Georgi, Howard. Physics 16 - Mechanics and Special Relativity (англ.)

from an even more distant galaxy (or quasar) so that we on earth would see the light coming at

us from several directions.

5

This has actually been seen, ﬁrst in radio-telescope observations, and

more recently and spectacularly in Hubble Space Telescope images and other visual data. Below

is one such image.

In this Hubble photo, around the bright oval galaxies of a distant cluster, we see the wispy arcs

of a still more distant object, focused and distorted by the gravitational lens produced by the dark

matter of the cluster. One can use this phenomenon to study the distribution of the dark matter.

Below is the result of such a study of another cluster of galaxies - a plot of the matter density in

a cluster, showing high peaks at the visible galaxies, but an enormous background of dark matter

between them.

6

5

or even, if the geometry is just right, from an “Einstein ring.”

6

from a Bell-Labs new release in 1997.

11

If the dark matter really is some kind of matter, there are indirect but fairly convincing argu-

ments that it cannot be made out of normal stuff. The evidence comes from the fact that we have a

picture of how light nuclei are formed in the very early universe that works reasonably well, and a

lot more normal matter would mess up this nice picture (of what is called “primordial nucleosyn-

thesis” — for example, if there were enough baryons to account for all the dark matter, the theory

predicts that would be much less deuterium in the universe than we actually see!). So most people

assume that it is some kind of new particle! There are many ongoing experimental efforts to ﬁnd

it.

I am not going to talk about all this in detail. Here are a couple of links to reviews on the web:

www.astro.queensu.ca/∼dursi/dm-tutorial/dm0.html — An excellent site by a student at Queens

Univ in Canada.

astron.berkeley.edu/∼mwhite/darkmatter/dm.html — A brief but up-to-date introduction with

additional links.

12

lecture 25 (outline)

Topics:

Where are we now?

The shape of physics

Elementary particles and their masses

Symmetry and GUTs

States of the vacuum

Where are we now?

We have made it through all the difﬁcult material that I wanted to cover in the course. What I want

to do in the last few lectures is to have some fun with the early universe, and also to try to pull

things together a bit, and of course to answer the question we started with — Why is F = ma?

The shape of physics

I thought I should spend a little time talking about what physicists actually do today. The answer

is — a lot of very different things. Here is a very partial list. The ﬁrst three are ﬁelds that are

actively done in the Harvard Physics Department (which means we think that they are among the

most exciting things). But I have also included a sampling of subﬁelds that are done elsewhere

(including at other departments at Harvard).

sub-ﬁeld size(m) #

condensed

matter

10

−8

− 10

−1

≈ ∞

atomic 10

−10

1− ≈ 100

particle < 10

−13

≈ 1

astro 10

5

< ≈ ∞

geo 10

2

< ≈ ∞

.

plasma 10

−2

< ≈ ∞

nuclear 10

−12

≈ 1 − ∞

1

I’ve included here a couple of the parameters that one might use to characterize what the people in

these subﬁelds study. Other parameters (energy, temperature, pressure, etc) could be added. But

this will be enough to illustrate the points I wanted to make. There are two. One is the picture of

physics that I have discussed before and a kind of map of parameter space.

Discuss the nature of physical discovery.

The other point I wanted to make involves more philosophical questions. What is is Physics?

and How is it different from other sciences.

All science is either physics or stamp collecting.

—Ernest Rutherford.

No longer true, or at least if it is, stamp collecting has become much more interesting — but

there is an important idea here.

fact 6= mathematical model

a different level of metaphor

Elementary particles and their masses

e

−

µ

−

τ

−

d s b

u c t

W

±

Z

νs?

......................................................................................................

.

......................................................................................................

.

10

−15

10

−10

10

−5

1

10

−15

10

−10

10

−5

1

)

leptons

(

quarks

(

charged

neutrinos

force particles (photon, gluon and graviton are massless)

Mass (GeV) −→

←− λ (cm)

The diagram in the previous section summarizes the particles that we see at short distances that

have no internal structure at the distances we can probe. This seems to be a complicated mess.

There is, in fact, a tremendous amount of physics in this picture. I love it because much of it

was pieced together in the 70s, which I was a postdoc and young faculty member and had the fun

of contributing to our understanding of all this. It is quite frustrating that we are still completely

confused about where the speciﬁc values of these particle masses come from. It seems pretty

random. But let me give you a quick introduction to some of this.

