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Editor's note:
|
||
|
||
FAQ #19, on the Equivalence Principle and the radiation of
|
||
charged particles, has opened a proverbial can of worms. It has been
|
||
brought to my attention by people who's expertise far exceeds my own
|
||
that many serious issues are glossed over or perhaps even misrepresented
|
||
in the article as I had previously editted it.
|
||
I have attempted to address some of these concerns in a modified
|
||
article, but some of the problems are beyond my ability to handle. Please
|
||
note that the changes are entirely my own, not the original author's, and
|
||
are based upon my own, probably naive, new understanding of the issues based
|
||
upon input from concerned readers.
|
||
It has been suggested that I simply pull the article until I can
|
||
prove its assertions or give adequate references to accepted literature.
|
||
However, that is likely never to happen if it depends only upon my
|
||
(non-existent) expertise on GR. So I have warily decided to keep it in the
|
||
public view in the hope that it will act as an irritant, and perhaps
|
||
induce some qualified expert to produce some pearls of wisdom which I can
|
||
use to fix its obvious deficiencies.
|
||
|
||
The other articles which I have indicated have been modified for this
|
||
posting contain only cosmetic changes.
|
||
-Scott
|
||
|
||
--------------------------------------------------------------------------------
|
||
FREQUENTLY ASKED QUESTIONS ON SCI.PHYSICS - Part 1/2
|
||
--------------------------------------------------------------------------------
|
||
|
||
This Frequently Asked Questions List is posted monthly, at or near
|
||
the first of the month, to the Usenet newsgroup sci.physics in an attempt
|
||
to provide good answers to frequently asked questions and other reference
|
||
material which is worth preserving. If you have corrections or answers to
|
||
other frequently asked questions that you would like included in this
|
||
posting, send E-mail to sichase@csa2.lbl.gov (Scott I. Chase).
|
||
|
||
The FAQ is distributed to all interested parties whenever sufficient
|
||
changes have accumulated to warrant such a mailing. To request that your
|
||
address be added to the list, send mail to my address, above, and include
|
||
the words "FAQ Mailing List" in the subject header of your message. To
|
||
faciliate mailing, the FAQ is now being distributed as a multi-part posting.
|
||
|
||
If you are a new reader of sci.physics, please read item #1, below.
|
||
If you do not wish to read the FAQ at all, add "Frequently Asked Questions"
|
||
to your .KILL file.
|
||
|
||
A listing of new items can be found above the subject index, so
|
||
that you can quickly identify new subjects of interest. To locate old
|
||
items which have been updated since the last posting, look for the stars (*)
|
||
in the subject index, which indicate new material.
|
||
|
||
Items which have been submitted by a single individual are
|
||
attributed to the original author. All other contributors have been thanked
|
||
privately.
|
||
|
||
New Items: NONE
|
||
|
||
Index of Subjects
|
||
-----------------
|
||
1. An Introduction to Sci.Physics
|
||
2. Gravitational Radiation
|
||
3. Energy Conservation in Cosmology and Red Shift
|
||
4. Effects Due to the Finite Speed of Light
|
||
5. The Top Quark
|
||
6. Tachyons
|
||
7. Special Relativistic Paradoxes
|
||
(a) The Barn and the Pole
|
||
(b) The Twin Paradox
|
||
8. The Particle Zoo
|
||
9.*Olbers' Paradox
|
||
10. What is Dark Matter?
|
||
11. Hot Water Freezes Faster than Cold!
|
||
12.*Which Way Will my Bathtub Drain?
|
||
13. Why are Golf Balls Dimpled?
|
||
14. Why do Mirrors Reverse Left and Right?
|
||
15. What is the Mass of a Photon?
|
||
16. How to Change Nuclear Decay Rates
|
||
17. Baryogenesis - Why Are There More Protons Than Antiprotons?
|
||
18. Time Travel - Fact or Fiction?
|
||
19.*Gravity and the Radiation of Charged Particles
|
||
20. The Nobel Prize for Physics
|
||
21. Open Questions
|
||
22. Accessing and Using Online Physics Resources
|
||
|
||
********************************************************************************
|
||
Item 1. updated 4-AUG-1992 by SIC
|
||
|
||
An Introduction to Sci.Physics
|
||
------------------------------
|
||
|
||
Sci.Physics is an unmoderated newsgroup dedicated to the discussion
|
||
of physics, news from the physics community, and physics-related social
|
||
issues. People from a wide variety of non-physics backgrounds, as well
|
||
as students and experts in all areas of physics participate in the ongoing
|
||
discussions on sci.physics. Professors, industrial scientists, graduate
|
||
students, etc., are all on hand to bring physics expertise to bear on
|
||
almost any question. But the only requirement for participation is
|
||
interest in physics, so feel free to post -- but before you do, please do
|
||
the following:
|
||
|
||
(1) Read this posting, a.k.a., the FAQ. It contains good answers,
|
||
contributed by the readership, to some of the most frequently asked
|
||
questions.
|
||
|
||
(2) Understand "netiquette." If you are not sure what this means,
|
||
subscribe to news.announce.newusers and read the excellent discussion of
|
||
proper net behavior that is posted there periodically.
|
||
|
||
(3) Be aware that there is another newsgroup dedicated to the discussion of
|
||
"alternative" physics. It is alt.sci.physics.new-theories, and is the
|
||
appropriate forum for discussion of physics ideas which are not widely
|
||
accepted by the physics community. Sci.Physics is not the group for such
|
||
discussions. A quick look at items posted to both groups will make the
|
||
distinction apparent.
