Natural Science Forum / Physics / Particle Physics / February 2005

if there were an internal clock it wouldn't care how fast it is moving
anyway.  the clock moving along with the particle would show the same decay
time as any other clock moving along with a muon at any other speed.  the
decay time change is as measured by an observer that has a clock in their
own frame moving at a different speed than the muon.

its all relative.

As none has been in there, there is only a mathematical
description and it is explained with potential
walls that can be tunneled with a certain probability,
meaning a mean decay time in its own time frame.

Similar to : how long can a paper bag hold when half
filled with water.

Rene

No internal parts.  Yes, it's a euphemism for something with a time
dependence.  Muons decay with some half-life, and when they go faster the
half-life (measured by the observer in his lab) gets longer.

It is a euphemism and nobody knows for what. The answer would require a
special hypothesis or theory all its own. SR (special relativity)
provided no such thing itself.

Einstein's SR is a principle theory that never tried to provide
mechanical explanations per se for the behavior of physical systems.
However, that doesn't mean that people can't invent mechanical
explanations using SR as a constraint on the behavior of the mechanical
models.

Physics is all about the behavior of inanimate physical systems over
time. The muon behaves in accordance to the predictions of SR. That is
a lot to say for SR right there.

No one theory does everything. Long ago I learned to praise SR for what
it can do rather than curse it for what it cannot do.
Hope that helps.

Patrick

The question makes sense, insofar as there is a question as to the
sanity of the proposer.
Here are the facts.
The muon has a perfecty normal half life, I know that you understand
that.
Now, from a statistical evaluation, too many muons travel too far within
the lifetime of the muon.
So what is going on?

Let's place a time bomb in a car set to blow the car to smithereens in 1
hour, and drive it for 1000 miles. If it goes the distance, we'd  would
normally say that the car must have travelled at 1000 mph at least.

But now we place a limitation on the speed of the car, 500 mph, and it
still goes
the distance. The only [?] possible explanation is that time, for the
car, must have stretched, or "dilated". The timer of the bomb slowed,
because the speed of the car cannot exceed 500 mph. Not only that, but
the distance the car has to travel
is also greatly reduced, from the car's perspective.
So, if you have a theory that time dilates, you also have a theory that
the speed
of the car is limited to 500 mph.

So now you are faced with a fact.
The car travels 1000 miles without blowing up on the way.

And a theory: The car cannot exceed 500 mph.
This "proves" the theory [???].

Consider the circularity here.
We have already ascertained the speed of the muon to be less than c from
the Lorentz Transforms which produced the time dilation in the first
place.

Proof:

"Required: the motion of the point relatively to the system K. If with
the help of the equations of transformation developed in § 3 "
Reference http://www.fourmilab.ch/etexts/einstein/specrel/www/

The speed of the car was limited by two factors.
1) The theory said so.
2) Incredulity.

It is simply too incredible to believe that a car can travel at 1000
mph. Forget science.  Forget the "car" has wings and is actually built
by Lockheed and not by Ford. Trust the theory.
Never mind that muons are much more light-like than they are
matter-like.
Nothing can go faster than light. The speed of the car is limited. So
there must be time dilation.

Here's how to work it out, with very simple numbers. I shall prove that
the speed
of mosquitoes is exactly 5 fps for everyone, and nothing can exceed the
speed of a mosquito. I shall of course use Einstein's method. Algebra is
a wonderful tool.

Much of this story is credited to Daryl McCullough, only the ladder
was added by me.  It explains the origins of Einstein's Special
Relativity
for those having difficulty grasping the subject.

Sam and Joe are housepainters, and are walking along the street at 3 fps
in still air carrying a 32 ft long ladder between them, Joe leading the
way. Sam is carrying some paint cans and Joe has the brushes and
rollers.

At some point along their journey a mosquito name Albert buzzes past
Sam's
ear. Sam swats at it, but drops a can of red paint as he does so.

Albert flies along the ladder from Sam to Joe at a constant speed
of 5 fps. When it reaches Joe, Joe also swats at it, but drops a paint

roller. Albert, still hungry but not liking the smell of Joe's cigar,
flies back along the ladder toward Sam, again with a constant speed of
5 fps in the still air. Upon reaching Sam, once again Sam tries to swat
the
wee beastie but drops a can of green paint. He yells as the mosquito
bites
him and this startles Joe, who drops a paint brush.

