### Do colliding black holes destroy information?

*I must premise this post with the fact that my formal physics education ended at sixth form and that these topics are gathered from online articles, discussion with friends and a lot of youtube videos. I hope that if there are errors, they are with my axioms and not my logic but the statements I make and sources I reference should be taken with a pinch of salt.*

I have recently been reading about the entropic theory of gravity and the idea that gravity could be an emergent property is fascinating! I have thought for a long time, talking about it at school, that there might be some fundamental logical underlying physics. What I mean by that is that if you look deep enough into maths you can derive the laws of physics. That the constants have to be that way because they are derived from mathematical laws rather than observation. This is my version of the “Mathematical Universe Hypothesis”. That the universe isn’t just described by mathematics it is mathematics.

I don’t believe this to be true necessarily but I maintain that it would be nice if it were true.

If gravity is an emergent property, something that we give a name to not because there is something fundamental about it but because systems of a particular setup will result in it. Much like how we have names for things like temperature and pressure. There is nothing fundamental about these concepts, they are emergent effects from underlying laws.

The idea, however, is based on a theory of black holes which attempts to solve the problem of Hawking radiation and the loss of information. In writing this post I thought I should give a brief overview of how we reach the problem. However, in checking my memory I discovered quite a few facts which I thought was worth adding, making the history section longer than intended.

Feel free to skip to past the history.

## A brief history of gravity

I was going to start this brief history of the understanding of gravity with Kepler's laws, however, when refreshing my memory about them on Wikipedia I noticed it mentioned that they refine Copernicus’ laws of planetary motion in a way that would fit the arc of history I was going to try and portray here. Delving deeper into a Wikipedia spiral (Depth-first search) I noticed a mention of Aristarchus of Samos. Someone I hadn’t heard of before, delving deeper, he is touted to be the first recorded theorist of a heliocentric model. In true Wikipedia style though the reliability of particular information shows it’s true colours:

>So the theory was most likely developed independently yet attributed. Sure.

Starting now at Nicolaus Copernicus, who postulated that:

- Orbits are a circle
- The sun is at the centre
- The speed of the planet in the orbit is constant

This is not always true however if one assumes a simple case where a single planet is orbiting the sun starting in a circular orbit, all three as we will see agree with Kepler's generalisation, which applies when the orbit is not circular.

Johannes Kepler provided a generalisation to these rules which allowed for non-circular orbits to be modelled.

- The planetary orbit is an ellipse (A circle can be considered a special case).
- The Sun is not at the centre but at a focal point of the elliptical orbit.
- The area speed swept by the planet is constant (For a circle this is equivalent to the actual speed being constant)

The area speed is perhaps a concept which you haven't heard of. If I had, I'd forgotten it purely remembering that Kepler's laws had something to do with area.

Illustration of Kepler's second law. The planet moves faster near the Sun, so the same area is swept out in a given time as at larger distances, where the planet moves more slowly. Source Wikipedia. |

This is a much better model, it matches the observation of planetary motion far better. It can still be considered a refinement of that original theory though.

It would be nice if we could predict what the area speed of a planet should be so that we could, given a hypothetical planet, predict it’s motion. This is what the next theory would bring us.

Sir Isaac Newton, a name everyone should know, showed using his law of gravitation that Kepler's laws were correct (Therefore transitively proving the special case where Copernicus could be correct) but in addition explained tides, the trajectories of comets and unified that the force acting to make things fall was the same force which governs celestial bodies.

*The force between two point masses is proportional to the product of the masses and inversely proportional to the square of the distance between them.*

This theory also has some interesting implications and in part lead to the creation of chaos theory by the introduction of the n-body problem. The theory introduces the idea that all massful objects assert a force on all other massful particles. This is an incredible generalisation but leads to a computational problem.
Not only does this lead to a formulation which is O(n^{2}) in the number of particles but as it turns out a small change in the starting conditions for any system with more than two masses can lead to large differences in the result. This led to a very public blunder for Henri PoincarÃ©. The problem was with his initial submission to an 1887 prize in honour of Oscar II, Kind of Sweden's 60^{th} birthday, to solve the 3 body problem. The submission contained an error first stating there existed stability before conceding that there didn't. PoincarÃ©'s correction formed the creation of chaos theory to study what could be better understood about systems which exhibit this behaviour.