|
||
|
||
(4) Read the responses already posted in the thread to which you want to
|
||
contribute. If a good answer is already posted, or the point you wanted
|
||
to make has already been made, let it be. Old questions have probably been
|
||
thoroughly discussed by the time you get there - save bandwidth by posting
|
||
only new information. Post to as narrow a geographic region as is
|
||
appropriate. If your comments are directed at only one person, try E-mail.
|
||
|
||
(5) Get the facts right! Opinions may differ, but facts should not. It is
|
||
very tempting for new participants to jump in with quick answers to physics
|
||
questions posed to the group. But it is very easy to end up feeling silly
|
||
when people barrage you with corrections. So before you give us all a
|
||
physics lesson you'll regret - look it up.
|
||
|
||
(6) Be prepared for heated discussion. People have strong opinions about
|
||
the issues, and discussions can get a little "loud" at times. Don't take it
|
||
personally if someone seems to always jump all over everything you say.
|
||
Everyone was jumping all over everybody long before you got there! You
|
||
can keep the discussion at a low boil by trying to stick to the facts.
|
||
Clearly separate facts from opinion - don't let people think you are
|
||
confusing your opinions with scientific truth. And keep the focus of
|
||
discussion on the ideas, not the people who post them.
|
||
|
||
(7) Tolerate everyone. People of many different points of view, and widely
|
||
varying educational backgrounds from around the world participate in this
|
||
newsgroup. Respect for others will be returned in kind. Personal
|
||
criticism is usually not welcome.
|
||
|
||
********************************************************************************
|
||
Item 2.
|
||
|
||
Gravitational Radiation updated: 4-May-1992 by SIC
|
||
-----------------------
|
||
|
||
Gravitational Radiation is to gravity what light is to
|
||
electromagnetism. It is produced when massive bodies accelerate. You can
|
||
accelerate any body so as to produce such radiation, but due to the feeble
|
||
strength of gravity, it is entirely undetectable except when produced by
|
||
intense astrophysical sources such as supernovae, collisions of black
|
||
holes, etc. These are quite far from us, typically, but they are so
|
||
intense that they dwarf all possible laboratory sources of such radiation.
|
||
|
||
Gravitational waves have a polarization pattern that causes objects
|
||
to expand in one direction, while contracting in the perpendicular
|
||
direction. That is, they have spin two. This is because gravity waves are
|
||
fluctuations in the tensorial metric of space-time.
|
||
|
||
All oscillating radiation fields can be quantized, and in the case
|
||
of gravity, the intermediate boson is called the "graviton" in analogy
|
||
with the photon. But quantum gravity is hard, for several reasons:
|
||
(1) The quantum field theory of gravity is hard, because gauge
|
||
interactions of spin-two fields are not renormalizable. See Cheng and Li,
|
||
Gauge Theory of Elementary Particle Physics (search for "power counting").
|
||
(2) There are conceptual problems - what does it mean to quantize
|
||
geometry, or space-time?
|
||
|
||
It is possible to quantize weak fluctuations in the gravitational
|
||
field. This gives rise to the spin-2 graviton. But full quantum gravity
|
||
has so far escaped formulation. It is not likely to look much like the
|
||
other quantum field theories. In addition, there are models of gravity
|
||
which include additional bosons with different spins. Some are the
|
||
consequence of non-Einsteinian models, such as Brans-Dicke which has a
|
||
spin-0 component. Others are included by hand, to give "fifth force"
|
||
components to gravity. For example, if you want to add a weak repulsive
|
||
short range component, you will need a massive spin-1 boson. (Even-spin
|
||
bosons always attract. Odd-spin bosons can attract or repel.) If
|
||
antigravity is real, then this has implications for the boson spectrum as
|
||
well.
|
||
|
||
The spin-two polarization provides the method of detection. All
|
||
experiments to date use a "Weber bar." This is a cylindrical, very
|
||
massive, bar suspended by fine wire, free to oscillate in response to a
|
||
passing graviton. A high-sensitivity, low noise, capacitive transducer
|
||
can turn the oscillations of the bar into an electric signal for analysis.
|
||
So far such searches have failed. But they are expected to be
|
||
insufficiently sensitive for typical radiation intensity from known types
|
||
of sources.
|
||
|
||
A more sensitive technique uses very long baseline laser
|
||
interferometry. This is the principle of LIGO (Laser Interferometric
|
||
Gravity wave Observatory). This is a two-armed detector, with
|
||
perpendicular laser beams each travelling several km before meeting to
|
||
produce an interference pattern which fluctuates if a gravity wave distorts
|
||
the geometry of the detector. To eliminate noise from seismic effects as
|
||
well as human noise sources, two detectors separated by hundreds to
|
||
thousands of miles are necessary. A coincidence measurement then provides
|
||
evidence of gravitational radiation. In order to determine the source of
|
||
the signal, a third detector, far from either of the first two, would be
|
||
necessary. Timing differences in the arrival of the signal to the three
|
||
detectors would allow triangulation of the angular position in the sky of
|
||
the signal.
|
||
|
||
The first stage of LIGO, a two detector setup in the U.S., has been
|
||
approved by Congress in 1992. LIGO researchers have started designing a
|
||
prototype detector, and are hoping to enroll another nation, probably in
|
||
Europe, to fund and be host to the third detector.
|
||
|
||
The speed of gravitational radiation (C_gw) depends upon the
|
||
specific model of Gravitation that you use. There are quite a few
|
||
competing models (all consistent with all experiments to date) including of
|
||
course Einstein's but also Brans-Dicke and several families of others.
