What is the Shape of the Universe

1
What is the Shape of the Universe?

• By
• D.N. Seppala-Holtzman
• St. Josephs College
• faculty.sjcny.edu/holtzman

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What is the Shape of the Universe ?

• Nobody knows
• There are many, many theories
• Here, we present a simple argument, from basic
  principles, that the universe is a hypersphere

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A Hypersphere??!!!

• Yes, a hypersphere
• Specifically, a 3-dimensional sphere in Euclidean
  4-dimensional space
• Dont panic

4
Some Basic Background

• Higher dimensional space
• Points
• Distance
• Spheres
• Balls

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Dimensions

• Most people feel comfortable with dimensions 1, 2
  and 3
• Mathematicians are comfortable in higher
  dimensions because they dont try to visual them
  they reason by analogy

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Dimension Zero

• Dimension zero consists of a single point
• There is no notion of distance because the notion
  is vacuous in this dimension

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Dimensions A 1st Attempt
D1
D3
D2
D4 ????
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Dimension One R1

• Can be thought of as an infinite line
• There is a one-to-one correspondence between each
  point on the line and a real number

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Dimension Two R2

• Consists of two, mutually orthogonal infinite
  lines called axes

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Dimension Two

• Each point corresponds to an ordered pair of real
  numbers (x1 , x2)
• The first real number, x1 , corresponds to a
  point on the horizontal axis while the second
  number, x2 , corresponds to a point on the
  vertical axis
• These values are the (signed) distances from the
  origin (point where the axes meet) on each axis

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Dimension Three R3

• Consists of 3 mutually orthogonal axes

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Dimension Three

• Each point corresponds to an ordered triple of
  real numbers (x1 , x2, x3)
• These values correspond to the (signed) distances
  from the origin on each of the three axes

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Higher Dimensions

• We continue by analogy
• Each point in Rn is just an ordered n-tuple of
  real numbers(x1, x2, x3, x4, . , xn)
• Thus, your Social Security Number could be
  considered to be a point in R9
• Your phone number (with area code) could be
  considered to be a point in R10

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Distance in R1

• The distance between two points, x and y, is
  defined to be d (x , y)

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Distance in R2

• The distance between two points, x(x1,x2) and
• y (y1,y2) is defined to be d (x , y)

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Distance in R3

• For two points, x and y, in R3 we define d (x ,
  y)

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Distance in Rn

• Generalizing, we define the distance between two
  points in Rn to be the square root of the sum of
  the squares of the differences of the
  coordinates. Thus d (x , y)

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Spheres

• Now that we have, in each dimension, an
  understanding of what points are and a formula
  for the distance between them, we have the
  necessary ingredients for spheres

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n-Spheres

• An n-sphere, denoted by Sn, is an n-dimensional
  subset of Rn1
• An n-sphere of radius r with center at a fixed
  point, c, is just the set of points, x, in Rn1
  which are distance r from c. Thus
• Sn x in Rn1 d (x , c) r

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The 0-sphere S0

• S0 is a subset of R1
• S0 consists of just 2 points on the real line,
  namely the point r units to the left of c and the
  point r units to the right of c
• c - r c c r

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The 1-sphere S1

• S1 is a subset of R2
• It is precisely those points in the plane some
  fixed distance, r, from a given center, c
• S1 is just the familiar circle

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The 2-sphere S2

• S2 is a subset of R3
• It is precisely those points in space some fixed
  distance, r, from a given center, c
• S2 is just the familiar sphere

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The 3-Sphere S3

• How are we to picture this?
• Once again, we resort to reasoning by analogy
• This will require us to compile a short list of
  properties of all spheres. This will facilitate
  generalization
• We will also need to introduce the notion of an
  n-Ball

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The n-Ball Bn

• The n-ball is an n-dimensional subset of Rn
  consisting of all points within distance r of
  some fixed point, c
• Bn x in Rn d (x , c) r
• ( Compare Sn x in Rn1 d (x , c) r )

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The 1-Ball B1

• B1 is just a line segment connecting the 2 points
  that make up S0
• Thus, the boundary of B1 is S0

