life expectancy

Recently I stumbled upon this math problem:
The positive integer N has a finite value but is unknown to us.
We are looking for a function f(X) which minimizes the sum
E = |f(0) - N| + |f(1) - N| + |f(2) - N| + ... + |f(N-1) - N|
for almost all N.

Notice that we do not know the value of N, so f(X) cannot depend on it, which eliminates the trivial solution f(X) = N among others.
In order to illustrate what I am looking for, consider as first example
1] f(X) = 0, which results in E = N².
However, there is a better solution
2] f(X) = X, which results in E = (N+1)*N/2, which is less for almost all N (except N=1).

Unfortunately, I do not know the best solution f(X) and this is where you are invited to leave a comment to help me out.
But I do have strong evidence (i.e. a numerical test up to large values of N) that
3] f(X) = X + sqrt(X) is an even better solution.

So what is the motivation for this problem and why the title for this blog post?
Consider a process or phenomenon which exists already for X years and we try to estimate the total lifetime N without any further information. So our estimate can only depend on X and we try to find a function which minimizes the total estimation error E as described above; every year we make an estimate f(X) which is wrong by the amount |f(X) - N| and we try to minimize the sum of those errors. In some sense this is a variation of the infamous 'doomsday argument'.

It is now obvious why 1] is a bad solution and 2] is much better. Btw the function f(X) = 2*X would give the same total error E minus a small constant, so whether we assume that the process ends immediately or estimate that it will last twice as long (as it already did) makes no significant difference.

Btw the solution 3] creates a paradox: The best estimate for the life expectancy seems to depend on what units one uses: We get a different result if we calculate with days rather than years.


added later: If I assume f(X) = c*X, perhaps motivated by that paradox, then I can show that c=sqrt(2) minimizes E for large enough N. However, the function
4] f(X) = sqrt(2)*X + sqrt(X) seems to be an even better candidate, but I do not know how to determine a,b to minimize E for f(X) = a*X + b*sqrt(X) or determine the general solution for f(X).


added later: There are better ways to illustrate the problem; e.g. a box contains an unknown number N of candies. You take out one after another and at each step X you have to guess N. At the end there is a penalty proportional to the sum of your wrong guesses.
Perhaps a "deeper" example considers a Turing machine, which performed already X steps and we try to guess after how many steps N it halts.
The reason these examples are "better" is that there is no change of physical units (e.g. from years to days) that would affect N.

no black holes?

Laura Mersini-Houghton and Harald Pfeiffer published a paper with numerical results suggesting that black holes may not really exist (see also this earlier result). As one would expect, several pop. sci. webpages have already picked this story up.

The paper is of course not a general proof, but describes a particular model using certain assumptions; it considers the spherically symmetric collapse of pressure-less dust and it makes simplifying assumptions about the Hawking radiation: The energy tensor for the Hawking radiation is taken from earlier calculations for (static) black holes, proportional to 1/R^2, and I don't think this is justified if one wants to prove that black holes do not exist. Further it is assumed that most of the radiation is generated by the collapsing body itself (*) and finally assumptions are made about the heat transfer function C which I cannot follow (yet).

The resulting differential equations are numerically integrated until a shell-crossing singularity appears, in other words a naked singularity (presumably an artifact of the model assumptions, i.e. perfect spherical symmetry, so it is only slightly embarrassing in a paper which wants to remove black hole singularities).
The behavior of the dust suggests a rebound near the horizon, but it is too bad the full evolution is unknown, because it raises interesting questions.
What happens to the pressure-less dust in the long run? Will it collapse again after the rebound, perhaps infinitely often?
What does the final state (including Hawking radiation and the "influx" of negative energy) actually look like?

I am sure this paper will generate several responses and eventually more realistic calculations will follow.
Until then I remain skeptical that this result will actually hold in general.


(*) I admit that I do not understand this passage in the earlier paper: "Hawking radiation is produced by the changing gravitational field of the collapsing star, i.e. prior to the black hole formation [..]. Otherwise the surface gravity of the black hole κ, and the temperature of Hawking radiation would increase with time..."
I thought the standard picture is that the "influx" is at the event horizon (not the collapsing body) and the temperature does indeed increase with time...


added later: Supposedly William Unruh was more direct and he thinks that the paper is nonsense.

any good answers to this one?

At the Strings 2014 conference, Piotr Bizon talked about the gravitational turbulent instability of AdS₅.
I became aware of this issue more than three years go and I have to admit that I still do not really understand what it means. As I see it, turbulence is one of the big unsolved problems in physics mostly due to the fact that it prevents us from neatly separating energy scales; the opposite of the clean separation which enables renormalization a la Wilson.
So what does it mean that this turbulence instability shows up on one side of the famous AdS/CFT correspondence?

my derivation of the Born rule

I just read (parts of) Sean Carroll's derivation of the Born rule, but I do not find it very convincing, because there is a much simpler, straightforward derivation available to resolve this problem of "self-locating uncertainty".

1) We shall use a "hardcore" many worlds interpretation, assuming that the world splits into a quasi-infinite number of branches at any time, which realizes all possible outcomes of quantum theory. We assume that those branches are all equally real and a simple counting argument shows that the Born rule does not hold for almost all of those branches. It follows that we do not live in one of those generic branches, which solves the first part of our self-location problem.

2) It is reasonable to assume that some of those infinitely many branches contain at least one quantum computer capable of simulating human life. Those computers will have to simulate quantum theory, but we can further assume that they will only keep one branch at a time in order to save resources. It is straightforward to assume that they are programmed to use the Born rule to select this branch randomly.

3) We observe the Born rule to great precision and it follows that we are the human beings simulated in one of those quantum computers. This finally resolves the self-location problem.

I would add that (some of) the simulated human beings will use the Copenhagen interpretation to explain what they experience; i.e. an interpretation which emphasizes the importance of the observer and her 'conscious experience'. Obviously, the simulated human beings are unaware that their 'conscious experience' is indeed a side effect of the procedure which selects the simulated branch randomly.