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From: email@example.com (SCOTT I CHASE)
Subject: Sci.Physics Frequently Asked Questions (3/4) - General Physics
Date: 6 Sep 1994 13:33 PST
Organization: Lawrence Berkeley Laboratory - Berkeley, CA, USA
Sender: firstname.lastname@example.org (SCOTT I CHASE)
Expires: Sat, 1 October 1994 00:00:00 GMT
Summary: This posting contains a list of Frequently Asked Questions
(and their answers) about physics, and should be read by anyone who
wishes to post to the sci.physics.* newsgroups.
Keywords: Sci.physics FAQ General
News-Software: VAX/VMS VNEWS 1.50
FREQUENTLY ASKED QUESTIONS ON SCI.PHYSICS - Part 3/4
EFFECTS DUE TO THE FINITE SPEED OF LIGHT updated 13-JUN-1994 by SIC
---------------------------------------- original by Scott I. Chase
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
J. Terrell, Phys Rev. _116_, 1041 (1959). For a textbook
discussion, see Marion's _Classical Dynamics_, Section 10.5.
Ping-Kang Hsiung, Robert H. Thibadeau, and Robert H. P. Dunn,
Pixel, vol.1 no.1 (Jan/Feb 1990), "Ray Tracing Relativity". This
article has beautiful computer-generated pictures of Terrell rotation
and other effects of "SR photography".
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 17-NOV-1993 by CDF
--------------------------- 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.)
Lord Rayleigh, "On the Irregular Flight of a Tennis Ball", _Scientific
Papers I_, p. 344
Briggs Lyman J., "Effect of Spin and Speed on the Lateral Deflection of
a Baseball; and the Magnus Effect for Smooth Spheres", Am. J. Phys. _27_,
589 (1959). [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.]
R. Watts and R. Ferver, "The Lateral Force on a Spinning Sphere" Aerodynamics
of a Curveball", Am. J. Phys. _55_, 40 (1986)
updated 9-DEC-1993 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 [but not to bound state
beta decay, in which the outgoing electron is captured by the daughter
nucleus into a tightly bound orbital -SIC] 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.
original by Blair P. Houghton
What is a Dippy Bird, and how is it used?
The Anatomy and Habits of a Dippy Bird:
1. The armature: The body of the bird is a straight tube attached to two
bulbs, approximately the same size, one at either end. The tube flows into
the upper bulb, like the neck of a funnel, and extends almost to the bottom
of the lower bulb, like the straw in a soda.
2. The pivot: At about the middle of the tube is clamped a transverse
bar, which allows the apparatus to pivot on a stand (the legs). The bar is
bent very slightly concave dorsally, to unbalance the bird in the forward
direction (thus discouraging dips to the rear). The ends of the pivot have
downward protrusions, which hit stops on the stand placed so that the bird
is free to rock when in a vertical position, but can not quite rotate
enough to be horizontal during a dip.
3. The wick: The upper bulb is coated in fuzzy material, and has extended
from it a beak, made of or covered in the same material.
4. The tail. The tail has no significant external features, except that
it should not be insulated (skin-oil deposited on the bird's glass parts
from handling will insulate it and can affect its operation).
5. The guts: The bird is partially filled with a somewhat carefully
measured amount of a fluid with suitable lack of viscosity and density and
a low latent heat of evaporation (small d(energy)/d(mass), ld). For water,
ld is 2250 kJ/kg; for methylene chloride, ld is 406; for mercury, ld is a
wondrous 281; ethyl alcohol has an ld of 880, more than twice that of MC.
Boiling point is not important, here; evaporation and condensation take
place on the surface of a liquid at any temperature.
6. The frills: Any hats, eyes, feathers, or liquid coloring have been
added purely for entertainment value. (An anecdote: as it stood pumping in
the Arizona sun on my kitchen windowsill for several days, the rich,
Kool-Aid red of my bird's motorwater faded to a pale peach. I have since
retired him to the mantelpiece in the family room).
7. Shreddin': The bird is operated by getting the head wet, taking care
not to make it so wet that it drips down the tube. (Water on the bottom
bulb will reverse the thermodynamic processes.) The first cycle will
take somewhat longer than the following cycles. If you can keep water
where the bird can dip it, the bird will dip for as long as the ambient
humidity remains favorable.
