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    Splitting Black Holes - Take 2
    By Johannes Koelman | December 15th 2012 11:22 AM | 13 comments | Print | E-mail | Track Comments
    About Johannes

    I am a Dutchman, currently living in India. Following a PhD in theoretical physics (spin-polarized quantum systems*) I entered a Global Fortune

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    So, is it or is it not possible? Can black holes be split?

    In my last blog I demonstrated that the laws of thermodynamics forbid the splitting of a single black hole. However, I also demonstrated the same laws don't forbid the splitting of a pair of black holes into many.

    This leaves the door wide open for splitting black holes by smashing them together.

    Or does it?

    I got some good reactions to my 'splitting black holes' blog post. I am happy to see that most of the Hammock Physicist readers who reacted consider such a process, although thermodynamically not forbidden, still to be deeply unphysical. You guys are doing better than the commenters at Physics Stack Exchange! Over there, the claim that multiple black holes can be split went unchallenged.

    The conclusive argument against black hole splitting is surprisingly simple and provides us with a nice instance of the thoughts behind a given law being more useful than the law itself.

    A simple example will make it perfectly obvious that the second law of thermodynamics (the requirement of entropy non-decrease) is only part of the story.


    Colliding Trays


    Imagine trays containing colored balls. Let's say we have two trays: one containing eight slots filled with four red and four yellow balls, and another tray with the same number of slots filled with four green and four blue balls. We smash these trays together. This leads to a chaotic process in which the balls redistribute over the trays. Suppose the outcome is one tray filled with four red and four green balls, and the other tray with four yellow and four blue balls.

    Would you be surprised?




    Of course you would be. Yet, I could come up with the argument that the entropy has not decreased. There are a number of ways (70 ways, to be precise) to distribute an equal number of balls of two contrasting colors over eight slots. This count is obviously independent on the choice of colors, as long as these provides a means to distinguish the two groups of balls. Therefore, the initial state can be realized in 70x70 or 4,900 ways, and the final state can be realized in exactly the same number of ways.

    With entropy being nothing else than the logarithm of the number of realizations compatible with the large-scale physical description of the system, it follow that the entropy has not changed. Before, as well as after the collision, the entropy is given by Ln(4,900) = 8.5.

    As the entropy hasn't decreased, I can argue that the outcome of the collision is in accordance with the second law of thermodynamics, and is therefore entirely feasible.

    This, obviously, makes no sense.

    This simple example makes it immediately clear how to resolve the issue. Key is that the entropy of the post-collision state could have been much higher. In fact, distributing four red, four green, four blue and four yellow balls over sixteen slots can be done in 63,063,000 ways. As we don't control the detailed configuration that is produced, we should be very surprised if we observe a state that can be realized in only 4,900 out of a total of 63,063,000 ways. In other words, we should expect a post-collision entropy close to Ln(63,063,000) = 18.0, and not a post-collision entropy of Ln(4,900) = 8.5.


    Colliding Black Holes


    Now back to black hole collisions. Does the same argument work for these?

    You bet. Also for black hole collisions, to determine what is a statistically favorable collision outcome requires a comparison between alternative results. If there is an outcome that can be realized in overwhelmingly more ways than any of the alternatives, that is the outcome that will result.

    Let's see how this works out. As an example we take two black holes of 3N Planck masses each. We consider two alternative scenarios:

    A) 'splitting': 3N + 3N --> 4N + N + N

    B) 'merging': 3N + 3N --> 6N

    We don't need Einstein's general relativity, we don't need to consider curved four-dimensional spaces, and we even don't need any serious math. All we need as input is the fact that a black hole containing M Planck masses has entropy S=4πM2. From this we deduce that the initial state has total entropy S=72πN2, and can be realized in eS = e72πN2 ways.

    The end products from scenario A) has exactly the same entropy as the initial state (S=72πN2) and can therefore also be realized in e72πN2 ways. For large N, the number of realizations of this final product is spectacularly large. Even for a very modest N=5 there are e1,800π = 102,456 realizations, a figure that dwarfs astronomical numbers like Googol (10100). For N=20 the number of realizations is a staggering 1039,294.

    Although these are numbers beyond comprehension, scenario A) does not represent the statistically favorable transition. This is because scenario B) leads to twice the entropy S=144πN2, encompassing a number of micro states that is the square of the number obtained under scenario A). For N=20 this corresponds to 1078,588 micro states, a figure overwhelmingly larger than the corresponding figure for scenario A).

    The conclusion is that although non entropy-decreasing black hole splitting reactions can be defined, these are not realizable from a statistical physics perspective.

    No black hole collision will form a spray of mini black holes. Not now, not on the 21st of this month, and not in four billion years when our galaxy collides with Andromeda. Sorry to disappoint you. I'm afraid you have to place your bets on another Armageddon.

    Comments

    question about black holes and gravity: thinking about how so much matter can be squeezed into a singularity, the answer (if I understand correctly) is that the matter just stacks upon itself down an infinitely long weird space-time well. In a sense, the last item in masks the ones below. But doesn't this impact the gravitational heft of the black hole ? The matter at the bottom of the well, it seems, should behave gravitationally as if it is farther away from the surface that we experience. If so, the black hole should "weigh" less than the sum of what went into it, so in effect matter would appear to be lost, gravitationally speaking. And if that's the case (long shot), does it impact the calculations of dark energy ? The assumption has been that matter is not destroyed, but what if it is being "masked" by the black hole, so that over time as more matter falls into black holes there is less gravitational energy restricting the expansion of the universe ?