Using this formula, predictions for the orbit of Uranus were being made but with some error. Urbain Le Verrier and John Couch Adams were convinced that this was caused by an, as of yet, undiscovered body. Both men worked on trying to reverse engineer the observations in order to predict the orbit and mass of this unknown body.

Neptune was discovered to be within 1 degree of Urbain Le Verrier’s predicted location! I find the idea truly astounding, that simply using the law of gravity and the laws of motion, you could predict the existence of a planet, but that is what happened.

The next theory comes in a similar way to the prediction of Neptune, discrepancies in where a planet should be based on Newton's law of gravity. This time Mercury was misbehaving and this too was observed by Urbain Le Verrier but was not solved until Einstein's theory of general relativity.

This is where the maths of the topic exceeds my ability quite a bit. The relevant equation is:

This relies heavily on tensor calculus, a topic which I have not yet managed to master however my understanding of the theory's derivation which I haphazardly tried to explain at a dinner party is as follows.

From this theory too, you can derive Newton's law of gravitation (By transitivity the other two), albeit by making some very particular assumptions, much like we did between Kepler and Copernicus. The derivation is above me but is explained in this Quora answer where the author pushes his own book on the topic.

“Under the assumptions of a static spacetime and small curvature... and low velocities.”

There is a lot to be said about this theory because it introduces so many new elements to gravity including removing it from being a force and instead of showing that it is a property of emerging from objects trying to follow straight lines in a curved space-time.

## Setting up the problem

General relativity has some issues, one prediction which until observed caused quite some controversy; the black hole.

I didn’t realise that before Einstein the idea of a body with a gravity so high that not even light could escape existed. However, in checking the ideas for this post I discovered that John Michell in a letter published in November 1784, proposed the idea using Newtonian mechanics that a body with the density of our sun but with a radius 500 times as big would have a surface escape velocity greater than that of the speed of light. He even correctly noted that such non-radiating bodies would be detectable through their gravitational effects on objects around them.

The black holes predicted by general relativity, however, can be much smaller as they are instead the result of the density of matter rather than the amount.

General relativity is still today our best understanding of gravity, it explains almost all of what we observe in exquisite detail. There seems to be no more refinement needed to the orbits of planets in our solar system. There are cracks, however, elsewhere which indicate that GR too is not complete. Observations on the scale of the galaxy show that stars seem to be travelling faster than they would in an orbit described by GR using estimations of the mass of the galaxy from what we can observe. Not a small amount either. Introducing the required extra mass needed to make the equations stack up requires 5 times more mass we can’t see than that which we can. It is postulated that this extra mass comes from dark matter particles, so-called because we can’t observe them. These particles are assumed to not interact with matter other than gravitationally. Something that makes detecting them nigh-on-impossible.

That is not the problem I intend to get around to discussing in this post, however. The problem I want to explore comes about from looking at black holes. These are considered interesting places to think about because they are so extreme. They are also one of the few places where gravity acts on small scales, scales small enough for quantum mechanics to come into play.

My maths is again not good enough to understand the derivations in quantum mechanics but what is relevant to understand here is that uncertainty is fundamental and that you cannot be certain about both the energy of a particle and time. This is represented by the following equation:

This is derived from Heisenberg's uncertainty principle but has the effect of allowing virtual particles to exist. How long these virtual particles can exist depends on their size. The larger they are the less time they can exist for. When we apply this to the boundary of a black hole, at the point where light can no longer escape, the so-called event horizon, the maths I am told leads to one of the virtual particles dropping into the black hole and the other being emitted as radiation. This can be seen in the animation below. I never understood why it is always the mass reducing of the pair of particles that drops into the black hole but rewatching this video, in the hope of improving my explanation, it seems the answer has to do with the way Hawking described the waves representing these particles through time.

This mathematical description of what can be thought of as the temperature of the black hole was first described by Stephen Hawking and is therefore referred to as Hawking radiation. There are much better explanations elsewhere. PBS Space Time is a great channel to help with understanding these topics at a high level without delving into the maths. I hope at the very least I haven't made the topic more confusing.

Hawking radiation leads to a thermodynamic view of black holes as black body radiators (objects which do not reflect light but instead emit light only as a result of their temperature). The realisation that black holes can have temperature meant they aren't the zero entropy objects they had been assumed to be.

