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- Subject: sci.physics Frequently Asked Questions (Part 2 of 4)
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- Summary: This posting contains a list of Frequently Asked Questions
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- --------------------------------------------------------------------------------
- FREQUENTLY ASKED QUESTIONS ON SCI.PHYSICS - Part 2/4
- --------------------------------------------------------------------------------
- Item 6.
-
- Gravitational Radiation updated 20-May-1992 by SIC
- ----------------------- original by Scott I. Chase
-
- Gravitational Radiation is to gravity what light is to
- electromagnetism. It is produced when massive bodies accelerate. You can
- accelerate any body so as to produce such radiation, but due to the feeble
- strength of gravity, it is entirely undetectable except when produced by
- intense astrophysical sources such as supernovae, collisions of black
- holes, etc. These are quite far from us, typically, but they are so
- intense that they dwarf all possible laboratory sources of such radiation.
-
- Gravitational waves have a polarization pattern that causes objects
- to expand in one direction, while contracting in the perpendicular
- direction. That is, they have spin two. This is because gravity waves are
- fluctuations in the tensorial metric of space-time.
-
- All oscillating radiation fields can be quantized, and in the case
- of gravity, the intermediate boson is called the "graviton" in analogy
- with the photon. But quantum gravity is hard, for several reasons:
- (1) The quantum field theory of gravity is hard, because gauge
- interactions of spin-two fields are not renormalizable. See Cheng and Li,
- Gauge Theory of Elementary Particle Physics (search for "power counting").
- (2) There are conceptual problems - what does it mean to quantize
- geometry, or space-time?
-
- It is possible to quantize weak fluctuations in the gravitational
- field. This gives rise to the spin-2 graviton. But full quantum gravity
- has so far escaped formulation. It is not likely to look much like the
- other quantum field theories. In addition, there are models of gravity
- which include additional bosons with different spins. Some are the
- consequence of non-Einsteinian models, such as Brans-Dicke which has a
- spin-0 component. Others are included by hand, to give "fifth force"
- components to gravity. For example, if you want to add a weak repulsive
- short range component, you will need a massive spin-1 boson. (Even-spin
- bosons always attract. Odd-spin bosons can attract or repel.) If
- antigravity is real, then this has implications for the boson spectrum as
- well.
-
- The spin-two polarization provides the method of detection. Most
- experiments to date use a "Weber bar." This is a cylindrical, very
- massive, bar suspended by fine wire, free to oscillate in response to a
- passing graviton. A high-sensitivity, low noise, capacitive transducer
- can turn the oscillations of the bar into an electric signal for analysis.
- So far such searches have failed. But they are expected to be
- insufficiently sensitive for typical radiation intensity from known types
- of sources.
-
- A more sensitive technique uses very long baseline laser
- interferometry. This is the principle of LIGO (Laser Interferometric
- Gravity wave Observatory). This is a two-armed detector, with
- perpendicular laser beams each travelling several km before meeting to
- produce an interference pattern which fluctuates if a gravity wave distorts
- the geometry of the detector. To eliminate noise from seismic effects as
- well as human noise sources, two detectors separated by hundreds to
- thousands of miles are necessary. A coincidence measurement then provides
- evidence of gravitational radiation. In order to determine the source of
- the signal, a third detector, far from either of the first two, would be
- necessary. Timing differences in the arrival of the signal to the three
- detectors would allow triangulation of the angular position in the sky of
- the signal.
-
- The first stage of LIGO, a two detector setup in the U.S., has been
- approved by Congress in 1992. LIGO researchers have started designing a
- prototype detector, and are hoping to enroll another nation, probably in
- Europe, to fund and be host to the third detector.
-
- The speed of gravitational radiation (C_gw) depends upon the
- specific model of Gravitation that you use. There are quite a few
- competing models (all consistent with all experiments to date) including of
- course Einstein's but also Brans-Dicke and several families of others.
- All metric models can support gravity waves. But not all predict radiation
- travelling at C_gw = C_em. (C_em is the speed of electromagnetic waves.)
-
- There is a class of theories with "prior geometry", in which, as I
- understand it, there is an additional metric which does not depend only on
- the local matter density. In such theories, C_gw != C_em in general.
-
- However, there is good evidence that C_gw is in fact at least
- almost C_em. We observe high energy cosmic rays in the 10^20-10^21 eV
- region. Such particles are travelling at up to (1-10^-18)*C_em. If C_gw <
- C_em, then particles with C_gw < v < C_em will radiate Cerenkov
- gravitational radiation into the vacuum, and decelerate from the back
- reaction. So evidence of these very fast cosmic rays is good evidence that
- C_gw >= (1-10^-18)*C_em, very close indeed to C_em. Bottom line: in a
- purely Einsteinian universe, C_gw = C_em. However, a class of models not
- yet ruled out experimentally does make other predictions.
-
- A definitive test would be produced by LIGO in coincidence with
- optical measurements of some catastrophic event which generates enough
- gravitational radiation to be detected. Then the "time of flight" of both
- gravitons and photons from the source to the Earth could be measured, and
- strict direct limits could be set on C_gw.
-
- For more information, see Gravitational Radiation (NATO ASI -
- Les Houches 1982), specifically the introductory essay by Kip Thorne.
-
- ********************************************************************************
- Item 7.
-
- IS ENERGY CONSERVED IN GENERAL RELATIVITY? original by Michael Weiss
- ------------------------------------------ and John Baez
-
- In special cases, yes. In general--- it depends on what you mean
- by "energy", and what you mean by "conserved".
-
- In flat spacetime (the backdrop for special relativity) you can
- phrase energy conservation in two ways: as a differential equation, or as
- an equation involving integrals (gory details below). The two formulations
- are mathematically equivalent. But when you try to generalize this to
- curved spacetimes (the arena for general relativity) this equivalence
- breaks down. The differential form extends with nary a hiccup; not so the
- integral form.
-
- The differential form says, loosely speaking, that no energy is
- created in any infinitesimal piece of spacetime. The integral form says
- the same for a finite-sized piece. (This may remind you of the
- "divergence" and "flux" forms of Gauss's law in electrostatics, or the
- equation of continuity in fluid dynamics. Hold on to that thought!)
-
- An infinitesimal piece of spacetime "looks flat", while the effects
- of curvature become evident in a finite piece. (The same holds for curved
- surfaces in space, of course). GR relates curvature to gravity. Now, even
- in Newtonian physics, you must include gravitational potential energy to
- get energy conservation. And GR introduces the new phenomenon of
- gravitational waves; perhaps these carry energy as well? Perhaps we need
- to include gravitational energy in some fashion, to arrive at a law of
- energy conservation for finite pieces of spacetime?
-
- Casting about for a mathematical expression of these ideas,
- physicists came up with something called an energy pseudo-tensor. (In fact,
- several of 'em!) Now, GR takes pride in treating all coordinate systems
- equally. Mathematicians invented tensors precisely to meet this sort of
- demand--- if a tensor equation holds in one coordinate system, it holds in
- all. Pseudo-tensors are not tensors (surprise!), and this alone raises
- eyebrows in some circles. In GR, one must always guard against mistaking
- artifacts of a particular coordinate system for real physical effects.
- (See the FAQ entry on black holes for some examples.)
-
- These pseudo-tensors have some rather strange properties. If you
- choose the "wrong" coordinates, they are non-zero even in flat empty
- spacetime. By another choice of coordinates, they can be made zero at any
- chosen point, even in a spacetime full of gravitational radiation. For
- these reasons, most physicists who work in general relativity do not
- believe the pseudo-tensors give a good *local* definition of energy
- density, although their integrals are sometimes useful as a measure of
- total energy.
-
- One other complaint about the pseudo-tensors deserves mention.
- Einstein argued that all energy has mass, and all mass acts
- gravitationally. Does "gravitational energy" itself act as a source of
- gravity? Now, the Einstein field equations are
-
- G_{mu,nu} = 8pi T_{mu,nu}
-
- Here G_{mu,nu} is the Einstein curvature tensor, which encodes
- information about the curvature of spacetime, and T_{mu,nu} is the
- so-called stress-energy tensor, which we will meet again below. T_{mu,nu}
- represents the energy due to matter and electromagnetic fields, but
- includes NO contribution from "gravitational energy". So one can argue
- that "gravitational energy" does NOT act as a source of gravity. On the
- other hand, the Einstein field equations are non-linear; this implies that
- gravitational waves interact with each other (unlike light waves in
- Maxwell's (linear) theory). So one can argue that "gravitational energy"
- IS a source of gravity.
