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Sci.Physics FAQ (Frequently Asked Questions) 1/2
FREQUENTLY ASKED QUESTIONS ON SCI.PHYSICS - Part 1/2
This Frequently Asked Questions List is posted monthly, on or near
the first of the month, to the USENET newsgroups sci.physics.research,
sci.physics and alt.sci.physics.new-theories 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 firstname.lastname@example.org (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. Please
send your request from the exact address you would like to use for receipt
of the FAQ. To faciliate mailing, the FAQ is now being distributed as a
If you are a new reader of the Physics newsgroups, 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
New Items: 2. The Care and Feeding of Kill Files
Index of Subjects
1. An Introduction to the Physics Newsgroups on USENET
2.*The Care and Feeding of Kill Files
3. Gravitational Radiation
4. Energy Conservation in Cosmology and Red Shift
5. Effects Due to the Finite Speed of Light
6. The Top Quark
8. Special Relativistic Paradoxes and Puzzles
(a) The Barn and the Pole
(b) The Twin Paradox
(c) The Superluminal Scissors
9. The Particle Zoo
10. Olbers' Paradox
11. What is Dark Matter?
12. Hot Water Freezes Faster than Cold!
13. Why are Golf Balls Dimpled?
14. How to Change Nuclear Decay Rates
15. Some Frequently Asked Questions About Black Holes
16. Below Absolute Zero - What Does Negative Temperature Mean?
17. Which Way Will my Bathtub Drain?
18. Why do Mirrors Reverse Left and Right?
19. What is the Mass of a Photon?
20. Why Do Stars Twinkle While Planets Do Not?
21. Baryogenesis - Why Are There More Protons Than Antiprotons?
22. Time Travel - Fact or Fiction?
23. The EPR Paradox and Bell's Inequality Principle
24. The Nobel Prize for Physics
25. Open Questions
26. Accessing and Using Online Physics Resources
Item 1. updated 17-JUN-1993 by SIC
An Introduction to the Physics Newsgroups on USENET
The USENET hierarchy contains three newsgroups dedicated to the
discussion of physics and physics-related topics. These are sci.physics,
sci.physics.research, and alt.sci.physics.new-theories.
Sci.Physics is an unmoderated newsgroup dedicated to the discussion
of physics, news from the physics community, and physics-related social
issues. Sci.Physics.Research is a moderated newgroup designed to offer an
environment with less traffic and more opportunity for discussion of
serious topics in physics among experts and beginners alike. The current
moderators of sci.physics.research are John Baez (email@example.com),
William Johnson(firstname.lastname@example.org), Cameron Randale (Dale) Bass
(email@example.com.Virginia.edu), and Lee Sawyer (firstname.lastname@example.org).
Alt.sci.physics.new-theories is an open forum for discussion of any
topics related to conventional or unconventional physics. In this
context, "unconventional physics" includes any ideas on physical science,
whether or not they are widely accepted by the mainstream physics community.
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 and sci.physics.research. 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
(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
(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) Don't post textbook problems in the hope that someone will do your
homework for you. Do you own homework; it's good for you. On the other
hand, questions, even about elementary physics, are always welcome. So
if you want to discuss the physics which is relevent to your homework,
feel free to do so. Be warned that you may still have plenty of
work to do, trying to figure out which of the many answers you get are
(7) 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.
(8) 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.
The Care and Feeding of Kill Files updated 28-SEP-1993 by SIC
With most newsreaders, it is possible for you to selectively ignore
articles with certain title words, or by a certain author. This feature is
implemented as a "kill file," which contains instructions to your
newsreader about how to filter out unwanted articles. The exact details on
how to specify articles you want to ignore varies from program to program,
so you should check the documentation for your particular newsreader. Some
examples are given below for a few common newsreaders. If your newsreader
does not support kill files, you may want to consider upgrading to one that
does. Some of the more popular newsreaders that support kill files are rn,
trn, nn, xrn, gnews, and gnus.