2

at “low”

energies

below

1000 GeV

charge particle

electric photon

color×2 gluons×8

2 gluons for charges - 2D space of color triangle

6 gluons for transitions between colors - SU (3)

••

•

...............................................

.

...............................................

.

..

.

..

.

..

.

..

.

e

+

d

−1/3

R

d

−1/3

G

d

−1/3

B

.

..

.

..

.

..

.

Electric and color charges add to zero

3

at “high”

energies

above 1000 GeV

charge particle

electric photon

0

electroweak W

+

, W

−

, Z

0

color×2 gluons×8

••

...............................................

.

...............................................

.

W

+

W

−

W

±

for transitions between ﬂavors - SU(2)

(ν

e

, e

−

), (u, d), · · · - symmetry in doublets at short distances when masses from peculiar vac-

uum state are unimportant - weak force at longer distances

W

±

and Z were understood long before they were seen because of “weak interactions” such

as neutron decay

n[udd] → p[uud] + e

−

or neutrino production in the sun

p + p → D + e

+

+ ν

and neutral current neutrino scattering

ν + p → ν + p

at very

high

energies

???????

charge particle

electric photon

0

electroweak W

+

, W

−

, Z

0

color×2 gluons×8

Xs × 12

X

s for transitions between antiquarks and leptons or between quarks and antiquarks -

SU

(5)

4

e

+

¯ν

d

−1/3

R

d

−1/3

G

d

−1/3

B

.

...............................

.

..

.

..

.

..

.

Antineutrino at the center, but displaced in the 4th dimension.

4-simplex - 4-d analog of a tetrahedron - SU(5)

at very

high

energies

???????

charge particle

electric photon

0

electroweak W

+

, W

−

, Z

0

color×2 gluons×8

Xs × 12

Xs for transitions between antiquarks and leptons or between quarks and antiquarks - SU (5)

couplings may unify at ≈ 10

15

GeV

if this is right, Xs should produce proton decay

i.e. p → e

+

π

0

not seen yet

However, we know one thing very well from our detailed of the properties of these particles

and forces. The very existence of all these particle masses depends on the peculiar state of the

vacuum.

I hope that this sounds like a very odd statement. The vacuum is the state with nothing in it.

How can “nothing” affect things. I think that one of the most remarkable facts about our current

understanding is that it is based on the idea that the vacuum is not trivial at all.

My colleague Sidney Coleman has a beatuiful phrase to suggest that it is not surprising that the

vacuum can have an important inﬂuence. “We live in the vacuum as a ﬁsh swims in the sea.”

5

The point is that the “nothing” in the vacuum refers only to what we can control. We have

learned in recent years that even when we take out everything that moves, there is still a lot of

structure left.

Think of the ﬁsh. For the most part, ﬁsh are interested in the other things that are swimming

around in the water trying to eat them. They probably take for granted the properties of the water

in which they swim. But you can imagine ﬁsh scientists that do experiments on the water itself.

There are properties that do not involve any change in the structure of the water - density, viscosity,

wetness, and all that. This is sort of analogous to determining the speed of light in the vacuum. But

some such experiments can be quite interesting, particularly if the bear on fundamental questions

like symmetry. Our ﬁsh scientist can also do experiments that involve the “phase” of the water.

By feeding energy into a small region, they may get the water to boil and and study the gas phase.

Or by cooling a region, they may get a region of ice. This kind of experimention is not without

dangers. Kurt Vonnegut’s “Cat’s Cradle” is a brilliant book about what might happen if there were

a solid phase of water more stable - that is with lower energy - than the liquid phase at the same

temperature and pressure. In that case, if some small region of the ocean were somehow pushed

into the more stable phase, this region would grow and take over the whole ocean, with disasterous

ecological consequences.

Of particular interest for us (unlike the ﬁsh) are the properties that our vacuum state has with

respect to the various forces that act on matter. The particular vacuum state that supports all these

curious masses “screens” the charges associated with some of these forces and gives mass to the

corresponding force particles.

The is a very useful analog - inside a superconductor - electromagnetic force is screened be-

cause charge carriers can move around inﬁnitely easily and surround any charge with opposite

charges. The result is that inside a superconductor, there is no long range force between charged

particles. All that is left is a force that falls off over the distance that it takes - “photon” has a mass

but at short distances below the atomic scale, the photon is still massless.