|
||
All metric models can support gravity waves. But not all predict radiation
|
||
travelling at C_gw = C_em. (C_em is the speed of electromagnetic waves.)
|
||
|
||
There is a class of theories with "prior geometry", in which, as I
|
||
understand it, there is an additional metric which does not depend only on
|
||
the local matter density. In such theories, C_gw != C_em in general.
|
||
|
||
However, there is good evidence that C_gw is in fact at least
|
||
almost C_em. We observe high energy cosmic rays in the 10^20-10^21 eV
|
||
region. Such particles are travelling at up to (1-10^-18)*C_em. If C_gw <
|
||
C_em, then particles with C_gw < v < C_em will radiate Cerenkov
|
||
gravitational radiation into the vacuum, and decelerate from the back
|
||
reaction. So evidence of these very fast cosmic rays good evidence that
|
||
C_gw >= (1-10^-18)*C_em, very close indeed to C_em. Bottom line: in a
|
||
purely Einsteinian universe, C_gw = C_em. However, a class of models not
|
||
yet ruled out experimentally does make other predictions.
|
||
|
||
A definitive test would be produced by LIGO in coincidence with
|
||
optical measurements of some catastrophic event which generates enough
|
||
gravitational radiation to be detected. Then the "time of flight" of both
|
||
gravitons and photons from the source to the Earth could be measured, and
|
||
strict direct limits could be set on C_gw.
|
||
|
||
For more information, see Gravitational Radiation (NATO ASI -
|
||
Les Houches 1982), specifically the introductory essay by Kip Thorne.
|
||
|
||
********************************************************************************
|
||
Item 3.
|
||
|
||
ENERGY CONSERVATION IN COSMOLOGY AND RED SHIFT updated: 10-May-1992 by SIC
|
||
----------------------------------------------
|
||
|
||
IS ENERGY CONSERVED IN OUR UNIVERSE? NO
|
||
|
||
Why? Every conserved quantity is the result of some symmetry of
|
||
nature. This is known as Noether's theorem. For example, momentum
|
||
conservation is the result of translation invariance, because position is
|
||
the variable conjugate to momentum. Energy would be conserved due to
|
||
time-translation invariance. However, in an expanding or contracting
|
||
universe, there is no time-translation invariance. Hence energy is not
|
||
conserved. If you want to learn more about this, read Goldstein's
|
||
Classical Mechanics, and look up Noether's theorem.
|
||
|
||
DOES RED-SHIFT LEAD TO ENERGY NON-CONSERVATION: SOMETIMES
|
||
|
||
There are three basic cosmological sources of red-shifted light:
|
||
(1) Very massive objects emitting light
|
||
(2) Very fast objects emitting light
|
||
(3) Expansion of the universe leading to CBR (Cosmic Background
|
||
Radiation) red-shift
|
||
|
||
About each:
|
||
(1) Light has to climb out the gravitational well of a very massive object.
|
||
It gets red-shifted as a result. As several people have commented, this
|
||
does not lead to energy non-conservation, because the photon had negative
|
||
gravitational potential energy when it was deep in the well. No problems
|
||
here. If you want to learn more about this read Misner, Thorne, and
|
||
Wheeler's Gravitation, if you dare.
|
||
|
||
(2) Fast objects moving away from you emit Doppler shifted light. No
|
||
problems here either. Energy is only one part a four-vector, so it
|
||
changes from frame to frame. However, when looked at in a Lorentz
|
||
invariant way, you can convince yourself that everything is OK here too.
|
||
If you want to learn more about this, read Taylor and Wheeler's
|
||
Spacetime Physics.
|
||
|
||
(3) CBR has red-shifted over billions of years. Each photon gets redder
|
||
and redder. And the energy is lost. This is the only case in which
|
||
red-shift leads to energy non-conservation. Several people have speculated
|
||
that radiation pressure "on the universe" causes it to expand more quickly,
|
||
and attempt to identify the missing energy with the speed at which the
|
||
universe is expanding due to radiation pressure. This argument is
|
||
completely specious. If you add more radiation to the universe you add
|
||
more energy, and the universe is now more closed than ever, and the
|
||
expansion rate slows.
|
||
|
||
If you really MUST construct a theory in which something like
|
||
energy is conserved (which is dubious in a universe without
|
||
time-translation invariance), it is possible to arbitrarily define things
|
||
so that energy has an extra term which compensates for the loss. However,
|
||
although the resultant quantity may be a constant, it is of questionable
|
||
value, and certainly is not an integral associated with time-invariance, so
|
||
it is not what everyone calls energy.
|
||
|
||
********************************************************************************
|
||
Item 4.
|
||
|
||
EFFECTS DUE TO THE FINITE SPEED OF LIGHT updated 28-May-1992 by SIC
|
||
----------------------------------------
|
||
|
||
There are two well known phenomena which are due to the finite
|
||
speed of electromagnetic radiation, but are essentially classical in
|
||
nature, requiring no other facts of special relativity for their
|
||
understanding.
|
||
|
||
(1) Apparent Superluminal Velocity of Galaxies
|
||
|
||
A distant object can appear to travel faster than the speed of
|
||
light relative to us, provided that it has some component of motion towards
|
||
us as well as perpendicular to our line of sight. Say that on Jan. 1 you
|
||
make a position measurement of galaxy X. One month later, you measure it
|
||
again. Assuming you know it's distance from us by some independent
|
||
measurement, you derive its linear speed, and conclude that it is moving
|
||
faster than the speed of light.
