accretion or evaporation).

Below is a detailed technical summary of the work.

1. The Problem: Static vs. Dynamical Spacetimes

Most existing literature on black hole shadows focuses on static spacetimes (like the Schwarzschild metric), where time-translation invariance allows for conserved quantities like energy ( $E$ ). However, real astrophysical black holes are dynamical: they grow by accreting matter or shrink via radiation/evaporation.

In dynamical, spherically symmetric spacetimes (such as the Vaidya metric), time-translation symmetry is broken. This leads to two major mathematical challenges:

Non-conservation of energy: The canonical energy of a photon is no longer constant along its geodesic.
Complexity of motion: The second-order equations of motion cannot be easily reduced to first-order forms, making it difficult to analytically locate the photon sphere and calculate the resulting shadow.

2. Methodology: Force Decomposition and Adiabatic Approximation

The authors employ an analytical approach using ingoing Eddington-Finkelstein (EF) coordinates, which are well-behaved at the event horizon. Their methodology follows three main steps:

Force-Decomposed Radial Equation: They derive a radial acceleration equation that explicitly separates the motion into three distinct physical components:
1. Induced term ( $a_i$ ): A new term driven by the time-variation of mass ( $\dot{M}$ ).
2. Centrifugal term ( $a_c$ ): The standard outward force due to angular momentum.
3. General Relativistic (GR) correction ( $a_g$ ): A term representing purely relativistic curvature effects.
Energy-Flux Relation: They derive a gauge-invariant law, $\frac{d(E^2)}{dv} = -\frac{\Lambda}{r}$ , where $\Lambda$ is a generalized flux. This describes exactly how the photon's energy changes as it moves through a changing gravitational field.
Adiabatic Regime: To make the problem tractable, they assume an "adiabatic" limit where the mass changes slowly ( $\dot{M}$ is small). This allows them to treat the photon sphere as a quasi-static structure that shifts slightly over time.

3. Key Contributions

Identification of the Induced Acceleration: The paper introduces $a_i = -\epsilon (\dot{M}/r)\dot{v}^2$ . This term is crucial because it shows that accretion ( $\dot{M} > 0$ ) acts as an additional force that modifies the orbital stability of light.
First-Order Shift Formula: They provide an explicit analytical formula for the shift in the photon-sphere radius ( $r_{ph}$ ) due to mass variation.
Link between Mass Flux and Shadow Angle: They establish a direct mathematical connection between the rate of mass change and the observable angular size of the black hole shadow.

4. Results and Findings

The authors apply their framework to the Vaidya spacetime (a standard model for a radiating/accreting black hole) and obtain the following results:

Accretion ( $\dot{M} > 0$ ): The induced term causes the photon sphere to expand ( $r_{ph}$ increases). Consequently, the critical impact parameter ( $b_{crit}$ ) increases, leading to a larger observable shadow angle.
Mass Loss/Evaporation ( $\dot{M} < 0$ ): The photon sphere contracts, and the observable shadow shrinks.
Quantitative Scaling: For a distant observer, the shadow angle $\omega_{sh}$ scales linearly with the mass $M(v)$ in the adiabatic limit. The authors provide numerical tables demonstrating that a 5% increase in mass results in a proportional ~5% increase in the shadow angle.

5. Significance

This paper is significant for both theoretical and observational astrophysics:

Theoretical Refinement: It refines the "force-balance" concept of photon orbits, extending it from static to dynamical settings.
Observational Benchmark: As technologies like the Event Horizon Telescope (EHT) continue to improve, astronomers will need to distinguish between different black hole models. This paper provides "observer-ready" analytical expressions that can be used to predict how a shadow might change over time if a black hole is undergoing rapid accretion or mass loss.
Foundation for Future Work: The framework serves as a baseline for more complex studies involving rotating (Kerr-Vaidya) black holes, plasma effects, and non-adiabatic (rapid) gravitational collapse.