a student wrote:
> Ed Hanna wrote:
> > It was proposed in part 1 that a probability of motion within a
> > discrete space-time can range from 0% to 100% (postulate 3).
> >
> > In ordinary life we add speeds in a linear fashion:
> >
> > S1 + S2 = S3
> >
> > But if motion within a discrete space-time lattice were a probability
> > as proposed in postulate 3, then we need a different formula, because
> > (independent) probabilities add like this:
> >
> > P1 + P2 - (P1 * P2) = P3 (sorry if this is too obvious)
> >
>
> It is an interesting addition law. One could denote addition by &, with
> u & v := u + v - uv = 1 - (1-u)(1-v).
> This has the desirable properties
> u & 0 = u, u & 1 = 1, u & v = v & u, (u & v) & w = u & (v & w).
> Also have u & v ~ u + v for u,v << 1.
>
> However, there is a problem with adding left and right motions (and
> more generally a problem extending to vectors, above and beyond the
> isotropy problem): Suppose a spaceship moves to the left at 90% the
> maximum speed of unity, and fires a particle to the left at 100% the
> maximum speed. The addition law then gives
> w = u & v = -0.9 & (-1) = -2.8,
> which is a problem! More generally, one doesn't have the property
> (-u) & (-v) = -(u & v),
> or even u + (-1) = -1.
dear A student,
Thank you for your analysis.
I hasten to clarify that this "interesting addition law" is the
standard textbook rule for adding independent probabilities, and can be
found in any elementary statistics textbook. If motion within a
discrete space-time lattice were probabilistic rather than
deterministic (postulate 3), it ought to be subject to statistical
analysis.
I understand the limitations that you have noted, and I agree with
them. However, this textbook rule for adding probabilities is
originally intended to be valid in the domain where the two
probabilities range from 0 to 1. None of the statistical textbooks
I've seen allow for probabilities with negative values, so I'm not sure
how they should be handled. I'd rather consider that a spaceship
moving to the left, does not have a negative probability of motion, so
much as it has a positive probability of motion, but in the opposite
direction. I'll need to ponder on this.
Meanwhile, in my previous post I was expressing surprise that an
off-the-shelf statistical rule would be applicable in this context
whenever a probability of motion equal to 100% is involved. That is,
100% probability of motion, when added to *any* speed is still a
constant 100% and therefore appears to be a constant 100% when viewed
from that speed's frame of reference. This seems to be the same as
saying that given a discrete space-time lattice with probability of
motion and no gaps (postulates 1, 3, and 2), there would be a maximum
speed equal to 100% probability of motion, and that speed would be a
constant 100% when viewed from any frame of reference, regardless of
*its* speed.
The major limitation of the rule u & v := u + v - uv is the addition of
probability of motion with objects moving at less than 100%.
This rule works fine when adding the chance of getting heads from one
coin (50%) plus the chance of getting heads from a second coin (also
50%) for a total of .5 + .5 - .25 = .75 chance of getting heads by
flipping both coins. It does not work very well when used for adding
speeds in the real world.
For example, absent any further analysis, if a spaceship moves to the
right with 50% probability of motion, and it fires a missile to the
right with its own additional 50% probability of motion, the textbook
probability addition rule w = u + v - uv would yield .5 + .5 - .25 =
..75 for the speed of the missile, which we know to be contrary to
experimental results.
The interesting thing is that the textbook rule for adding probability
of motion seems to work when one of the two being added is 100% (and
presumably massless), but when we try to apply the textbook rule to
probability of motion with objects moving at less than 100% (presumable
objects with mass), it seems to fail.
Taking this failure to accommodate objects with mass as our clue, we
could try to incorporate mass into the textbook probability addition
rule, as well as explicitly substituting "velocity" in place of
"probability of motion".
If we start with the textbook probability addition rule:
w = u + v - uv (1)
and multiply this rule by some initial mass (m) giving us:
mw = mu + mv - muv (2)
then we immediately run into a problem because now the rule has units
of
momentum = momentum + momentum - energy. It is physically
impossible to subtract energy from momentum because of the mismatched
units. On the other hand, the energy lost on the right side of the
equation must balance somehow, and the only units on the left side that
can gain are mass and velocity. Fortunately, we know that energy
divided by c^2 becomes mass, because if e=mc^2, then e/c^2=m.
This means that the (-muv) energy term on the right could become a
(+muv/c^2) mass term on the left.
Equation 2 can be recast to reflect the conservation of momentum as:
m'w' = mu + mv (3)
where m' and w' are the final mass and velocity after the energy from
the right becomes its mass equivalent on the left. Since the original
mass on the left (m) increases by this mass equivalent (muv/c^2) then
we should be able to substitute (m+muv/c^2) for m' to yield:
(m+muv/c^2)w' = mu + mv (4)
where:
m(1+uv/c^2)w' = mu + mv (5)
or:
(1+uv/c^2)w' = u + v (6)
and thus:
w' = (u + v)/(1+uv/c^2) (7)
for any given mass (m) which has now dropped out of the equation.
This third emergent property of discrete space-time (equation 7 and
mass increase with velocity) seems to correspond to the Lorentz
equation for the addition of velocity in continuum physics. Also, mass
is not constant, but increases with increased velocity. Conversely,
the Lorentz equation for adding velocity and its accompanying increase
of mass with velocity seems to be a natural consequence inherent with
the addition of probabilities of motion within a discrete space-time
lattice.
Unfortunately, while this gives us a third testable prediction from a
discrete space-time model, once again it is not sufficiently different
from continuum physics to allow for a concrete test that will
distinguish between the two.
I'm not sure if this solution satisfies what Alan meant when he wrote
about trying to solve "the velocity addition problem in the same way as
Einstein or, perhaps, in some novel way."
Regards,
Ed Hanna