-
- In certain special cases, energy conservation works out with fewer
- caveats. The two main examples are static spacetimes and asymptotically
- flat spacetimes.
-
- Let's look at four examples before plunging deeper into the math.
- Three examples involve redshift, the other, gravitational radiation.
-
- (1) Very fast objects emitting light.
-
- According to *special* relativity, you will see light coming from a
- receding object as redshifted. So if you, and someone moving with the
- source, both measure the light's energy, you'll get different answers.
- Note that this has nothing to do with energy conservation per se. Even in
- Newtonian physics, kinetic energy (mv^2/2) depends on the choice of
- reference frame. However, relativity serves up a new twist. In Newtonian
- physics, energy conservation and momentum conservation are two separate
- laws. Special relativity welds them into one law, the conservation of the
- *energy-momentum 4-vector*. To learn the whole scoop on 4-vectors, read a
- text on SR, for example Taylor and Wheeler (see refs.) For our purposes,
- it's enough to remark that 4-vectors are vectors in spacetime, which most
- people privately picture just like ordinary vectors (unless they have
- *very* active imaginations).
-
- (2) Very massive objects emitting light.
-
- Light from the Sun appears redshifted to an Earthbound astronomer.
- In quasi-Newtonian terms, we might say that light loses kinetic energy as
- it climbs out of the gravitational well of the Sun, but gains potential
- energy. General relativity looks at it differently. In GR, gravity is
- described not by a "potential" but by the "metric" of spacetime. But "no
- problem", as the saying goes. The Schwarzschild metric describes spacetime
- around a massive object, if the object is spherically symmetrical,
- uncharged, and "alone in the universe". The Schwarzschild metric is both
- static and asymptotically flat, and energy conservation holds without major
- pitfalls. For further details, consult MTW, chapter 25.
-
- (3) Gravitational waves.
-
- A binary pulsar emits gravitational waves, according to GR, and one
- expects (innocent word!) that these waves will carry away energy. So its
- orbital period should change. Einstein derived a formula for the rate of
- change (known as the quadrapole formula), and in the centenary of
- Einstein's birth, Russell Hulse and Joseph Taylor reported that the binary
- pulsar PSR1913+16 bore out Einstein's predictions within a few percent.
- Hulse and Taylor were awarded the Nobel prize in 1993.
-
- Despite this success, Einstein's formula remained controversial for
- many years, partly because of the subtleties surrounding energy
- conservation in GR. The need to understand this situation better has kept
- GR theoreticians busy over the last few years. Einstein's formula now
- seems well-established, both theoretically and observationally.
-
- (4) Expansion of the universe leading to cosmological redshift.
-
- The Cosmic Background Radiation (CBR) has red-shifted over billions
- of years. Each photon gets redder and redder. What happens to this
- energy? Cosmologists model the expanding universe with
- Friedmann-Robertson-Walker (FRW) spacetimes. (The familiar "expanding
- balloon speckled with galaxies" belongs to this class of models.) The FRW
- spacetimes are neither static nor asymptotically flat. Those who harbor no
- qualms about pseudo-tensors will say that radiant energy becomes
- gravitational energy. Others will say that the energy is simply lost.
-
- It's time to look at mathematical fine points. There are many to
- choose from! The definition of asymptotically flat, for example, calls for
- some care (see Stewart); one worries about "boundary conditions at
- infinity". (In fact, both spatial infinity and "null infinity" clamor for
- attention--- leading to different kinds of total energy.) The static case
- has close connections with Noether's theorem (see Goldstein or Arnold). If
- the catch-phrase "time translation symmetry implies conservation of energy"
- rings a bell (perhaps from quantum mechanics), then you're on the right
- track. (Check out "Killing vector" in the index of MTW, Wald, or Sachs and
- Wu.)
-
- But two issues call for more discussion. Why does the equivalence
- between the two forms of energy conservation break down? How do the
- pseudo-tensors slide around this difficulty?
-
- We've seen already that we should be talking about the
- energy-momentum 4-vector, not just its time-like component (the energy).
- Let's consider first the case of flat Minkowski spacetime. Recall that the
- notion of "inertial frame" corresponds to a special kind of coordinate
- system (Minkowskian coordinates).
-
- Pick an inertial reference frame. Pick a volume V in this frame,
- and pick two times t=t_0 and t=t_1. One formulation of energy-momentum
- conservation says that the energy-momentum inside V changes only because of
- energy-momentum flowing across the boundary surface (call it S). It is
- "conceptually difficult, mathematically easy" to define a quantity T so
- that the captions on the Equation 1 (below) are correct. (The quoted
- phrase comes from Sachs and Wu.)
-
- Equation 1: (valid in flat Minkowski spacetime, when Minkowskian
- coordinates are used)
-
- t=t_1
- / / /
- | | |
- | T dV - | T dV = | T dt dS
- / / /
- V,t=t_0 V,t=t_1 t=t_0
-
- p contained p contained p flowing out through
- in volume V - in volume V = boundary S of V
- at time t_0 at time t_1 during t=t_0 to t=t_1
-
- (Note: p = energy-momentum 4-vector)
-
- T is called the stress-energy tensor. You don't need to know what
- that means! ---just that you can integrate T, as shown, to get
- 4-vectors. Equation 1 may remind you of Gauss's theorem, which deals
- with flux across a boundary. If you look at Equation 1 in the right
- 4-dimensional frame of mind, you'll discover it really says that the
- flux across the boundary of a certain 4-dimensional hypervolume is
- zero. (The hypervolume is swept out by V during the interval t=t_0
- to t=t_1.) MTW, chapter 7, explains this with pictures galore. (See
- also Wheeler.)
-
- A 4-dimensional analogue to Gauss's theorem shows that Equation 1
- is equivalent to:
-
- Equation 2: (valid in flat Minkowski spacetime, with Minkowskian
- coordinates)
-
- coord_div(T) = sum_mu (partial T/partial x_mu) = 0
-
- We write "coord_div" for the divergence, for we will meet another
- divergence in a moment. Proof? Quite similar to Gauss's theorem: if
- the divergence is zero throughout the hypervolume, then the flux
- across the boundary must also be zero. On the other hand, the flux
- out of an infinitesimally small hypervolume turns out to be the
- divergence times the measure of the hypervolume.
-
- Pass now to the general case of any spacetime satisfying Einstein's
- field equation. It is easy to generalize the differential form of
- energy-momentum conservation, Equation 2:
-
- Equation 3: (valid in any GR spacetime)
-
- covariant_div(T) = sum_mu nabla_mu(T) = 0
-
- (where nabla_mu = covariant derivative)
-
- (Side comment: Equation 3 is the correct generalization of Equation 1 for
- SR when non-Minkowskian coordinates are used.)
-
- GR relies heavily on the covariant derivative, because the
- covariant derivative of a tensor is a tensor, and as we've seen, GR loves
- tensors. Equation 3 follows from Einstein's field equation (because
- something called Bianchi's identity says that covariant_div(G)=0). But
- Equation 3 is no longer equivalent to Equation 1!
-
- Why not? Well, the familiar form of Gauss's theorem (from
- electrostatics) holds for any spacetime, because essentially you are
- summing fluxes over a partition of the volume into infinitesimally small
- pieces. The sum over the faces of one infinitesimal piece is a divergence.
- But the total contribution from an interior face is zero, since what flows
- out of one piece flows into its neighbor. So the integral of the
- divergence over the volume equals the flux through the boundary. "QED".
-
- But for the equivalence of Equations 1 and 3, we would need an
- extension of Gauss's theorem. Now the flux through a face is not a scalar,
- but a vector (the flux of energy-momentum through the face). The argument
- just sketched involves adding these vectors, which are defined at different
- points in spacetime. Such "remote vector comparison" runs into trouble
- precisely for curved spacetimes.