Now it's your turn. I'll give the answers further down, but take a
moment
to do the calculations for yourself.

1) How many seconds did it take for Albert to fly from Sam to Joe?
2) How many seconds did it take for Albert to fly from Joe to Sam?
3) How far is it between the red paint can and the roller?
4) How far is it between the green paint can and the roller?

(Answers below)
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Assume the speed of the mosquito is c = 5 fps.
The speed of Sam and Joe is v = 3 fps, given.

We then must have a distance along the road for Joe of
32ft + vt, and  for the mosquito, a distance of ct.

Solving for t,
ct = 32 + vt
ct - vt = 32
t(c-v) = 32
t = 32 /(c-v) = 32/(5 - 3) = 16 seconds
So the answer to  Q.1) is 16 seconds.

The mosquito coming back is going to meet Sam going forward,
so it flies along the 32 feet of the ladder in time
t = 32/(c+v) = 32/8 = 4 seconds.

The answer to Q.2) is therefore 4 seconds.

The distance from the dropped red paint can to the dropped roller
is just ct, or 5 * 16 = 80 feet, so the answer to Q.3) is 80 ft.
Or we could do it by vt + 32 = 3 * 16 + 32 = 80, once again.
(Remember Joe had a 32 ft head start over the mosquito)

Coming back, Albert again flies at 5 fps but this time
for only 4 seconds, so it reaches the green paint can 20 feet
from the roller, which is the answer to Q.4)

So, as Sam sees it, Albert takes 16 seconds to reach Joe, flying at
5-3 = 2 fps, and 4 seconds to return, flying along the ladder at
5+3 = 8 fps.

Now we think like Einstein with his mosquito brain. Sam wants to know
when the mosquito reached Joe.

He isn't able to see the mosquito, its too small at 32 feet away,
so he guesses that since it went 32 ft each way, and took 20 seconds to
fly
away and back again, it must have reached Joe after 10 seconds = 1/2 of
20.

So we explain it carefully. First we label the red paint can "A" and the
dropped roller "B".  We write:

If at the point A of space there is a clock, an observer called Sam at
the
red paint can determine the time values of events in the immediate
proximity of the red paint can by finding the positions of the hands
which
are simultaneous with these events. If there is at the point B of space
another clock in all respects resembling the one at the red paint can,
it
is possible for an observer Joe at the dropped roller to determine the
time
values of events in the immediate neighbourhood of the roller at B. But
it
is not possible without further assumption to compare, in respect of
time,
an event at the A with an event at the dropped roller. We have so far
defined only an "A time" and a "B time." We have not defined a common
"time" for the red paint can and the dropped roller, for the latter
cannot
be defined at all unless we establish by definition that the "time"
required by a mosquito to travel from the red paint can to the dropped
roller equals the "time" it requires to travel from B to the red paint
can,
A.

Now, we want to do this algebraically, because tomorrow Joe and Sam
might
be carrying a different length of ladder, and we want a general
solution.

So we write:
If we place x'=x-vt, it is clear that a point at rest in the system
ladder
must have a system of values x', y, z, independent of time.

What that means is the ladder's length is x', so that 32  =  80 - 3 *
16,
and doesn't change as time passes. Did you think it would? Well, we'll
have
to see.  Maybe if we water it, it might grow.

According to Einstein, we are to assume the speed of the mosquito is
independent of the speed of Sam (which is fair enough) and also we are
to
assume that the time for the mosquito to make the round trip (20
seconds)
when divided by 2 is equal to the time it took to reach Joe, 16 seconds.

We don't know yet about the 16 seconds, we can only write it
algebraically
and pretend it is 10 seconds.
It is actually written as  x'/(c-v) [or 32/(5-3) in real numbers].

Now we say:

From the origin of system ladder let a mosquito be emitted at the time
tau0
along the ladder to x' (the other end of the ladder), and at the time
tau1
be reflected thence (that just means go back) to the origin of the
co-ordinates (which we are deliberately vague about as to whether we
mean Sam on the ladder or the red paint can), arriving there at the time
tau2; we then
must have (don't you just love that phrase, "then must have" ?)