It was only while recently browsing a little more on general relativity (I've been trying to understand this lecture series since 2009) that I discovered that the entropy I learnt about in physics and Shannon's entropy my computer science are one and the same. What that means is that what the rules around physical entropy are stating is actually talking about information. The realisation that these two entropies are the same is one of the reasons that information is now seen as a fundamental element in modern physics.

One of the problems caused by Hawking radiation is that information is potentially lost into the black hole. The information being lost into the black hole is not the problem so much as, if the black hole can evaporate through Hawking radiation then the information is truly lost from the Universe. This is unless the information is somehow encoded in the radiation that the black hole emits. This is called the black hole information paradox and is where hologram theory comes in. A great explanation of this problem can be found in this video by the physicist Sabine Hossenfelder.

## Holograms

Hologram theory states that the information of the particles absorbed by a black hole is stored in a hologram on the event horizon of the black hole. The amount of information which can be stored is proportional to the surface area of the black hole and not the volume.

This is where I initially thought there was a contradiction and this is partially based off of a statement made in the book “The case against reality” which takes the analogy whereby the amount of information that can be stored in a volume is analogous to making a hard drive with that volume. In that analogy the author states that one company is releasing a hard drive made of small spheres within a volume and another makes one the same size but using up all the volume, thinking that this will lead you to a larger storage capacity.
Using the idea of surface area’s representing the total storage, the company using the smaller spheres can store more information than the company using a single sphere. This is even though the single sphere contains more volume.

You can see from the diagram below, even though A takes up less volume in total, there are gaps between the spheres, it has a much larger total surface area than B.

I then became confused by taking this analogy too far.

If the maximum information/entropy that can possibly be stored in a given space is represented by a black hole filling that space, and the amount of information is proportional to the surface area, surely you can fill the space with little black holes that haven't merged and store more information!

- The largest entropy state that can exist in a volume is a black hole the size of the volume
- A volume containing lots of small black holes contains more information than a black hole the size of that volume

Both of these things can't be true. I even tried to reduce this to formal logic to show it couldn't be true. Keeping A and B the same as in the image above, and assuming there is a limit on the total volume throughout:

My logic is not flawed, there is a contradiction, however, it's caused by a false axiom. Set off by the hard drive analogy I had assumed that all black holes have the same density (Density used here to mean volume enclosed by event horizon over mass).

To my surprise, this is not the case!

In fact, the density of a black hole falls as the black hole grows. Once I had finally bothered to look up the formula for the radius of the event horizon for a given mass I could see that if you put lots of the small black holes into a volume, such that you might be able to store more information than the black hole the size of some bounding sphere, you would actually be describing a single black hole. The formula is the Schwarzschild radius and can is shown below.

Putting the small black holes so close together even without touching would be in fact a larger black hole! Since you can’t describe the arrangement of matter within a black hole as soon as the density within the bounding sphere becomes that of a black hole that size, it is a black hole that size.

This solves what I had considered the contradiction which arises mainly from my taking the hard drive analogy too far. The resulting size of the black hole from A on the diagram would be far larger than the black hole described in B. It did raise a second question though:

If density falls as the mass increases for a black hole and assuming the distribution of mass inside the black hole doesn’t matter (Excuse the pun); for any volume of non-zero density there exists a black hole which would contain it. For this I work through the equations for Schwarzschild radius, density and volume of a sphere:

Putting these equations together, we can work out the size of the bounding sphere, for any given density, for which that density would result in a black hole.

This results in a formula for any given density, a radius of a black hole for which that density should result in. This, of course, assumes the density continues indefinitely but so long as there isn't an edge, beyond which there is no more matter, or at least an asymptotic drop off in the density of matter, this should result in a black hole.

To me the conclusion from this is that either:

- There is a point in space at which there is no more matter
- The universe is in fact inside a black hole
- This use of the Schwarzschild radius in this way is outside of its valid range

Either way, it’s an interesting exploration into gravity. I think the answer is most likely something to do with a valid range in which you can use the equations. If this theoretical universe sized black hole is outside of the cosmic horizon, which back of the envelope calculations show it would be, by a long way, it falls under things which it doesn't make sense to talk about. Because nothing outside that boundary could have any observable effect from our perspective on anything within that horizon.

https://physics.aps.org/articles/v14/143

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