|
||
|
||
What have you forgotten? Let's say that on Jan. 1, the object is D
|
||
km from us, and that between Jan. 1 and Feb. 1, the object has moved d km
|
||
closer to us. You have assumed that the light you measured on Jan. 1 and
|
||
Feb. 1 were emitted exactly one month apart. Not so. The first light beam
|
||
had further to travel, and was actually emitted (1 + d/c) months before the
|
||
second measurement, if we measure c in km/month. The object has traveled
|
||
the given angular distance in more time than you thought. Similarly, if
|
||
the object is moving away from us, the apparent angular velocity will be
|
||
too slow, if you do not correct for this effect, which becomes significant
|
||
when the object is moving along a line close to our line of sight.
|
||
|
||
Note that most extragalactic objects are moving away from us due to
|
||
the Hubble expansion. So for most objects, you don't get superluminal
|
||
apparent velocities. But the effect is still there, and you need to take
|
||
it into account if you want to measure velocities by this technique.
|
||
|
||
References:
|
||
|
||
Considerations about the Apparent 'Superluminal Expansions' in
|
||
Astrophysics, E. Recami, A. Castellino, G.D. Maccarrone, M. Rodono,
|
||
Nuovo Cimento 93B, 119 (1986).
|
||
|
||
Apparent Superluminal Sources, Comparative Cosmology and the Cosmic
|
||
Distance Scale, Mon. Not. R. Astr. Soc. 242, 423-427 (1990).
|
||
|
||
(2) Terrell Rotation
|
||
|
||
Consider a cube moving across your field of view with speed near
|
||
the speed of light. The trailing face of the cube is edge on to your line
|
||
of sight as it passes you. However, the light from the back edge of that
|
||
face (the edge of the square farthest from you) takes longer to get to your
|
||
eye than the light from the front edge. At any given instant you are
|
||
seeing light from the front edge at time t and the back edge at time
|
||
t-(L/c), where L is the length of an edge. This means you see the back
|
||
edge where it was some time earlier. This has the effect of *rotating* the
|
||
*image* of the cube on your retina.
|
||
|
||
This does not mean that the cube itself rotates. The *image* is
|
||
rotated. And this depends only on the finite speed of light, not any other
|
||
postulate or special relativity. You can calculate the rotation angle by
|
||
noting that the side face of the cube is Lorentz contracted to L' =
|
||
L/gamma. This will correspond to a rotation angle of arccos(1/gamma).
|
||
|
||
It turns out, if you do the math for a sphere, that the amount of
|
||
apparent rotation exactly cancels the Lorentz contraction. The object
|
||
itself is flattened, but then you see *behind* it as it flies by just
|
||
enough to restore it to its original size. So the image of a sphere is
|
||
unaffected by the Lorentz flattening that it experiences.
|
||
|
||
Another implication of this is that if the object is moving at
|
||
nearly the speed of light, although it is contracted into an
|
||
infinitesimally thin pancake, you see it rotated by almost a full 90
|
||
degrees, so you see the complete backside of the object, and it doesn't
|
||
disappear from view. In the case of the sphere, you see the transverse
|
||
cross-section (which suffers no contraction), so that it still appears to
|
||
be exactly a sphere.
|
||
|
||
That it took so long historically to realize this is undoubtedly
|
||
due to the fact that although we were regularly accelerating particle beams
|
||
in 1959 to relativistic speeds, we still do not have the technology to
|
||
accelerate any macroscopic objects to speeds necessary to reveal the
|
||
effect.
|
||
|
||
References: J. Terrell, Phys Rev. _116_, 1041 (1959). For a textbook
|
||
discussion, see Marion's _Classical Dynamics_, Section 10.5.
|
||
|
||
********************************************************************************
|
||
Item 5.
|
||
|
||
TOP QUARK updated: 10-May-1992 by SIC
|
||
---------
|
||
|
||
The top quark is the hypothetical sixth fundamental strongly
|
||
interacting particle (quark). The known quarks are up (u), down (d),
|
||
strange (s), charm (c) and bottom (b). The Standard Model requires quarks
|
||
to come in pairs in order to prevent mathematical inconsistency due to
|
||
certain "anomalous" Feynman diagrams, which cancel if and only if the
|
||
quarks are paired. The pairs are (d,u),(s,c) and (b,?). The missing
|
||
partner of the b is called "top".
|
||
|
||
In addition, there is experimental evidence that the b quark has an
|
||
"isodoublet" partner, which is so far unseen. The forward-backward
|
||
asymmetry in the reaction e+ + e- -> b + b-bar and the absence of
|
||
flavor-changing neutral currents in b decays imply the existence of the
|
||
isodoublet partner of the b. ("b-bar", pronounced "bee bar", signifies the
|
||
b antiquark.)
|
||
|
||
The mass of the top quark is restricted by a variety of
|
||
measurements. Due to radiative corrections which depend on the top quark
|
||
circulating as a virtual particle inside the loop in the Feynman diagram,
|
||
a number of experimentally accessible processes depend on the top quark
|
||
mass. There are about a dozen such measurements which have been made so
|
||
far, including the width of the Z, b-b-bar mixing (which historically gave
|
||
the first hints that the top quark was very massive), and certain aspects
|
||
of muon decay. These results collectively limit the top mass to roughly
|
||
140 +/- 30 GeV. This uncertainty is a "1-sigma" error bar.
