<p>The detail you seek is contained in <a href="https://iopscience.iop.org/article/10.1088/0004-637X/777/2/170" rel="noreferrer">Johansson (2014)</a>.</p>
<p>The photon ring around a black hole is not the event horizon. It is the projection of unstable photon orbits that are able to loop around the black hole more than once before heading in our direction. This ring is circular and of radius <span class="math-container">$3r_s/2$</span> for a non-spinning black hole; which is then projected to <span class="math-container">$3\sqrt{3}r_s/2$</span> at the observer. The situation for spinning black holes is much more complex. The radius of the unstable photon orbit <em>does</em> depend on the spin, but the projection of this to the observer maintains almost the same radius as in the non-spinning case. The rings are predicted to be almost circular, but off-centre. Only when the spin approaches its maximal value is appreciable non-circularity expected.</p>
<p>Johansson also discusses the no-hair theorem and how violations of this would affect unstable circular photon orbits and the projected photon ring. These are investigated in a parametric way - e.g. by introducing a quadrupole moment term (that differs from the one implicit to the Kerr metric) to the metric, in addition to the spin. These calculations show that the introduction of these additional terms in the metric introduce changes in the projected photon ring radius and can make it asymmetric - much more so than spin alone.</p>
<p>In terms of motivation, it seems that alternate gravity theories (to General Relativity) admit the possibility of spacetime metrics for spinning objects that depend on more than just their mass and spin (e.g. <a href="https://arxiv.org/abs/1110.2768" rel="noreferrer">Bambi et al. 2010</a>).</p>

CC BY-SA 4.0
2022-04-19T11:52:34.053Z
Thank you for the detailed answer.
Richard
CC BY-SA 4.0
2022-04-19T10:26:01.690Z
2022-04-19T18:19:21.400Z
2022-04-19T18:19:21.400Z
ProfRob
<p>The recent publications of the images of black hole shadow by the <a href="https://en.wikipedia.org/wiki/Event_Horizon_Telescope" rel="noreferrer">Event Horizon Telescope</a> collaboration prompts us to study black hole images in more details. According to the <a href="https://en.wikipedia.org/wiki/No-hair_theorem" rel="noreferrer">no-hair theorem</a>, the black hole space-time can be completely described in terms of only three parameters, namely, the mass, spin and charge of the black hole. However, astrophysically relevant black holes are considered to be neutral due to the existence of plasma environments that neutralizes the charge.  In the <a href="https://eventhorizontelescope.org/testing-general-relativity" rel="noreferrer">official website</a> of the collaboration, they write:</p>
<p><strong>Query 1:</strong></p>
<blockquote>
<p>The diameter of the shadow is proportional to the mass of the black hole and is mostly insensitive to the value of the black hole spin.</p>
</blockquote>
<p>We know that the location of the event horizon is directly related to the spin of the black hole, and its radius is given by <span class="math-container">$r_\mathrm{h}=1+\sqrt{1-a^2}$</span>, where <span class="math-container">$a$</span> is the spin parameter of the black hole. So, how could the diameter of the shadow insensitive to the black hole spin?</p>
<p><strong>Query 2:</strong></p>
<blockquote>
<p>General relativity predicts that the shadow of a black hole should be circular, but a black hole that violates the no-hair theorem could have a prolate or oblate shadow.</p>
</blockquote>
<p>I couldn't understand the reason behind this statement. What is the possible reason for this deviation from circularity depending on the validity of the no-hair theorem? Any links to relevant research papers would be sufficient.</p>

CC BY-SA 4.0
2022-04-19T11:59:55.577Z
[Divergent reflections around the photon sphere of a black hole](https://www.nature.com/articles/s41598-021-93595-w) has plots & equations for black hole photon trajectories, both Schwarzschild and Kerr.
PM 2Ring
CC BY-SA 4.0
2022-04-23T22:58:31.637Z
How did you come to use "… insensitive to the value of (anything)…"?
Robbie Goodwin
<black-hole><general-relativity><event-horizon-telescope>
Doubt regarding size and shape of black hole images published by the Event Horizon Telescope (EHT)
Richard
# Doubt regarding size and shape of black hole images published by the Event Horizon Telescope (EHT)

(Asked by: username_0 on 2022-04-19)

The recent publications of the images of black hole shadow by the [Event Horizon Telescope](https://en.wikipedia.org/wiki/Event_Horizon_Telescope) collaboration prompts us to study black hole images in more details. According to the [no-hair theorem](https://en.wikipedia.org/wiki/No-hair_theorem), the black hole space-time can be completely described in terms of only three parameters, namely, the mass, spin and charge of the black hole. However, astrophysically relevant black holes are considered to be neutral due to the existence of plasma environments that neutralizes the charge. In the [official website](https://eventhorizontelescope.org/testing-general-relativity) of the collaboration, they write:

**Query 1:**

> The diameter of the shadow is proportional to the mass of the black hole and is mostly insensitive to the value of the black hole spin.

We know that the location of the event horizon is directly related to the spin of the black hole, and its radius is given by $r\_\mathrm{h}=1+\sqrt{1-a^2}$, where $a$ is the spin parameter of the black hole. So, how could the diameter of the shadow insensitive to the black hole spin?

**Query 2:**

> General relativity predicts that the shadow of a black hole should be circular, but a black hole that violates the no-hair theorem could have a prolate or oblate shadow.

I couldn't understand the reason behind this statement. What is the possible reason for this deviation from circularity depending on the validity of the no-hair theorem? Any links to relevant research papers would be sufficient.

**Tags:** black-hole, general-relativity, event-horizon-telescope
> **Comments:**
> **username_1**: [Divergent reflections around the photon sphere of a black hole](https://www.nature.com/articles/s41598-021-93595-w) has plots & equations for black hole photon trajectories, both Schwarzschild and Kerr.  
> **username_2**: How did you come to use "… insensitive to the value of (anything)…"?  

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## 1 answers

### Answer by username_3 on 2022-04-19. Score: 17 | Accepted answer

The detail you seek is contained in [Johansson (2014)](https://iopscience.iop.org/article/10.1088/0004-637X/777/2/170).

The photon ring around a black hole is not the event horizon. It is the projection of unstable photon orbits that are able to loop around the black hole more than once before heading in our direction. This ring is circular and of radius $3r\_s/2$ for a non-spinning black hole; which is then projected to $3\sqrt{3}r\_s/2$ at the observer. The situation for spinning black holes is much more complex. The radius of the unstable photon orbit *does* depend on the spin, but the projection of this to the observer maintains almost the same radius as in the non-spinning case. The rings are predicted to be almost circular, but off-centre. Only when the spin approaches its maximal value is appreciable non-circularity expected.

Johansson also discusses the no-hair theorem and how violations of this would affect unstable circular photon orbits and the projected photon ring. These are investigated in a parametric way - e.g. by introducing a quadrupole moment term (that differs from the one implicit to the Kerr metric) to the metric, in addition to the spin. These calculations show that the introduction of these additional terms in the metric introduce changes in the projected photon ring radius and can make it asymmetric - much more so than spin alone.

In terms of motivation, it seems that alternate gravity theories (to General Relativity) admit the possibility of spacetime metrics for spinning objects that depend on more than just their mass and spin (e.g. [Bambi et al. 2010](https://arxiv.org/abs/1110.2768)).

> **Comments:**
> **username_0**: Thank you for the detailed answer.  

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