Come on, how does it really work?
Short answer: Thermodynamics plus Mechanics.
Medium answer (and essential clues): Evaporative cooling on the outside;
pV=nRT, evaporation/condensation, and gravity on the inside.
Initially the system is at equilibrium, with T equal in both
chambers and pV/n in each compensating for the fluid levels. Evaporation
of water outside the head draws heat from inside it; the vapor inside
condenses, reducing pV/RT. This imbalances the pressures, so the vapor in
the abdomen pushes down, which pushes fluid up the thorax, which reduces V
in the head. Since p is decreasing in the abdomen, evaporation occurs,
increasing n, and drawing heat from outside the body.
The rising fluid raises the CM above the pivot point; the hips are
slightly concave dorsally, so the bird dips forward. Tabs on the legs and
the pivot maintain the angle at full dip, for drainage. The amount of
fluid is set so that at full dip the lower end of the tube is exposed to
the vapor. (The tube reaches almost to the bottom of the abdomen, like a
straw in a soda, but flows into the head like the neck of a funnel.) A
bubble of vapor rises in the tube and fluid drains into the abdomen.
The rising bubble transfers heat to the head and the falling fluid
releases gravitational potential energy as heat into the rising bubble and
the abdomen. The CM drops below the pivot point and the bird bobs up. The
system is thus reset; it's not quite at equilibrium, but is close enough
that the process can repeat this chain of events.
The beak acts as a wick, if allowed to dip into a reservoir of
water, to keep the head wet, although it is not necessary for the bird to
drink on every dip.
Is that all there is to know about dippy birds?
Of course not. Research continues to unravel these unanswered
questions about the amazing dippy-bird:
1. All of the energy gained by the rising fluid is returned to the system
when the fluid drops; where does this energy go, in what proportions, and
how does this affect the rate at which the bird operates?
2. The heat that evaporates the water comes from both the surrounding air
and the inside of the head; but, in what proportion?
3. Exactly what should the fluid be? Methylene Chloride is an excellent
candidate, since it's listed in the documentation for recent birds sold by
Edmund Scientific Corp. (trade named Happy Drinking Bird), and because its
latent heat of evaporation (ld) is 406 kJ/kg, compared to 2250 kJ/kg for
water (a 5.5:1 ratio of condensed MC to evaporated water, if all
water-evaporating heat comes from inside the bird). Ethanol, at 880 kJ/kG,
is only half as efficient. Mercury would likewise be a good prospective
choice, having an ld of 281 kJ/kG (8:1!), but is expensive and dangerous,
and its density would require careful redesign and greater quality control
in the abdomen and pivot-stops to ensure proper operation at full dip; this
does, however, indicate that the apparatus could be made in miniature,
filled with mercury, and sold through a catalog-store such as The Sharper
Image as a wildly successful yuppie desk-toy (Consider the submission of
this FAQ entry to be prior art for patent purposes).
4. Does ambient temperature have an effect on operation aside from the
increase in rate of evaporation of water? I.e., if the temperature and
humidity can be controlled independently such that the rate of evaporation
can be kept constant, what effect does such a change in ambient temperature
and humidity have on the operation of the bird? Is the response transient,
permanent, or composed of both?
Dippy Bird Tips:
They have real trouble working at all in humid climates (like
around the U. of Md., where I owned my first one), but can drive you bats
in dry climates (aside from the constant hammering, it's hard to keep the
water up to a level where the bird can get at it...). The evaporation of
water from the head depends on the diffusibility of water vapor into the
atmosphere; high partial pressures of water vapor in the atmosphere
translate to low rates of evaporation.
If you handle your bird, clean the glass with alcohol or Windex
or Dawn or something; the oil from your hands has a high specific heat,
which damps the transfer of heat, and a low thermal conductivity, which
attenuates the transfer of heat. Once it's clean, grasp the bird only by
the legs or the tube, which are not thermodynamically significant, or
wear rubber gloves, just like a real EMT.
The hat is there for show; the dippy bird operates okay with or
without it, even though it may reduce the area of evaporation slightly.