|
||
|
||
Direct searches for the top quark have been performed, looking for
|
||
the expected decay products in both p-p-bar and e+e- collisions. The best
|
||
current limits on the top mass are:
|
||
(1) From the absence of Z -> t + t-bar, M(t) > M(Z)/2 = 45 GeV.
|
||
This is a "model independent" result, depending only on the fact that the
|
||
top quark should be weakly interacting, coupling to the Z with sufficient
|
||
strength to have been detected at the current resolution of the LEP
|
||
experiments which have cornered the market on Z physics in the last several
|
||
years.
|
||
(2) From the absence of top quark decay products in the reaction p
|
||
+ p-bar -> t + t-bar -> hard leptons + X at Fermilab's Tevatron collider,
|
||
the CDF (Collider Detector at Fermilab) experiment. Each top quark is
|
||
expect to decay into a W boson and a b quark. Each W subsequently decays
|
||
into either a charged lepton and a neutrino or two quarks. The cleanest
|
||
signature for the production and decay of the t-t-bar pair is the presence
|
||
of two high-transverse-momentum (high Pt) leptons (electron or muon) in the
|
||
final state. Other decay modes have higher branching ratios, but have
|
||
serious experimental backgrounds from W bosons produced in association with
|
||
jets. The current lower limit on M(t) from such measurements is 91 GeV
|
||
(95% confidence), 95 GeV (90% confidence). However, these limits assume
|
||
that the top quark has the expected decay products in the expected branching
|
||
ratios, making these limits "model dependent," and consequently not as
|
||
"hard" as the considerably lower LEP limit of ~45 GeV.
|
||
|
||
The future is very bright for detecting the top quark. LEP II, the
|
||
upgrade of CERN's e+e- collider to E >= 2*Mw = 160 GeV by 1994, will allow
|
||
a hard lower limit of roughly 90 GeV to be set. Meanwhile, upgrades to
|
||
CDF, start of a new experiment, D0, and upgrades to the accelerator
|
||
complex at Fermilab have recently allowed higher event rates and better
|
||
detector resolution, should allow production of standard model top quarks of
|
||
mass < 150 GeV in the next two years, and even higher mass further in the
|
||
future, at high enough event rate to identify the decays and give rough mass
|
||
measurements.
|
||
|
||
References: Phys. Rev. Lett. _68_, 447 (1992) and the references therein.
|
||
|
||
********************************************************************************
|
||
Item 6.
|
||
|
||
Tachyons updated: 4-May-1992 by SIC
|
||
--------
|
||
|
||
There was a young lady named Bright,
|
||
Whose speed was far faster than light.
|
||
She went out one day,
|
||
In a relative way,
|
||
And returned the previous night!
|
||
|
||
-Reginald Buller
|
||
|
||
|
||
It is a well known fact that nothing can travel faster than the
|
||
speed of light. At best, a massless particle travels at the speed of light.
|
||
But is this really true? In 1962, Bilaniuk, Deshpande, and Sudarshan, Am.
|
||
J. Phys. _30_, 718 (1962), said "no". A very readable paper is Bilaniuk
|
||
and Sudarshan, Phys. Today _22_,43 (1969). I give here a brief overview.
|
||
|
||
Draw a graph, with momentum (p) on the x-axis, and energy (E) on
|
||
the y-axis. Then draw the "light cone", two lines with the equations E =
|
||
+/- p. This divides our 1+1 dimensional space-time into two regions. Above
|
||
and below are the "timelike" quadrants, and to the left and right are the
|
||
"spacelike" quadrants.
|
||
|
||
Now the fundamental fact of relativity is that E^2 - p^2 = m^2.
|
||
(Let's take c=1 for the rest of the discussion.) For any non-zero value of
|
||
m (mass), this is an hyperbola with branches in the timelike regions. It
|
||
passes through the point (p,E) = (0,m), where the particle is at rest. Any
|
||
particle with mass m is constrained to move on the upper branch of this
|
||
hyperbola. (Otherwise, it is "off-shell", a term you here in association
|
||
with virtual particles - but that's another topic.) For massless particles,
|
||
E^2 = p^2, and the particle moves on the light-cone.
|
||
|
||
These two cases are given the names tardyon (or bradyon in more
|
||
modern usage) and luxon, for "slow particle" and "light particle". Tachyon
|
||
is the name given to the supposed "fast particle" which would move with v>c.
|
||
|
||
Now another familiar relativistic equation is E =
|
||
m*[1-(v/c)^2]^(-.5). Tachyons (if they exist) have v > c. This means that
|
||
E is imaginary! Well, what if we take the rest mass m, and take it to be
|
||
imaginary? Then E is negative real, and E^2 - p^2 = m^2 < 0. Or, p^2 -
|
||
E^2 = M^2, where M is real. This is a hyperbola with branches in the
|
||
spacelike region of spacetime. Tachyons are constrained to move on this
|
||
hyperbola.
|
||
|
||
You can now deduce many interesting properties of tachyons. For
|
||
example, they accelerate (p goes up) if they lose energy (E goes down).
|
||
Futhermore, a zero-energy tachyon is "transcendent," or infinitely fast.
|
||
This has profound consequences. For example, let's say that there are
|
||
electrically charged tachyons. Since they move faster than the speed of
|
||
light in the vacuum, they produce Cerenkov radiation. This lowers their
|
||
energy, and they accelerate. So any charged tachyon in the region of
|
||
spacetime where you might choose to put a "charged tachyon detector" will
|
||
quickly accelerate off to the edge of the universe, to be lost forever.
|
||
You will never find a charged tachyon, whether they exist or not.