Let's say that you wish to `kill' all posts made by a certain user.
Using the `rn' or `trn' newsreader, you would type a [CTRL]-K while in read
mode to begin editing the kill file, and then type the following:
This will look for articles that come with "From: email@example.com" in
the header, junk them, and then display the subject lines of titles that
just got zapped.
For names of Subject titles, you would type something like this:
/: *The Big Bang Never Happened/:j
/: *Space Potatoes Have Inertia/:j
When finished, save the kill file in the normal manner for the editor
In trn 3.0 and higher you can use the faster command
to kill all of username's postings. In trn change the 'j' to ',' to kill
all the replies as well. Note the '\' to escape the '.'. This is needed in
any search string in a kill file (although they usually work if you
forget). Also in [t]rn you can simply hit K to automatically killfile the
current subject without directly editing the file.
For the `nn' newsreader, type a capital K when viewing the contents
of a newsgroup. nn will then ask you a few questions on whether it is a
Subject or a Name, duration of time that the posts are to be killed, etc.
Simply answer the questions accordingly.
There's a lot more to it, of course, when you become proficient.
You can kill all articles cross-posted to specific groups, for example, or
kill any article with a particular name or phrase appearing anywhere in the
header. A good primer is in the "rn KILL file FAQ" which appears
periodically in news.answers. You should also check the man pages for your
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
The spin-two polarization provides the method of detection. Most
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
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 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.
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
(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
(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.
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
(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.
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
References: J. Terrell, Phys Rev. _116_, 1041 (1959). For a textbook
discussion, see Marion's _Classical Dynamics_, Section 10.5.
TOP QUARK updated: 18-APR-1993 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
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
(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) and D0 experiments. 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 published 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. Unpublished results
from CDF and D0 now claim lower top mass limits of 113 GeV and 103 GeV for
the respective detectors, presumably at 95% confidence. These numbers
will probably change by the time they make it into print.
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. There have already been a few unpublished "candidate" events
from CDF and D0, which, if verified, would be the first direct evidence of
the top quark, with mass in the vacinity of 150 GeV.
References: Phys. Rev. Lett. _68_, 447 (1992) and the references therein.
Tachyons updated: 22-MAR-1993 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!
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
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 hear 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. The energy and momentum of a tachyon must
satisfy this relation.
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 were
electrically charged tachyons. Since they would move faster than the speed
of light in the vacuum, they should produce Cerenkov radiation. This would
*lower* their energy, causing them to accelerate more! In other words,
charged tachyons would probably lead to a runaway reaction releasing an
arbitrarily large amount of energy. This suggests that coming up with a
sensible theory of anything except free (noninteracting) tachyons is likely
to be difficult. Heuristically, the problem is that we can get spontaneous
creation of tachyon-antitachyon pairs, then do a runaway reaction, making
the vacuum unstable. To treat this precisely requires quantum field theory,
which gets 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).
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.
How about using tachyons to transmit information faster than the
speed of light, in violation of Special Relativity? It's worth noting
that when one considers the relativistic quantum mechanics of tachyons, the
question of whether they "really" go faster than the speed of light becomes
much more touchy! In this framework, tachyons are *waves* that satisfy a
wave equation. Let's treat free tachyons of spin zero, for simplicity.
We'll set c = 1 to keep things less messy. The wavefunction of a single
such tachyon can be expected to satisfy the usual equation for spin-zero
particles, the Klein-Gordon equation:
(BOX + m^2)phi = 0
where BOX is the D'Alembertian, which in 3+1 dimensions is just
BOX = (d/dt)^2 - (d/dx)^2 - (d/dy)^2 - (d/dz)^2.
The difference with tachyons is that m^2 is *negative*, and m is
To simplify the math a bit, let's work in 1+1 dimensions, with
coordinates x and t, so that
BOX = (d/dt)^2 - (d/dx)^2
Everything we'll say generalizes to the real-world 3+1-dimensional case.