Ditto the feathers and the eyes.
Chemical data from Gieck, K., _Engineering Formulas_, 3d. Ed.,
McGraw-Hill, 1979, as translated by J. Walters, B. Sc.
I've also heard that SciAm had an "Amateur Scientist" column on
this technology a few years ago. Perhaps someone who understands how a
library works could look up the yr and vol...
Kool-Aid is a trademark of some huge corporation that makes its
money a farthing at a time...
Below Absolute Zero - What Does Negative Temperature Mean? updated 24-MAR-1993
---------------------------------------------------------- by Scott I. Chase
Questions: What is negative temperature? Can you really make a system
which has a temperature below absolute zero? Can you even give any useful
meaning to the expression 'negative absolute temperature'?
Answer: Absolutely. :-)
Under certain conditions, a closed system *can* be described by a
negative temperature, and, surprisingly, be *hotter* than the same system
at any positive temperature. This article describes how it all works.
Step I: What is "Temperature"?
To get things started, we need a clear definition of "temperature."
Our intuitive notion is that two systems in thermal contact should exchange
no heat, on average, if and only if they are at the same temperature. Let's
call the two systems S1 and S2. The combined system, treating S1 and S2
together, can be S3. The important question, consideration of which
will lead us to a useful quantitative definition of temperature, is "How will
the energy of S3 be distributed between S1 and S2?" I will briefly explain
this below, but I recommend that you read K&K, referenced below, for a
careful, simple, and thorough explanation of this important and fundamental
With a total energy E, S has many possible internal states
(microstates). The atoms of S3 can share the total energy in many ways.
Let's say there are N different states. Each state corresponds to a
particular division of the total energy in the two subsystems S1 and S2.
Many microstates can correspond to the same division, E1 in S1 and E2 in
S2. A simple counting argument tells you that only one particular division
of the energy, will occur with any significant probability. It's the one
with the overwhelmingly largest number of microstates for the total system
S3. That number, N(E1,E2) is just the product of the number of states
allowed in each subsystem, N(E1,E2) = N1(E1)*N2(E2), and, since E1 + E2 =
E, N(E1,E2) reaches a maximum when N1*N2 is stationary with respect to
variations of E1 and E2 subject to the total energy constraint.
For convenience, physicists prefer to frame the question in terms
of the logarithm of the number of microstates N, and call this the entropy,
S. You can easily see from the above analysis that two systems are in
equilibrium with one another when (dS/dE)_1 = (dS/dE)_2, i.e., the rate of
change of entropy, S, per unit change in energy, E, must be the same for
both systems. Otherwise, energy will tend to flow from one subsystem to
another as S3 bounces randomly from one microstate to another, the total
energy E3 being constant, as the combined system moves towards a state of
maximal total entropy. We define the temperature, T, by 1/T = dS/dE, so
that the equilibrium condition becomes the very simple T_1 = T_2.
This statistical mechanical definition of temperature does in fact
correspond to your intuitive notion of temperature for most systems. So
long as dS/dE is always positive, T is always positive. For common
situations, like a collection of free particles, or particles in a harmonic
oscillator potential, adding energy always increases the number of
available microstates, increasingly faster with increasing total energy. So
temperature increases with increasing energy, from zero, asymptotically
approaching positive infinity as the energy increases.
Step II: What is "Negative Temperature"?
Not all systems have the property that the entropy increases
monotonically with energy. In some cases, as energy is added to the system,
the number of available microstates, or configurations, actually decreases
for some range of energies. For example, imagine an ideal "spin-system", a
set of N atoms with spin 1/2 one a one-dimensional wire. The atoms are not
free to move from their positions on the wire. The only degree of freedom
allowed to them is spin-flip: the spin of a given atom can point up or
down. The total energy of the system, in a magnetic field of strength B,
pointing down, is (N+ - N-)*uB, where u is the magnetic moment of each atom
and N+ and N- are the number of atoms with spin up and down respectively.
Notice that with this definition, E is zero when half of the spins are
up and half are down. It is negative when the majority are down and
positive when the majority are up.
The lowest possible energy state, all the spins will point down,
gives the system a total energy of -NuB, and temperature of absolute zero.