    Can' t be excluded the matter falling into the black hole later goes to supply a tissue of micro black holes surrounding the big one ... in fact if a "field is delineated" or has boundaries because in that volume a feature is altered.
    A black hole can be superconducting also because where there is not 'motion of particles" you might have absolute zero and then with a supercold, the superconducting property....
    The mass of a black hole is rigid or motionless as a result of the terrible gravitation .
    It is therefore possible that the superconductivity is very special, like "instantaneous" ... would explain the 'quantum entanglement! I hope it is enough clear ....

    Johannes Koelman
    @Georges -- that's a fair question, and - not surprisingly - a question that has been asked multiple times. The short answer is that Einstein's equation for gravity is very clear about this: gravity doesn't attenuate when particles dive deep under the black hole's horizon. This can be understood by the hand-waving argument that the 'deep well' is a well in time, not in space. The 'direction of time' for a particle passing the horizon is pointed inwards.
    Mr Koelman, maybe it's a problem of perception, but I would say that the Mikado toy model has returned under a different prospective!
    If the gravitation originates somehow by entropy it still not exactly clear, but for me, Mikado toy model reproduces many aspects of it.
    About the figure with colored balls in this article, I would say that is very closed to a new representation of a Feynman diagram, where is visible the clash electron-positron and two gamma rays ...
    Also, I would say that you can sum it all up by saying that the 'entropy may be a phenomenon that involves matter, anti-matter and radiation. Because :
    without this phenomenon, embedded in the foundations of reality, one could never speak of increasing chaos or disorder, I mean , without the 'rise of the items or alteration of motion.
    The new theoretical frontier that offers microscopicblacks holes now it seems to take a lot, but it is not a reinterpretation of the "torsion field"?
    Another question a micro-black hole can swirl faster than light?
    If so we might think that the reality "contains" two different regimes?
    One made of linear motion and one rotational motions, perpetually separated by those ones rotating faster than light?
    A Dual-Motion Universe theory?
    I want to remind that the different nature of linear motion , take you straight to a Galileo Galilei' job ...
    Soon I'll explain how to interpret the fine structure constant but laugh so hard you have to prepare well.
    See you soon. Giacomo.

    As I understand it, a solution with lower entropy will be chosen if the solution with maximal entropy is excluded for other reasons. So, for BH splitting to occur, there should be an interaction that is strong enough to create black holes, but that excludes the creation of a single BH.

    Say, the combined single BH would have to much angular momentum.

    I my very naive vision, I see two BHs moving head on with (close to) the speed of light. They miss each other by a separation outside of the combined horizon in their center of mass. In this scenario, a combined BH could end up with too much angular momentum while the interaction could still be very close to a combined horizon. Excess angular momentum could then be shed by spinning off small BHs.

    Obviously, this is a rather classical view of BHs, and there might not be an interaction at all.

    Johannes Koelman
    Rob -- would suggest to change your first sentence into something like: "a solution with lower entropy can emerge if its initial state is microscopically different from all the initial states leading to maximal entropy".
    "a solution with lower entropy can emerge...."

    That is probably what I said. However, I have difficulty in parsing that sentence.

    My point is indeed that the "initial state" is such that the "maximal entropy state" cannot be reached from it. That could be because, say, the total angular moment of the initial state is larger than is possible for a single black hole of that mass, which would be the maximal entropy state.

    Johannes Koelman
    The point is that angular momentum and charge (next to mass the only two remaining internal macroscopic variables for black holes) constrain the number of micro states particularly for lighter black holes. So adding angular momentum (or charge) to the picture actually makes the argument against black hole splitting stronger.
    "So adding angular momentum (or charge) to the picture actually makes the argument against black hole splitting stronger. "

    My question is along these lines. If two black hole circle each other, or have a close encounter at near-light speed, the angular momentum of the system could be larger than can be carried in a single, extremal, black hole with the mass of both combined, when it is assumed that the encounter was not close enough to encircle both holes in a single horizon.

    Normally, the excess angular momentum is radiated away with gravitational waves.

    My question is then, could there be constellations where angular momentum is too high for a single black hole, but the encounter is close enough for some kind of (theoretical) interaction that would transfer mass/energy from one black hole to the other? Maybe creating new, smaller, black holes to get rid of the excess angular momentum. (I am really fantasizing here)

    Mr Koelman I wrote a couple of comments and I have had no consideration, I'm used to, but we would keep asking her what he thinks of superconductivity of a black hole?
    Best things . giacomo.

    i am sorry for the ugly translation.

    Very interesting concept.

    I've always been interested in the mechanics of how black holes operate, but I haven't done didn't much research on this topic.

    The concept that black holes can be split as long as they collide with one another is quite fascinating.

    I will have to go back and read your first blog post so that I can grasp a better understanding on this.

    Thanks for sharing!

    I’m not a science major but space is an amazingly interesting thing to learn about.

    How do they collide if, there is no space between them?