-
- The mathematician Levi-Civita invented the standard solution to
- this problem, and dubbed it "parallel transport". It's easy to picture
- parallel transport: just move the vector along a path, keeping its
- direction "as constant as possible". (Naturally, some non-trivial
- mathematics lurks behind the phrase in quotation marks. But even
- pop-science expositions of GR do a good job explaining parallel transport.)
- The parallel transport of a vector depends on the transportation path; for
- the canonical example, imagine parallel transporting a vector on a sphere.
- But parallel transportation over an "infinitesimal distance" suffers no
- such ambiguity. (It's not hard to see the connection with curvature.)
-
- To compute a divergence, we need to compare quantities (here
- vectors) on opposite faces. Using parallel transport for this leads to the
- covariant divergence. This is well-defined, because we're dealing with an
- infinitesimal hypervolume. But to add up fluxes all over a finite-sized
- hypervolume (as in the contemplated extension of Gauss's theorem) runs
- smack into the dependence on transportation path. So the flux integral is
- not well-defined, and we have no analogue for Gauss's theorem.
-
- One way to get round this is to pick one coordinate system, and
- transport vectors so their *components* stay constant. Partial derivatives
- replace covariant derivatives, and Gauss's theorem is restored. The energy
- pseudo-tensors take this approach (at least some of them do). If you can
- mangle Equation 3 (covariant_div(T) = 0) into the form:
-
- coord_div(Theta) = 0
-
- then you can get an "energy conservation law" in integral form.
- Einstein was the first to do this; Dirac, Landau and Lifshitz, and
- Weinberg all came up with variations on this theme. We've said
- enough already on the pros and cons of this approach.
-
- We will not delve into definitions of energy in general relativity
- such as the Hamiltonian (amusingly, the energy of a closed universe always
- works out to zero according to this definition), various kinds of energy
- one hopes to obtain by "deparametrizing" Einstein's equations, or
- "quasilocal energy". There's quite a bit to say about this sort of thing!
- Indeed, the issue of energy in general relativity has a lot to do with the
- notorious "problem of time" in quantum gravity.... but that's another can
- of worms.
-
- References (vaguely in order of difficulty):
-
- Clifford Will, "The renaissance of general relativity", in "The New
- Physics" (ed. Paul Davies) gives a semi-technical discussion of the
- controversy over gravitational radiation.
- Wheeler, "A Journey into Gravity and Spacetime". Wheeler's try at
- a "pop-science" treatment of GR. Chapters 6 and 7 are a
- tour-de-force: Wheeler tries for a non-technical explanation of
- Cartan's formulation of Einstein's field equation. It might be
- easier just to read MTW!)
- Taylor and Wheeler, "Spacetime Physics".
- Goldstein, "Classical Mechanics".
- Arnold, "Mathematical Methods in Classical Mechanics".
- Misner, Thorne, and Wheeler (MTW), "Gravitation", chapters 7, 20,
- and 25
- Wald, "General Relativity", Appendix E. This has the Hamiltonian
- formalism and a bit about deparametrizing, and chapter 11
- discusses energy in asymptotically flat spacetimes.
- H. A. Buchdahl, "Seventeen Simple Lectures on General Relativity Theory"
- Lecture 15 derives the energy-loss formula for the binary star, and
- criticizes the derivation.
- Sachs and Wu, "General Relativity for Mathematicians", chapter 3
- John Stewart, "Advanced General Relativity". Chapter 3 ("Asymptopia")
- shows just how careful one has to be in asymptotically flat spacetimes
- to recover energy conservation. Stewart also discusses the Bondi-Sachs
- mass, another contender for "energy".
- Damour, in "300 Years of Gravitation" (ed. Hawking and Israel). Damour
- heads the "Paris group", which has been active in the theory of
- gravitational radiation.
- Penrose and Rindler, "Spinors and Spacetime", vol II, chapter 9. The
- Bondi-Sachs mass generalized.
- J. David Brown and James York Jr., "Quasilocal energy in general
- relativity", in "Mathematical Aspects of Classical Field Theory".
-
- ********************************************************************************
- Item 8.
-
- Olbers' Paradox updated: 24-JAN-1993 by SIC
- --------------- original by Scott I. Chase
-
- Why isn't the night sky as uniformly bright as the surface of the
- Sun? If the Universe has infinitely many stars, then it should be. After
- all, if you move the Sun twice as far away from us, we will intercept
- one-fourth as many photons, but the Sun will subtend one-fourth of the
- angular area. So the areal intensity remains constant. With infinitely
- many stars, every angular element of the sky should have a star, and the
- entire heavens should be as bright as the sun. We should have the
- impression that we live in the center of a hollow black body whose
- temperature is about 6000 degrees Centigrade. This is Olbers' paradox.
- It can be traced as far back as Kepler in 1610. It was rediscussed by
- Halley and Cheseaux in the eighteen century, but was not popularized as
- a paradox until Olbers took up the issue in the nineteenth century.
-
- There are many possible explanations which have been considered.
- Here are a few:
- (1) There's too much dust to see the distant stars.
- (2) The Universe has only a finite number of stars.
- (3) The distribution of stars is not uniform. So, for example,
- there could be an infinity of stars, but they hide behind one
- another so that only a finite angular area is subtended by them.
- (4) The Universe is expanding, so distant stars are red-shifted into
- obscurity.
- (5) The Universe is young. Distant light hasn't even reached us yet.
-
- The first explanation is just plain wrong. In a black body, the
- dust will heat up too. It does act like a radiation shield, exponentially
- damping the distant starlight. But you can't put enough dust into the
- universe to get rid of enough starlight without also obscuring our own Sun.
- So this idea is bad.
-
- The premise of the second explanation may technically be correct.
- But the number of stars, finite as it might be, is still large enough to
- light up the entire sky, i.e., the total amount of luminous matter in the
- Universe is too large to allow this escape. The number of stars is close
- enough to infinite for the purpose of lighting up the sky. The third
- explanation might be partially correct. We just don't know. If the stars
- are distributed fractally, then there could be large patches of empty space,
- and the sky could appear dark except in small areas.
-
- But the final two possibilities are are surely each correct and
- partly responsible. There are numerical arguments that suggest that the
- effect of the finite age of the Universe is the larger effect. We live
- inside a spherical shell of "Observable Universe" which has radius equal to
- the lifetime of the Universe. Objects more than about 15 billion years
- old are too far away for their light ever to reach us.
-
- Historically, after Hubble discovered that the Universe was
- expanding, but before the Big Bang was firmly established by the discovery
- of the cosmic background radiation, Olbers' paradox was presented as proof
- of special relativity. You needed the red-shift (an SR effect) to get rid
- of the starlight. This effect certainly contributes. But the finite age
- of the Universe is the most important effect.
-
- References: Ap. J. _367_, 399 (1991). The author, Paul Wesson, is said to
- be on a personal crusade to end the confusion surrounding Olbers' paradox.
-
- _Darkness at Night: A Riddle of the Universe_, Edward Harrison, Harvard
- University Press, 1987
-
- ********************************************************************************
- Item 9.
-
- What is Dark Matter? updated 11-MAY-1993 by SIC
- -------------------- original by Scott I. Chase
-
- The story of dark matter is best divided into two parts. First we
- have the reasons that we know that it exists. Second is the collection of
- possible explanations as to what it is.
-
- Why the Universe Needs Dark Matter
- ----------------------------------
-
- We believe that that the Universe is critically balanced between
- being open and closed. We derive this fact from the observation of the
- large scale structure of the Universe. It requires a certain amount of
- matter to accomplish this result. Call it M.
-
- We can estimate the total BARYONIC matter of the universe by
- studying Big Bang nucleosynthesis. This is done by connecting the observed
- He/H ratio of the Universe today to the amount of baryonic matter present
- during the early hot phase when most of the helium was produced. Once the
- temperature of the Universe dropped below the neutron-proton mass difference,
- neutrons began decaying into protons. If the early baryon density was low,
- then it was hard for a proton to find a neutron with which to make helium
- before too many of the neutrons decayed away to account for the amount of
- helium we see today. So by measuring the He/H ratio today, we can estimate
- the necessary baryon density shortly after the Big Bang, and, consequently,
- the total number of baryons today. It turns out that you need about 0.05 M
- total baryonic matter to account for the known ratio of light isotopes. So
- only 1/20 of the total mass of the Universe is baryonic matter.