There is only one configuration of the system at this energy, i.e., all the
spins must point down. The entropy is the log of the number of
microstates, so in this case is log(1) = 0. If we now add a quantum of
energy, size uB, to the system, one spin is allowed to flip up. There are
N possibilities, so the entropy is log(N). If we add another quantum of
energy, there are a total of N(N-1)/2 allowable configurations with two
spins up. The entropy is increasing quickly, and the temperature is rising
However, for this system, the entropy does not go on increasing
forever. There is a maximum energy, +NuB, with all spins up. At this
maximal energy, there is again only one microstate, and the entropy is
again zero. If we remove one quantum of energy from the system, we allow
one spin down. At this energy there are N available microstates. The
entropy goes on increasing as the energy is lowered. In fact the maximal
entropy occurs for total energy zero, i.e., half of the spins up, half
So we have created a system where, as we add more and more energy,
temperature starts off positive, approaches positive infinity as maximum
entropy is approached, with half of all spins up. After that, the
temperature becomes negative infinite, coming down in magnitude toward
zero, but always negative, as the energy increases toward maximum. When the
system has negative temperature, it is *hotter* than when it is has
positive system. If you take two copies of the system, one with positive
and one with negative temperature, and put them in thermal contact, heat
will flow from the negative-temperature system into the positive-temperature
Step III: What Does This Have to Do With the Real World?
Can this system ever by realized in the real world, or is it just a
fantastic invention of sinister theoretical condensed matter physicists?
Atoms always have other degrees of freedom in addition to spin, usually
making the total energy of the system unbounded upward due to the
translational degrees of freedom that the atom has. Thus, only certain
degrees of freedom of a particle can have negative temperature. It makes
sense to define the "spin-temperature" of a collection of atoms, so long as
one condition is met: the coupling between the atomic spins and the other
degrees of freedom is sufficiently weak, and the coupling between atomic
spins sufficiently strong, that the timescale for energy to flow from the
spins into other degrees of freedom is very large compared to the timescale
for thermalization of the spins among themselves. Then it makes sense to
talk about the temperature of the spins separately from the temperature of
the atoms as a whole. This condition can easily be met for the case of
nuclear spins in a strong external magnetic field.
Nuclear and electron spin systems can be promoted to negative
temperatures by suitable radio frequency techniques. Various experiments
in the calorimetry of negative temperatures, as well as applications of
negative temperature systems as RF amplifiers, etc., can be found in the
articles listed below, and the references therein.
Kittel and Kroemer,_Thermal Physics_, appendix E.
N.F. Ramsey, "Thermodynamics and statistical mechanics at negative
absolute temperature," Phys. Rev. _103_, 20 (1956).
M.J. Klein,"Negative Absolute Temperature," Phys. Rev. _104_, 589 (1956).
Which Way Will my Bathtub Drain? updated 16-MAR-1993 by SIC
-------------------------------- original by Matthew R. Feinstein
Question: Does my bathtub drain differently depending on whether I live
in the northern or southern hemisphere?
Answer: No. There is a real effect, but it is far too small to be relevant
when you pull the plug in your bathtub.
Because the earth rotates, a fluid that flows along the earth's
surface feels a "Coriolis" acceleration perpendicular to its velocity.
In the northern hemisphere low pressure storm systems spin counterclockwise.
In the southern hemisphere, they spin clockwise because the direction
of the Coriolis acceleration is reversed. This effect leads to the
speculation that the bathtub vortex that you see when you pull the plug
from the drain spins one way in the north and the other way in the south.
But this acceleration is VERY weak for bathtub-scale fluid
motions. The order of magnitude of the Coriolis acceleration can be
estimated from size of the "Rossby number" (see below). The effect of the
Coriolis acceleration on your bathtub vortex is SMALL. To detect its
effect on your bathtub, you would have to get out and wait until the motion
in the water is far less than one rotation per day. This would require
removing thermal currents, vibration, and any other sources of noise. Under
such conditions, never occurring in the typical home, you WOULD see an
effect. To see what trouble it takes to actually see the effect, see the
reference below. Experiments have been done in both the northern and
southern hemispheres to verify that under carefully controlled conditions,
bathtubs drain in opposite directions due to the Coriolis acceleration from
the Earth's rotation.
