How real is real? (Part 1)

In an idle moment I was messing around with some equations.

I took

• The equation relating the entropy of a black hole to the surface area of its event horizon,
• The equation for its Schwarzschild radius, to calculate the area, and
• The equation relating the entropy of a system to the number of its (accessible, equally probable) microstates,

to work out the number of microstates of the smallest possible black hole, just out of interest.

By setting the Schwarzschild radius of the black hole to a Planck length, as giving (intuitively) the smallest possible black hole, you end up with the number of microstates being e^π, and not one, which I had (intuitively) thought would be the minimum possible number of microstates. I thought this would be the "unit" black hole as it were, with a single (integer) microstate, and was not expecting a transcendental number instead of an integer number of microstates.

What did this mean? Certainly, e^π is equal to -1^-i, which neatly expresses everything in terms of unity, but leaves me none the wiser regarding physical interpretation. However, I remember conversations with people much smarter than me about how we cannot rely on physical or geometrical interpretations, that these are inevitably naive. Indeed, we would be paralysed, crippled, in our scientific endeavours, if we always had to relate our subject to something that can be comprehended by analogy with our limited everyday experience, even if such analogies are a source of awe and enthusiasm. The motivations and the manifestations of our scientific endeavours must always remain distinct. 

Nevertheless I asked my friend Aidan Keane what the physical meaning of this was, because he is actually a proper published cosmologist and black hole expert whereas I am just some schmuck with a degree in astronomy and physics. Chin stroking and head scratching ensued on both his part and mine and then he pointed out that e^π - otherwise known as Gelfond's constant - is equal to sum of the volumes of all even-dimensional unit (hyper-)spheres.

Now we are getting somewhere, I thought: a geometrical interpretation.

So maybe there in an inconsistency in the approach to area and volume, maybe the problem is my naive approach to calculating the surface area of the black hole, and maybe adopting an approach that includes higher dimensions would make it all cancel out to give unity instead of e^π. After all, the event horizon is in a sense not part of the universe: we can observe objects approaching it but never actually reaching it in all reference frames other than the reference frame of the event horizon itself, due to relativistic time dilation. It is misleading to think of the microstates of the black hole in the same way one thinks of the microstates of a thermodynamic system hypothetically contained within the event horizon in the same sense a ball contains air. The event horizon is the black hole. There is nothing beyond it. It is the end of time and edge of space. We are discussing the microstates of the event horizon itself, whose dimensionality transcends the observable universe.

But what's special about even dimensionality then? Apart from turning the argument of the gnarly gamma function in the expression for the volume of an n-dimensional unit sphere into an integer. "Collapsing tesseracts" in the movie Interstellar?

Anyway, the whole thing got me thinking again about some ideas I had about finite state machines twenty years ago back when I was writing a master's dissertation on random number generation. My thinking then about the states of a finite state machine are equally applicable to the microstates of a black hole here, I think.

Let's consider a system which at any given time can be in one of a finite number of possible states, and makes a transition from one state to another as a consequence of an interaction with its surroundings corresponding to an exchange of information between the system and its surroundings: input and output. 

Let's imagine the possible transitions do not include all possible pairs of states though. Each state has a list of possible successor states which includes some, but not all, other states. The possible predecessor states are therefore similarly limited to some, but not all, other states.

All states may be equally accessible and equally probable over time, given a sufficiently long (random) sequence of transitions, satisfying the requirements of a macroscopic observable like entropy based on the total number of accessible, equally probable microstates of the system.  Crucially, however, the probability of an individual "eventual" state after a number of transitions depends on the initial state if the (random) sequence of transitions is not sufficiently long. (N.B. I am using the term "eventual" rather than "final" because I am reserving the term "final" for states from which the system cannot make a transition, states which are their own successor). This difference in probability is because not all states are possible successor (or predecessor) states of each other. Given an initial state, the list of possible eventual states only increases to include all states after a sufficient number of transitions.  Similarly, given the eventual state, the list of possible initial states with which the sequence of transitions commenced only includes all states for sufficiently long sequences of transitions.    

So we have one entropy, a macroscopic entropy, related to the number of states that are accessible to the system after a "sufficiently long" sequence of transitions, and we have another entropy, a sort of "incomplete" or "interim" entropy, related to the number of states that are accessible only after a sequence of transitions that is not "sufficiently long", that is, a number of states that is less than the total number of states reflected in the macroscopic, observable entropy. This still holds even if the system is in thermal equilibrium with its surroundings and the macroscopic entropy is constant.

We can consider a "forward" and a "backward" incomplete or interim entropy related to the number of possible eventual and initial states for each intermediate state. That is, we can calculate this incomplete entropy working backwards from some intermediate state to a list of possible initial states, or working forwards to a list of possible eventual states, for a given sequence of transitions. The forward and backward incomplete or interim entropies of intermediate states may not be equal because they arise from different lengths of sequences of transitions. That is, the intermediate state may be “closer” to one or other of the initial and eventual state in the sequence of transitions.

