Warning: This blog post is neither entertaining nor informative. I recommend that you just skip it.
Once again we are floating in a Newtonian universe described by its microstate S(t), but of course we are just ordinary observers who do not necessarily know it.
At t0 we perform a little experiment, the equivalent of tossing a coin, with two possible outcomes H(ead) or T(ail); At t0+D we know the result.
The omnipotent demon watches the outcome too and in the case of T reverts the Newtonian universe to
its previous state S(t0-D), which explains the title of this blog post; In the case of H she does nothing.
However, the omnipotent demon is not infinitely patient and therefore the universe would loop only N times through the same state(s), then it continues.
We know about all this, because the omnipotent demon was so nice to inform us about it in advance.
Now we try to calculate the probability for H and there are two different ways to do it:
i) We do not know the microstate and both H and T are equally likely for what we know about S(t0).
The fact that the demon will play some tricks at t0+D does not change anything, thus p(H) = 1/2.
ii) There are the following possible cases for this silly game: This is the 1st time we observe the experiment
and the outcome is H, which we denote as 1H. Then there is 1T and also 2T, 3T, ... NT.
Notice that there cannot be a 2H, 3H etc. because the universe is deterministic, if T was seen the first time
it must be seen the 2nd time etc.
Since we cannot distinguish between those cases, we have the probability p(H) = 1/(N+1).
Obviously, this has some similarity with the sleeping beauty problem. But I am not asking which of the two
probabilities is 'correct' and I am not even interested if this little thought experiment tells us anything
about a relationship between time and probability. I am asking a different question.
How do you understand the limit D -> 0 ?
Told you so...
The field equations of general relativity can be separated into
hyperbolic evolution equations and elliptic constraints [ADM].
The evolution equations propagate an initial field configuration
'forward' in time, similar to other theories of classical fields.
However, due to the constraints one cannot choose the initial
configuration freely and this is very different from other classical
fields. In some sense the constraints 'connect' spacelike points and
thus one could call general relativity 'holistic' if this would
not be such an abused word.
We don't really know what the quantum theory of gravitation is,
but one would assume that the classical theory reflects the properties
of the underlying quantum theory and indeed the Wheeler-deWitt equation
is nothing but the operator version of one of the constraints.
I think one needs to keep this in mind when discussing thermodynamics
of general relativity, the information loss problem or the entropy of
black holes. E.g. if one specifies the metric near the horizon of a
(near spherically symmetric) black hole, the constraints already
determine the 3-geometry; Therefore I do not find it surprising that
counting microstates provides for a holographic result which differs
substantially from the naive expectation.
I would also think that an approach to the information
loss problem which emphasizes locality as 'conservative' is misguided.