Monday, January 18, 2010

Lecture 3

Lecture 3: Extra Dimensions, Branes, and Parallel Universes

I- Introduction to Branes
A- History of Branes
A brane is a domain that has fewer dimensions than the higher-dimensional space that surrounds or borders it.

The idea of branes came before string theory. Physicists derived similar objects that extend infinitely far in only some directions using Einstein's theory of relativity.
Particle physicists suggested brane-like surfaces that would trap particles.
String theory branes were the first branes proposed that could also trap forces.

Branes imply that even though our universe may have more than 3 spatial dimensions, particles and forces are trapped on (or confined to) such lower dimensional surfaces.

B- What do Branes imply?
Extra small dimensions aren't necessarily curled up, they could be intervals: bounded between two "walls"

Branes can be inside a space or at a boundary but either way they will trap the particles and forces along them.

Branes could have any number of dimensions, as long as the number is less than the number of dimensions in the full higher-dimensional space of which they are a part.

Consider light particles confined to a 3-dimensional brane: light rays would spread out only along the brane; that is, light would behave as it would in a three-dimensional universe, making it impossible to distinguish the number of true dimensions of space.

Anything confined to the 3-d brane would appear to be 3-dimensional.

C- Branes and forces
Although matter and forces could be stuck on the brane, gravity wouldn't be.
By general relativity, gravity is woven into the framework of space and time so gravity holds in all dimensions of space and time.

In fact, gravity would be the only force that could communicate between the world on the brane and the higher-dimensional world beyond the brane.

Are there other particles/forces not confined to the brane? Possibly.

(From Warped Passages by Lisa Randall. Sorry for the image quality.)

II- The Multiverse
A multiverse describes a space with more than one brane, that is, multiple universes.
It's possible that the universe contains multiple branes that only interact via gravity, or don't interact at all (if the branes were too far apart to ever communicate with each other).

(Warped Passages)

Particles on distinct branes could be ENTIRELY different than the ones we know; these universes could be made up of a completely different set of forces and set of matter particles.

Branes could be parallel or intersect. If they intersected, particles could be shared between intersecting branes.
The only force that would necessarily be shared is gravity, although the strength of gravity could vary, that is, the gravitational constant could be different.

III- Brane Theory
An interesting feature of branes is that even though physics is set up so that the laws should be the same at any point in space, branes don't respect these symmetries.

We will not go into this but branes actually play a huge role in string theory and are just as important as the strings!
In fact, in addition to fundamental strings, some force particles may be a result of strings attached to branes; that is, it is not strings alone that make up everything.

It was branes that allowed the different versions of string theory to become a single inclusive theory (M-theory).

IV- Gravity and Branes
A- Open and closed strings
If strings are like Leonard Susskind described them (open), there should exist closed strings; the strings should be able to close into loops.
Closed loops correspond to the graviton (the particle associated with gravity).
This is what singles out gravity as a force: since the strings are closed, they are not confined to branes.
Therefore gravitons can travel throughout the entire space while all other force particles are confined to the brane.

B- Why is Gravity so weak?
The fact that gravity is distributed throughout 9 or 10 spatial dimensions whereas all of the other forces may only interact on a 3 dimensional brane could account for gravity's being so much weaker than the other forces.

Another possibility is warped Geometry:
In relativity, space and time are warped by matter and energy, Randall and Raman applied this idea in an extra-dimensional context and found a configuration in which spacetime warps so severely that gravity could be strong in one region of space and weak elsewhere.
Warped geometry also allows an invisible extra dimension infinite in size that could be hidden by the distortions in space.

