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Item 7. Special Relativistic Paradoxes - part (c)
The Superluminal Scissors updated 31-MAR-1993
-------------------------
A Gedankenexperiment:
Imagine a huge pair of scissors, with blades one light-year long.
The handle is only about two feet long, creating a huge lever arm,
initially open by a few degrees. Then you suddenly close the scissors.
This action takes about a tenth of a second. Doesn't the contact point
where the two blades touch move down the blades *much* faster than the
speed of light? After all, the scissors close in a tenth of a second, but
the blades are a light-year long. That seems to mean that the contact
point has moved down the blades at the remarkable speed of 10 light-years
per second. This is more than 10^8 times the speed of light! But this
seems to violate the most important rule of Special Relativity - no signal
can travel faster than the speed of light. What's going on here?
Explanation:
We have mistakenly assumed that the scissors do in fact close when
you close the handle. But, in fact, according to Special Relativity, this
is not at all what happens. What *does* happen is that the blades of the
scissors flex. No matter what material you use for the scissors, SR sets a
theoretical upper limit to the rigidity of the material. In short, when
you close the scissors, they bend.
The point at which the blades bend propagates down the blade at
some speed less than the speed of light. On the near side of this point,
the scissors are closed. On the far side of this point, the scissors
remain open. You have, in fact, sent a kind of wave down the scissors,
carrying the information that the scissors have been closed. But this wave
does not travel faster than the speed of light. It will take at least one
year for the tips of the blades, at the far end of the scissors, to feel
any force whatsoever, and, ultimately, to come together to completely close
the scissors.
As a practical matter, this theoretical upper limit to the rigidity
of the metal in the scissors is *far* higher than the rigidity of any real
material, so it would, in practice, take much much longer to close a real
pair of metal scissors with blades as long as these.
One can analyze this problem microscopically as well. The
electromagnetic force which binds the atoms of the scissors together
propagates at the speeds of light. So if you displace some set of atoms in
the scissor (such as the entire handles), the force will not propagate down
the scissor instantaneously, This means that a scissor this big *must*
cease to act as a rigid body. You can move parts of it without other parts
moving at the same time. It takes some finite time for the changing forces
on the scissor to propagate from atom to atom, letting the far tip of the
blades "know" that the scissors have been closed.
Caveat:
The contact point where the two blades meet is not a physical
object. So there is no fundamental reason why it could not move faster
than the speed of light, provided that you arrange your experiment correctly.
In fact it can be done with scissors provided that your scissors are short
enough and wide open to start, very different conditions than those spelled
out in the gedankenexperiment above. In this case it will take you quite
a while to bring the blades together - more than enough time for light to
travel to the tips of the scissors. When the blades finally come together,
if they have the right shape, the contact point can indeed move faster
than light.
Think about the simpler case of two rulers pinned together at an
edge point at the ends. Slam the two rulers together and the contact point
will move infinitely fast to the far end of the rulers at the instant
they touch. So long as the rulers are short enough that contact does not
happen until the signal propagates to the far ends of the rulers, the
rulers will indeed be straight when they meet. Only if the rulers are
too long will they be bent like our very long scissors, above, when they
touch. The contact point can move faster than the speed of light, but
the energy (or signal) of the closing force can not.
An analogy, equivalent in terms of information content, is, say, a
line of strobe lights. You want to light them up one at a time, so that
the `bright' spot travels faster than light. To do so, you can send a
_luminal_ signal down the line, telling each strobe light to wait a
little while before flashing. If you decrease the wait time with
each successive strobe light, the apparent bright spot will travel faster
than light, since the strobes on the end didn't wait as long after getting
the go-ahead, as did the ones at the beginning. But the bright spot
can't pass the original signal, because then the strobe lights wouldn't
know to flash.
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Item 8.
The Particle Zoo updated 9-OCT-1992 by SIC
---------------- original by Matt Austern
If you look in the Particle Data Book, you will find more than 150
particles listed there. It isn't quite as bad as that, though...
The particles are in three categories: leptons, mesons, and
baryons. Leptons are particle that are like the electron: they are
spin-1/2, and they do not undergo the strong interaction. There are three
charged leptons, the electron, muon, and tau, and three neutral leptons, or
neutrinos. (The muon and the tau are both short-lived.)
Mesons and baryons both undergo strong interactions. The
difference is that mesons have integral spin (0, 1,...), while baryons have
half-integral spin (1/2, 3/2,...). The most familiar baryons are the
proton and the neutron; all others are short-lived. The most familiar
meson is the pion; its lifetime is 26 nanoseconds, and all other mesons
decay even faster.