-
- Unfortunately, the best estimates of the total mass of everything
- that we can see with our telescopes is roughly 0.01 M. Where is the other
- 99% of the stuff of the Universe? Dark Matter!
-
- So there are two conclusions. We only see 0.01 M out of 0.05 M
- baryonic matter in the Universe. The rest must be in baryonic dark matter
- halos surrounding galaxies. And there must be some non-baryonic dark matter
- to account for the remaining 95% of the matter required to give omega, the
- mass of the Universe, in units of critical mass, equal to unity.
-
- For those who distrust the conventional Big Bang models, and don't
- want to rely upon fancy cosmology to derive the presence of dark matter,
- there are other more direct means. It has been observed in clusters of
- galaxies that the motion of galaxies within a cluster suggests that they
- are bound by a total gravitational force due to about 5-10 times as much
- matter as can be accounted for from luminous matter in said galaxies. And
- within an individual galaxy, you can measure the rate of rotation of the
- stars about the galactic center of rotation. The resultant "rotation
- curve" is simply related to the distribution of matter in the galaxy. The
- outer stars in galaxies seem to rotate too fast for the amount of matter
- that we see in the galaxy. Again, we need about 5 times more matter than
- we can see via electromagnetic radiation. These results can be explained
- by assuming that there is a "dark matter halo" surrounding every galaxy.
-
- What is Dark Matter
- -------------------
-
- This is the open question. There are many possibilities, and
- nobody really knows much about this yet. Here are a few of the many
- published suggestions, which are being currently hunted for by
- experimentalists all over the world. Remember, you need at least one
- baryonic candidate and one non-baryonic candidate to make everything
- work out, so there there may be more than one correct choice among
- the possibilities given here.
-
- (1) Normal matter which has so far eluded our gaze, such as
- (a) dark galaxies
- (b) brown dwarfs
- (c) planetary material (rock, dust, etc.)
-
- (2) Massive Standard Model neutrinos. If any of the neutrinos are massive,
- then this could be the missing mass. On the other hand, if they are
- too heavy, as the purported 17 KeV neutrino would have been, massive
- neutrinos create almost as many problems as they solve in this regard.
-
- (3) Exotica (See the "Particle Zoo" FAQ entry for some details)
-
- Massive exotica would provide the missing mass. For our purposes,
- these fall into two classes: those which have been proposed for other
- reasons but happen to solve the dark matter problem, and those which have
- been proposed specifically to provide the missing dark matter.
-
- Examples of objects in the first class are axions, additional
- neutrinos, supersymmetric particles, and a host of others. Their properties
- are constrained by the theory which predicts them, but by virtue of their
- mass, they solve the dark matter problem if they exist in the correct
- abundance.
-
- Particles in the second class are generally classed in loose groups.
- Their properties are not specified, but they are merely required to be
- massive and have other properties such that they would so far have eluded
- discovery in the many experiments which have looked for new particles.
- These include WIMPS (Weakly Interacting Massive Particles), CHAMPS, and a
- host of others.
-
- References: _Dark Matter in the Universe_ (Jerusalem Winter School for
- Theoretical Physics, 1986-7), J.N. Bahcall, T. Piran, & S. Weinberg editors.
- _Dark Matter_ (Proceedings of the XXIIIrd Recontre de Moriond) J. Audouze and
- J. Tran Thanh Van. editors.
-
- ********************************************************************************
- Item 10.
-
- Some Frequently Asked Questions About Black Holes updated 02-FEB-1995 by MM
- ------------------------------------------------- original by Matt McIrvin
-
- Contents:
-
- 1. What is a black hole, really?
- 2. What happens to you if you fall in?
- 3. Won't it take forever for you to fall in? Won't it take forever
- for the black hole to even form?
- 4. Will you see the universe end?
- 5. What about Hawking radiation? Won't the black hole evaporate
- before you get there?
- 6. How does the gravity get out of the black hole?
- 7. Where did you get that information?
-
- 1. What is a black hole, really?
-
- In 1916, when general relativity was new, Karl Schwarzschild worked
- out a useful solution to the Einstein equation describing the evolution of
- spacetime geometry. This solution, a possible shape of spacetime, would
- describe the effects of gravity *outside* a spherically symmetric,
- uncharged, nonrotating object (and would serve approximately to describe
- even slowly rotating objects like the Earth or Sun). It worked in much the
- same way that you can treat the Earth as a point mass for purposes of
- Newtonian gravity if all you want to do is describe gravity *outside* the
- Earth's surface.
-
- What such a solution really looks like is a "metric," which is a
- kind of generalization of the Pythagorean formula that gives the length of
- a line segment in the plane. The metric is a formula that may be used to
- obtain the "length" of a curve in spacetime. In the case of a curve
- corresponding to the motion of an object as time passes (a "timelike
- worldline,") the "length" computed by the metric is actually the elapsed
- time experienced by an object with that motion. The actual formula depends
- on the coordinates chosen in which to express things, but it may be
- transformed into various coordinate systems without affecting anything
- physical, like the spacetime curvature. Schwarzschild expressed his metric
- in terms of coordinates which, at large distances from the object,
- resembled spherical coordinates with an extra coordinate t for time.
- Another coordinate, called r, functioned as a radial coordinate at large
- distances; out there it just gave the distance to the massive object.
-
- Now, at small radii, the solution began to act strangely. There
- was a "singularity" at the center, r=0, where the curvature of spacetime
- was infinite. Surrounding that was a region where the "radial" direction
- of decreasing r was actually a direction in *time* rather than in space.
- Anything in that region, including light, would be obligated to fall toward
- the singularity, to be crushed as tidal forces diverged. This was separated
- >from the rest of the universe by a place where Schwarzschild's coordinates
- blew up, though nothing was wrong with the curvature of spacetime there.
- (This was called the Schwarzschild radius. Later, other coordinate systems
- were discovered in which the blow-up didn't happen; it was an artifact of
- the coordinates, a little like the problem of defining the longitude of the
- North Pole. The physically important thing about the Schwarzschild radius
- was not the coordinate problem, but the fact that within it the direction
- into the hole became a direction in time.)
-
- Nobody really worried about this at the time, because there was no
- known object that was dense enough for that inner region to actually be
- outside it, so for all known cases, this odd part of the solution would not
- apply. Arthur Stanley Eddington considered the possibility of a dying star
- collapsing to such a density, but rejected it as aesthetically unpleasant
- and proposed that some new physics must intervene. In 1939, Oppenheimer
- and Snyder finally took seriously the possibility that stars a few times
- more massive than the sun might be doomed to collapse to such a state at
- the end of their lives.
-
- Once the star gets smaller than the place where Schwarzschild's
- coordinates fail (called the Schwarzschild radius for an uncharged,
- nonrotating object, or the event horizon) there's no way it can avoid
- collapsing further. It has to collapse all the way to a singularity for
- the same reason that you can't keep from moving into the future! Nothing
- else that goes into that region afterward can avoid it either, at least in
- this simple case. The event horizon is a point of no return.
-
- In 1971 John Archibald Wheeler named such a thing a black hole,
- since light could not escape from it. Astronomers have many candidate
- objects they think are probably black holes, on the basis of several kinds
- of evidence (typically they are dark objects whose large mass can be
- deduced from their gravitational effects on other objects, and which
- sometimes emit X-rays, presumably from infalling matter). But the
- properties of black holes I'll talk about here are entirely theoretical.
- They're based on general relativity, which is a theory that seems supported
- by available evidence.
-
- 2. What happens to you if you fall in?
-
- Suppose that, possessing a proper spacecraft and a self-destructive
- urge, I decide to go black-hole jumping and head for an uncharged,
- nonrotating ("Schwarzschild") black hole. In this and other kinds of hole,
- I won't, before I fall in, be able to see anything within the event
- horizon. But there's nothing *locally* special about the event horizon;
- when I get there it won't seem like a particularly unusual place, except
- that I will see strange optical distortions of the sky around me from all
- the bending of light that goes on. But as soon as I fall through, I'm
- doomed. No bungee will help me, since bungees can't keep Sunday from
- turning into Monday. I have to hit the singularity eventually, and before
- I get there there will be enormous tidal forces-- forces due to the
- curvature of spacetime-- which will squash me and my spaceship in some
- directions and stretch them in another until I look like a piece of
- spaghetti. At the singularity all of present physics is mute as to what
- will happen, but I won't care. I'll be dead.