Now - regardless of m, any solution is a linear combination, or
superposition, of solutions of the form
phi(t,x) = exp(-iEt + ipx)
where E^2 - p^2 = m^2. When m^2 is negative there are two essentially
different cases. Either |p| >= |E|, in which case E is real and
we get solutions that look like waves whose crests move along at the
rate |p|/|E| >= 1, i.e., no slower than the speed of light. Or |p| <
|E|, in which case E is imaginary and we get solutions that look waves
that amplify exponentially as time passes!
We can decide as we please whether or not we want to consider the second
sort of solutions. They seem weird, but then the whole business is
weird, after all.
1) If we *do* permit the second sort of solution, we can solve the
Klein-Gordon equation with any reasonable initial data - that is, any
reasonable values of phi and its first time derivative at t = 0. (For
the precise definition of "reasonable," consult your local
mathematician.) This is typical of wave equations. And, also typical
of wave equations, we can prove the following 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, localized disturbances do not spread with speed faster
than the speed of light! This seems to go against our notion that
tachyons move faster than the speed of light, but it's a mathematical
fact, known as "unit propagation velocity".
2) If we *don't* permit the second sort of solution, we can't solve the
Klein-Gordon equation for all reasonable initial data, but only for initial
data whose Fourier transforms vanish in the interval [-|m|,|m|]. By the
Paley-Wiener theorem this has an odd consequence: it becomes
impossible to solve the equation for initial data that vanish outside
some interval [-L,L]! In other words, we can no longer "localize" our
tachyon in any bounded region in the first place, so it becomes
impossible to decide whether or not there is "unit propagation
velocity" in the precise sense of part 1). Of course, the crests of
the waves exp(-iEt + ipx) move faster than the speed of light, but these
waves were never localized in the first place!
The bottom line is that you can't use tachyons to send information
faster than the speed of light from one place to another. Doing so would
require creating a message encoded some way in a localized tachyon field,
and sending it off at superluminal speed toward the intended receiver. But
as we have seen you can't have it both ways - localized tachyon disturbances
are subluminal and superluminal disturbances are nonlocal.
Item 8. 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
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
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
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 8. 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.
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
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)
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 .
v(t) = -- (x1(t), x2(t), x3(t)) = -- x(t)
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
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. Special Relativistic Paradoxes - part (c)
The Superluminal Scissors updated 31-MAR-1993
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?
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
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.
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
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.
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
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
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
-1 | e | mu | tau |
0 | nu(e) |nu(mu) |nu(tau)|
------------------------- + antiparticles
-1/3 | down |strange|bottom |
2/3 | up | charm | top |
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!
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
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
(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
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
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.
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.
"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.
Why are Golf Balls Dimpled? updated 14-May-1992 by SIC
--------------------------- original by Craig DeForest
The dimples, paradoxically, *do* increase drag slightly. But they
also increase `Magnus lift', that peculiar lifting force experienced by
rotating bodies travelling through a medium. Contrary to Freshman physics,
golf balls do not travel in inverted parabolas. They follow an 'impetus
(golfer) * *
* * <-- trajectory
\O/ * *
| * *
This is because of the combination of drag (which reduces
horizontal speed late in the trajectory) and Magnus lift, which supports
the ball during the initial part of the trajectory, making it relatively
straight. The trajectory can even curve upwards at first, depending on
conditions! Here is a cheesy diagram of a golf ball in flight, with some
F(drag) <--- O -------> V
\----> (sense of rotation)
The Magnus force can be thought of as due to the relative drag on
the air on the top and bottom portions of the golf ball: the top portion is
moving slower relative to the air around it, so there is less drag on the
air that goes over the ball. The boundary layer is relatively thin, and
air in the not-too-near region moves rapidly relative to the ball. The
bottom portion moves fast relative to the air around it; there is more drag
on the air passing by the bottom, and the boundary (turbulent) layer is
relatively thick; air in the not-too-near region moves more slowly relative
to the ball. The Bernoulli force produces lift. (alternatively, one could
say that `the flow lines past the ball are displaced down, so the ball is
The difficulty comes near the transition region between laminar
flow and turbulent flow. At low speeds, the flow around the ball is
laminar. As speed is increased, the bottom part tends to go turbulent
*first*. But turbulent flow can follow a surface much more easily than
As a result, the (laminar) flow lines around the top break away
from the surface sooner than otherwise, and there is a net displacement
*up* of the flow lines. The magnus lift goes *negative*.