Most of those 150+ particles are mesons and baryons, or,
collectively, hadrons. The situation was enormously simplified in the
1960s by the "quark model," which says that hadrons are made out of
spin-1/2 particles called quarks. A meson, in this model, is made out of a
quark and an anti-quark, and a baryon is made out of three quarks. We
don't see free quarks (they are bound together too tightly), but only
hadrons; nevertheless, the evidence for quarks is compelling. Quark masses
are not very well defined, since they are not free particles, but we can
give estimates. The masses below are in GeV; the first is current mass
and the second constituent mass (which includes some of the effects of the
binding energy):
Generation: 1 2 3
U-like: u=.006/.311 c=1.50/1.65 t=91-200/91-200
D-like: d=.010/.315 s=.200/.500 b=5.10/5.10
In the quark model, there are only 12 elementary particles, which
appear in three "generations." The first generation consists of the up
quark, the down quark, the electron, and the electron neutrino. (Each of
these also has an associated antiparticle.) These particles make up all of
the ordinary matter we see around us. There are two other generations,
which are essentially the same, but with heavier particles. The second
consists of the charm quark, the strange quark, the muon, and the muon
neutrino; and the third consists of the top quark, the bottom quark, the
tau, and the tau neutrino. (The top has not been directly observed; see
the "Top Quark" FAQ entry for details.) These three generations are
sometimes called the "electron family", the "muon family", and the "tau
family."
Finally, according to quantum field theory, particles interact by
exchanging "gauge bosons," which are also particles. The most familiar on
is the photon, which is responsible for electromagnetic interactions.
There are also eight gluons, which are responsible for strong interactions,
and the W+, W-, and Z, which are responsible for weak interactions.
The picture, then, is this:
FUNDAMENTAL PARTICLES OF MATTER
Charge -------------------------
-1 | e | mu | tau |
0 | nu(e) |nu(mu) |nu(tau)|
------------------------- + antiparticles
-1/3 | down |strange|bottom |
2/3 | up | charm | top |
-------------------------
GAUGE BOSONS
Charge Force
0 photon electromagnetism
0 gluons (8 of them) strong force
+-1 W+ and W- weak force
0 Z weak force
The Standard Model of particle physics also predict the
existence of a "Higgs boson," which has to do with breaking a symmetry
involving these forces, and which is responsible for the masses of all the
other particles. It has not yet been found. More complicated theories
predict additional particles, including, for example, gauginos and sleptons
and squarks (from supersymmetry), W' and Z' (additional weak bosons), X and
Y bosons (from GUT theories), Majorons, familons, axions, paraleptons,
ortholeptons, technipions (from technicolor models), B' (hadrons with
fourth generation quarks), magnetic monopoles, e* (excited leptons), etc.
None of these "exotica" have yet been seen. The search is on!
REFERENCES:
The best reference for information on which particles exist, their
masses, etc., is the Particle Data Book. It is published every two years;
the most recent edition is Physical Review D Vol.45 No.11 (1992).
There are several good books that discuss particle physics on a
level accessible to anyone who knows a bit of quantum mechanics. One is
_Introduction to High Energy Physics_, by Perkins. Another, which takes a
more historical approach and includes many original papers, is
_Experimental Foundations of Particle Physics_, by Cahn and Goldhaber.
For a book that is accessible to non-physicists, you could try _The
Particle Explosion_ by Close, Sutton, and Marten. This book has fantastic
photography.
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Item 9.
Olbers' Paradox updated: 24-JAN-1993 by SIC
---------------
Why isn't the night sky as uniformly bright as the surface of the
Sun? If the Universe has infinitely many stars, then it should be. After
all, if you move the Sun twice as far away from us, we will intercept
one-fourth as many photons, but the Sun will subtend one-fourth of the
angular area. So the areal intensity remains constant. With infinitely
many stars, every angular element of the sky should have a star, and the
entire heavens should be a bright as the sun. We should have the
impression that we live in the center of a hollow black body whose
temperature is about 6000 degrees Centigrade. This is Olbers' paradox.
It can be traced as far back as Kepler in 1610. It was rediscussed by
Halley and Cheseaux in the eighteen century, but was not popularized as
a paradox until Olbers took up the issue in the nineteenth century.
There are many possible explanations which have been considered.
Here are a few:
(1) There's too much dust to see the distant stars.
(2) The Universe has only a finite number of stars.
(3) The distribution of stars is not uniform. So, for example,
there could be an infinity of stars, but they hide behind one
another so that only a finite angular area is subtended by them.
(4) The Universe is expanding, so distant stars are red-shifted into
obscurity.
(5) The Universe is young. Distant light hasn't even reached us yet.
The first explanation is just plain wrong. In a black body, the
dust will heat up too. It does act like a radiation shield, exponentially
damping the distant starlight. But you can't put enough dust into the
universe to get rid of enough starlight without also obscuring our own Sun.
So this idea is bad.
The premise of the second explanation may technically be correct.
But the number of stars, finite as it might be, is still large enough to
light up the entire sky, i.e., the total amount of luminous matter in the
Universe is too large to allow this escape. The number of stars is close
enough to infinite for the purpose of lighting up the sky. The third
explanation might be partially correct. We just don't know. If the stars
are distributed fractally, then there could be large patches of empty space,
and the sky could appear dark except in small areas.
But the final two possibilities are are surely each correct and
partly responsible. There are numerical arguments that suggest that the
effect of the finite age of the Universe is the larger effect. We live
inside a spherical shell of "Observable Universe" which has radius equal to
the lifetime of the Universe. Objects more than about 15 billions years
old are too far away for their light ever to reach us.
