<p>How would you add angular momentum? </p>

<p>The only way to do that is to drop something <em>with</em> angular momentum (relative to the centre of mass of the BH and the thing, if you want to get picky) into the BH. Could be a spinning object, or an object fired in at a oblique angle, or circularly polarised light or even suitably polarised gravity waves.  </p>

<p>Whatever you do, you increase both the mass and the angular momentum of the BH. Now the limit on the spin of a black hole is expressed in terms of a relation between these two quantities and the Schwarzschild radius of the black hole (which is determined by the mass) and it turns out that whatever you add, and however you add it (if you're firing something in obliquely, it has to actually <em>hit</em> the BH so it can't be too ar from the centre of mass) you cannot increase the angular momentum enough relative to the increase in mass to break that limit.</p>

<p>The equations you have to look at are on <a href="https://en.wikipedia.org/wiki/Kerr_metric" rel="nofollow noreferrer">wikipedia</a> see especially <a href="https://en.wikipedia.org/wiki/Kerr_metric#Overextreme_Kerr_solutions" rel="nofollow noreferrer">this section</a>. One way of expressing the limit is given there as 
<span class="math-container">$GM^2 &lt; Jc$</span>  where <span class="math-container">$J$</span> is the angular momentum. Suppose you have a black hole just under this limit and add an object of mass <span class="math-container">$m$</span> angular momentum just under <span class="math-container">$Gm^2/c$</span> velocoty just under <span class="math-container">$c$</span> and on a trajectory that just grazes the Schwarzchild radius of the hole at <span class="math-container">$\frac{2GM}{c^2}$</span>. The angular moment of the result is just under <span class="math-container">$$GM^2/c + 2GMmc/c^2 + Gm^2/c$$</span> where the terms are respectively the original angular moment of the hole, the angular moment from the grazing impact and the angular momentum of the added body. This works out to be (just under) <span class="math-container">$$G(M+m)^2/c$$</span>, the limit for the (now-enlarged) black hole.</p>

CC BY-SA 4.0
2019-04-22T00:27:34.670Z
What if I shoot quickly rotating objects to pass close to the event horizon of the black hole without being captured by it? Wouldn't the framedragging effect from these objects spin up the black hole without adding to its mass?
cuckoo
CC BY-SA 4.0
2019-04-21T10:55:22.573Z
2019-04-21T10:55:22.573Z
Steve Linton
<p>I am aware that a "naked singularity" is not possible. However, what would happen if I added angular momentum to a black hole near the maximal angular momentum limit? Does it asymptote?</p>

CC BY-SA 4.0
2019-04-21T07:13:28.287Z
Can you add angular momentum without adding more mass?
ProfRob
<black-hole><astrophysics>
What happens if I were to add angular momentum to a maximally spinning black hole?
virchau13
# What happens if I were to add angular momentum to a maximally spinning black hole?

(Asked by: username_0 on 2019-04-21)

I am aware that a "naked singularity" is not possible. However, what would happen if I added angular momentum to a black hole near the maximal angular momentum limit? Does it asymptote?

**Tags:** black-hole, astrophysics
> **Comments:**
> **username_1**: Can you add angular momentum without adding more mass?  

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## 1 answers

### Answer by username_2 on 2019-04-21. Score: 2

How would you add angular momentum?

The only way to do that is to drop something *with* angular momentum (relative to the centre of mass of the BH and the thing, if you want to get picky) into the BH. Could be a spinning object, or an object fired in at a oblique angle, or circularly polarised light or even suitably polarised gravity waves.

Whatever you do, you increase both the mass and the angular momentum of the BH. Now the limit on the spin of a black hole is expressed in terms of a relation between these two quantities and the Schwarzschild radius of the black hole (which is determined by the mass) and it turns out that whatever you add, and however you add it (if you're firing something in obliquely, it has to actually *hit* the BH so it can't be too ar from the centre of mass) you cannot increase the angular momentum enough relative to the increase in mass to break that limit.

The equations you have to look at are on [wikipedia](https://en.wikipedia.org/wiki/Kerr_metric) see especially [this section](https://en.wikipedia.org/wiki/Kerr_metric#Overextreme_Kerr_solutions). One way of expressing the limit is given there as
$GM^2 < Jc$ where $J$ is the angular momentum. Suppose you have a black hole just under this limit and add an object of mass $m$ angular momentum just under $Gm^2/c$ velocoty just under $c$ and on a trajectory that just grazes the Schwarzchild radius of the hole at $\frac{2GM}{c^2}$. The angular moment of the result is just under $$GM^2/c + 2GMmc/c^2 + Gm^2/c$$ where the terms are respectively the original angular moment of the hole, the angular moment from the grazing impact and the angular momentum of the added body. This works out to be (just under) $$G(M+m)^2/c$$, the limit for the (now-enlarged) black hole.

> **Comments:**
> **username_3**: What if I shoot quickly rotating objects to pass close to the event horizon of the black hole without being captured by it? Wouldn't the framedragging effect from these objects spin up the black hole without adding to its mass?  

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