-
- For ordinary black holes of a few solar masses, there are actually
- large tidal forces well outside the event horizon, so I probably wouldn't
- even make it into the hole alive and unstretched. For a black hole of 8
- solar masses, for instance, the value of r at which tides become fatal is
- about 400 km, and the Schwarzschild radius is just 24 km. But tidal
- stresses are proportional to M/r^3. Therefore the fatal r goes as the cube
- root of the mass, whereas the Schwarzschild radius of the black hole is
- proportional to the mass. So for black holes larger than about 1000 solar
- masses I could probably fall in alive, and for still larger ones I might
- not even notice the tidal forces until I'm through the horizon and doomed.
-
- 3. Won't it take forever for you to fall in? Won't it take forever
- for the black hole to even form?
-
- Not in any useful sense. The time I experience before I hit the
- event horizon, and even until I hit the singularity-- the "proper time"
- calculated by using Schwarzschild's metric on my worldline -- is finite.
- The same goes for the collapsing star; if I somehow stood on the surface of
- the star as it became a black hole, I would experience the star's demise in
- a finite time.
-
- On my worldline as I fall into the black hole, it turns out that
- the Schwarzschild coordinate called t goes to infinity when I go through
- the event horizon. That doesn't correspond to anyone's proper time,
- though; it's just a coordinate called t. In fact, inside the event
- horizon, t is actually a *spatial* direction, and the future corresponds
- instead to decreasing r. It's only outside the black hole that t even
- points in a direction of increasing time. In any case, this doesn't
- indicate that I take forever to fall in, since the proper time involved is
- actually finite.
-
- At large distances t *does* approach the proper time of someone who
- is at rest with respect to the black hole. But there isn't any
- non-arbitrary sense in which you can call t at smaller r values "the proper
- time of a distant observer," since in general relativity there is no
- coordinate-independent way to say that two distant events are happening "at
- the same time." The proper time of any observer is only defined locally.
-
- A more physical sense in which it might be said that things take
- forever to fall in is provided by looking at the paths of emerging light
- rays. The event horizon is what, in relativity parlance, is called a
- "lightlike surface"; light rays can remain there. For an ideal
- Schwarzschild hole (which I am considering in this paragraph) the horizon
- lasts forever, so the light can stay there without escaping. (If you
- wonder how this is reconciled with the fact that light has to travel at the
- constant speed c-- well, the horizon *is* traveling at c! Relative speeds
- in GR are also only unambiguously defined locally, and if you're at the
- event horizon you are necessarily falling in; it comes at you at the speed
- of light.) Light beams aimed directly outward from just outside the
- horizon don't escape to large distances until late values of t. For
- someone at a large distance from the black hole and approximately at rest
- with respect to it, the coordinate t does correspond well to proper time.
-
- So if you, watching from a safe distance, attempt to witness my
- fall into the hole, you'll see me fall more and more slowly as the light
- delay increases. You'll never see me actually *get to* the event horizon.
- My watch, to you, will tick more and more slowly, but will never reach the
- time that I see as I fall into the black hole. Notice that this is really
- an optical effect caused by the paths of the light rays.
-
- This is also true for the dying star itself. If you attempt to
- witness the black hole's formation, you'll see the star collapse more and
- more slowly, never precisely reaching the Schwarzschild radius.
-
- Now, this led early on to an image of a black hole as a strange
- sort of suspended-animation object, a "frozen star" with immobilized
- falling debris and gedankenexperiment astronauts hanging above it in
- eternally slowing precipitation. This is, however, not what you'd see. The
- reason is that as things get closer to the event horizon, they also get
- *dimmer*. Light from them is redshifted and dimmed, and if one considers
- that light is actually made up of discrete photons, the time of escape of
- *the last photon* is actually finite, and not very large. So things would
- wink out as they got close, including the dying star, and the name "black
- hole" is justified.
-
- As an example, take the eight-solar-mass black hole I mentioned
- before. If you start timing from the moment the you see the object half a
- Schwarzschild radius away from the event horizon, the light will dim
- exponentially from that point on with a characteristic time of about 0.2
- milliseconds, and the time of the last photon is about a hundredth of a
- second later. The times scale proportionally to the mass of the black
- hole. If I jump into a black hole, I don't remain visible for long.
-
- Also, if I jump in, I won't hit the surface of the "frozen star."
- It goes through the event horizon at another point in spacetime from
- where/when I do.
-
- (Some have pointed out that I really go through the event horizon a
- little earlier than a naive calculation would imply. The reason is that my
- addition to the black hole increases its mass, and therefore moves the
- event horizon out around me at finite Schwarzschild t coordinate. This
- really doesn't change the situation with regard to whether an external
- observer sees me go through, since the event horizon is still lightlike;
- light emitted at the event horizon or within it will never escape to large
- distances, and light emitted just outside it will take a long time to get
- to an observer, timed, say, from when the observer saw me pass the point
- half a Schwarzschild radius outside the hole.)
-
- All this is not to imply that the black hole can't also be used for
- temporal tricks much like the "twin paradox" mentioned elsewhere in this
- FAQ. Suppose that I don't fall into the black hole-- instead, I stop and
- wait at a constant r value just outside the event horizon, burning
- tremendous amounts of rocket fuel and somehow withstanding the huge
- gravitational force that would result. If I then return home, I'll have
- aged less than you. In this case, general relativity can say something
- about the difference in proper time experienced by the two of us, because
- our ages can be compared *locally* at the start and end of the journey.
-
- 4. Will you see the universe end?
-
- If an external observer sees me slow down asymptotically as I fall,
- it might seem reasonable that I'd see the universe speed up
- asymptotically-- that I'd see the universe end in a spectacular flash as I
- went through the horizon. This isn't the case, though. What an external
- observer sees depends on what light does after I emit it. What I see,
- however, depends on what light does before it gets to me. And there's no
- way that light from future events far away can get to me. Faraway events
- in the arbitrarily distant future never end up on my "past light-cone," the
- surface made of light rays that get to me at a given time.
-
- That, at least, is the story for an uncharged, nonrotating black
- hole. For charged or rotating holes, the story is different. Such holes
- can contain, in the idealized solutions, "timelike wormholes" which serve
- as gateways to otherwise disconnected regions-- effectively, different
- universes. Instead of hitting the singularity, I can go through the
- wormhole. But at the entrance to the wormhole, which acts as a kind of
- inner event horizon, an infinite speed-up effect actually does occur. If I
- fall into the wormhole I see the entire history of the universe outside
- play itself out to the end. Even worse, as the picture speeds up the light
- gets blueshifted and more energetic, so that as I pass into the wormhole an
- "infinite blueshift" happens which fries me with hard radiation. There is
- apparently good reason to believe that the infinite blueshift would imperil
- the wormhole itself, replacing it with a singularity no less pernicious
- than the one I've managed to miss. In any case it would render wormhole
- travel an undertaking of questionable practicality.
-
- 5. What about Hawking radiation? Won't the black hole evaporate
- before you get there?
-
- (First, a caveat: Not a lot is really understood about evaporating
- black holes. The following is largely deduced from information in Wald's
- GR text, but what really happens-- especially when the black hole gets very
- small-- is unclear. So take the following with a grain of salt.)
-
- Short answer: No, it won't. This demands some elaboration.
-
- From thermodynamic arguments Stephen Hawking realized that a black
- hole should have a nonzero temperature, and ought therefore to emit
- blackbody radiation. He eventually figured out a quantum- mechanical
- mechanism for this. Suffice it to say that black holes should very, very
- slowly lose mass through radiation, a loss which accelerates as the hole
- gets smaller and eventually evaporates completely in a burst of radiation.
- This happens in a finite time according to an outside observer.
-
- But I just said that an outside observer would *never* observe an
- object actually entering the horizon! If I jump in, will you see the black
- hole evaporate out from under me, leaving me intact but marooned in the
- very distant future from gravitational time dilation?