V- Parallel Universes
The Universe: Parallel Universes
(44 minutes)

VI- Different sizes of infinity
In the parallel universes video, the claim is that since there may be an infinite number of universes, any possible universe one could imagine would necessarily exist. We could imagine a universe where Bush had never been president; where some world wars had never taken place, but other ones had; one in which your hero or icon was your best friend; one in which your dreams were a reality; and so on.
While there may be an infinite number of parallel universes, and in fact, Brane worlds allow such a possibility, it is mathematically sloppy to say that this implies that any conceived universe can and must exist. The reason for this is that there are different sizes of infinity. That is, some infinities are larger than other infinities.

A- How do we measure infinity?
Mathematicians measure the size of any set of elements with something called cardinality.

A set is a collection of elements. Sets can be finite or infinite in size and can include numbers, colors, shapes, objections, operations, you name it.
For example the set {1, 2, 3} has three elements 1, 2, and 3. This notation is pretty standard. Likewise, the set {x, y, z} has three elements. Meanwhile, the set {x, y, z, w} has four elements. If we want to compare sets, we can consider a map from one set to another; that is, we can associate each element in one set with a single element of another set.

Let's take the sets A= {1, 2, 3} and B= {x, y, z} and say 1 --> x, 2 --> z, 3--> y. Every element in set A is mapped to a unique element in B, so our map is valid. But there is more going on. Notice that every element in the set B is mapped to. That is, for each element in B: x, y, and z, there is some element that maps to it. In math, such a map is called "onto" or "sujective." Note also that no two elements in A map to the same element in B. If 1 and 2 both mapped to x our map would still be valid, but, in this case, they don't. We call such a map "one-to-one" or "injective", meaning that each element maps to a unique element. If a map satisfies both of these properties (it is injective and surjective), we say it is bijective, and consequently the sets are the same size! This is certainly true of our example: both sets have three elements.

Let's consider two infinite sets and compare their sizes.
Let N = {1, 2, 3, 4, ...}, that is every positive whole number, and let
E = {2, 4, 6, 8, ...}, or every positive even number.

Are these sets the same size?

We let x denote an arbitrary element in the set N. Our map will be given by the function f(x) = 2x. To find out whether the sets are the same size, let's check if the map is injective and surjective. If it's not, maybe there's another map that is. If it is, then we can stop there: our sets are the same size.

First we check if the function is onto (surjective). Can we find an element in E that is not mapped to; that is, is there a positive even number for which there is no whole number that can be multiplied to two to obtain that number. Try to think of one!
14? Nope, 7 maps to 14.
1028? Nope, 514 maps to 1028.
There is in fact no number in E that is not twice some number in N. Try to think of how you might verify this. A trick is to think of a function that maps from E to N.

Next, let's check to see that our function is one-to-one (injective). Are there any two numbers in the set N that map to the same number in E? Let's suppose there are. We'll call them x, and y, and we'll suppose that they both map to an element k in E. Then we have 2x = k = 2y. Division by 2 tells us that x = y, so in fact these must be the same number. The function is injective!

This means that even though N seems like it might be a bigger set, these sets are actually the same size.

But what about another infinite set, such as the real numbers? The real numbers, R, is the set of all positive numbers, negative numbers, whole numbers, fractions, and irrational numbers (numbers that can't be expressed as fractions such as pi (3.1415...)). There is no bijective map from N to R because there cannot be a map that is onto. For any map, we can always find a real number that is not mapped to. These infinite sets are not the same size!

B- What does this imply about parallel universes?
We can see that since there are different sizes of infinity, there is not always a one to one correspondence between elements in infinite sets.
If we let
P = {all possible universes}, and
A = {all universes that have ever existed or will ever exist},

the fact that both of these sets are infinite does not imply that any universe we can conceive of must exist. Although, it is possible that we can think of an infinite number of universes that do exist. Some possible universes must exist, in fact an infinite number of possible universes must exist. But we cannot be sure that all possible universes must exist. If you can think of a possible universe. Maybe one in which you are reading this with four eyes, there is no way to be sure whether this universe could or could not exist.

A video on branes (not shown in class):
Additional sources:
For more detail, I highly recommend Lisa Randall's book Warped Passages.

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