A Strange Duel
Consider two duelists, separated about 200m and pointing rifles of the
exact same type at each other.
At time tF one duelist fires, aiming precisely at the rifle
of the other and a short moment later, at tI (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
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(tI) being the
exact middle of the barrel of the second rifle.
Assume that we allow for a displacement d from the ideal trajectory x(tI),
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 tI. 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 tF?
Obviously, if we calculate backwards from t0 to tF,
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
The obvious answer would be: "You told us the the first gun was
fired at tF; 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 tF.
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
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 tF 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
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
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
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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