The dimples aid the rapid formation of a turbulent boundary layer
around the golf ball in flight, giving more lift. Without 'em, the ball
would travel in more of a parabolic trajectory, hitting the ground sooner.
(and not coming straight down.)
References: Perhaps the best (and easy-to-read) reference on this effect is
a paper in American Journal of Physics by one Lyman Briggs, c. 1947.
Briggs was trying to explain the mechanism behind the `curve ball' in
baseball, using specialized apparatus in a wind tunnel at the NBS. He
stumbled on the reverse effect by accident, because his model `baseball'
had no stitches on it. The stitches on a baseball create turbulence in
flight in much the same way that the dimples on a golf ball do.
updated 4-SEP-1992 by SIC
Original by Bill Johnson
How to Change Nuclear Decay Rates
"I've had this idea for making radioactive nuclei decay faster/slower than
they normally do. You do [this, that, and the other thing]. Will this work?"
Short Answer: Possibly, but probably not usefully.
"One of the paradigms of nuclear science since the very early days
of its study has been the general understanding that the half-life, or
decay constant, of a radioactive substance is independent of extranuclear
considerations." (Emery, cited below.) Like all paradigms, this one is
subject to some interpretation. Normal decay of radioactive stuff proceeds
via one of four mechanisms:
* Emission of an alpha particle -- a helium-4 nucleus -- reducing
the number of protons and neutrons present in the parent nucleus
by two each;
* "Beta decay," encompassing several related phenomena in which a
neutron in the nucleus turns into a proton, or a proton turns into
a neutron -- along with some other things including emission of
a neutrino. The "other things", as we shall see, are at the bottom
of several questions involving perturbation of decay rates;
* Emission of one or more gamma rays -- energetic photons -- that
take a nucleus from an excited state to some other (typically
ground) state; some of these photons may be replaced by
"conversion electrons," of which more shortly; or
*Spontaneous fission, in which a sufficiently heavy nucleus simply
breaks in half. Most of the discussion about alpha particles will
also apply to spontaneous fission.
Gamma emission often occurs from the daughter of one of the other decay
modes. We neglect *very* exotic processes like C-14 emission or double
beta decay in this analysis.
"Beta decay" refers most often to a nucleus with a neutron excess,
which decays by converting a neutron into a proton:
n ----> p + e- + anti-nu(e),
where n means neutron, p means proton, e- means electron, and anti-nu(e)
means an antineutrino of the electron type. The type of beta decay which
involves destruction of a proton is not familiar to many people, so
deserves a little elaboration. Either of two processes may occur when this
kind of decay happens:
p ----> n + e+ + nu(e),
where e+ means positron and nu(e) means electron neutrino; or
p + e- ----> n + nu(e),
where e- means a negatively charged electron, which is captured from the
neighborhood of the nucleus undergoing decay. These processes are called
"positron emission" and "electron capture," respectively. A given nucleus
which has too many protons for stability may undergo beta decay through
either, and typically both, of these reactions.