|
||
|
||
However, tachyons are not entirely invisible. You can imagine that
|
||
you might produce them in some exotic nuclear reaction. If they are
|
||
charged, you could "see" them by detecting the Cerenkov light they produce
|
||
as they speed away faster and faster. Such experiments have been done. So
|
||
far, no tachyons have been found. Even neutral tachyons can scatter off
|
||
normal matter with experimentally observable consequences. Again, no such
|
||
tachyons have been found.
|
||
|
||
Once you move away from relativistic kinematics and start talking
|
||
about the quantum field theory or particle physics of tachyons, things get
|
||
much more complicated. It is not easy to summarize results here. However,
|
||
one reasonably modern reference is _Tachyons, Monopoles, and Related
|
||
Topics_, E. Recami, ed. (North-Holland, Amsterdam, 1978).
|
||
|
||
One little-publicized fact is that in the framework of field
|
||
theory, one CANNOT transmit information faster than the speed of light with
|
||
tachyons. Since this may be controversial let us be more precise.
|
||
It's easiest to begin by looking at the wave equation for a free
|
||
scalar particle, the so-called Klein-Gordon equation:
|
||
|
||
(BOX + m^2)phi = 0
|
||
|
||
where BOX is the D'Alembertian, which in 1+1 dimensions is just
|
||
|
||
BOX = (d/dt)^2 - (d/dx)^2.
|
||
|
||
(For four-dimensional space-time just throw in -(d/dy)^2 -(d/dz)^2.)
|
||
In field theory, noninteracting massive particles (tardyons) are
|
||
described by this equation with the mass m being real. Non-interacting
|
||
tachyons would be described by this equation with m imaginary.
|
||
Regardless of m, any solution is a linear combination, or superposition,
|
||
of solutions of the form
|
||
|
||
exp(-iEt + ipx)
|
||
|
||
where E^2 - p^2 = m^2. By actually solving the equation this way, one
|
||
notices a strange thing. If the solution phi and its time derivative
|
||
are zero outside the interval [-L,L] when t = 0, they will be zero
|
||
outside the interval [-L-|t|, L+|t|] at any time t. In other words,
|
||
disturbances do not spread with speed faster than 1 (the speed of
|
||
light).
|
||
|
||
However, there are lots of problems with tachyons in quantum field
|
||
theory. A lot of mathematically rigorous work on quantum field theory
|
||
uses the Garding-Wightman axioms for quantum fields. These rule out
|
||
tachyons for other reasons because they require that all states satisfy
|
||
E^2 - p^2 >= 0. This allows one to define the vacuum as the state
|
||
minimizing E^2 - p^2 (required by these axioms to be unique). As
|
||
described above, theories with tachyons violate this axiom. In fact, if
|
||
one has a bunch of tachyons around, one can make E^2 - p^2 as negative
|
||
as you like. Heuristically, this is bad because it means that the
|
||
vacuum is unstable: spontaneous creation of tachyon-antitachyon pairs
|
||
will tend to occur, reducing the total energy of the system.
|
||
|
||
********************************************************************************
|
||
Item 7. Special Relativistic Paradoxes - part (a)
|
||
|
||
The Barn and the Pole updated 4-AUG-1992 by SIC
|
||
--------------------- original by Robert Firth
|
||
|
||
These are the props. You own a barn, 40m long, with automatic
|
||
doors at either end, that can be opened and closed simultaneously by a
|
||
switch. You also have a pole, 80m long, which of course won't fit in the
|
||
barn.
|
||
|
||
Now someone takes the pole and tries to run (at nearly the speed of
|
||
light) through the barn with the pole horizontal. Special Relativity (SR)
|
||
says that a moving object is contracted in the direction of motion: this is
|
||
called the Lorentz Contraction. So, if the pole is set in motion
|
||
lengthwise, then it will contract in the reference frame of a stationary
|
||
observer.
|
||
|
||
You are that observer, sitting on the barn roof. You see the pole
|
||
coming towards you, and it has contracted to a bit less than 40m. So, as
|
||
the pole passes through the barn, there is an instant when it is completely
|
||
within the barn. At that instant, you close both doors. Of course, you
|
||
open them again pretty quickly, but at least momentarily you had the
|
||
contracted pole shut up in your barn. The runner emerges from the far door
|
||
unscathed.
|
||
|
||
But consider the problem from the point of view of the runner. She
|
||
will regard the pole as stationary, and the barn as approaching at high
|
||
speed. In this reference frame, the pole is still 80m long, and the barn
|
||
is less than 20 meters long. Surely the runner is in trouble if the doors
|
||
close while she is inside. The pole is sure to get caught.
|
||
|
||
Well does the pole get caught in the door or doesn't it? You can't
|
||
have it both ways. This is the "Barn-pole paradox." The answer is buried
|
||
in the misuse of the word "simultaneously" back in the first sentence of
|
||
the story. In SR, that events separated in space that appear simultaneous
|
||
in one frame of reference need not appear simultaneous in another frame of
|
||
reference. The closing doors are two such separate events.
|
||
|
||
SR explains that the two doors are never closed at the same time in
|
||
the runner's frame of reference. So there is always room for the pole. In
|
||
fact, the Lorentz transformation for time is t'=(t-v*x/c^2)/sqrt(1-v^2/c^2).
|
||
It's the v*x term in the numerator that causes the mischief here. In the
|
||
runner's frame the further event (larger x) happens earlier. The far door
|
||
is closed first. It opens before she gets there, and the near door closes
|
||
behind her. Safe again - either way you look at it, provided you remember
|
||
that simultaneity is not a constant of physics.