Coriolis accelerations are significant when the Rossby number is
SMALL. So, suppose we want a Rossby number of 0.1 and a bathtub-vortex
length scale of 0.1 meter. Since the earth's rotation rate is about
10^(-4)/second, the fluid velocity should be less than or equal to
2*10^(-6) meters/second. This is a very small velocity. How small is it?
Well, we can take the analysis a step further and calculate another, more
famous dimensionless parameter, the Reynolds number.
The Reynolds number is = L*U*density/viscosity
Assuming that physicists bathe in hot water the viscosity will be
about 0.005 poise and the density will be about 1.0, so the Reynolds Number
is about 4*10^(-2).
Now, life at low Reynolds numbers is different from life at high
Reynolds numbers. In particular, at low Reynolds numbers, fluid physics is
dominated by friction and diffusion, rather than by inertia: the time it
would take for a particle of fluid to move a significant distance due to an
acceleration is greater than the time it takes for the particle to break up
due to diffusion.
The same effect has been accused of responsibility for the
direction water circulates when you flush a toilet. This is surely
nonsense. In this case, the water rotates in the direction which the pipe
points which carries the water from the tank to the bowl.
Reference: Trefethen, L.M. et al, Nature 207 1084-5 (1965).
Why do Mirrors Reverse Left and Right? updated 04-MAR-1994 by SIC
-------------------------------------- original by Scott I. Chase
The simple answer is that they don't. Look in a mirror and wave
your right hand. On which side of the mirror is the hand that waved? The
right side, of course.
Mirrors DO reverse In/Out. Imaging holding an arrow in your hand.
If you point it up, it will point up in the mirror. If you point it to the
left, it will point to the left in the mirror. But if you point it toward
the mirror, it will point right back at you. In and Out are reversed.
If you take a three-dimensional, rectangular, coordinate system,
(X,Y,Z), and point the Z axis such that the vector equation X x Y = Z is
satisfied, then the coordinate system is said to be right-handed. Imagine
Z pointing toward the mirror. X and Y are unchanged (remember the arrows?)
but Z will point back at you. In the mirror, X x Y = - Z. The image
contains a left-handed coordinate system.
This has an important effect, familiar mostly to chemists and
physicists. It changes the chirality, or handedness of objects viewed in
the mirror. Your left hand looks like a right hand, while your right hand
looks like a left hand. Molecules often come in pairs called
stereoisomers, which differ not in the sequence or number of atoms, but
only in that one is the mirror image of the other, so that no rotation or
stretching can turn one into the other. Your hands make a good laboratory
for this effect. They are distinct, even though they both have the same
components connected in the same way. They are a stereo pair, identical
except for "handedness".
People sometimes think that mirrors *do* reverse left/right, and
that the effect is due to the fact that our eyes are aligned horizontally
on our faces. This can be easily shown to be untrue by looking in any
mirror with one eye closed!
Reference: _The Left Hand of the Electron_, by Isaac Asimov, contains
a very readable discussion of handedness and mirrors in physics.
updated 16-MAR-1992 by SIC
Original by John Blanton
Why Do Stars Twinkle While Planets Do Not?
Stars, except for the Sun, although they may be millions of miles
in diameter, are very far away. They appear as point sources even when
viewed by telescopes. The planets in our solar system, much smaller than
stars, are closer and can be resolved as disks with a little bit of
magnification (field binoculars, for example).
Since the Earth's atmosphere is turbulent, all images viewed up
through it tend to "swim." The result of this is that sometimes a single
point in object space gets mapped to two or more points in image space, and
also sometimes a single point in object space does not get mapped into any
point in image space. When a star's single point in object space fails to
map to at least one point in image space, the star seems to disappear
temporarily. This does not mean the star's light is lost for that moment.
It just means that it didn't get to your eye, it went somewhere else.
Since planets represent several points in object space, it is
highly likely that one or more points in the planet's object space get
mapped to a points in image space, and the planet's image never winks out.
