astronomy 350 cosmology

May 13, 2003 Lynn Cominsky - Cosmology A350

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Professor Lynn Cominsky

Department of Physics and Astronomy

Offices: Darwin 329A and NASA EPO

(707) 664-2655

Best way to reach me: [email protected]

Astronomy 350Cosmology


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Group 14

Mike LightnerEmily OgdenDonald Siemon


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Extra credit

You can earn one extra point for each Space Mystery that you try out and evaluate at http://mystery.sonoma.edu

Evaluation forms are found at: http://mystery.sonoma.edu/resources/teachers/evaluation.html

The mysteries are: Live! From 2-alpha Alien Bandstand Starmarket

Evaluation forms must be turned in by 5/27/03 (day of final exam)


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Big Bang Timeline

We are here

Today’s lecture


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Big Bang?


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Distance in 3 Dimensions

We all experience three spatial dimensions (usually referred to as x, y and z)

Distances in three dimensions are easily

found from an extension of the 2D Pythagorean

theorem for right triangles

a2 + b2 = c2

In 3D: d = a2 + b2 + c2


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Vectors

Vectors are used to mathematically represent quantities which have both size and direction

This vector d has components (a, b, c) in

the (x, y, z) directions and magnitude d

d =abc


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Vector Fields

Vector fields are physical quantities which have magnitudes and directions at each point

This is a 2D vector field where the direction at

each point is given by

What is the magnitude (length) of the vector at each point?


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Tensors

It is difficult to visualize a tensor

This is a visualization of the stresses in a 3D material when force is applied at two points on the top surface

The stress tensor is a 3 x 3 matrix of numbers

Push here


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Tensors

Einstein unified the 3 spatial dimensions with the dimension of time to make a four dimensional space time (x, y, z, ct) in which gravity is defined by a 4x4 tensor

Components of the tensor down the diagonal

are the coefficients in

d2 = g11x2 + g22y2 + g33z2 + g44c2t2

They are (1,1,1,-1) for flat space


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Tensor Fields The electromagnetic field is another example of

a 4D tensor field. It has 4x4 components, which tell you the magnitude of the components in 3 different directions for both the electric and magnetic field.

0 Bz -By -iEx

-Bz 0 Bx -iEy

By -Bx 0 -iEz

iEx iEy iEz 0


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Flatland

It’s hard to visualize 4 dimensions, so let’s start out by examining the lives of the characters in Edwin A. Abbott’s Flatland

Rank in Flatland is a function of increasing

symmetry:

A woman, soldier, workman,merchant, professional man,

gentleman, nobleman, high priest


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Flatland

What do they see when a 3D being (Lord Sphere) comes to visit?

3D cross-sections of Lord Sphere float through the 2D

world of Flatland


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Troubles in Flatland

It’s hard to eat in a 2D world!

It is also impossible to tie your shoes! Why?

A digestive tract cuts a 2D being in half!


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A Square and his wife alone in their 2D house, when Lord Sphere drops in from the third dimension

There is no privacy in 2D from a 3D being!


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A 3D being would be able to change the symmetry of a 2D resident or help him escape from jail!

The 3D being can lift the 2D resident up out

of Flatland!


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How do Flatlanders know the shape of their Universe?

A flat plane (with edges) is an open 2D Universe

Is there a closed 2D Universe? A Moebius strip is a 2D closed

universe

movie


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Exploring Geometries

Take the newspaper Cut a long skinny strip Twist one end of the strip once and tape

together Congratulations – you have just made a

Moebius strip! How many sides does this have? Try drawing

on it to see. What happens to it when you cut it all around

the strip direction?


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What would happen if Flatlanders walked all the way around a closed 2D world?

They would be mirror-reversed!

Flat torus – another example of a closed 2D world


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The 3D Universe

Open (<1) Hyperbolic geometry

Flat (=1) Euclidean geometry

Closed (>1) Spherical geometry


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Infinite Universe?

Is the Universe infinite or just really, really, really big?

Some scientists (like Janna Levin) prefer to think of the Universe as finite but unbounded. An example of such a space is a 3D torus.

With such a topology, we could see the backs of our heads, if we could see far enough in one direction


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Curved Space

This is not an infinite series of reflections, but is caused by light traveling all the way around the hyperdonut

A hyperdonut is one example of a curved space in 3D


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3D Torus games

Play game here


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The 4D Universe

Many cosmologists believe that our Universe is a 4D hypersphere

This is a 3D movie projection of a 4D hypersurface

movie


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Geometry in the 4th dimension

A 2D square is created by moving a line in a perpendicular direction

A 3D cube is created by moving a square in a perpendicular direction


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A Flatlander can only visualize a cube, if it is unfolded in 2D

If you move a 3D cube in a fourth perpendicular direction, you get a hypercube

A 3D being can only visualize a hypercube by unfolding it in 3D into a tesseract


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Christus Hypercubus was painted by Salvador Dali in 1955 –it features a tesseract

A 4D hypercube is bounded by 8 3D cubes, has 16 corners and a volume L4


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Here is another 2D projection of a 4D hypercube

At each face, you can see a cube in different directions as you change your perspective

d2 = x2 + y2 + z2 + w2


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Troubles in Spaceland

Thieves from the fourth dimension could steal things from locked safes (or operate without cutting you open!)