½(tau0 + tau2) = tau1,
or ½([midmorning + 0] + [midmorning + 20]) = [midmorning + 16], which is
curious to say the least, since Sam and Joe could be doing this in the
late afternoon for all the difference it would make.
But ok, Einstein wanted to be complete, so I guess its fine.

But our hero and physics wizard isn't satisfied with this. Oh no, we
need
to include the length of the ladder as well, or we won't have any
spacetime
to prattle on about later so that people will see just how smart we are.

Here is Einstein's equation:
½[tau(0,0,0,t)+tau(0,0,0,t+x'/(c-v)+x'/(c+v))] = tau(x',0,0,t+x'/(c-v))

You can read about it at
   http://www.fourmilab.ch/etexts/einstein/specrel/www/
 (in Section 3)

Putting in the mosquito numbers,

½[tau(0,0,0,t)+tau(0,0,0,t+32/(5-3)+32/(5+3))] = tau(32,0,0,t+32/(5-3))
½[tau(0,0,0,t)+tau(0,0,0,t+20)] = tau(32,0,0,t+16)

In agreement with experience (gotta love that phrase!) clearly!
(0,0,0,t)
is pretty meaningless, and we can drop the "t+" since we really don't
care
if Sam and Joe are walking in the midmorning or late afternoon.

So,
½ * tau(0,0,0,20) = tau(32,0,0,16).

There's some differentiation by Einstein to make himself look smart and
important, he has to show off all his skills if not his common sense,
because "common sense is the collection of prejudices acquired by age
eighteen", he tells us, and he eventually arrives at

tau  = (t-vx/c^2) / sqrt( 1 - v^2/c^2 )
xi   = (x-vt)     / sqrt( 1 - v^2/c^2 )
eta  = y
zeta = z.

That is what you get when you treat time as if it were a vector and mix
in
some distance.
We can forget y and z, the mosquito didn't fly up into a tree or into
the
ditch at the side.

We apply this to the equations derived:

tau = (16 - 3 * 80 / 25)  /  sqrt (1 - 3^2/5^2)
   = (6.4) / 0.8
   = 8 seconds

xi = 32 / sqrt (1 - 9 / 25)
  = 40 feet

Sanity check:

c = 40 ft / 8 seconds  = 5 fps. Yep, that's the right speed for Albert.

So...
We are standing at the roadside watching Sam and Joe carry a 40 ft
ladder
that they think is a 32 ft ladder, because the speed of mosquitoes is 5
fps
in all inertial frames of reference.

It must be right, its only algebra after all is said and done.

So now you should be able to fully understand Special Relativity, all
you need do is replace the speed of the mosquito with the speed of
light,
have Sam and Joe run at the relativistic speed of 0.6c, the algebra is
perfect, and who needs common sense anyway?

Just remember that 40 ft ladders shrink to 32 ft ladders when you run
with
them at 180,000 km/sec, and you'll be as smart as Einstein the cretin.

For myself, I'll keep the collection of prejudices I acquired by the
time I was eighteen.

Oh, wait. I said I'd prove that nothing can exceed the speed of a
mosquito.

Let's suppose a butterfly travels at speed w = 10 fps, racing the
mosquito
and getting to the other end of the ladder in half the time, 8 seconds.
From Einstein (reference above)

V  = (c+w)/(1 + w/c)
    = (5 +10) / (1 + 10/5)
    = 15 / 3
    = 5 fps.

So the butterfly can never exceed the speed of a mosquito, QED.
Simple really.
As with the muon, time stretches for the butterfly's internal clock.

tau = ( 8 - 10/25) / sqrt( 1 - 100 / 25)
     = ( 8 - 0.4 ) / sqrt ( 1-4)
     = 7.6 / i * 1.7321

Oh dear.... the internal clock has stretched out into the complex plane

Hmmm...
Let's try it this way.

tau = (t- vx/c^2)   / sqrt (1 - v^2/c^2)

Since v = x/t,  (80/8 in this case) then x = vt, so

tau   = ( t - v * vt / c^2) / sqrt (1 - v^2/c^2)

        = t (1 - v^2/c^2) / sqrt (1 - v^2/c^2)

         = t * sqrt (1 - v^2/c^2)

         = 8 * sqrt ( 1 - 100/25)

         = 8 * i * 1.7321.....