Each individual ray is twinkling away as badly as any star, but when all of
those individual rays are viewed together, the next effect is averaged out
to something considerably steadier.
The result is that stars tend to twinkle, and planets do not.
Other extended objects in space, even very far ones like nebulae, do not
twinkle if they are sufficiently large that they have non-zero apparent
diameter when viewed from the Earth.
TIME TRAVEL - FACT OR FICTION? updated 07-MAR-1994
------------------------------ original by Jon J. Thaler
We define time travel to mean departure from a certain place and
time followed (from the traveller's point of view) by arrival at the same
place at an earlier (from the sedentary observer's point of view) time.
Time travel paradoxes arise from the fact that departure occurs after
arrival according to one observer and before arrival according to another.
In the terminology of special relativity time travel implies that the
timelike ordering of events is not invariant. This violates our intuitive
notions of causality. However, intuition is not an infallible guide, so we
must be careful. Is time travel really impossible, or is it merely another
phenomenon where "impossible" means "nature is weirder than we think?" The
answer is more interesting than you might think.
THE SCIENCE FICTION PARADIGM:
The B-movie image of the intrepid chrononaut climbing into his time
machine and watching the clock outside spin backwards while those outside
the time machine watch the him revert to callow youth is, according to
current theory, impossible. In current theory, the arrow of time flows in
only one direction at any particular place. If this were not true, then
one could not impose a 4-dimensional coordinate system on space-time, and
many nasty consequences would result. Nevertheless, there is a scenario
which is not ruled out by present knowledge. This usually requires an
unusual spacetime topology (due to wormholes or strings in general
relativity) which has not not yet seen, but which may be possible. In
this scenario the universe is well behaved in every local region; only by
exploring the global properties does one discover time travel.
It is sometimes argued that time travel violates conservation laws.
For example, sending mass back in time increases the amount of energy that
exists at that time. Doesn't this violate conservation of energy? This
argument uses the concept of a global conservation law, whereas
relativistically invariant formulations of the equations of physics only
imply local conservation. A local conservation law tells us that the
amount of stuff inside a small volume changes only when stuff flows in or
out through the surface. A global conservation law is derived from this by
integrating over all space and assuming that there is no flow in or out at
infinity. If this integral cannot be performed, then global conservation
does not follow. So, sending mass back in time might be alright, but it
implies that something strange is happening. (Why shouldn't we be able to
do the integral?)
One case where global conservation breaks down is in general
relativity. It is well known that global conservation of energy does not
make sense in an expanding universe. For example, the universe cools as it
expands; where does the energy go? See FAQ article #4 - Energy
Conservation in Cosmology, for details.
It is interesting to note that the possibility of time travel in GR
has been known at least since 1949 (by Kurt Godel, discussed in , page
168). The GR spacetime found by Godel has what are now called "closed
timelike curves" (CTCs). A CTC is a worldline that a particle or a person
can follow which ends at the same spacetime point (the same position and
time) as it started. A solution to GR which contains CTCs cannot have a
spacelike embedding - space must have "holes" (as in donut holes, not holes
punched in a sheet of paper). A would-be time traveller must go around or
through the holes in a clever way.
The Godel solution is a curiosity, not useful for constructing a
time machine. Two recent proposals, one by Morris, et al.  and one by
Gott , have the possibility of actually leading to practical devices (if
you believe this, I have a bridge to sell you). As with Godel, in these
schemes nothing is locally strange; time travel results from the unusual
topology of spacetime. The first uses a wormhole (the inner part of a
black hole, see fig. 1 of ) which is held open and manipulated by
electromagnetic forces. The second uses the conical geometry generated by
an infinitely long string of mass. If two strings pass by each other, a
clever person can go into the past by traveling a figure-eight path around
the strings. In this scenario, if the string has non-zero diameter and
finite mass density, there is a CTC without any unusual topology.
With the demonstration that general relativity contains CTCs,
people began studying the problem of self-consistency. Basically, the
problem is that of the "grandfather paradox:" What happens if our time
traveller kills her grandmother before her mother was born? In more
readily analyzable terms, one can ask what are the implications of the
quantum mechanical interference of the particle with its future self.