There is no privacy in 3D from a 4D being!


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Visitors from the 4th dimension

Try the “digustoscope” to see yourself as a 4D being in a 3D world!

Do powerful beings such as a Cosmic

Creator (or the Devil) live in the

Fourth Dimension?


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Angels and Devils

This 2D exercise from U Wash helps you to visualize the effects of different geometries

But first, let’s see how 2D beings would see a 3D object passing through their world (e.g. Flatland by Abbott)

Cube moviesSphere movies


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Geometry in higher dimensions

Here is 2D projection of a 5D hypercube

It obeys the same mathematical laws as objects in worlds with a lower number of dimensions

d2 = x2 + y2 + z2 + w2 +v2


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Physics in higher dimensions

Kaluza was the first to try to unify the fields of (Maxwell) electromagnetism and (Einstein) gravity by rewriting the laws of physics in 5D (or a 5x5 tensor)

In this theory, light was caused by a ripple in the 5th dimension


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Kaluza-Klein Theory

Theodor Kaluza’s original idea was refined by mathematician Oskar Klein

At each point in 4D spacetime, another curled up or “compactified” dimension is present, but it is so small that it is not observable

At each point in spacetime, a

curled up dimension exists


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Physics in Higher Dimensions

Grand Unified Theory expands the 5x5 tensor to include the Yang-Mills field, which describes the weak and strong interactions in N dimensions

The resultant tensor has 4+N dimensions

N is 5 for the standard model of particle physics


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Supersymmetry allows particles with different types of spin (fermions and bosons) to interchange

Each particle has a supersymmetric partner called a “sparticle”

No “sparticles” have yet been detected

The WIMP (weakly interacting massive

particle) is the lightest “sparticle”


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Supergravity theory includes supersymmetry as well as interactions with matter (gravity)

It requires an 11D Kaluza-Klein theory

However, it still does not have enough complexity to explain all the interactions that we see in the Standard Model

Meanwhile, searches for “sparticles”

continue at higher energies


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Superstrings

Strings are little closed loops that are 1020 times smaller than a proton

Strings vibrate at different frequencies Each resonant vibration frequency creates a

different particle Matter is composed of harmonies from vibrating

strings – the Universe is a string symphony

“String theory is twenty-first century physics that fell accidentally into the twentieth

century” - Edward Witten


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Superstrings

Strings can execute many different motions through spacetime

But, there are only certain sets of motions that are self-consistent

Gravity is a natural consequence of a self-consistent string theory – it is not something that is added on later

Self-consistent string theories only exist in 10 or 26 dimensions – enough mathematical space to create all the particles and interactions that we

have observed


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Superstring Dimensions

Since we can observe only 3 spatial and 1 time dimensions, the extra 6 dimensions (in a 10D string theory) are curled up to a very small size

The shape of the curled up dimensions is known mathematically as a Calbi-Yau space


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Superstring Universe

At each point in 3D space, the extra dimensions exist in unobservably small Calabi-Yau shapes


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Superstring Interactions

Strings interact by joining and splitting

2 strings joined split into 2


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Superstring Interactions

Strings annihilate and erupt repeatedly subject to the quantum mechanical uncertainty principle, just like particle pairs

2 strings

annihilation

2 virtual strings

eruption

2 strings


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Superstrings and Gravity

Gravitational force is represented by the exchange of closed strings

Even if an infinite number of string-like Feynman diagrams are added up, the theory stays finite


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Superstring Motions

Strings can have two different types of motions in a universe where some dimensions are curled up and others are extended

Sliding

Winding


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Superstring Theories

There are at least five different versions of string theory, which seem to have different properties

As physicists began to understand the mathematics, the different versions of the theories began to resemble each other (“duality”)

In 1995, Edward Witten showed how all five versions were really different mathematical representations of the same underlying theory

This new theory is known as M-theory (for Mother or Membrane)


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M-Theory

Unification of five different types of superstring theory into one theory called M-theory

M-theory has 11 dimensions


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Some questions

Can we find the underlying physical principles which have led to us to string theory?

Does the correct string (or membrane) theory have 10 or 11 dimensions?

Will we ever be able to find evidence for the curled up dimensions?

Is string theory really the long-sought “Theory of Everything”?

Will any non-physicists ever be able to understand string theory?


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Print Resources

Elegant Universe by Brian Greene (Norton) Hyperspace by Michio Kaku (Anchor Books) Fourth Dimension by Rudy Rucker (Houghton Mifflin) Surfing through Hyperspace by Clifford A. Pickover (Oxford)


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Web Resources

VROOM visualization of 4 dimensions http://www.evl.uic.edu/EVL/VROOM/HTML/PROJECTS/02Sandin.html

Ned Wright’s Cosmology Tutorial http://www.astro.ucla.edu/~wright/cosmolog.htmFourth dimension web site

http://www.math.union.edu/~dpvc/math/4D/welcome.html


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Web Resources

Michio Kaku’s web site http://www.mkaku.org E. Lowry’s EM Field in Spacetime http://www.ultranet.com/~eslowry/elmag Visualizing tensor fields http://www.nas.nasa.gov/Pubs/TechReports/RelatedPapers/StanfordTensorFieldVis/CGA93/abstract.html

Exploring the Shape of Space http://www.geometrygames.org/ESoS/index.html

astronomy 350 cosmology

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