Post by Michael LevinSorry if this is a stupid question - I'm a biologist with an amateur
interest in physics. I was listening to an audio lecture by Richard Wolfson
and he was talking about the experiment where, due to relativistic effects,
the lifetime of muons is extended due to their rapid motion (0.9c or
something like that) and the resulting time dilation. He was making the
point that it isn't just clocks that are affected by time dilation, but
everything (time itself). I think I follow all this stuff so far. But he
keeps talking about "the muon's internal clock". I am sure this must be a
euphemism for something, but what? What's this internal clock? Does a muon
have internal components of some sort which decide when it's to decay? It
would seem that the relativistic explanation for what's going on models the
muon as a physical clock-like process (which can be slowed). But, what is
the relationship between quantum mechanics' assertion that decay is in
principle unpredictable (presumably meaning, not controlled by
mechanistic
factors) and this effect where it would seem that some process counts time
for the muon? I hope my question makes sense; I'd like to know if anything
is known about how these two theories intersect in this case. Does the muon
have internal components which decide when to decay? If so, is it different
from "point" particles like electrons (and if yes, can the same sort of
relativistic experiment be done with them)? Any thoughts would be
appreciated!
--
Mike Levin
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" ?)
œ(tau0 + tau2) = tau1,
or œ([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:
œ[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,
œ[tau(0,0,0,t)+tau(0,0,0,t+32/(5-3)+32/(5+3))] = tau(32,0,0,t+32/(5-3))
œ[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,
œ * 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