## Monday, January 18, 2010

### Lecture 4

Today's lecture is in two parts: First we will take a closer look at relativity, and then we will discuss a theory called supersymmetry which is an integral part of string theory.

Part One: Relativity
I - Special Relativity
A- What is relativity?
Galileo's principle of Relativity was that two observers moving uniformly relative to one another must formulate the laws of nature in exactly the same way.
No observer can distinguish between absolute rest and absolute motion by appealing to any law of nature;
there is no such thing as absolute motion, but only relative motion (of one observer with respect to another).

Examples:
Imagine two people in space floating towards each other. Who is moving? It could be that both are moving, or that only one is moving? Each person may think she is stationary while the other is moving. In fact, there is no objective truth, other than that they are both moving relative to one another.

Consider yourself in a speeding car. You are going somewhere around 50 mph. And so is the t-shirt you are wearing. But relative to you, the t-shirt is not in fact moving, and this is what you experience.

B- Einstein's Relativity
Einstein made relativity a universal principle that applies to all physical phenomena: electricity, magnetism, light, etc.

From this assumption, and the assumption that the speed of light is a constant, Einstein derived his special theory of relativity: a description of space and time as a fabric of interwoven dimensions, a description of events with respect to observers.

Special relativity describes linear motion. That means anything that is a constant speed. A change in speed or direction corresponds to an acceleration and is within the realm of general relativity.
It is general relativity therefore that describes gravity, which cannot be distinguished from acceleration.

C- Spacetime
Usually we consider an event with respect to time. One could plot her height on the x-axis, and time on the y-axis. Or one could plot spatial position with respect to time, possibly choosing to include all three spatial coordinates, thus giving four coordinates (t, x, y, z).
Since we can't visualize a 4-dimensional space, it's typical to consider a 2-dimensional spacetime (t,z).

The speed of light is approximately c = 3 x10^8 meters/second.

We normalize c to 1, so that any other speed is some percentage of c.

Let's look at spacetime with two spatial directions. The boundary given by the speed of light is called a light cone. A photon (or light particle) must travel somewhere on this cone, and any thing else will travel inside the cone.

Newton's second law is F= ma (i.e., force is mass times acceleration).
If we had a constant force k = ma or a = k/m.
If we take the derivative of both sides, we get the rate of change of acceleration: k/m.
Hence v(t) = k/m *t + b (some constant)
This linear function increases to infinity, violating the condition that nothing can travel faster than the speed of light.

The reason this condition still holds is that mass is a function of velocity, and is not constant (although for all of the speeds we experience, it may as well be).

As an object's velocity grows larger and larger, it must be harder and harder to increase the velocity still further.
In fact, mass must become infinite as the velocity approaches the speed of light. (Light particles are massless).
The function of mass with respect to velocity, must look something like this:

Since mass depends on velocity, mass too is relative; mass is not an absolute quantity.

II- General Relativity
Since special relativity describes linear motion, we can use it to describe accelerated motion.

Something that accelerates changes speed.

As opposed to a linear function (like the blue one), many functions are not linear. However, if we zoom in on them, they would appear linear.

If we look at the slope of a function at any instant (the derivative), we always get a linear line. This line corresponds to the rate of change in speed; at any instant, motion is linear.

This means that special relativity can be used to describe all motion; it can be extended to general relativity.

By general relativity, gravity is a result of the warping of space and time by mass.

It is general relativity that allows us to understand and consider black holes, regions of space where gravitational attraction is so strong that not even light can escape.

Primary source: The Geometry of Spacetime by James Callahan (a Smith professor!)

Part Two: Supersymmetry
I- History
Supersymmetry is an important part of string theory because there is no way string theory could possibly describe the known universe without it.
Supersymmetry is a physical theory in its own right and has been around longer than string theory. It is possible that supersymmetry holds and string theory doesn't, but less likely that string theory is true and supersymmetry isn't.

II- What is symmetry exactly?
Just as space and time are symmetrical under rotation and translation (by Relativity), supersymmetry suggests that particles may possess a powerful symmetry.

If something is symmetrical, that means you can interchange the parts and be left with an identical object.
Consider the letter "Y". It has a reflective symmetry. The mirror image of Y cannot be distinguished from Y itself. The letter "O" has even more symmetries. "O" can be rotated or flipped to one's liking and still appear the same.

III- Bosons and Fermions
Supersymmetry describes a type of symmetry: a symmetry between two classes of particles: bosons and fermions.

Bosons and fermions are in fact the only two types of particles. All particles are either bosons or fermions.