Different phases may have dramatically different physical properties.

6

lectur

e 25

Topics:

Where are we?

Running Hubble backwards

Relativistic cosmology

Back to the big bang

Thermal equilibrium

The hot bang and the CMBR

Where are we?

Having done a lightning introduction to particle physics before break and an introduction to the

Hubble expansion in the last lecture, we are now going to work our way back towards the beginning

of the universe. This is sort of fun, for its own sake, but it will also, I claim, get us closer to an

answer to the question we asked at the very beginning of the course — why is Newton’s third law

a formula for acceleration. We won’t get there today, but we should make it by Thursday.

Running Hubble backwards

Let us start by going back to the laws we derived to describe the evolution of the Hubble expansion.

If we put ourselves at the origin and take ~r to be the position of some distant galaxy, the Hubble

law can be written as

˙

~r = H(t) ~r (1)

where H(t) is the Hubble parameter. Then, assuming that we can use Newton’s theory of gravity

and ignore relativity, we found that gravity slows the Hubble expansion

¨

~r = −~r

4π

3

G

ρ(t) (2)

Then by considering the conservation of the mass inside a sphere with radius |~r | (still ignoring

relativity), we found that we could integrate (2) to obtain

1

2

(~

v )

2

−

4π G ρ(~r )

2

3

=

1

2

C (3)

where

the constant C depends on the initial conditions. Again, this derivation was valid for a

universe dominated by slowly moving matter, so long as the cosmological principle is satisﬁed.

It is conventional and convenient to get rid of the vectors in (1)-(3). We can do this by deﬁning

an (arbitrary) distance scale a between points in the universe. For example, we could take a = |~r |,

the distance between us and the distant galaxy. But the notion is more general, as it has to be

because we want to use it to describe the universe at earlier times, before we or even galaxies were

around. a is just an arbitrary scale that measures the relative size of the universe. The value of a

doesn’t matter at all, but if a doubles, that means that the distance between (far apart) things in the

universe has doubled!

Using a = |~r | and that fact that the direction of ~r doesn’t change with time, we can write

~r = a ˆr

˙

~r = ˙a ˆr

¨

~r = ¨a ˆr (4)

1

and

rewrite (1)-(3) as

˙a

a

= H(t) (5)

¨a

a

= −

4π

3

G

ρ(t) (6)

and

˙a

2

a

2

= H

2

=

8π

G ρ

3

+

C

a

2

(7)

No

w for simplicity, let us assume that C = 0. As I have said before, there are theoretical

reasons (that we will discuss next time) to think that this is a good approximation, and some ob-

servational support for the detailed picture that emerges. We can then follow the Hubble evolution

backwards towards the big bang, at least for a while.

For C = 0, we can take the square root of (7) and write

˙a

a

=

s

8π

G ρ

3

(8)

Note

that have taken the positive sign, corresponding to expansion.

To determine the time evolution of a, we need a relation between a and ρ. This relation depends

on what kind of stuff our universe is made of at the time of interest. For slowly moving matter,

we can simply use the fact ρ a

3

is a constant, because the density is inversely proportional to the

volume. Thus

0 =

d

dt

(ρ

a

3

) = ˙ρ a

3

+ 3ρ a

2

˙a ⇒

˙a

a

= −

1

3

˙ρ

ρ

or ˙ρ = −3ρ ˙a/a (9)

Thus

we can write (8) as

−

1

3

˙ρ

ρ

=

s

8π

G ρ

3

(10)

This

is a differential equation that we can solve for ρ (or course, the result is only meaningful as

long as our assumptions are valid):

˙ρ = −

√

24π

G ρ

3/2

(11)

ρ

−1/2

=

Z

dρ

ρ

3/2

= −

√

24π

G

Z

dt = −

√

24π

G (t − t

0

) (12)

ρ

−1/2

=

√

6π

G (t − t

0

) (13)

ρ =

1

6π

G (t − t

0

)

2

(14)

where t

0

is a constant set by initial conditions. As expected, the density decreases as the universe

expands and increases as we run the tape backwards. Since ρ a

3

remains constant as t increases,

(14) implies that a grows as

a ∝ (t − t

0

)

2/3

(15)

2

Georgi, Howard. Physics 16 - Mechanics and Special Relativity (англ.)

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