-
- You won't, and the reason is that the discussion above only applies
- to a black hole that is not shrinking to nil from evaporation. Remember
- that the apparent slowing of my fall is due to the paths of outgoing light
- rays near the event horizon. If the black hole *does* evaporate, the delay
- in escaping light caused by proximity to the event horizon can only last as
- long as the event horizon does! Consider your external view of me as I
- fall in.
-
- If the black hole is eternal, events happening to me (by my watch)
- closer and closer to the time I fall through happen divergingly later
- according to you (supposing that your vision is somehow not limited by the
- discreteness of photons, or the redshift).
-
- If the black hole is mortal, you'll instead see those events happen
- closer and closer to the time the black hole evaporates. Extrapolating,
- you would calculate my time of passage through the event horizon as the
- exact moment the hole disappears! (Of course, even if you could see me,
- the image would be drowned out by all the radiation from the evaporating
- hole.) I won't experience that cataclysm myself, though; I'll be through
- the horizon, leaving only my light behind. As far as I'm concerned, my
- grisly fate is unaffected by the evaporation.
-
- All of this assumes you can see me at all, of course. In practice
- the time of the last photon would have long been past. Besides, there's
- the brilliant background of Hawking radiation to see through as the hole
- shrinks to nothing.
-
- (Due to considerations I won't go into here, some physicists think
- that the black hole won't disappear completely, that a remnant hole will be
- left behind. Current physics can't really decide the question, any more
- than it can decide what really happens at the singularity. If someone ever
- figures out quantum gravity, maybe that will provide an answer.)
-
- 6. How does the gravity get out of the black hole?
-
- Purely in terms of general relativity, there is no problem here. The
- gravity doesn't have to get out of the black hole. General relativity
- is a local theory, which means that the field at a certain point in
- spacetime is determined entirely by things going on at places that can
- communicate with it at speeds less than or equal to c. If a star
- collapses into a black hole, the gravitational field outside the
- black hole may be calculated entirely from the properties of the star
- and its external gravitational field *before* it becomes a black hole.
- Just as the light registering late stages in my fall takes longer and
- longer to get out to you at a large distance, the gravitational
- consequences of events late in the star's collapse take longer and
- longer to ripple out to the world at large. In this sense the black
- hole *is* a kind of "frozen star": the gravitational field is a fossil
- field. The same is true of the electromagnetic field that a black
- hole may possess.
-
- Often this question is phrased in terms of gravitons, the hypothetical
- quanta of spacetime distortion. If things like gravity correspond to the
- exchange of "particles" like gravitons, how can they get out of the
- event horizon to do their job?
-
- Gravitons don't exist in general relativity, because GR is not a
- quantum theory. They might be part of a theory of quantum gravity
- when it is completely developed, but even then it might not be best to
- describe gravitational attraction as produced by virtual gravitons.
- See the FAQ on virtual particles for a discussion of this.
-
- Nevertheless, the question in this form is still worth asking, because
- black holes *can* have static electric fields, and we know that these
- may be described in terms of virtual photons. So how do the virtual
- photons get out of the event horizon? Well, for one thing, they can
- come from the charged matter prior to collapse, just like classical
- effects. In addition, however, virtual particles aren't confined to
- the interiors of light cones: they can go faster than light!
- Consequently the event horizon, which is really just a surface that
- moves at the speed of light, presents no barrier.
-
- I couldn't use these virtual photons after falling into the hole to
- communicate with you outside the hole; nor could I escape from the
- hole by somehow turning myself into virtual particles. The reason is
- that virtual particles don't carry any *information* outside the light
- cone. See the FAQ on virtual particles for details.
-
- 7. Where did you get that information?
-
- The numbers concerning fatal radii, dimming, and the time of the
- last photon came from Misner, Thorne, and Wheeler's _Gravitation_ (San
- Francisco: W. H. Freeman & Co., 1973), pp. 860-862 and 872-873. Chapters 32
- and 33 (IMHO, the best part of the book) contain nice descriptions of some
- of the phenomena I've described.
-
- Information about evaporation and wormholes came from Robert Wald's
- _General Relativity_ (Chicago: University of Chicago Press, 1984). The
- famous conformal diagram of an evaporating hole on page 413 has resolved
- several arguments on sci.physics (though its veracity is in question).
-
- Steven Weinberg's _Gravitation and Cosmology_ (New York: John Wiley
- and Sons, 1972) provided me with the historical dates. It discusses some
- properties of the Schwarzschild solution in chapter 8 and describes
- gravitational collapse in chapter 11.
-
- ********************************************************************************
- Item 11.
-
- The Solar Neutrino Problem original by Bruce Scott
- -------------------------- updated 5-JUN-1994 by SIC
-
- The Short Story:
-
- Fusion reactions in the core of the Sun produce a huge flux of
- neutrinos. These neutrinos can be detected on Earth using large underground
- detectors, and the flux measured to see if it agrees with theoretical
- calculations based upon our understanding of the workings of the Sun and
- the details of the Standard Model (SM) of particle physics. The measured
- flux is roughly one-half of the flux expected from theory. The cause of the
- deficit is a mystery. Is our particle physics wrong? Is our model of the
- Solar interior wrong? Are the experiments in error? This is the "Solar
- Neutrino Problem."
-
- There are precious few experiments which seem to stand in
- disagreement with the SM, which can be studied in the hope of making
- breakthroughs in particle physics. The study of this problem may yield
- important new insights which may help us go beyond the Standard Model.
- There are many experiments in progress, so stay tuned.
-
- The Long Story:
-
- A middle-aged main-sequence star like the Sun is in a
- slowly-evolving equilibrium, in which pressure exerted by the hot gas
- balances the self-gravity of the gas mass. Slow evolution results from the
- star radiating energy away in the form of light, fusion reactions occurring
- in the core heating the gas and replacing the energy lost by radiation, and
- slow structural adjustment to compensate the changes in entropy and
- composition.
-
- We cannot directly observe the center, because the mean-free path
- of a photon against absorption or scattering is very short, so short that
- the radiation-diffusion time scale is of order 10 million years. But the
- main proton-proton reaction (PP1) in the Sun involves emission of a
- neutrino:
-
- p + p --> D + positron + neutrino(0.26 MeV),
-
- which is directly observable since the cross-section for interaction with
- ordinary matter is so small (the 0.26 MeV is the average energy carried
- away by the neutrino). Essentially all the neutrinos make it to the Earth.
- Of course, this property also makes it difficult to detect the neutrinos.
- The first experiments by Davis and collaborators, involving large tanks of
- chloride fluid placed underground, could only detect higher-energy
- neutrinos from small side-chains in the solar fusion:
-
-
- PP2: Be(7) + electron --> Li(7) + neutrino(0.80 MeV),
- PP3: B(8) --> Be(8) + positron + neutrino(7.2 MeV).
-
- Recently, however, the GALLEX experiment, using a gallium-solution detector
- system, has observed the PP1 neutrinos to provide the first unambiguous
- confirmation of proton-proton fusion in the Sun.
-
- There is a "neutrino problem", however, and that is the fact that
- every experiment has measured a shortfall of neutrinos. About one- to
- two-thirds of the neutrinos expected are observed, depending on
- experimental error. In the case of GALLEX, the data read 80 units where 120
- are expected, and the discrepancy is about two standard deviations. To
- explain the shortfall, one of two things must be the case: (1) either the
- temperature at the center is slightly less than we think it is, or (2)
- something happens to the neutrinos during their flight over the
- 150-million-km journey to Earth. A third possibility is that the Sun
- undergoes relaxation oscillations in central temperature on a time scale
- shorter than 10 Myr, but since no-one has a credible mechanism this
- alternative is not seriously entertained.
-
- (1) The fusion reaction rate is a very strong function of the temperature,
- because particles much faster than the thermal average account for most of
- it. Reducing the temperature of the standard solar model by 6 per cent
- would entirely explain GALLEX; indeed, Bahcall has recently published an
- article arguing that there may be no solar neutrino problem at all.
- However, the community of solar seismologists, who observe small
- oscillations in spectral line strengths due to pressure waves traversing
- through the Sun, argue that such a change is not permitted by their
- results.