Oh dear, we seem to be stuck with complex time... Never mind,  let's
accept it
anyway, and work out the shrivelled length.

xi  =  40 ft /  i * 1.7321

Oh look, we've complex length as well. Still, the speed of the muon....
oops, butterfly, I should have said,  is going to be xi/tau, and that
cancels the imaginary part to give 40/8  * i * 1.7321 / i * 1.7321   = 5
fps!! Yippee, we've "proven" the speed of the butterfly cannot exceed
the speed of the mosquito, even if it gets to the other end of the
ladder first.

Well, do carry on, any relativist will assure you it works, I'm sure.
 80 ft did you say? what 80 ft?

With that in mind, now listen to the garbage the relativists will feed
you.

Remember that the analogue of the speed of light is the speed of the
mosquito,
the analogue of the muon is the butterfly, the analogue of Special
Relativity is the faeces of the male bovine and that Einstein was
anencephalous.

Have a good day.
Androcles

The standard model of particle physics is the currently-accepted theory
of the interactions of elementary particles; it accurately describes
about 99% of what we know about particle interactions. The standard
model is a quantum-mechanical gauge theory of strong, weak, and
electromagnetic interactions.

The muon decay is (essentially 100% of the time):
    mu- => e- + nu_ebar + nu_mu
In the standard model there are several Feynman diagrams that contribute
to this process, involving the weak intermediate bosons. Those bosons
have masses about 800 times that of a muon and about 150,000 times the
mass of an electron. The incredibly slow rate of muon decay is basically
due tho this large disparity in these masses ("incredibly slow" is about
2.2 microseconds).

The "clock" involved in muon decay is related to the various parameters
of those Feynman diagrams, including their coupling constants, masses,
and the resulting phase space for the decay. As the underlying theory is
Lorentz invariant, the decay lifetime of moving muons behaves according
to the usual time dilation of SR.

The underlying process is stochastic. That is, its rate is determined by
the amplitude computed from the Feynman diagrams, but the amplitude for
the process only determines a probability of decaying.

Yes and no. This is a quantum theory, and the number of "internal
components" present is not well defined. Basically, most of the time
(>>99%) there is just a muon present, but there is nonzero probability
of other particles being present -- these are virtual particles. Decay
occurs when these virtual particles happen to emerge in such a way that
permits them to propagate macroscopic distances (called "on mass shell")
-- for muon decay there is only a very tiny probability of this happening.

Electrons are stable, meaning there is no set of particles into which
they are allowed to decay. All less massive particles have zero charge,
but the electron is charged; conservation of charge prevents this decay
(as do several more subtle conservation rules).

Tom Roberts    tjroberts@lucent.com

It is just a colloquial way of saying that, in gthe frame in which the
clock is stationary, the probability per unit time of decaying is a
constant, the reciprocal of which we refer to as the mean life of a
muon.

What's this internal clock?  Does a muon

No, except insofar as the deacy probability per unit time remains
constant.

The muon is also a point particle.
And the electron never decays.

Franz

Simply addressing the point as to whether the time dilation is real (as
viewed in a frame where the particle is moving), note that g-2
experiments store muons in storage rings (where they precess). Thus,
both the speed of the muons is known (the muon bunches would proceed
from station to station on the ring at a rate that is measurable), and
the lifetime in the laboratory is known (by the decay rate from the
population dN/dt). Thus the time dilation in this case is NOT simply
inferred by the distance traveled.

PD

The famous Mt. Washington experiment you refer to has been discredited
by the several of the neutrino oscillation researseachers.  In their
work the production heights have been carefully studied and it seem
more muons are produced below Mt. Washington's peak than above it.

...a much better explanation than a faulty internal clock Eh? ;-)

Sue...

http://www.google.com/search?hl=en&q=muon+production+height&btnG=Google+Search

Traveling at 0.9c there would be an observed time dilation (increased
half-life to decay) of 2.294X,

1/sqrt{1-[(v^2)/(c^2)]}

Conversely, time in the particle frame would appear to pass only
0.4359X as fast as in the external observers' frame when clocks are
locally compared.  That word "locally" is vital.  There is no anomaly
in either or any reference frame until clocks are *locally* compared
by sharing the same spacetime.  If clocks are not local, lightspeed
delay in information propagation results in relativity.

