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History of Higher Dimensions

- Riemann’s geometry
- The popularity of the fourth dimension
- The Kaluza-Klein Theory
- Supergravity

Riemann’s Geometry

- Opposed Euclidian Geometry
- Euclid based geometry was based on common sense, not logic.
- “It is obvious that a point has no dimension at all. A line has one dimension: length. A plane has two dimensions: length and breadth. A solid has three dimensions: length, breadth, and height. And there it stops. Nothing has four dimensions.”

Riemann’s idea

- Force was an application of Geometry.
- Results from a fourth spatial dimension
- Curves in this dimension lead to forces.

Riemann’s Problem

- There was no current mathematics that could accurately describe this geometry.
- Riemann had to develop his own geometry.
- Riemann’s Metric Tensor

4-Dimensional Tensor

Riemann’s Tensor

- Contains all the information necessary to describe a mathematically curved space in N dimensions.
- A 4-Dimensional Tensor contains 10 numbers.
- g12 = g21

The Popular Fourth Dimension

- After Riemann’s presentation, scientists thought more of the fourth dimension.
- In 1877, the trial of psychic Henry Slade put the fourth dimension in the public mind.
- Magic, Ghosts, God seen as residing in the fourth dimension.

The Kazula-Klein Theory

- United Einstein’s theory of gravity with Maxwell’s theory of light.
- Did so by introducing a fifth dimension past Einstein’s fourth temporal dimension.

Problems with Kazula-Klein

- Physicists in the 1920s weren’t convinced the fifth dimension existed.
- Fifth dimension as “rolled up” in a tiny circle of the Planck length left it untestable.
- Energy required was the Planck Energy (1019 GeV)
- Swept aside in the discovery of Quantum Mechanics.

Expanding Kazula-Klein to N Dimensions

- In 1960, Scientists extended Kazula-Klein to introduce symmetries into physics.
- “If the wave function of a particle vibrates along this surface [of a hypersphere], it will inherit this SU(N).”

Supergravity

- Introduced with supersymmetry.
- Allowed for shuffling of fermions [Quarks and leptons: particles of half-integral spins (1/2, 3/2, 5/2…)] and bosons (Photons and “gravitons”: quanta with integral spins) while keeping the equation intact.
- Has interesting properties:
- a x b = -b x a
- a x a = -a x a
- a x a = 0 even when a = 0

Supersymmetry

- All particles have super partners, sparticles.
- Gravitons and gravitinos.
- Leptons and sleptons.
- Quarks and squarks.

Supergravity Theory

- Expands Kazula-Klein to 11 dimensions.

Problems with Supergravity

- Couldn’t find sparticles.
- Huge energy required to test.
- Nonrenormalizable.
- When trying to calculate with it, you got meaningless infinities.

Superstrings

- The universe consists of strings of 10 and 26 dimensions that vibrate.
- These vibrations result in forces and matter.
- As strings move in space-time they can:
- Break into smaller strings
- Collide with other strings to form longer strings.

Heterotic String

- David Gross, Emil Martinec, Jeffrey Harvey, and Ryan Rohm.
- Superstring theory already contains Einstein’s theory of gravity, and will not work without it.
- The graviton is the smallest vibration of a closed string.
- String theory is self-consistant.

Heterotic String

- Consists of a closed string that has two different vibrations:
- Clockwise vibrations in 10-dimensional space
- Counterclockwise vibrations in 26-dimentional space (where 16 have been compacted).
- A hybrid theory since it includes 10 and 26-dimensional space.
- Contains a E(8) x E(8) symmetry.

Discovery of the Superstring theory

- Accidentally discovered in 1968.
- Gabriel Veneziano and Mahiko Suzuki came across the Euler beta function.
- Found it fit almost all properties required to describe strong interactions of elementary particles.

Why Ten and Twenty-Six?

- When calculating how strings break and form in N-dimensional space, meaningless terms pop up and destroy the properties of the theory.
- When using Ramanujan modular functions, the number 24 keeps popping up, and when it is generalized 24 becomes 8.
- Scientists add two more dimensions when counting vibrations appearing in relativistic theory

So… Why 10 and 26?

- We don’t know…
- Unfortunately, we don’t understand the underlying principle of Superstrings since the theory was discovered in reverse of normal.
- We started with a Theory instead of building up a Theory from what we had like what happens normally.
- Also waiting for mathematics to catch up to help solve the Superstring equations.

Creation

- Superstring theory is at heart a theory of creation.
- Started as ten dimensional, this supersymmetric space was also unstable.
- Broke into two parts, a 4-dimensional part that grows infinitely and a 6-dimensional part that shrunk infinitesimally.

The Beginning of the Universe (Quoted from Hyperspace)

- 10-43 seconds: The ten-dimensional universe breaks down to a four- and six-dimensional universe. The six-dimensional universe collapses down to 10-32 cm in size. The four dimensional universe inflates rapidly. The temperature is 1032ºK.
- 10-35 seconds: The GUT force breaks; the strong force is no longer united with the electroweak interactions. SU(3) breaks off from the GUT symmetry. A small speck in the larger universe becomes inflated by a factor of 1050, eventually becoming our visible universe.

The Beginning of the Universe (Quoted from Hyperspace)

- 10-9 seconds: The temperature is now 1015ºK, and the electroweak symmetry breaks into SU(2) and U(1).
- 10-3 seconds: Quarks begin to condense into neutrons and protons. The temperature is roughly 1014ºK.
- 3 minutes: The protons and neutrons are now condensing into stable nuclei. The energy of random collisions is no longer powerful enough to break up the nucleus of the emerging nuclei. Space is still opaque to light because ions do not transmit light well.

The Beginning of the Universe (Quoted from Hyperspace)

- 300,000 years: Electrons begin to condense around nuclei. Atoms begin to form. Because light is no longer scattered or absorbed as much, the universe becomes transparent to light. Outer space becomes black.
- 3 billion years: The first quasars appear.
- 5 billion years: The first galaxies appear.
- 10-15 billion years: The solar system is born. A few billion years after that, the first forms of life appear on earth.

Sources

- Kaku, Michio. Hyperspace. New York: 1994.
- http://www.pbs.org/wgbh/nova/elegant/everything.html
- http://superstringtheory.com/index.html
- http://www.cssh.qc.ca/ecoles/simon/museedesenfants.quebec/Peintres/Picasso/Posters/expo.htm
- http://www.artlex.com/ArtLex/m/movement.html
- http://physics.weber.edu/carroll/honors-time/art.htm
- http://www.math.uiowa.edu/~goodman/algebrabook.dir/polymodels.html
- http://www.pballew.net/arithme9.html
- http://tena4.vub.ac.be/beyondstringtheory/string.html

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