# How can a black hole merger create a black hole with an event horizon surface area larger than the sum of the two original areas?

Many popular and professional science sites said something about Stephen Hawking's black hole area theorem being proven observationally, finally, not just mathematically, to 95% confidence. For example, this article and this article.

The theorem says that black holes' surface areas are directly related to their entropies, and can therefore never decrease, only (possibly) increase...

These articles say that the faster the spin of a black hole, the smaller its area... and that ADDING mass should INCREASE its spin, thereby DECREASING its surface area.

But adding mass obviously increases its mass, thereby increasing its surface area. Also, Hawking's own Hawking Radiation theory says that black holes should SHRINK over time. But increasing entropy should increase its area, as said before...?

So... Is there tension between increasing and decreasing the area of a black hole?

In principle, how can the area of the remnant black hole be larger than the total area of the pre-merger black holes?

Also, if we spot a large black hole we've never seen before, and don't know it's history, how can the equations of Kerr and Schwarzschild apply? Wouldn't they be wildly inaccurate if the black hole has gone through dramatic mergers, etc. that we don't know about?

The theorem says that black holes' surface areas are directly related to their entropies, and can therefore never decrease, only (possibly) increase...

The logic is really the other way around. The Hawking area theorem says that the area can only increase. This leads to the idea that we can form an analogy between area and entropy, or possibly find some fundamental link between area and entropy. The Hawking area theorem doesn't invoke any assumptions about entropy or thermodynamics. It's just a theorem in general relativity, and its main assumption is an energy condition, which is roughly speaking a statement that matter can't have negative mass-energy.

These articles say that the faster the spin of a black hole, the smaller its area... and that ADDING mass should INCREASE its spin, thereby DECREASING its surface area.

The total angular momentum of the system is conserved. It doesn't go up or down. If you start with a black hole with spin $$a_1$$ and drop in something else (infalling matter or another black hole) with spin $$a_2$$, then the result is a black hole with spin $$a_3=a_1+a_2$$. Because the spins are vectors, the magnitude of $$a_3$$ can be either greater than or less than the magnitude of $$a_1$$.

The area of the event horizon for a Kerr black hole is $$A=8\pi m(m+\sqrt{m^2-a^2})$$, where $$m$$ is the mass and $$a$$ is the spin. If you compare areas for a fixed mass, then increasing the spin decreases the area. However, the mass is not fixed. What the Hawking area theorem tells us is that the increase in mass is always enough to more than compensate for any increase in spin.

Also, Hawking's own Hawking Radiation theory says that black holes should SHRINK over time. But increasing entropy should increase its area, as said before...?

The Hawking area theorem is a theorem in classical gravity. Hawking radiation is a nonclassical phenomenon.

Also, if we spot a large black hole we've never seen before, and don't know it's history, how can the equations of Kerr and Schwarzschild apply? Wouldn't they be wildly inaccurate if the black hole has gone through dramatic mergers, etc. that we don't know about?

This is substantially what the no-hair theorems are about.

• Angular momentum conservation for a compact binary is $\mathbf{J} = \mathbf{L} + \mathbf{S}$ where $\mathbf{L}$ is the binary orbital angular momentum and $\mathbf{S}$ is the total spin vector. How is the equation $a_3 = a_1 + a_2$ justified (where $a$ is the dimensionless spin)? Aug 1 at 20:44
• @DaddyKropotkin: Same equation, different notation.
– user15381
Aug 2 at 12:20
• Can you explain what happened to $\mathbf{L}$ then? The orbital angular momentum before inspiral/merger is not the same as after insprial/merger... Aug 2 at 12:30

But ADDING mass should INCREASE its spin, thereby DECREASING its surface area..

No. That does not need to be the case. If a mass spins it depends on the state of the added mass how the spin ends up. It can spin faster, the spin can stay the same, or it can spin slower. The final state of the merger can include any of these possibilities. Though the non-spinning state is highly unlikely to occur. That chance is even zero. Both holes, before the merge, are described by the Kerr metric, if we consider the holes isolated. When together you can't just superimpose the metrics. General relativity is non-linear and the superpositio priniple doesn't hold. The final black hole can again be described by Kerr. Regardless its history. As entropy never decreases its size must have increased.

Seen from our perspective all matter ends up on the event horizon. If we consider two separated BHs the the matter is distributed over two horizons. After the merger a new BH has come into existence. With a bigger surface area as the separate ones obviously. If this horizon had a surface that was less than the two separate ones added together, the entropy of the matter would be less than the two separate entropies added. Entropy never decreases. So the new surface area must be twice as big armt least.

Compare this with two containers filled with gas (equal temperature and pressure) that are separated. If the separation is removed, the entropy of the new container will be the sum of the separate containers (the entropy is linear proportional to the volume). The entropy can only be reduced if an external agent performs work. There is no such agent for black holes. Hence the entropy of the constituent matter becomes the sum of both separate ones or increases: the surface area of the new event horizon becomes equal or bigger than the separate areas. This happens theoretically and has a nice interpretation for the entropy to be situated on the surface of the hole. This also means that the entropy for the inside matter can't exceed that of the surface: the holographic principle.The last happens if there are no container walls, so to speak. The same happens when two black holes merge.

The Hawking radiation will obviously. Per unit area less particles are formed at the boundary. There are simply less virtual pairs pulled into reality by the gravitational tidal force because this tidal force is smaller for the bigger hole (I merely use the virtual pair metaphore here as Hawking used this too). For us it looks like matter frozen at the horizon radiates less intense. But the total entropy of all that frozen matter has increased. In a freely falling frame (falling through one of both horizons) the entropy of all matter inside the merged hole has increased after the merger. Both perspectives are equally valid.