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Time's Arrow & Archimedes' Point : New Directions for the Physics of Time, by Huw Price tackles a very difficult question, "what is time?" in a way that can be understood by an ordinary person.  The author writes that the book is written for physicists who are non-philosophers and for philosophers who are non-physicists.  Since it's written for people who are either non-philosophers or non-physicists it stands to reason that people who are both can enjoy the book as well.  I agree, and I should know.

The purpose of this book is to debunk the notion that the universe is asymmetric with respect to time.  It shows that supposed asymmetry is an artifact of our viewpoint in two ways:  first, we see things from this point in time because here we are (or I should say now we are): creatures in time and second, we're so used to seeing things as asymmetric in time that we unwittingly make time-asymmetrical assumptions.  Mr. Price attempts to combat these two problems first by imagining a viewpoint outside of time and second by using this viewpoint to keenly spot and remove time-asymmetrical assumptions.

Mr. Price shifts our viewpoint away from this universe within time to an imaginary "place" outside of time.  From this viewpoint, the universe is four-dimensional: three spatial dimensions and one time dimension.  He calls this view the block universe.  From this viewpoint, the author says, it's easier to spot those time-asymmetrical assumptions.

For example, a rationale for entropy increasing as described by the Second Law of Thermodynamics is based on the time-asymmetrical notion that velocities of molecules in a gas are uncorrelated before they collide but correlated afterwards, even if they never collide again.  To remove this asymmetry, Price offers the view that entropy is about probability.  Although all "micro-states" (states that describe the position of every molecule of a gas, for example) are equally probable, some "macro-states" (states that describe how many molecules are in a chamber, for example) are more probable than others because there are more micro-states that give rise to these more probable macro-states.  This probability view removes the time-asymmetry.  In this view, from a point in time of low entropy, the entropy will become higher in both directions of time.  So why does it still seem to us that entropy increases only forwards in time?

I think it's because even in the block universe view, we have not entirely eliminated human bias.

I think we can and should take another step back -- waaay back -- to look at the universe in a way that goes a step further in eliminating human bias.  Let me try to set it up for you.

Here we are in a universe that is special in that it contains us.  Even if we look at it as a block universe it's still special in this way.  How probable is it that the universe would unfold in a way that it should contain us?  Intuitively, we would say the probability is low.  (That's why people believe in God -- but that just "moves the lump in the carpet" as Price would say -- how likely is it that God would form?)  But it's our intuition that gets us into trouble.  We only know of one universe, so it's very hard to say how probable it is that our universe would have formed this way.  To even ask the question meaningfully, we must be able to spell out the universe of universes (which I'll call the multiverse) from which our universe is drawn.

So let's take a step back to the "multiverse block view" in which we imagine from a point of view outside of time all possible universes as analogous to micro-states, and our universe as one of those micro-states.  The subset in the multiverse of universes like ours in which we exist (a thorny issue in itself: what is "we", but I'll gloss over it for now) represents a macro-state (set of micro-states) which I'll call "U".  Now we can meaningfully ask the question: how probable is this macro-state, U, in the multiverse?  Or to put it another way, what is the ratio of the cardinality of U to the cardinality of the multiverse?  I suggest the probability is very low indeed, because for every micro-state in U there are zillions of possible universes that aren't in U.  Thinking about the multiverse block view seems to confirm our intuition that our universe is "improbable".

Now, back to entropy...  A few paragraphs ago I pointed out that when viewed as probability, entropy tends to get larger in both directions of time.  I say "tends" because the possibility still exists that entropy could spontaneously reach a low point -- that is, at some time the entropy could be lower than at the neighboring times.  In any one universe, the probability of this happening is low -- extremely low, but it is not zero.

At this point I've talked about two very low probabilities: the probability of a universe (in the multiverse) being in the set U and the probability of a spontaneous low point in entropy in any one universe.  Now let's consider these probabilities together.  What is the probability of a spontaneous low point in entropy in a universe that belongs to set U?  Using shorthand, I would ask what is p(E) given U?   There's a handy formula for this one: p(E) given U = p(E and U) / p(U).

If E and U were independent of one another then p(E and U) would be p(E)*p(U) so p(E) given U becomes simply p(E).  But E and U are not independent.  In a universe that has no spontaneous low point in entropy, there would be insufficient order amidst the chaos to give rise to "us" -- in other words, a spontaneous low point in entropy increases a universe's chances of belonging to U.  Let's suppose that there must be a spontaneous low point in entropy -- and possibly some other conditions -- in order for a universe to be part of U.  In that case, p(E and U) is simply p(U).  Now p(E) given U = p(U)/p(U) = 1.

Now I think I've explained why it's reasonable to suppose that there has been a spontaneous low point in entropy in our universe.  Do you agree?  Fine, then let's move on...

Why does time move forwards and not backwards?

I think it's because time is what we experience when we move from lower to higher entropy -- the unwinding of the clock in two senses: increased entropy and the passing of time as the clock ticks.  If we imagine beings on the "other side" (in time) of the entropy low-point we would imagine time flowing the other way for them -- toward increased entropy.  And they would be asking each other: "Why does entropy increase with time?"