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What is Dark Energy? February 12, 2015

On The List That Cannot Be Named, Joshua is perhaps the best exemplar of all our extended cocktail conversations held over the last twenty years by e-mail.  He recently had occasion to explain to a bright young man the current state of understanding of the nature of dark energy.  I asked him if I could reprint it here and (after redaction) I do so now:

Looking randomly through old email for something else, I stumbled on this attempt at "general relativity and beyond, for the bright 10th grader".  I’m rather pleased with it, and thought some folks on [The List Which Cannot Be Named] might enjoy it, as an example of "teaching science the student can’t actually do yet, without telling any lies."  The usual practice of science popularization, of course, is "tell entertaining lies that remind people who actually know of the science, while convincing the layman that he almost understands it."  I think the first way is better, and find it frustrating that no one else seems to agree.

Joshua

{BEGIN EARNEST YOUNG MAN’S HEARTFELT APPEAL]
> I’m a student in Mrs. [REDACTED]’s class and a member of QuarkNet, and
> I’m having some trouble researching for my physics expo project. I  also     > think we met for Shabbat dinner at the [REDACTED]’s a few weeks ago.
> My project is on Dark Matter and Dark Energy, and while I had no
> trouble finding sources that discussed why we know they exist, when it
> came to actually defining what Dark Energy is, most sources just said
> "we don’t know what it is but here are some wrong ideas ….". I was
> wondering if you could tell me what physicists today actually think
> Dark Energy is and why it’s causing the universe to accelerate.

