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What is Dark Energy? *February 12, 2015*

*Posted by stuffilikenet in Awesome, Brain, Brilliant words, Geek Stuff, Science, Uncategorizable.*

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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 MANâ€™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

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