Event Horizon

[Pugdaddy]

I just listened to a Reith Lecture on Black Holes from Steven Hawking. His reply to a question was " if two black holes collide and merge to form a single Black Hole, the area of the event horizon around the resulting Black Hole is greater than the sum of the two event horizons of the original Black Holes". The area of the surface of a Black hole is representative of the total information in the Black Hole. How can there be more information. In the recent gravitation wave detection, about 3 solar masses of mass/energy was radiated away in the resulting gravitation wave. So where is the extra information coming from?

[elfmotat]

When he's talking about information, he's speaking of entropy which is a measure of the system's hidden information (information which has been "course-grained" away). Total information, if you can keep track of it, is always conserved. Entropy is not conserved. The amount of "stuff" present doesn't need to change for the entropy of the "stuff" to increase. For example, take the picture below:

 

 

At the top, two types of gas molecules are present in a box but are separated by a barrier. At the bottom, the barrier has been removed and the gasses have been allowed to spread out and mix together. The bottom box's entropy is greater than the top box's, just from allowing the gasses to mix.

 

In geometric units (c=G=1), the area of the event horizon (of a non-rotating BH) is [math]A=16 \pi M^2[/math], and its entropy is [math]S = A/4 = 4 \pi M^2[/math]. If we were to assume a perfect collision between two black holes of mass [math]m_1[/math] and [math]m_2[/math], where no energy is lost, the new combined BH would have a mass [math]M=m_1 + m_2[/math]. Its entropy would therefore be [math] S = 4 \pi (m_1^2 + m_2^2 + 2 m_1 m_2)[/math], whereas the entropy of each BH before collision would be [math]S_1 = 4 \pi m_1^2[/math] and [math]S_2 = 4 \pi m_2^2[/math]. We therefore see that [math]S > S_1+S_2[/math]. The entropy generated from their combination is [math]S_{gen} = 8 \pi m_1 m_2[/math].

Edited April 26, 2016 by elfmotat