Hi, [REDACTED].  Yes, certainly — I remember you from Linda [REDACTED]’s lovely Thanksgiving shabbat.
The basic idea of dark energy is a bit mathematical, and is therefore
going to require some handwaving, unless you want to skip ahead and
actually learn differential geometry.
(Aside:  calculus is all about limits and derivatives and integrals,
and Newtonian mechanics and calculus grew up together and were made
for each other.  Vector calculus extends those ideas to functions
that have direction as well as magnitude, so it’s all about gradients
and divergence and curl — Maxwell’s electromagnetism and vector
calculus were made for each other in the late 19c.  The next step
beyond that is doing geometry and calculus on "manifolds", which are
n-dimensional spaces that are curved, so that when you carry a vector
around a loop it doesn’t always come back pointing the same way.  If
you have ever seen a Foucault pendulum in a museum, this is the point
they are trying to get across; the old riddle about the man who walks
a mile south, shoots a bear, walks a mile east, then a mile north
and is back where he started — what color is the bear? white, of
course, because he must be at the north pole — is another simple
example.  Differential geometry, or "calculus on manifolds" as some
people call it, is the math that drives Einstein gravity, because to
study curved spacetime you need this kind of machinery.  But we can
talk about it, without actually teaching you to do problems.  If you
want to get a headstart on the actual math, I can point you at some
good texts, but really you should focus on getting one-dimensional
calculus well mastered before investing too much effort beyond it.)
So, the Einstein equations look like T_mn = 8 pi G_mn, where T_mn is
the local density of "stress-energy", or mass, energy, pressure,
energy flow, and momentum flow, at a given point.  That is, roughly,
T_00 is a function of position and time that gives the density of
mass and energy at that point at that instant, and the other 15
components, m=0..3 and n=0..3, are things like "how much x-momentum
is flowing through that point in the z-direction" and so on.  This is
sort of complicated because it has so many components, and you’re
not used to thinking about exotic things like momentum flow as a
source of gravity just like mass.  Partly this is because we are
used to speeds very much slower than c, so that in "natural" c=1
units (for example, time in nanoseconds and distance in feet, or
time in seconds and distance in light-seconds — that’s seven times
around the earth, or most of the way to the moon!) all the other T_mn’s
are basically zero for any matter less drastic than the interior of
an atomic nucleus or a neutron star.  Anyway, you can think of T_mn
as a complete description of the "stuff" at a given point at a given
moment in time.
G_mn is a complicated thing that describes the curvature of spacetime
at a given point; it turns out that you can completely describe the
curvature of a manifold by taking a vector, carrying it around a
circle, and seeing how it changes when it comes back.  I can
calculate G_mn for any set of coordinates you give me, but not every
different set of coordinates gives a different G_mn; for example, the
x-y plane is flat, and it’s still the same plane and just as flat
even if I describe it in polar coordinates as r, theta instead of
x, y.  This is a subtle point, because I can take a radial vector,
say pointing in the +r direction at r=1,theta=0 (that’s on the x-axis
in x-y coordinates), and carry it to r=1,theta=90 (on the y-axis)
where it is now pointing in the -theta direction!  So if I believed
the coordinates, I’d be tempted to say "this radial vector turned into
a circumferential vector when I moved it; the plane must be curved!"
But notice that if I bring the vector back where I found it, no matter
what path I walk to get there, the vector will once again be radial.
My polar bear hunter in the riddle was sort of a cheat, because he was
relying on the coordinates for his directions; when he walked "east"
a mile, he was going in a circle around the pole and he knew it.  But
let him do a 6,000 mile triangle instead of a 1 mile triangle, and
things get more interesting:  he walks down to the equator, along a
"straight" meridian — we can see that it’s curved in three-dimensional
space, but considering the two-dimensional surface of the earth as its
own thing, a meridian is straight, bending neither to right nor to
left — then along the "straight" equator, and back along a "straight"
meridian.  The existence of a triangle with three right angles proves
that the earth’s surface is curved; the "local" proof that doesn’t
rely on standing out in space and watching from afar is that if our
hunter carried a gyroscope or Foucault pendulum with him, he’d be able
to see that his coordinates turned by 90 degrees as he went around the
big triangle.
There is a thing called the Riemann tensor that completely describes
the curvature of any manifold; it’s written R_abcd, and it means,
roughly, if I carry a vector pointing in the "a" direction around a
small loop in the "c-d" plane, how much will it now point in the "b"
direction instead.  Since the two-dimensional surface of the earth
has only one "c-d" plane (longitude and latitude) to walk around on,
the curvature of a 2D manifold can be given by a single number.  For
3D manifolds, there are 6 numbers, for 4D, there are 20 — geometry
is complicated!  Of the 20 components of the Riemann tensor that gives
the curvature of spacetime, there are six that describe how much
space is "spreading" or "shrinking" over time, and 14 that describe
how much it is "shearing" in various directions without changing volume.
The six volume-changing components are called the Ricci tensor, given
by R_ab = (1/4)(R_a0b0 + R_a1b1 + R_a2b2 + R_a3b3).  And there is a
sort of "average" of the Ricci tensor called the Ricci scalar, which
is R = (1/4)(R_00 + R_11 + R_22 + R_33), which is a single number
that describes the curvature of spacetime as well as a single number
possibly could.  The Einstein tensor G_mn mentioned above is actually
G_mn = R_mn – (1/2)R.  I told you it was complicated, but don’t get
too intimidated by the preceding two paragraphs; really this is all
just bookkeeping.  The physics is coming now, so pay attention.
Einstein, after ten years of hard work 1905 to 1915, came up with a way
to generalize his theory of flat spacetime and constant motion into
a theory of curved spacetime, gravity and accelerated motion.  (Hence,
the name "general" relativity.  Special relativity is high-school
simple from a mathematical standpoint, once you’ve had the incredible
conceptual leap Einstein achieved in spring 1905; general relativity
needs the machinery of Riemann tensors, which wasn’t well understood
by physicists at that point in time, and requires some upper-division
college math even today.)  What he was trying to say was, "spacetime
is curved by stuff, wherever there is stuff; elsewhere, it may have
ripples caused by stuff in other places."  The final mathematical
formulation of that is T_mn = 8 pi G_mn.  The "stress-energy tensor"
T_mn is "stuff", and G_mn is "the part of curvature that actually
changes the volume of things."  For a practical example, if I put a
million fireflies in a sphere all around the earth, a mile up, and
suddenly stopped their wings and let them fall, the sphere would get
smaller as they fell, and that’s because there is mass (the earth)
inside the sphere.  If I put a million fireflies in a perfect sphere
formation next to the earth, say fifty thousand miles over the north
pole (and not in a moving orbit, just hovering there) and let them
fall, the ones closer to the earth would fall faster so the sphere
would elongate like a football, and the ones out to each side would
