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FREQUENTLY ASKED QUESTIONS ON SCI.PHYSICS - Part 2/4
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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.
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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".
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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
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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.
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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".
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(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.
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