|
||
|
||
References: Taylor and Wheeler's _Spacetime Physics_ is the classic.
|
||
Feynman's _Lectures_ are interesting as well.
|
||
|
||
********************************************************************************
|
||
Item 7. Special Relativistic Paradoxes - part (b)
|
||
|
||
The Twin Paradox updated 17-AUG-1992 by SIC
|
||
---------------- original by Kurt Sonnenmoser
|
||
|
||
A Short Story about Space Travel:
|
||
|
||
Two twins, conveniently named A and B, both know the rules of
|
||
Special Relativity. One of them, B, decides to travel out into space with
|
||
a velocity near the speed of light for a time T, after which she returns to
|
||
Earth. Meanwhile, her boring sister A sits at home posting to Usenet all
|
||
day. When A finally comes home, what do the two sisters find? Special
|
||
Relativity (SR) tells A that time was slowed down for the relativistic
|
||
sister, B, so that upon her return to Earth, she knows that B will be
|
||
younger than she is, which she suspects was the the ulterior motive of the
|
||
trip from the start.
|
||
|
||
But B sees things differently. She took the trip just to get away
|
||
from the conspiracy theorists on Usenet, knowing full well that from her
|
||
point of view, sitting in the spaceship, it would be her sister, A, who
|
||
was travelling ultrarelativistically for the whole time, so that she would
|
||
arrive home to find that A was much younger than she was. Unfortunate, but
|
||
worth it just to get away for a while.
|
||
|
||
What are we to conclude? Which twin is really younger? How can SR
|
||
give two answers to the same question? How do we avoid this apparent
|
||
paradox? Maybe twinning is not allowed in SR? Read on.
|
||
|
||
Paradox Resolved:
|
||
|
||
Much of the confusion surrounding the so-called Twin Paradox
|
||
originates from the attempts to put the two twins into different frames ---
|
||
without the useful concept of the proper time of a moving body.
|
||
|
||
SR offers a conceptually very clear treatment of this problem.
|
||
First chose _one_ specific inertial frame of reference; let's call it S.
|
||
Second define the paths that A and B take, their so-called world lines. As
|
||
an example, take (ct,0,0,0) as representing the world line of A, and
|
||
(ct,f(t),0,0) as representing the world line of B (assuming that the the
|
||
rest frame of the Earth was inertial). The meaning of the above notation is
|
||
that at time t, A is at the spatial location (x1,x2,x3)=(0,0,0) and B is at
|
||
(x1,x2,x3)=(f(t),0,0) --- always with respect to S.
|
||
|
||
Let us now assume that A and B are at the same place at the time t1
|
||
and again at a later time t2, and that they both carry high-quality clocks
|
||
which indicate zero at time t1. High quality in this context means that the
|
||
precision of the clock is independent of acceleration. [In principle, a
|
||
bunch of muons provides such a device (unit of time: half-life of their
|
||
decay).]
|
||
|
||
The correct expression for the time T such a clock will indicate at
|
||
time t2 is the following [the second form is slightly less general than the
|
||
first, but it's the good one for actual calculations]:
|
||
|
||
t2 t2 _______________
|
||
/ / / 2 |
|
||
T = | d\tau = | dt \/ 1 - [v(t)/c] (1)
|
||
/ /
|
||
t1 t1
|
||
|
||
where d\tau is the so-called proper-time interval, defined by
|
||
|
||
2 2 2 2 2
|
||
(c d\tau) = (c dt) - dx1 - dx2 - dx3 .
|
||
|
||
Furthermore,
|
||
d d
|
||
v(t) = -- (x1(t), x2(t), x3(t)) = -- x(t)
|
||
dt dt
|
||
|
||
is the velocity vector of the moving object. The physical interpretation
|
||
of the proper-time interval, namely that it is the amount the clock time
|
||
will advance if the clock moves by dx during dt, arises from considering
|
||
the inertial frame in which the clock is at rest at time t --- its
|
||
so-called momentary rest frame (see the literature cited below). [Notice
|
||
that this argument is only of a heuristic value, since one has to assume
|
||
that the absolute value of the acceleration has no effect. The ultimate
|
||
justification of this interpretation must come from experiment.]
|
||
|
||
The integral in (1) can be difficult to evaluate, but certain
|
||
important facts are immediately obvious. If the object is at rest with
|
||
respect to S, one trivially obtains T = t2-t1. In all other cases, T must
|
||
be strictly smaller than t2-t1, since the integrand is always less than or
|
||
equal to unity. Conclusion: the traveling twin is younger. Furthermore, if
|
||
she moves with constant velocity v most of the time (periods of
|
||
acceleration short compared to the duration of the whole trip), T will
|
||
approximately be given by ____________
|
||
/ 2 |
|
||
(t2-t1) \/ 1 - [v/c] . (2)
|
||
|
||
The last expression is exact for a round trip (e.g. a circle) with constant
|
||
velocity v. [At the times t1 and t2, twin B flies past twin A and they
|
||
compare their clocks.]
|
||
|
||
Now the big deal with SR, in the present context, is that T (or
|
||
d\tau, respectively) is a so-called Lorentz scalar. In other words, its
|
||
value does not depend on the choice of S. If we Lorentz transform the
|
||
coordinates of the world lines of the twins to another inertial frame S',
|
||
we will get the same result for T in S' as in S. This is a mathematical
|
||
fact. It shows that the situation of the traveling twins cannot possibly
|
||
lead to a paradox _within_ the framework of SR. It could at most be in
|
||
conflict with experimental results, which is also not the case.
