In
Search of a Consistent Theory of Time Travel for Fun and Recreation
Basic
Concepts before any time travel considerations arise
Can you think of
better way to remind yourself of how much physics you never knew, and
at the same time how badly you understood all the physics that you
thought you at least somewhat understood. Me neither.
Let's start with our
undefined terms, these are primitive notions1,
intutive concepts that I choose not to define, either because I'm too
lazy, or because the rabbit holes discovered in defining them don't
appear to be as interesting as the rabbit holes one can discover by
leaving these primitive and moving on, but arguelby both. While were
at it, we'll pretend we can use UTC as an absolute time system, which
is obviously wrong, but we'll pretend it doesn't matter since we're
taking things slow.
We can now define a
whole class of undefined terms such as B(p,t) as the physical body of
person p, at time t. This for instance takes into
consideration that over time our bodies tend to change. So for fun
we'll pretend that B(p,t) is a well defined intutive term.
Let P(t) be the set
of all protons, nuetrons, and electrons that exist at time t.
And for the fun of it, we
could pretend that its actually well-defined.
Let
BE(p,t) = { x : x ∊P(t)
and with 99.99% probability or above x ∊
B(p,t)}
Inspiration
as follows. We'll pretend the Heisenburg Uncertainty Principal
implies that we can't know with complete certainty that at any given
time, any given proton, nuetron, or electron is part of our physical
bodies, so it becomes convienient to consider the electrons,
nuetrons, and protons, that we are 99.99% sure of as making up our
bodies at some particle time. The remaining inspiration for the
relationship is from the old television series, Get
Smart.
The
BE relationship may be refered to as our apparent physical body.
Let
C(p,t1,t2)
= | BE(p,t1)
⋂ BE(p,t2)
│
/ │BE(p,t1)
⋃ BE(p,t2)│
Intutively we can consider C has
a change function here, measuring how much our apparent physical
bodies change from one time to another. For example one second from
now, the electrons, photons, and neutrons that make up our physical
body, might be nearly the same as the ones we have now with
relatively few exceptions, or for example C(John,now,now+1 second)
rounded to 25 decimal places is probably 1.0000000000000000000000000.
See How many atoms are in the human body?2
I'm just guessing at the point, but I doubt our physical essense
is changing much from one second to the next. In other words,
C(p,t,t+1 second) is always pretty close to 1, at least after the
first second of our conception.
Though I don't have any handy
references, I think I've heard or read from several sources that over
a period of seven years, that perhaps our apparent physical bodies
change completely. That is, C(p,t,t+7 years) is pretty close to zero.
As should now be obviously
apparent, I've put in enough information that certain things can be
checked. Ignoring leap years, and any round off error my choice of
Linux calculators introduces, I can compute that, 7 years is
220,752,000 seconds, .99999995^220752000 is 0.000016085, where as
.9999999^220752000 is 0, and .99999999^2207520000 is 0.11. Or in
laymen's terms, if one were to assume our bodies change completely in
seven years, then from one second to the next our bodies rounded to 7
decimal places does not change (e.g. C(John,t,t+second) = 1), but
rounded to 8 decimal places there is some decernible change, and not
the 26 decimal places I was initially guessing.
Now here's where it gets
interesting, and we haven't even messed things up with time travel
considerations.
Since B(p,t) is an undefined
initutive term, it doesn't technically make sense to consider what
B(p,t) ⋂ P(t) means, but if we consider a physical body, to be the
sum of its physical parts, then with our deepest apologies to the
quarks, leptons, and boson of existence, we could take as an axiom
B(p,t) = B(p,t) ⋂ P(t). And since the goal of having fun, out ranks
most other goals, we take this axiom until such time as it can be
demonstrated that its introduction leads to false results otherwise
not achieved without.
We can conclude that BE(p,t) ⊂
B(p,t) for any (p,t) in the domain of the relations.
If we assume the quantum
mechanics correctly implies, that for any x ∊ P(t) we cannot be
completely certain that x ∊ B(p,t), and likewise cannot be
completely certain that x ∉ B(p,t).
If so, then Prob(BE(p,t) =
B(p,t)) is non-zero, but more importantly Prob(BE(p,t) = ∅) is
non-zero as well. Or intutively this could be taken as meaning, that
any given point in time, we may be exactly what we seem to be, and
that any given point in time, we may have nothing in common with what
we seem to be.
