# Black holes in a head-on collision

Assume two uncharged non-rotating black holes traveling straight at each other with no outside forces acting on the system. What is thought to happen to the kinetic energy of these two masses when they collide? Is the excess energy lost through gravitational radiation? What would the effect of these gravity waves be on matter or energy that they encounter?

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Some energy is lost to gravitational radiation. Some probably ends up in the final black hole (i.e., $m_{\rm final}$ could be greater than the sum of the two initial masses). Figuring out the proportions of these two is not trivial, I would imagine.

The gravitational waves striking matter would not have any terribly dramatic effect. Once you're a decent way away from the black holes, the strain amplitude is low. They'd shake things around a bit.

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During the collision of two black holes gravitational radiation is emitted. After the collision process the gravitational radiation eventually dies off and a single black hole remains.

To treat this 2-body system quantitatively is a major task for numerical relativity, which has seen some big breakthrough after the work by Frans Pretorius. It is quite impractical, if not impossible, to deal with the general relativistic 2-body problem analytically.

The gravitational radiation emitted in such a process - from the inspiralling phase through the merger to the ringdown of the final state black hole - has a characteristic profile that may facilitate the discovery of corresponding gravity waves with an experiment like (advanced) LIGO.

It is possible to give a simple upper bound on the maximum of energy of the gravitational radiation emitted in such a process using Hawking's area theorem.

Suppose for simplicity that the initial state consists approximately of two Schwarzschild black holes with equal masses $m$ and the final state approximately of a single black hole with mass $M$.

The energy $E$ of the gravitational radiation emitted during the collision is given by $E=2m-M$. So you get a maximal amount of gravity waves if $M$ is as small as possible, $M=M_{\rm min}$.

Hawking's area theorem implies that the area of the final black hole is bigger or equal to the sum of the areas of the two initial black holes. Using area $\propto$ mass$^2$ leads to the inequality $M^2 \geq 2m^2$. The final black hole mass becomes minimal if this inequality is saturated, $M_{\rm min}=\sqrt{2}m$.

In conclusion, the maximal amount of energy radiated away in the collision of two equal mass Schwarzschild black holes is given by

$E_{\rm max} = 2m-\sqrt{2}m = (2-\sqrt{2}) m \approx 0.29 \times 2m$

So in the "best case" scenario you can extract at most 29% of the initial energy in the form of gravitational radiation.

You can play the same game with Schwarzschild black holes of different masses or Kerr-Newman black holes.

As to your second question, the gravitational waves emitted in such a process are not different (apart from their characteristic pattern alluded to above) from gravitational waves emitted by other sources. You may check the Wikipage for an animation of how the two polarizations of a gravitational wave distort a ring of particles.

EDIT:

While writing this answer Lawrence B. Crowell posted his answer, which also contains the upper bound on gravitational radiation derived above.

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When two black hole collide the horizon area of the resulting black hole must exceed the sum of the areas of the two initial black holes. This places an upper limit on the amount of gravitational radiation which is produced. If the two initial black holes have equal masses $M$ then the finial black hole must obey $$4\pi(2M_f)^2~=~16\pi(M_f)^2\ge~4\pi (2M) ^2~+~4\pi (2M)^2~=~32\pi M^2$$ and so $M_f~\ge~\sqrt{2}M$. The amount of gravitational radiation emitted is then $E~\le~(2~-~\sqrt{2})M$ which is $.585M$

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