Both fundamental particles such as the electron and quarks, and composite objects such as a proton or the nucleus of an atom can be classified distinctly as either a boson or a fermion.

The classification is given by the particle's "spin." I use quotes because the particles are not actually spinning, but interestingly enough, they move as if they were. This property therefore called intrinsic spin. It is a fixed property for any particle. Further it's value is either a whole number 0, 1, or 2 (known particles), or a multiple of a half such as 1/2 or 3/2.

What does spin mean? If a particle has a spin 1, that means the particle would rotate exactly once before returning to its original state.

Remark: A graviton has spin 2.

The particles are then classified as having either integer spin (0,1,2) as BOSONS or half-integer spin (1/2, 3/2) as FERMIONS. The fact that particles can only have integer or half integer spins is yet another example of quantization, roughly, an idea in Quantum physics that some things come in units.

Protons, neutrons, and electrons are all fermions (and in fact, all have spin 1/2). In fact, all familiar matter seems to have spin 1/2.

Bosons and fermions behave differently.

By something called the Pauli exclusion principle, two fermions of the same type can never occupy the same space. This is why there is only one electron for each orbit around the nucleus of an atom.

This principle is intuitive because since all familiar matter is made up of fermions, we experience this principle first-hand daily. It is the reason the matter around you can hold a solid structure; that you will not fall through the seat of your chair. It is also the reason you can't walk through walls: the electrons in your body cannot occupy the same space as those in the wall.

Bosons on the other hand can and will be found in the same place, which seems to totally go against our intuition. However, consider light. Light is a bosonic particle and light can indeed pass through light.

Also, of the known particles, bosons are always force carrying particles, whereas fermions are always matter particles.

Supersymmetry exchanges bosons and fermions!

IV- Superstring Theory
It's only theoretical and we don't have proof, but it has great potential as a theory. For one,
in order to use string theory to reproduce the fundamental particles in the Standard Model, supersymmetry must be incorporated into the theory.
String theory that incorporates supersymmetry is called "Superstring theory"

Superstring theory eliminates the tachyon (a problem particle that could travel faster than the speed of light). Phew.

Supersymmetry also has the potential to solve another problem called the Hierarchy problem, although we will not discuss this problem here.

V- Supersymmetry
In supersymmetric theory, every existing particle can be paired with a partner particle with the same mass and charge that interacts via the same forces, but is of the opposite class. That is, a boson can be replaced with a fermion.

Two particles that are fundamentally different can be exchanged and physics cannot tell the difference!

This is an amazing idea because it could mean that matter particles and force particles are symmetrical (or interchangeable) by supersymmetry transformations.

Supersymmetry would be nicer if it paired known particles but the standard model does not contain equal numbers of fermions and bosons, and certainly doesn't contained matching pairs as would be desired. If supersymmetry is true, the universe must contain at least twice the number of known particles, which means we've got plenty more particles to find!

We would expect that if there were particles out there with the same mass and charge as the ones we know of, that we would be able to observe them, however, because of something called broken symmetry, it is expected that these superpartners would actually have to be much more massive and therefore could only be produced at super high energies (in particle accelerators).

Review on a lot of what we've covered in an interview with Ed Witten (creator of M-theory): http://www.youtube.com/watch?v=iLZKqGbNfck&feature=related

Wikipedia article on special relativity: http://en.wikipedia.org/wiki/Introduction_to_special_relativity#Reference_frames_and_Lorentz_transformations:_relativity_revisited

An important part of general relativity is the equivalence principle: http://en.wikipedia.org/wiki/Equivalence_principle

I HOPE YOU ALL ENJOYED THE COURSE!

String theory topics not explored in this class that you may wish to pursue on your own:
(nearly all of these are a bit more advanced than what we covered)
The Higgs mechanism
broken symmetries
uncertainty principle
other Grand Unification Theories (GUTs)
virtual particles
Kaluza-klein modes
tachyons

QUIZ

1. What are the fundamental particles of all known matter as far as we know?

2. What are the four fundamental forces? Which ones are currently part of a single theory, and how are those forces unified?

3. What are some of the ideas involved in Quantum mechanics. Explain any or all of these to the best of your ability.

4. What is the difference between special relativity and general relativity?

5. What does relativity imply about how gravity works?

6. What is a string?

7. What are three of the problems in modern physics that string theory may solve?

8. What are some of the reasons we may not be aware of extra dimensions?

9. Why would gravity not be confined to a brane?

10. What is supersymmetry?