A Strange Duel

2007-09-16

Consider two duelists, separated about 200m and pointing rifles of the exact same type at each other.
At time t_F one duelist fires, aiming precisely at the rifle of the other and a short moment later, at t_I (the
I stands for insertion), the bullet enters the rifle of the opponent without recognizable damage to either
the rifle or the bullet.



Obviously, one would think that this is very unlikely, if not impossible.
Let's freeze the scene at time t₀, when the bullet is halfway between the two gunmen. If we know the
position x(t₀) and momentum p(t₀) of the bullet, we can calculate the trajectory x(t), with x(t_I) being the
exact middle of the barrel of the second rifle.

Assume that we allow for a displacement d from the ideal trajectory x(t_I), with d approximately 1/1000
of a millimeter to guarantee a damage free entry (I took this value from typical tolerance limits for bullets).
If D = 100m is half the distance between the duelists and L = 0.01m the length of the bullet, then we can
estimate the maximum uncertainty u in the position of the tip of the bullet allowed at time t₀.



Simple geometry suggests u/L = (d/D) or u = 10^-10m which is on the order of the diameter of a single atom.
In other words, this little calculation confirms what we already knew: it is very unlikely that the bullet will
enter the 2nd rifle at t_I. If we use a reasonable value for the uncertainty u (after all the bullet travels through air
and interacts with the environment) and calculate the trajectory forward, we find that it is most likey that the
bullet will miss the barrel of the 2nd gun and we hope also the duelist.



But why are we not surprised that the bullet came from the first rifle at time t_F?
Obviously, if we calculate backwards from t₀ to t_F, we find that the same incredible precision is needed to
ensure that the bullet came from the 1st barrel and not from somewhere else. How can we assume the bullet
observed at t₀ was actually fired from the first gun at time t_F?



The obvious answer would be: "You told us the the first gun was fired at t_F; In general, we know that bullets
are fired from guns. Within the thought experiment the duelists have a memory and know that the bullet was
fired from the gun at t_F."
This is all well and fine, but strongly believing in the time-reversible laws of physics we can easily turn this argument
around and conclude that our memories must be wrong. Calculating the trajectory x(t) backwards in time and using
the same argument as above about the incredible small limit of tolerance u, one would have to conclude that it is
much more likely that the bullet actually emerged from outer space and our memories about guns being fired are false.

So how do physicists deal with this issue? Usually, by assuming boundary conditions which eliminate the 'absurd'
solutions. If we place the duelists in a laboratory and assume that the laboratory walls are absorbers of bullets (but
not emitters), then we can calculate from t₀ backwards to t_F and eliminate all trajectories which would lead us to the
conclusion that the bullet emerged somehow from those walls. A physicist will also assume reasonable initial conditions
(e.g. both duelists were alive and normal human beings before the experiment).



But notice the word 'assume' in the above statements. The physicist in our thought experiment assumes that the walls
of her laboratory are absorbers and not emitters of bullets. Again, if she would strongly believe in her 'backward'
calculation, then the obvious conclusion would be that something strange has happened inside those walls.

In general, there is (more or less) only one way how reality can match with our memories. But there are many ways
how our memories could be false and our experiences fake, either by accident or conspiracies. E.g. there is (more or less)
only one way how Lee Harvey Oswald could have fired the 'magic bullet' to kill president John F. Kennedy. But there
are millions of ways how this murder could have happened differently: A conspiracy of the CIA, the Mafia or both; Aliens
from outer space, with or without connection to Cuba; Maybe JFK was really a fake actor or maybe it was all a simulated
event in a simulated universe or perhaps just a bad dream and never really happened...
Once we allow for only a small probability for one of those 'absurd' scenarios, we have to assume a similar probability
for infinitely many conspiracy theories. The probability that our memories and experiences are explained by reality, as
we normally assume, would then be close to zero.

But once we believe that our memories correspond to reality, we will refrain from 'calculating backwards' as depicted
above and use our 'knowledge' of the past to eliminate 'unphysical' solutions from our calculation.
In other words, we will treat the past differently from the future.

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