-
- (2) A mechanism (called MSW, after its authors) has been proposed, by which
- the neutrinos self-interact to periodically change flavor between electron,
- muon, and tau neutrino types. Here, we would only expect to observe a
- fraction of the total, since only electron neutrinos are detected in the
- experiments. (The fraction is not exactly 1/3 due to the details of the
- theory.) Efforts continue to verify this theory in the laboratory. The MSW
- phenomenon, also called "neutrino oscillation", requires that the three
- neutrinos have finite and differing mass, which is also still unverified.
-
- To use explanation (1) with the Sun in thermal equilibrium
- generally requires stretching several independent observations to the
- limits of their errors, and in particular the earlier chloride results must
- be explained away as unreliable (there was significant scatter in the
- earliest ones, casting doubt in some minds on the reliability of the
- others). Further data over longer times will yield better statistics so
- that we will better know to what extent there is a problem. Explanation (2)
- depends of course on a proposal whose veracity has not been determined.
- Until the MSW phenomenon is observed or ruled out in the laboratory, the
- matter will remain open.
-
- In summary, fusion reactions in the Sun can only be observed
- through their neutrino emission. Fewer neutrinos are observed than
- expected, by two standard deviations in the best result to date. This can
- be explained either by a slightly cooler center than expected or by a
- particle-physics mechanism by which neutrinos oscillate between flavors.
- The problem is not as severe as the earliest experiments indicated, and
- further data with better statistics are needed to settle the matter.
-
- References:
-
- [0] The main-sequence Sun: D. D. Clayton, Principles of Stellar Evolution
- and Nucleosynthesis, McGraw-Hill, 1968. Still the best text.
- [0] Solar neutrino reviews: J. N. Bahcall and M. Pinsonneault, Reviews of
- Modern Physics, vol 64, p 885, 1992; S. Turck-Chieze and I. Lopes,
- Astrophysical Journal, vol 408, p 347, 1993. See also J. N. Bahcall,
- Neutrino Astrophysics (Cambridge, 1989).
- [1] Experiments by R. Davis et al: See October 1990 Physics Today, p 17.
- [2] The GALLEX team: two articles in Physics Letters B, vol 285, p 376
- and p 390. See August 1992 Physics Today, p 17. Note that 80 "units"
- correspond to the production of 9 atoms of Ge(71) in 30 tons of
- solution containing 12 tons Ga(71), after three weeks of run time!
- [3] Bahcall arguing for new physics: J. N. Bahcall and H. A. Bethe,
- Physical Review D, vol 47, p 1298, 1993; against new physics: J. N.
- Bahcall et al, "Has a Standard Model Solution to the Solar Neutrino
- Problem Been Found?", preprint IASSNS-94/13 received at the National
- Radio Astronomy Observatory, 1994.
- [4] The MSW mechanism, after Mikheyev, Smirnov, and Wolfenstein: See the
- second GALLEX paper.
- [5] Solar seismology and standard solar models: J. Christensen-Dalsgaard
- and W. Dappen, Astronomy and Astrophysics Reviews, vol 4, p 267, 1992;
- K. G. Librecht and M. F. Woodard, Science, vol 253, p 152, 1992. See
- also the second GALLEX paper.
-
- ********************************************************************************
- Item 12.
-
- The Expanding Universe original by Michael Weiss
- ---------------------- updated 5-DEC-1994 by SIC
-
- Here are the answers to some commonly asked questions about exactly
- what it means to say that the Universe is expanding.
-
- (1) IF THE UNIVERSE IS EXPANDING, DOES THAT MEAN ATOMS ARE GETTING BIGGER?
- IS THE SOLAR SYSTEM EXPANDING?
-
- Mrs. Felix: Why don't you do your homework?
- Allen Felix: The Universe is expanding. Everything will fall
- apart, and we'll all die. What's the point?
- Mrs. Felix: We live in Brooklyn. Brooklyn is not expanding!
- Go do your homework.
-
- -from "Annie Hall" by Woody Allen.
-
- Mrs. Felix is right. Neither Brooklyn, nor its atoms, nor the solar
- system, nor even the galaxy, is expanding. The Universe expands
- (according to standard cosmological models) only when averaged over a very
- large scale.
-
- The phrase "expansion of the Universe" refers both to experimental
- observation and to theoretical cosmological models. Lets look at them one
- at a time, starting with the observations.
-
- Observation
- -----------
-
- The observation is Hubble's redshift law.
-
- In 1929, Hubble reported that the light from distant galaxies is
- redshifted. If you interpret this redshift as a Doppler shift, then the
- galaxies are receding according to the law:
-
- (velocity of recession) = H * (distance from Earth)
-
- H is called Hubble's constant; Hubble's original value for H was 550
- kilometers per second per megaparsec (km/s/Mpc). Current estimates range
- >from 40 to 100 km/s/Mpc. (Measuring redshift is easy; estimating distance
- is hard. Roughly speaking, astronomers fall into two "camps", some
- favoring an H around 80 km/s/Mpc, others an H around 40-55).
-
- Hubble's redshift formula does *not* imply that the Earth is in
- particularly bad oder in the universe. The familiar model of the universe
- as an expanding balloon speckled with galaxies shows that Hubble's alter
- ego on any other galaxy would make the same observation.
-
- But astronomical objects in our neck of the woods--- our solar
- system, our galaxy, nearby galaxies--- show no such Hubble redshifts.
- Nearby stars and galaxies *do* show motion with respect to the Earth
- (known as "peculiar velocities"), but this does not look like the
- "Hubble flow" that is seen for distant galaxies. For example, the
- Andromeda galaxy shows blueshift instead of redshift. So the verdict
- of observation is: our galaxy is not expanding.
-
- By the way, Hubble's constant, is not, in spite of its name,
- constant in time. In fact, it is decreasing. Imagine a galaxy D
- light-years from the Earth, receding at a velocity V = H*D. D is
- always increasing because of the recession. But does V increase? No.
- In fact, V is decreasing. (If you are fond of Newtonian analogies,
- you could say that "gravitational attraction" is causing this
- deceleration. But be warned: some general relativists would object
- strenuously to this way of speaking.) So H is going down over time.
- But it *is* constant over space, i.e., it is the same number for all
- distant objects as we observe them today.
-
- Theory
- ------
-
- The theoretical models are, typically, Friedmann-Robertson-Walker (FRW)
- spacetimes.
-
- Cosmologists model the universe using "spacetimes", that is to say,
- solutions to the field equations of Einstein's theory of general
- relativity. The Russian mathematician Alexander Friedmann discovered an
- important class of global solutions in 1923. The familiar image of the
- universe as an expanding balloon speckled with galaxies is a "movie
- version" of one of Friedmann's solutions. Robertson and Walker later
- extended Friedmann's work, so you'll find references to
- "Friedmann-Robertson-Walker" (FRW) spacetimes in the literature.
-
- FRW spacetimes come in a great variety of styles--- expanding,
- contracting, flat, curved, open, closed, .... The "expanding balloon"
- picture corresponds to just a few of these.
-
- A concept called the metric plays a starring role in general
- relativity. The metric encodes a lot of information; the part we care about
- (for this FAQ entry) is distances between objects. In an FRW expanding
- universe, the distance between any two "points on the balloon" does
- increase over time. However, the FRW model is NOT meant to describe OUR
- spacetime accurately on a small scale--- where "small" is interpreted
- pretty liberally!
-
- You can picture this in a couple of ways. You may want to think of
- the "continuum approximation" in fluid dynamics--- by averaging the motion
- of individual molecules over a large enough scale, you obtain a continuous
- flow. (Droplets can condense even as a gas expands.) Similarly, it is
- generally believed that if we average the actual metric of the universe
- over a large enough scale, we'll get an FRW spacetime.
-
- Or you may want to alter your picture of the "expanding balloon".
- The galaxies are not just painted on, but form part of the substance of the
- balloon (poetically speaking), and locally affect its "elasticity".
-
- The FRW spacetimes ignore these small-scale variations. Think of a
- uniformly elastic balloon, with the galaxies modelled as mere points.
- "Points on the balloon" correspond to a mathematical concept known as a
- *comoving geodesic*. Any two comoving geodesics drift apart over time, in
- an expanding FRW spacetime.