|
||
|
||
Of course the situation of the two twins is not symmetric, although
|
||
one might be tempted by expression (2) to think the opposite. Twin A is
|
||
at rest in one and the same inertial frame for all times, whereas twin B
|
||
is not. [Formula (1) does not hold in an accelerated frame.] This breaks
|
||
the apparent symmetry of the two situations, and provides the clearest
|
||
nonmathematical hint that one twin will in fact be younger than the other
|
||
at the end of the trip. To figure out *which* twin is the younger one, use
|
||
the formulae above in a frame in which they are valid, and you will find
|
||
that B is in fact younger, despite her expectations.
|
||
|
||
It is sometimes claimed that one has to resort to General
|
||
Relativity in order to "resolve" the Twin "Paradox". This is not true. In
|
||
flat, or nearly flat space-time (no strong gravity), SR is completely
|
||
sufficient, and it has also no problem with world lines corresponding to
|
||
accelerated motion.
|
||
|
||
References:
|
||
Taylor and Wheeler, _Spacetime Physics_ (An *excellent* discussion)
|
||
Goldstein, _Classical Mechanics_, 2nd edition, Chap.7 (for a good
|
||
general discussion of Lorentz transformations and other SR basics.)
|
||
|
||
********************************************************************************
|
||
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 particle 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.
|
||
|
||
********************************************************************************
|
||
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
|
||
|
||
********************************************************************************
|
||
Item 10.
|
||
|
||
What is Dark Matter? updated 11-May-1991 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.
|
||
|
||
You can estimate the total BARYONIC matter of the universe by
|
||
studying big bang nucleosynthesis. The more matter in the universe, the
|
||
more slowly the universe should have expanded shortly after the big bang.
|
||
The longer the "cooking time" allowed, the higher the production of helium
|
||
from primordial hydrogen. We know the He/H ratio of the universe, so we
|
||
can estimate how much baryonic matter exists in the universe. 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:
|
||
|
||
(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. Note that the possible 17 KeV tau
|
||
neutrino would give far too much mass creating almost as many problems as
|
||
it solves 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.
|
||
|
||
********************************************************************************
|
||
Item 11.
|
||
|
||
Hot Water Freezes Faster than Cold! updated 11-May-1992 by SIC
|
||
----------------------------------- original by Richard M. Mathews
|
||
|
||
You put two pails of water outside on a freezing day. One has hot
|
||
water (95 degrees C) and the other has an equal amount of colder water (50
|
||
degrees C). Which freezes first? The hot water freezes first! Why?
|
||
|
||
It is commonly argued that the hot water will take some time to
|
||
reach the initial temperature of the cold water, and then follow the same
|
||
cooling curve. So it seems at first glance difficult to believe that the
|
||
hot water freezes first. The answer lies mostly in evaporation. The effect
|
||
is definitely real and can be duplicated in your own kitchen.
|
||
|
||
Every "proof" that hot water can't freeze faster assumes that the
|
||
state of the water can be described by a single number. Remember that
|
||
temperature is a function of position. There are also other factors
|
||
besides temperature, such as motion of the water, gas content, etc. With
|
||
these multiple parameters, any argument based on the hot water having to
|
||
pass through the initial state of the cold water before reaching the
|
||
freezing point will fall apart. The most important factor is evaporation.
|
||
|
||
The cooling of pails without lids is partly Newtonian and partly by
|
||
evaporation of the contents. The proportions depend on the walls and on
|
||
temperature. At sufficiently high temperatures evaporation is more
|
||
important. If equal masses of water are taken at two starting
|
||
temperatures, more rapid evaporation from the hotter one may diminish its
|
||
mass enough to compensate for the greater temperature range it must cover
|
||
to reach freezing. The mass lost when cooling is by evaporation is not
|
||
negligible. In one experiment, water cooling from 100C lost 16% of its mass
|
||
by 0C, and lost a further 12% on freezing, for a total loss of 26%.
|
||
|
||
The cooling effect of evaporation is twofold. First, mass is
|
||
carried off so that less needs to be cooled from then on. Also,
|
||
evaporation carries off the hottest molecules, lowering considerably the
|
||
average kinetic energy of the molecules remaining. This is why "blowing on
|
||
your soup" cools it. It encourages evaporation by removing the water vapor
|
||
above the soup.
|
||
|
||
Thus experiment and theory agree that hot water freezes faster than
|
||
cold for sufficiently high starting temperatures, if the cooling is by
|
||
evaporation. Cooling in a wooden pail or barrel is mostly by evaporation.
|
||
In fact, a wooden bucket of water starting at 100C would finish freezing in
|
||
90% of the time taken by an equal volume starting at room temperature. The
|
||
folklore on this matter may well have started a century or more ago when
|
||
wooden pails were usual. Considerable heat is transferred through the
|
||
sides of metal pails, and evaporation no longer dominates the cooling, so
|
||
the belief is unlikely to have started from correct observations after
|
||
metal pails became common.
|
||
|
||
References:
|
||
"Hot water freezes faster than cold water. Why does it do so?",
|
||
Jearl Walker in The Amateur Scientist, Scientific American,
|
||
Vol. 237, No. 3, pp 246-257; September, 1977.
|
||
|
||
"The Freezing of Hot and Cold Water", G.S. Kell in American
|
||
Journal of Physics, Vol. 37, No. 5, pp 564-565; May, 1969.
|
||
|
||
********************************************************************************
|
||
END OF FAQ PART 1/2
|
||
|
||
|
||
|