The spacetime four vector is conserved.  An inertial frame of
reference that travels further though space travels less through
time.  However, let's match units.  If space is meters and time is
seconds it doesn't work.  (lightspeed)(time), ct, is meters.  Time is
a very "long" dimension.  It takes a whole lot of space to noticibly
bite into time.

Whether there is a clock present or not is irrelevant.  One could set
up the experiment, end all accelerations, and then locally build the
clock starting with smelting metal ore.  Makes no difference.

A very large collection of muons at rest has a well-determined
invarient decay half-life.  A single muon has only probabilities and a
broad envelope, so one accumulates observations.  Counting statistics
improve as sqrt(number of observations).  10,000 times the number of
observations gives you 100X narrower window, events vs. time interval
elapsed.

Quantum mechanics, Heisenberg Uncertaintly, and an energy barrier to
decay to tunnel through or leap over.

The process (clock) is invariant.  The mix of space and time in
spacetime is maleable by velocity (Special Relativity) and
acceleration (General Relativity).  It is observer-dependent.  Newton
was wrong.

Time is what a clock measures.  The mix of observed space and time in
spacetime depends on the observer.

Get in an untethered boat.  How fast are you moving?  Current, tides,
waves vs. the shore; rotation of the Earth about its axis vs. the
fixed stars; orbit of the Earth about the sun; orbit of the solar
system in the Milky Way... It depends.  As space is negotiable
depending on viewpoint, so is time.  However, time scales as
lightspeed.  Jiggling time gives you small results for great efforts.

I have to try my hand at answering this question.

In biology we can say that it is the DNA and the associated translational
mechanisms that give the cells their properties and behaviors. If a cell
loses its DNA it no longer behaves as a living cell.

We might be inclined to use analogous reasoning in relativity and think that
there is some mechanism in particles, atoms or pieces of matter that enforce
the results of relativity. This was the view of H.A. Lorentz who postulated
that their was some mechanism that caused atoms to compress in the direction
of motion through the ether. But, although Lorentz worked out some of the
essential mathematics, it is Einstein who gets the credit for the theory.

There is nothing in particles, atoms or matter that 'causes' special
relativity. Special relativity is really the geometry of flat spacetime.
Flat Spacetime has a geometry that is different than Euclidean geometry and
it is not caused by the particles that might be moving through it. It is a
geometry that is intrinsic to spacetime itself. That is what Einstein saw
and that is one reason he is famous.

Time dilation and the muon effect is one of the theorems or results that
comes out of spacetime geometry. I'm not going to try to present spacetime
geometry here, or derive time dilation here. It's not horribly difficult to
learn. Time dilation means that moving clocks are seen to run slower than
stationary clocks. But there is nothing different about the clocks, or any
mechanism within the clocks that makes them run slower. It is just a result
of spacetime geometry.

In ordinary Eucliden geometry we have the Pythagorean theorem. But we don't
think that if we looked at the points and lines in a geometrical diagram
closely enough through a microscope we would see a mechanism that 'caused'
the Pythagorean theorem to be true. It is just a result of the axioms of the
geometry. In the same way spacetime geometry has its postulates, which are
well verified by experiment, and time dilation is one of the results.

Geometry not mechanism.

David Park
djmp@earthlink.net
http://home.earthlink.net/~djmp/

Wolfson is correct in this.

Time dilation is an effect not caused by time but by the variance of an
object's time rates.

Whether a particle decays or not depends on its stability, so that can
be described as an "internal clock", I would think.

AE was one of the founders of qm and he was the sole inventor of
Relativity, but the decay process is not in conflict with SR, AFAIK.
The SR explanation in the Twin Paradox experiment and others the effect
of time dilation is in line with qm's "clock-like process which can be
slowed".  The Twins decay (age) but their rates of decay depend on the
speeds achieved during the trip of the experiment.

Obviously, this is an example where the two theories coincide and
support each other.

I hope this is a better answer to your questions than those you have
received so far.
TomGee