-
- At the scale of the Solar System, we get a pretty good
- approximation to the spacetime metric by using another solution to
- Einstein's equations, known as the Schwarzschild metric. Using evocative
- but dubious terminology, we can say this models the gravitational field of
- the Sun. (Dubious because what does "gravitational field" mean in GR, if
- it's not just a synonym for "metric"?) The geodesics in the Schwarzschild
- metric do NOT display the "drifting apart" behavior typical of the FRW
- comoving geodesics--- or in more familiar terms, the Earth is not drifting
- away from the Sun.
-
- The "true metric" of the universe is, of course, fantastically
- complicated; you can't expect idealized simple solutions (like the FRW and
- Schwarzschild metrics) to capture all the complexity. Our knowledge of the
- large-scale structure of the universe is fragmentary and imprecise.
-
- In old-fashioned, Newtonian terms, one says that the Solar System
- is "gravitationally bound" (ditto the galaxy, the local group). So the
- Solar System is not expanding. The case for Brooklyn is even clearer: it
- is bound by atomic forces, and its atoms do not typically follow geodesics.
- So Brooklyn is not expanding. Now go do your homework.
-
- References: (My thanks to Jarle Brinchmann, who helped with this list.)
-
- Misner, Thorne, and Wheeler, "Gravitation", chapters 27 and 29. Page 719
- discusses this very question; Box 29.4 outlines the "cosmic distance
- ladder" and the difficulty of measuring cosmic distances; Box 29.5 presents
- Hubble's work. MTW refer to Noerdlinger and Petrosian, Ap.J., vol. 168
- (1971), pp. 1--9, for an exact mathematical treatment of gravitationally
- bound systems in an expanding universe.
-
- M.V.Berry, "Principles of Cosmology and Gravitation". Chapter 2 discusses
- the cosmic distance ladder; chapters 6 and 7 explain FRW spacetimes.
-
- Steven Weinberg, "The First Three Minutes", chapter 2. A non-technical
- treatment.
-
- Hubble's original paper: "A Relation Between Distance And Radial
- Velocity Among Extra-Galactic Nebulae", Proc. Natl. Acad. Sci., Vol. 15,
- No. 3, pp. 168-173, March 1929.
-
- Sidney van den Bergh, "The cosmic distance scale", Astronomy & Astrophysics
- Review 1989 (1) 111-139.
-
- M. Rowan-Robinson, "The Cosmological Distance Ladder", Freeman.
-
- A new method has been devised recently to estimate Hubble's constant, using
- gravitational lensing. The method is described in:
-
- \O Gr\on and Sjur Refsdal, "Gravitational Lenses and the age of the
- universe", Eur. J. Phys. 13, 1992 178-183.
-
- S. Refsdal & J. Surdej, Rep. Prog. Phys. 56, 1994 (117-185)
-
- and H is estimated with this method in:
-
- H.Dahle, S.J. Maddox, P.B. Lilje, to appear in ApJ Letters.
-
- Two books may be consulted for what is known (or believed) about the
- large-scale structure of the universe:
-
- P.J.E.Peebles, "An Introduction to Physical Cosmology".
- T. Padmanabhan, "Structure Formation in the Universe".
-
- ======================================================================
-
- (2) WHAT CAUSES THE HUBBLE REDSHIFT? ARE THE LIGHT-WAVES "STRETCHED" AS
- THE UNIVERSE EXPANDS, OR IS THE LIGHT DOPPLER-SHIFTED BECAUSE DISTANT
- GALAXIES ARE MOVING AWAY FROM US?
-
- In a word: yes. In two sentences: the Doppler-shift explanation is
- a linear approximation to the "stretched-light" explanation. Switching
- >from one viewpoint to the other amounts to a change of coordinate systems
- in (curved) spacetime.
-
- A detailed explanation requires looking at Friedmann-Robertson-Walker
- (FRW) models of spacetime. The famous "expanding balloon speckled with
- galaxies" provides a visual analogy for one of these; like any analogy, it
- will mislead you if taken too literally, but handled with caution it can
- furnish some insight.
-
- Draw a latitude/longitude grid on the balloon. These define
- *co-moving* coordinates. Imagine a couple of speckles ("galaxies")
- imbedded in the rubber surface. The co-moving coordinates of the speckles
- don't change as the balloon expands, but the distance between the speckles
- steadily increases. In co-moving coordinates, we say that the speckles
- don't move, but "space itself" stretches between them.
-
- A bug starts crawling from one speckle to the other. A second
- after the first bug leaves, his brother follows him. (Think of the bugs as
- two light-pulses, or successive wave-crests in a beam of light.) Clearly
- the separation between the bugs will increase during their journey. In
- co-moving coordinates, light is "stretched" during its journey.
-
- Now we switch to a different coordinate system, this one valid only
- in a neighborhood (but one large enough to cover both speckles). Imagine a
- clear, flexible, non-stretching patch, attached to the balloon at one
- speckle. The patch clings to the surface of the balloon, which slides
- beneath it as the balloon inflates. (The bugs crawl along *under* the
- patch.) We draw a coordinate grid on the patch. In the patch coordinates,
- the second speckle recedes from the first speckle. And so in patch
- coordinates, we can regard the redshift as a Doppler shift.
-
- Is this visually appealing? I think so. However, this explanation
- glosses over one crucial point: the time coordinate. FRW spacetimes come
- fully-equipped with a specially distinguished time coordinate (called the
- co-moving or cosmological time). For example, a co-moving observer could
- set her clock by the average density of surrounding speckles, or by the
- temperature of the Cosmic Background Radiation. (From a purely
- mathematical standpoint, the co-moving time coordinate is singled out by a
- certain symmetry property.)
-
- We have many choices of time-coordinate to go with the
- space-coordinates drawn on our patch. Let's use cosmological time. Notice
- that this is *not* the choice usually made in Special Relativity: though
- the two speckles separate rapidly, their cosmological clocks remain
- synchronized. Bugs embarking on their journey from the "moving" speckle
- appear to crawl "upstream" against flowing space as they head towards the
- "home" speckle. The current diminishes as they approach home. (In other
- words, bug-speed is anisotropic in these coordinates.) These differences
- >from the usual SR picture are symptoms of a deeper fact: besides the
- obvious "spatial" curvature of the balloon's surface, FRW spacetimes have
- "temporal" curvature as well. Indeed, not all FRW spacetimes exhibit
- spatial curvature, but (with one exception) all have temporal curvature.
-
- You can work out the magnitude of the redshift using patch
- coordinates. I leave this as an exercise, with a couple of hints. (1)
- Since bug-speed is anisotropic far from the home speckle, consider also a
- patch attached to the "moving" speckle. Compute the initial distance
- between the bugs (the "wavelength") in both patch coordinate systems, using
- the standard *non-relativistic* Doppler formula for a stationary source,
- moving receiver. (2) Now think about how the bug-distance changes as the
- bugs journey to the home speckle (this time sticking with home patch
- coordinates). The bug-distance does *not* propagate unchanged. Consider
- instead the analog of the period of a lightwave: the time between
- bug-crossings of a grid line on the patch. This *does* propagate almost
- unchanged, *provided* the rate of balloon expansion stays pretty much the
- same throughout the bugs' perilous trek. The final result: the magnitude
- of the redshift, computed using Doppler's formula, agrees to first-order
- with magnitude computed using the "stretched-light" explanation. (To the
- cognoscenti: the assumptions are that Hx<<1 and (dH/dt)x<<1, where
- H(t)=dR(t)/dt, R(t) is the scale factor, t is cosmological time, and x is
- the average distance between the "speckles" (co-moving geodesics) during
- the course of the journey.)
-
- (This long-winded "proof of equivalence" between the Doppler and
- "stretched-light" explanations substitutes a paragraph of imagery for a
- half-page of calculus.)
-
- Let me close by emphasizing the word "approximation" from the first
- paragraph of this entry. The Doppler explanation fails for very large
- redshifts, for then we must consider how Hubble's "constant" changes over
- the course of the journey.
-
- References:
-
- Misner, Thorne, and Wheeler, "Gravitation", chapter 29.
-
- M.V.Berry, "Principles of Cosmology and Gravitation", chapter 6.
-
- Steven Weinberg, "The First Three Minutes", chapter 2, especially pp. 13
- and 30.
-
- ********************************************************************************
- END OF PART 2/4
-