However, even when the lengths of the sequence through which the system transits before and after the intermediate state are the same, the incomplete entropies (backward and forward) might not be equal. We can think in general without reference to specific initial or eventual states. The incomplete entropy arising from the number of possible initial (forward) and eventual (backward) states will not be equal if states tend to have more successors than predecessors (or vice versa: fan-in versus fan-out). The number of possible eventual (or initial) states will not be the same except within sets of states connected by sequences of transitions such that the number of successor and predecessor states are the same. 

One can consider a "residual" entropy for each intermediate state related to the difference between these two incomplete entropies, which is zero for sufficiently long sequences as both tend to the same macroscopic value, but which is not zero for shorter sequences where the rate at which successive states acquire either predecessors or successors differ as the sequence of transitions in which they are embedded increases in length. The residual entropy of a state is zero for sufficiently long transition sequences, corresponding to observable macroscopic entropy, since the backward and forward incomplete entropies are "completed" and equal to the macroscopic entropy and so cancel. So the residual entropy is globally zero, but may be locally non-zero if individual states have different numbers of successor and predecessor states (and may also be locally zero).

There is a another thing to consider. The state of the system changes as a result of an interaction between the system and its surroundings in a way that is not unphysical. This is where we stop talking about mathematics and start talking about physics. This is where our experimental observations about how one thing depends on another, our data, are involved. The state changes as a result of an interaction with the surroundings in a way that involves an exchange of information to which we have access.

This consideration is important because this is the reason why we would consider finite transition sequences that are not sufficiently long to result in a macroscopic observable entropy in the first place (without having to impose some non-physical, artificial constraint on the system in order to do so). 

Each transition is associated with an exchange of information between the system and its surroundings. The specific nature of that information selects pairs of state between which a corresponding transition can be made, like the input of a finite state machine. That is, there is a finite number of transitions (less than the total number of possible transitions) with associated pairs of states, to which a given input corresponds. For a given information exchange, there is a number of candidate pairs of states between which the system can have made a transition. This list of pairs does not include all pairs between which transitions can be made. There is therefore a list of possible initial states which does not include all states.

A subsequent information exchange resulting in another transition reduces this list of possible initial states of the sequence further, since the possible initial states entail a list of possible successors, some of which are eliminated by the observation of the subsequent input and related restriction of the list of possible transitions in the same way as before. This proceeds until the sequence uniquely identifies the initial state.

The importance of this is that each state is uniquely identifiable entirely on the basis of the properties of the system itself. The states do not require arbitrary labels. The input (and output) sequences that make these properties manifest do not need to have any special properties. They can be finite random sequences in the way discussed by Kolmogorov and Chaitin. The precise "values" of the input (e.g. binary digits) do not matter. The only thing that matters about the label is its uniqueness, not the characters with which it is written. The process can be repeated with different random sequences and the uniqueness will persist, since it arises from properties of the system, not the input submitted to it. We do not require any "divine intervention" to differentiate between the states: we do not require an arbitrary injection of information to make the system understandable by the application of arbitrary state labels. The system is completely self-contained. 

These finite sequences that uniquely identify the states of the system, the state labels that are intrinsic to the system itself, are subject to the arguments above regarding residual entropy, by definition, since they are based on limitation of states addressed by a sequence of transitions to a number less than the total number of possible states. This is the reason why these considerations regarding residual entropy were entertained in the first place: as a way to describe a kind of "dynamics" which the states will exhibit in so far as they can be uniquely identified.

Which brings us back to black holes. Earlier I reserved the term "final state" for a special purpose. We can consider a black hole to represent a final state, one from which the system will never exit, one whose only successors are itself. This could, in this model, correspond to a single unitary final state, or a collection of states all of which are connected to each other, so that they are all possible successors of each other. As transitions occur between them, the list of possible initial states does not decrease as the list of successor states is unrestricted, and the process by which an initial state can be uniquely identified by an intrinsic state label is truncated before it is complete and reaches a conclusion. We cannot uniquely identify any states in this structure. All we can know is the residual entropy of eventual states up to a certain horizon of this structure. The dimensionality of the structure is infinite since the states are connected to themselves.    

The exchange of information which prompts the transitions is lossless, by which I mean physically they happen at the speed of light (with no information lost in directions transverse to the direction in which information is propagated to describe inequalities between the energy density of magnetic and electric fields, for example). The trajectories along which transitions progress from one state of the system to the next are null geodesics and  their statistical description assumes a geometrical interpretation. The thermodynamic characterisation of a black hole in terms of a surface area indicates the possibility of a statistical or combinatorial formulation of General Relativity that may allow us to build a bridge with quantum theory. We should relate the geometry of space-time to its information content rather than its matter-energy content.

However, if space-time geometry is to be considered in terms of the residual entropy of the states in which we observe the cosmos, we are faced with a paradox. The residual entropy is not observable to the system's surroundings, where only macroscopic entropies are manifested. The universe can only be observed from within, where residual entropies are manifested. To describe the universe requires us to be part of the universe, in accordance with the (final) Anthropic Principle. Our direct experience of the universe, and our ability to describe it at the most fundamental level on the basis of that experience, without resort to divine intervention or any other arbitrary sources of information beyond the content of the universe itself, vouchsafe its, and our, reality.