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- Subject: sci.physics Frequently Asked Questions (Part 3 of 4)
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- --------------------------------------------------------------------------------
- FREQUENTLY ASKED QUESTIONS ON SCI.PHYSICS - Part 3/4
- --------------------------------------------------------------------------------
- Item 13.
-
- Apparent Superluminal Velocity of Galaxies updated 5-DEC-1994 by SIC
- ------------------------------------------ original by Scott I. Chase
-
- 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 its distance from us by some independent
- measurement, you derive its linear speed, and conclude that it is moving
- faster than the speed of light.
-
- What have you forgotten? Let's say that on Jan. 1, the object is D
- km from us, and that between Jan. 1 and Feb. 1, the object has moved d km
- closer to us. You have assumed that the light you measured on Jan. 1 and
- Feb. 1 were emitted exactly one month apart. Not so. The first light beam
- had further to travel, and was actually emitted (1 + d/c) months before the
- second measurement, if we measure c in km/month. The object has traveled
- the given angular distance in more time than you thought. Similarly, if
- the object is moving away from us, the apparent angular velocity will be
- too slow, if you do not correct for this effect, which becomes significant
- when the object is moving along a line close to our line of sight.
-
- Note that most extragalactic objects are moving away from us due to
- the Hubble expansion. So for most objects, you don't get superluminal
- apparent velocities. But the effect is still there, and you need to take
- it into account if you want to measure velocities by this technique.
-
- References:
-
- Considerations about the Apparent 'Superluminal Expansions' in
- Astrophysics, E. Recami, A. Castellino, G.D. Maccarrone, M. Rodono,
- Nuovo Cimento 93B, 119 (1986).
-
- Apparent Superluminal Sources, Comparative Cosmology and the Cosmic
- Distance Scale, Mon. Not. R. Astr. Soc. 242, 423-427 (1990).
-
-
- ********************************************************************************
- Item 14.
-
- Hot Water Freezes Faster than Cold! updated 11-May-1992 by SIC
- ----------------------------------- original by Richard M. Mathews
-
- You put two pails of water outside on a freezing day. One has hot
- water (95 degrees C) and the other has an equal amount of colder water (50
- degrees C). Which freezes first? The hot water freezes first! Why?
-
- It is commonly argued that the hot water will take some time to
- reach the initial temperature of the cold water, and then follow the same
- cooling curve. So it seems at first glance difficult to believe that the
- hot water freezes first. The answer lies mostly in evaporation. The effect
- is definitely real and can be duplicated in your own kitchen.
-
- Every "proof" that hot water can't freeze faster assumes that the
- state of the water can be described by a single number. Remember that
- temperature is a function of position. There are also other factors
- besides temperature, such as motion of the water, gas content, etc. With
- these multiple parameters, any argument based on the hot water having to
- pass through the initial state of the cold water before reaching the
- freezing point will fall apart. The most important factor is evaporation.
-
- The cooling of pails without lids is partly Newtonian and partly by
- evaporation of the contents. The proportions depend on the walls and on
- temperature. At sufficiently high temperatures evaporation is more
- important. If equal masses of water are taken at two starting
- temperatures, more rapid evaporation from the hotter one may diminish its
- mass enough to compensate for the greater temperature range it must cover
- to reach freezing. The mass lost when cooling is by evaporation is not
- negligible. In one experiment, water cooling from 100C lost 16% of its mass
- by 0C, and lost a further 12% on freezing, for a total loss of 26%.
-
- The cooling effect of evaporation is twofold. First, mass is
- carried off so that less needs to be cooled from then on. Also,
- evaporation carries off the hottest molecules, lowering considerably the
- average kinetic energy of the molecules remaining. This is why "blowing on
- your soup" cools it. It encourages evaporation by removing the water vapor
- above the soup.
-
- Thus experiment and theory agree that hot water freezes faster than
- cold for sufficiently high starting temperatures, if the cooling is by
- evaporation. Cooling in a wooden pail or barrel is mostly by evaporation.
- In fact, a wooden bucket of water starting at 100C would finish freezing in
- 90% of the time taken by an equal volume starting at room temperature. The
- folklore on this matter may well have started a century or more ago when
- wooden pails were usual. Considerable heat is transferred through the
- sides of metal pails, and evaporation no longer dominates the cooling, so
- the belief is unlikely to have started from correct observations after
- metal pails became common.
-
- References:
- "Hot water freezes faster than cold water. Why does it do so?",
- Jearl Walker in The Amateur Scientist, Scientific American,
- Vol. 237, No. 3, pp 246-257; September, 1977.
-
- "The Freezing of Hot and Cold Water", G.S. Kell in American
- Journal of Physics, Vol. 37, No. 5, pp 564-565; May, 1969.
-
- ********************************************************************************
- Item 15.
-
- 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
- trajectory':
-
- * *
- * *
- (golfer) * *
- * * <-- trajectory
- \O/ * *
- | * *
- -/ \-T---------------------------------------------------------------ground
-
- 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
- relevant vectors:
-
- F(magnus)
- ^
- |
- 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
- pushed up.')
-
- 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
- laminar flow.
-
- 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:
-
- 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)
-
- ********************************************************************************
- Item 16.
- 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.
-
- Long Answer:
-
- "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
- result.
-
- 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 more
- 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.
-
- ********************************************************************************
- Item 17.
- original by Blair P. Houghton
- (blair@world.std.com)
-
- 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.
-
- Long answer:
-
- 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.
-
- Bibliography:
- -------------
-
- 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...
-
- ********************************************************************************
- Item 18.
-
- 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."
- Actually various kinds of "temperature" appear in the literature of
- physics (e.g., kinetic temperature, color temperature). The relevant
- one here is the one from thermodynamics, in some sense the most
- fundamental.
-
- 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
- result.
-
- 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 on 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 pointing 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
- as well.
-
- 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
- down.
-
- 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 temperature. 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
- system.
-
- 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.
-
- References:
-
- 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).
-
- ********************************************************************************
- Item 19.
-
- 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).
-
- ********************************************************************************
- Item 20.
-
- 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. Imagine 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.
-
- ********************************************************************************
- Item 21.
- 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.
-
- ********************************************************************************
- Item 22.
-
- 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 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.
-
- CONSERVATION LAWS:
-
- 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 all right, but it
- implies that something strange is happening. (Why shouldn't we be able to
- do the integral?)
-
- GENERAL RELATIVITY:
-
- 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 #7 - 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 [1], 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. [2] and one by
- Gott [3], 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 [2]) 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.
-
- GRANDFATHER PARADOXES:
-
- 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 [5] 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 [4] 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.
-
- TACHYONS:
-
- 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
- travel.
-
- t
- 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.
-
- Another t
- 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.
-
- CONCLUSION:
-
- 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.
-
- REFERENCES:
-
- 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 hep-th@xxx.lanl.gov 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.
-
- ********************************************************************************
- Item 23.
-
- 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
- somewhat arbitrary.
-
- (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 the Universe go back in time forever, or only a finite
- amount?
-
- 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
- new physics?
-
- 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
- mechanism required?
-
- 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 their 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
- substructure?
-
- 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
- Quantum Electrodynamics?
-
- 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
-