converge as they fell so the waist of the football would get narrower.
The volume of the sphere of fireflies wouldn’t change, because there is
no stuff (that is, no gravitating earth) inside the sphere.  The
converging sphere is firefly worldlines coming together, described by
the Ricci tensor part of spacetime’s Riemann curvature.  The constant
volume elongating sphere is described by the other 14 components of
the Riemann tensor (which also have a name, the Weyl tensor).  What
the Einstein equation says is "there is Ricci curvature wherever there
is stuff; to keep the universe from tearing, this means there will be
Weyl ripples, but no Ricci curvature, in empty space where there is
no stuff."
After that long setup, what is dark energy?  Well, a few years after
Einstein published general relativity, some smart mathematicians
pointed out that there are no static solutions to the Einstein equation;
that is, a universe made of stars and galaxies that have always been
there and never expand or collapse is not possible.  At that time,
there was a mostly unexamined prejudice in favor of an eternal static
universe, and Einstein was concerned that this was evidence against the
whole idea of general relativity.  (The observation of gravitational
bending of starlight in 1919, however, convinced many people that he
must be substantially on the right track.)  So, in a minor paper,
Einstein asked the question, "what is the minimal change to my
equations that will preserve all the good features, yet allow a static
universe?"  Later, he remarked to a biographer that lacking the courage
to stand by his equations and boldly predict the Big Bang here was
"my greatest blunder," but what an interesting blunder it was!
It turns out that about the only thing you can do to the Einstein
equation that doesn’t break everything is to add a term that looks
like this:  T_mn = 8 pi G_mn + lambda g_mn.  The g_mn here is the
simplest and humblest tensor in differential geometry:  it’s what we
use to make dot products, and in flat coordinates it’s just the unit
matrix.  We call it the "metric tensor" or just the metric.  It
obviously wouldn’t do to say, "spacetime curves where there is stuff,
and empty space curves slightly toward the Willis Tower", nor even
"…and empty space curves uniformly inward in all directions, but only
as seen by observers who are stationary relative to Ken’s Diner."  If
you’re going to break the simplicity of "spacetime curves where there
is stuff, you had better at least do it in a way that looks the same
to all observers in empty space, no matter where or when they are and
how they are moving.  This is what that lambda term with the metric
tensor does.  The revised Einstein equation reads "spacetime curves
where there is stuff, and empty space curves slightly, by an amount
given by lambda, in a symmetric way for all observers."  It’s ugly,
but it’s not as ugly as other alternatives.  The amount lambda is
called "the cosmological constant", and of course if lambda=0 we are
right back to honest general relativity.
Now it turns out that the effect of a cosmological constant is to give
the universe "an itch it can’t scratch" — because it’s curvature that
is caused by spacetime itself, rather than by "stuff" in spacetime,
any physical effect lambda may cause will continue eternally as long
as there is spacetime.  This should be enough to give you a clue what
a cosmological constant will do:  it causes an empty universe to expand
or to contract, depending on the sign of lambda, at an exponential pace,
because the growth in spacetime is proportional to how much spacetime
there is.  So, you could just about imagine a universe with matter and
energy that acts gravitationally to pull it into a big crunch, balanced
by a cosmological constant that expands spacetime just enough so the
crunch never happens.  It’s a delicate balance, and an unsatisfactory
solution to the original, obsolete problem of how to make the universe
endure forever at a static size.
Now that we know the universe is expanding, however, we can still
distinguish between matter and energy (which slow the expansion) and
a cosmological constant (which speeds it up).  Looking back at old
supernovae, we can see how fast the universe was expanding at various
times in its early history, and if we graph this carefully we can find
out whether the universe is slowing or accelerating in its expansion.
This was successfully done in the late 1990s, and the surprise result
is that the universe has been accelerating for at least the last eight
billion years of its 13.7 billion year history.  So the cosmological
constant wasn’t such a useless idea after all!
Now, notice that there are two ways to think about the cosmological
constant.  The one I presented went with the equation
T_mn = 8 pi G_mn + lambda g_mn,
which I read informally as "stuff (on the left) equals curvature plus
a constant, so the constant is the curvature of empty spacetime."
But consider moving the constant to the other side:
T_mn – lambda g_mn = G_mn
and now it reads "stuff minus a constant equals curvature, so the
constant is a funny kind of stuff that empty spacetime is full of."
It’s really semantics whether lambda is a kind of stuff that lives in
empty space, or a kind of curvature that happens in the absence of
stuff.  As a particle physicist, I’d rather think of it as a kind of
stuff, because then I can try to come up with a particle theory that
includes it and explains why spacetime is full of it!  Also, once I
think of it as "stuff that is proportional to the metric and seems
to be everywhere," I can entertain the possibility that it really
isn’t all-pervasive and eternal and everywhere; maybe the universe
has lambda a function of temperature, or of the age of the universe,
or something, instead of a constant.  It’s a cosmological *constant*
only in the sense that it’s proportional to the metric, so it doesn’t
vary asymmetrically from place to place or from a stationary
observer to a moving one.  It still might be a real substance that
obeys those laws and that only exists when the conditions are right.
Dark energy, to finally answer your question, is the stuff we infer
from the fact that the universe is accelerating, which behaves enough
like a cosmological constant in the Einstein equations to give that
behavior, but which might or might not be a true constant property
of the vacuum.  By contrast, dark matter is ordinary stuff that is
located some places and not others, that behaves just like atoms or
photons or neutrinos or any other kind of matter, except that it isn’t
made of stars because it doesn’t glow, and it isn’t even made of
atoms because we’d see it glow in star nurseries (and because we know
how various isotopes formed just after the Big Bang, and the numbers
come out wrong if most of the atoms are unaccounted for).
You asked what my guess was about what dark matter and dark energy
actually are.  For dark matter, I have lots of good candidates;
I think the obvious answer is that if supersymmetry is right, there
are some heavy exotic particles predicted that would have been
formed in the early universe, that interact very weakly with atoms,
and that have about the right properties to be the dark matter.
For dark energy, I truly have no guess — it could be a fundamental
property of spacetime (in which case the geometric picture with the
cosmological constant on the right side of the equation makes more
sense than the particle physics picture with lambda on the left),
or it could be something more dynamic.  To understand it, I would
sort of have to understand "what empty space is," which effectively
means I need to understand everything about all the particles in
not-empty space, whereas to understand dark matter I just have to
understand one kind of particle among who knows how many.
Does that help?
Joshua