Boulware  shows that there is a problem - unitarity is violated. This is
related to the question of when one can do the global conservation integral
discussed above. It is an example of the "Cauchy problem" [1, chapter 7].
OTHER PROBLEMS (and an escape hatch?):
How does one avoid the paradox that a simple solution to GR has
CTCs which QM does not like? This is not a matter of applying a theory in
a domain where it is expected to fail. One relevant issue is the
construction of the time machine. After all, infinite strings aren't
easily obtained. In fact, it has been shown  that Gott's scenario
implies that the total 4-momentum of spacetime must be spacelike. This
seems to imply that one cannot build a time machine from any collection of
non-tachyonic objects, whose 4-momentum must be timelike. There are
implementation problems with the wormhole method as well.
Finally, a diversion on a possibly related topic.
If tachyons exist as physical objects, causality is no longer
invariant. Different observers will see different causal sequences. This
effect requires only special relativity (not GR), and follows from the fact
that for any spacelike trajectory, reference frames can be found in which
the particle moves backward or forward in time. This is illustrated by the
pair of spacetime diagrams below. One must be careful about what is
actually observed; a particle moving backward in time is observed to be a
forward moving anti-particle, so no observer interprets this as time
One reference | Events A and C are at the same
frame: | place. C occurs first.
| Event B lies outside the causal
| B domain of events A and C.
-----------A----------- x (The intervals are spacelike).
C In this frame, tachyon signals
| travel from A-->B and from C-->B.
| That is, A and C are possible causes
of event B.
reference | Events A and C are not at the same
frame: | place. C occurs first.
| Event B lies outside the causal
-----------A----------- x domain of events A and C. (The
| intervals are spacelike)
| C In this frame, signals travel from
| B-->A and from B-->C. B is the cause
| B of both of the other two events.
The unusual situation here arises because conventional causality
assumes no superluminal motion. This tachyon example is presented to
demonstrate that our intuitive notion of causality may be flawed, so one
must be careful when appealing to common sense. See FAQ article # 25 -
Tachyons, for more about these weird hypothetical particles.
The possible existence of time machines remains an open question.
None of the papers criticizing the two proposals are willing to
categorically rule out the possibility. Nevertheless, the notion of time
machines seems to carry with it a serious set of problems.
1: S.W. Hawking, and G.F.R. Ellis, "The Large Scale Structure of Space-Time,"
Cambridge University Press, 1973.
2: M.S. Morris, K.S. Thorne, and U. Yurtsever, PRL, v.61, p.1446 (1989).
--> How wormholes can act as time machines.
3: J.R. Gott, III, PRL, v.66, p.1126 (1991).
--> How pairs of cosmic strings can act as time machines.
4: S. Deser, R. Jackiw, and G. 't Hooft, PRL, v.66, p.267 (1992).
--> A critique of Gott. You can't construct his machine.
5: D.G. Boulware, University of Washington preprint UW/PT-92-04.
Available on the email@example.com bulletin board: item number 9207054.
--> Unitarity problems in QM with closed timelike curves.
6: "Nature", May 7, 1992
--> Contains a very well written review with some nice figures.
Open Questions updated 01-JUN-1993 by SIC
-------------- original by John Baez
While for the most part a FAQ covers the answers to frequently
asked questions whose answers are known, in physics there are also plenty
of simple and interesting questions whose answers are not known. Before you
set about answering these questions on your own, it's worth noting that
while nobody knows what the answers are, there has been at least a little,
and sometimes a great deal, of work already done on these subjects. People
have said a lot of very intelligent things about many of these questions.
So do plenty of research and ask around before you try to cook up a theory
that'll answer one of these and win you the Nobel prize! You can expect to
really know physics inside and out before you make any progress on these.
The following partial list of "open" questions is divided into two
groups, Cosmology and Astrophysics, and Particle and Quantum Physics.
However, given the implications of particle physics on cosmology, the
division is somewhat artificial, and, consequently, the categorization is
(There are many other interesting and fundamental questions in
fields such as condensed matter physics, nonlinear dynamics, etc., which
are not part of the set of related questions in cosmology and quantum
physics which are discussed below. Their omission is not a judgement
about importance, but merely a decision about the scope of this article.)