"Conversion electrons" are produced by the process of "internal
conversion," whereby the photon that would normally be emitted in gamma
decay is *virtual* and its energy is absorbed by an atomic electron. The
absorbed energy is sufficient to unbind the electron from the nucleus
(ignoring a few exceptional cases), and it is ejected from the atom as a
Now for the tie-in to decay rates. Both the electron-capture and
internal conversion phenomena require an electron somewhere close to the
decaying nucleus. In any normal atom, this requirement is satisfied in
spades: the innermost electrons are in states such that their probability
of being close to the nucleus is both large and insensitive to things in
the environment. The decay rate depends on the electronic wavefunctions,
i.e, how much of their time the inner electrons spend very near the
nucleus -- but only very weakly. For most nuclides that decay by electron
capture or internal conversion, most of the time, the probability of
grabbing or converting an electron is also insensitive to the environment,
as the innermost electrons are the ones most likely to get grabbed/converted.
However, there are exceptions, the most notable being the
the astrophysically important isotope beryllium-7. Be-7 decays purely
by electron capture (positron emission being impossible because of
inadequate decay energy) with a half-life of somewhat over 50 days. It has
been shown that differences in chemical environment result in half-life
variations of the order of 0.2%, and high pressures produce somewhat
similar changes. Other cases where known changes in decay rate occur are
Zr-89 and Sr-85, also electron capturers; Tc-99m ("m" implying an excited
state), which decays by both beta and gamma emission; and various other
"metastable" things that decay by gamma emission with internal conversion.
With all of these other cases the magnitude of the effect is less than is
typically the case with Be-7.
What makes these cases special? The answer is that one or another
of the usual starting assumptions -- insensitivity of electron wave
function near the nucleus to external forces, or availability of the
innermost electrons for capture/conversion -- are not completely valid.
Atomic beryllium only has 4 electrons to begin with, so that the "innermost
electrons" are also practically the *outermost* ones and therefore much
more sensitive to chemical effects than usual. With most of the other
cases, there is so little energy available from the decay (as little as a
few electron volts; compare most radioactive decays, where hundreds or
thousands of *kilo*volts are released), courtesy of accidents of nuclear
structure, that the innermost electrons can't undergo internal conversion.
Remember that converting an electron requires dumping enough energy into it
to expel it from the atom (more or less); "enough energy," in context, is
typically some tens of keV, so they don't get converted at all in these
cases. Conversion therefore works only on some of the outer electrons,
which again are more sensitive to the environment.
A real anomaly is the beta emitter Re-187. Its decay energy is
only about 2.6 keV, practically nothing by nuclear standards. "That this
decay occurs at all is an example of the effects of the atomic environment
on nuclear decay: the bare nucleus Re-187 [i.e., stripped of all orbital
electrons -- MWJ] is stable against beta decay and it is the difference of
15 keV in the total electronic binding energy of osmium [to which it decays
-- MWJ] and rhenium ... which makes the decay possible" (Emery). The
practical significance of this little peculiarity, of course, is low, as
Re-187 already has a half life of over 10^10 years.
Alpha decay and spontaneous fission might also be affected by
changes in the electron density near the nucleus, for a different reason.
These processes occur as a result of penetration of the "Coulomb barrier"
that inhibits emission of charged particles from the nucleus, and their
rate is *very* sensitive to the height of the barrier. Changes in the
electron density could, in principle, affect the barrier by some tiny
amount. However, the magnitude of the effect is *very* small, according to
theoretical calculations; for a few alpha emitters, the change has been
estimated to be of the order of 1 part in 10^7 (!) or less, which would be
unmeasurable in view of the fact that the alpha emitters' half lives aren't
known to that degree of accuracy to begin with.
All told, the existence of changes in radioactive decay rates due
to the environment of the decaying nuclei is on solid grounds both
experimentally and theoretically. But the magnitude of the changes is
nothing to get very excited about.
Reference: The best review article on this subject is now 20 years old: G.
T. Emery, "Perturbation of Nuclear Decay Rates," Annual Review of Nuclear
Science vol. 22, p. 165 (1972). Papers describing specific experiments are
cited in that article, which contains considerable arcane math but also
gives a reasonable qualitative "feel" for what is involved.
END OF FAQ PART 1/2
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