Historically, after Hubble discovered that the Universe was
expanding, but before the Big Bang was firmly established by the discovery
of the cosmic background radiation, Olbers' paradox was presented as proof
of special relativity. You needed the red-shift (an SR effect) to get rid
of the starlight. This effect certainly contributes. But the finite age
of the Universe is the most important effect.
References: Ap. J. _367_, 399 (1991). The author, Paul Wesson, is said to
be on a personal crusade to end the confusion surrounding Olbers' paradox.
_Darkness at Night: A Riddle of the Universe_, Edward Harrison, Harvard
University Press, 1987
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Item 10.
What is Dark Matter? updated 11-MAY-1993 by SIC
--------------------
The story of dark matter is best divided into two parts. First we
have the reasons that we know that it exists. Second is the collection of
possible explanations as to what it is.
Why the Universe Needs Dark Matter
----------------------------------
We believe that that the Universe is critically balanced between
being open and closed. We derive this fact from the observation of the
large scale structure of the Universe. It requires a certain amount of
matter to accomplish this result. Call it M.
We can estimate the total BARYONIC matter of the universe by
studying Big Bang nucleosynthesis. This is done by connecting the observed
He/H ratio of the Universe today to the amount of baryonic matter present
during the early hot phase when most of the helium was produced. Once the
temperature of the Universe dropped below the neutron-proton mass difference,
neutrons began decaying into protons. If the early baryon density was low,
then it was hard for a proton to find a neutron with which to make helium
before too many of the neutrons decayed away to account for the amount of
helium we see today. So by measuring the He/H ratio today, we can estimate
the necessary baryon density shortly after the Big Bang, and, consequently,
the total number of baryons today. It turns out that you need about 0.05 M
total baryonic matter to account for the known ratio of light isotopes. So
only 1/20 of the total mass of they Universe is baryonic matter.
Unfortunately, the best estimates of the total mass of everything
that we can see with our telescopes is roughly 0.01 M. Where is the other
99% of the stuff of the Universe? Dark Matter!
So there are two conclusions. We only see 0.01 M out of 0.05 M
baryonic matter in the Universe. The rest must be in baryonic dark matter
halos surrounding galaxies. And there must be some non-baryonic dark matter
to account for the remaining 95% of the matter required to give omega, the
mass of universe, in units of critical mass, equal to unity.
For those who distrust the conventional Big Bang models, and don't
want to rely upon fancy cosmology to derive the presence of dark matter,
there are other more direct means. It has been observed in clusters of
galaxies that the motion of galaxies within a cluster suggests that they
are bound by a total gravitational force due to about 5-10 times as much
matter as can be accounted for from luminous matter in said galaxies. And
within an individual galaxy, you can measure the rate of rotation of the
stars about the galactic center of rotation. The resultant "rotation
curve" is simply related to the distribution of matter in the galaxy. The
outer stars in galaxies seem to rotate too fast for the amount of matter
that we see in the galaxy. Again, we need about 5 times more matter than
we can see via electromagnetic radiation. These results can be explained
by assuming that there is a "dark matter halo" surrounding every galaxy.
What is Dark Matter
-------------------
This is the open question. There are many possibilities, and
nobody really knows much about this yet. Here are a few of the many
published suggestions, which are being currently hunted for by
experimentalists all over the world. Remember, you need at least one
baryonic candidate and one non-baryonic candidate to make everything
work out, so there there may be more than one correct choice among
the possibilities given here.
(1) Normal matter which has so far eluded our gaze, such as
(a) dark galaxies
(b) brown dwarfs
(c) planetary material (rock, dust, etc.)
(2) Massive Standard Model neutrinos. If any of the neutrinos are massive,
then this could be the missing mass. On the other hand, if they are
too heavy, like the purported 17 KeV neutrino would have been, massive
neutrinos create almost as many problems as they solve in this regard.
(3) Exotica (See the "Particle Zoo" FAQ entry for some details)
Massive exotica would provide the missing mass. For our purposes,
these fall into two classes: those which have been proposed for other
reasons but happen to solve the dark matter problem, and those which have
been proposed specifically to provide the missing dark matter.
Examples of objects in the first class are axions, additional
neutrinos, supersymmetric particles, and a host of others. Their properties
are constrained by the theory which predicts them, but by virtue of their
mass, they solve the dark matter problem if they exist in the correct
abundance.
Particles in the second class are generally classed in loose groups.
Their properties are not specified, but they are merely required to be
massive and have other properties such that they would so far have eluded
discovery in the many experiments which have looked for new particles.
These include WIMPS (Weakly Interacting Massive Particles), CHAMPS, and a
host of others.
References: _Dark Matter in the Universe_ (Jerusalem Winter School for
Theoretical Physics, 1986-7), J.N. Bahcall, T. Piran, & S. Weinberg editors.
_Dark Matter_ (Proceedings of the XXIIIrd Recontre de Moriond) J. Audouze and
J. Tran Thanh Van. editors.