Cosmology and Astrophysics
1. What happened at, or before the Big Bang? Was there really an initial
singularity? Of course, this question might not make sense, but it might.
Does the history of universe go back in time forever, or only a finite
2. Will the future of the universe go on forever or not? Will there be a
"big crunch" in the future? Is the Universe infinite in spatial extent?
3. Why is there an arrow of time; that is, why is the future so much
different from the past?
4. Is spacetime really four-dimensional? If so, why - or is that just a
silly question? Or is spacetime not really a manifold at all if examined
on a short enough distance scale?
5. Do black holes really exist? (It sure seems like it.) Do they really
radiate energy and evaporate the way Hawking predicts? If so, what happens
when, after a finite amount of time, they radiate completely away? What's
left? Do black holes really violate all conservation laws except
conservation of energy, momentum, angular momentum and electric charge?
What happens to the information contained in an object that falls into a
black hole? Is it lost when the black hole evaporates? Does this require
a modification of quantum mechanics?
6. Is the Cosmic Censorship Hypothesis true? Roughly, for generic
collapsing isolated gravitational systems are the singularities that might
develop guaranteed to be hidden beyond a smooth event horizon? If Cosmic
Censorship fails, what are these naked singularities like? That is, what
weird physical consequences would they have?
7. Why are the galaxies distributed in clumps and filaments? Is most of
the matter in the universe baryonic? Is this a matter to be resolved by
8. What is the nature of the missing "Dark Matter"? Is it baryonic,
neutrinos, or something more exotic?
Particle and Quantum Physics
1. Why are the laws of physics not symmetrical between left and right,
future and past, and between matter and antimatter? I.e., what is the
mechanism of CP violation, and what is the origin of parity violation in
Weak interactions? Are there right-handed Weak currents too weak to have
been detected so far? If so, what broke the symmetry? Is CP violation
explicable entirely within the Standard Model, or is some new force or
2. Why are the strengths of the fundamental forces (electromagnetism, weak
and strong forces, and gravity) what they are? For example, why is the
fine structure constant, which measures the strength of electromagnetism,
about 1/137.036? Where did this dimensionless constant of nature come from?
Do the forces really become Grand Unified at sufficiently high energy?
3. Why are there 3 generations of leptons and quarks? Why are there mass
ratios what they are? For example, the muon is a particle almost exactly
like the electron except about 207 times heavier. Why does it exist and
why precisely that much heavier? Do the quarks or leptons have any
4. Is there a consistent and acceptable relativistic quantum field theory
describing interacting (not free) fields in four spacetime dimensions? For
example, is the Standard Model mathematically consistent? How about
5. Is QCD a true description of quark dynamics? Is it possible to
calculate masses of hadrons (such as the proton, neutron, pion, etc.)
correctly from the Standard Model? Does QCD predict a quark/gluon
deconfinement phase transition at high temperature? What is the nature of
the transition? Does this really happen in Nature?
6. Why is there more matter than antimatter, at least around here? Is
there really more matter than antimatter throughout the universe?
7. What is meant by a "measurement" in quantum mechanics? Does
"wavefunction collapse" actually happen as a physical process? If so, how,
and under what conditions? If not, what happens instead?
8. What are the gravitational effects, if any, of the immense (possibly
infinite) vacuum energy density seemingly predicted by quantum field
theory? Is it really that huge? If so, why doesn't it act like an
enormous cosmological constant?
9. Why doesn't the flux of solar neutrinos agree with predictions? Is the
disagreement really significant? If so, is the discrepancy in models of
the sun, theories of nuclear physics, or theories of neutrinos? Are
neutrinos really massless?
The Big Question (TM)
This last question sits on the fence between the two categories above:
How do you merge Quantum Mechanics and General Relativity to create a
quantum theory of gravity? Is Einstein's theory of gravity (classical GR)
also correct in the microscopic limit, or are there modifications
possible/required which coincide in the observed limit(s)? Is gravity
really curvature, or what else -- and why does it then look like curvature?
An answer to this question will necessarily rely upon, and at the same time
likely be a large part of, the answers to many of the other questions above.
END OF FAQ PART 3/4