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The 2-Ball B2

• B2 is just the familiar disk
• The boundary of B2 is S1, the circle

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The 3-Ball B3

• B3 is just a solid 3-dimensional ball
• Its boundary is the 2-sphere, S2

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Balls and Spheres

• Thus, we see that the 0-Sphere is the boundary of
  the 1-Ball, the 1-Sphere is the boundary of the
  2-Ball and the 2-Sphere is the boundary of the
  3-Ball
• In general, in each dimension, Sn is the boundary
  of Bn1
• Bn1 consists of Sn together with its interior

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A Property of 1-Spheres

• S1 can be thought of as infinitely many copies of
  S0 with the pair of points (starting at the north
  pole as a single point a degenerate S0) getting
  farther and farther apart until they reach a
  maximum distance (at the equator), thereafter
  getting increasingly close until they come
  together at the south pole (again a degenerate
  S0)

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S1

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A Property of 2-Spheres

• S2 can be thought of as infinitely many copies of
  S1 with the circles (starting at the north pole)
  getting bigger and bigger until they reach a
  maximum radius (at the equator), thereafter
  getting increasingly small until they shrink to a
  point at the south pole

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S2
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Other Properties of S2

• The circumference of a circle with radius R on
  the surface of S2 is less than 2pR
• The maximal circumference for the circle is
  reached when its radius is ¼ of the circumference
  of the 2-sphere (i.e. at the equator)
• When the radius of the circle reaches ½ the
  circumference of S2, its circumference becomes
  zero

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The Earth as an Example

• Approx. Dist. Lat. Approx. 2pR
  
• From N.P. (R) Circum.
• 70 89N 440 440
• 1645 66.5N 10,016 10,330
• 6300 Eqtr 25,120 39,564
• 10955 66.5S 10,016 68,797
• 12,600 S.P. 0 79,128

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A Property of All Spheres

• Each Sn is made up of infinitely many copies of
  Sn-1. They start off as a degenerate sphere (a
  single point) at the north pole, increase in size
  until they reach a maximal radius at the equator
  of the Sn that contains them, thereafter
  shrinking until they, once again, become
  degenerate at the south pole

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Another Property of Spheres

• S1 can be thought of as two copies of B1 glued
  together at their common boundary which is just
  S0
• S2 can be thought of as two copies of B2 glued
  together at their common boundary which is just
  S1
• Sn can be thought of as two copies of Bn glued
  together at their common boundary which is just
  Sn-1

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Decomposing Spheres

• Thus we have seen two different decompositions of
  Sn
• One consists of infinitely many copies of Sn-1
• The other is comprised of two copies of Bn glued
  together along their common boundary which is
  Sn-1

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The Hypersphere I

• Using the first of these decompositions, we can
  view the hypersphere, S3, as infinitely many
  nested copies of S2

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The Hypersphere II

• Using the second of these decomposi-tions, we can
  view the hypersphere, S3, as two copies of B3
  glued together by identify-ing corresponding
  points on their common boundary, S2

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More Properties of Spheres

• Start at any point on a sphere and walk in any
  direction you will eventually end up where you
  started
• If you are standing at any point on a sphere and
  I am standing at the antipodal point, then any
  step you take will be a step in my direction

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How Does This Relate to the Universe?

• Putting all this together, leads one to the
  conclusion (at least naively) that the Universe
  is a hypersphere
• To reach this conclusion, we must assemble a few
  empirically determined and generally accepted
  facts about the Universe

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The Big Bang

• It is generally agreed that the Universe began
  with the Big Bang some 14 billion years ago
• The Universe has been expanding ever since
• Thus, no two galaxies can be farther apart than
  14 billion light years (1 light year? 6 trillion
  miles)

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Sphere of Stars

• Looking out into space, we are looking back in
  time
• If we look out in all directions a distance of r
  miles, we are looking at a 2-sphere of stars all
  r miles from us
• If r miles is 1 light year, for example, we are
  looking at these stars as they were 1 year ago

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Hubbles Law

• All galaxies are receding from each other at a
  rate which is proportional to their distance
  apart
• Current best estimates say that a galaxy that is
  1 billion light years away from us is receding at
  the rate of 1/14 of a light year each year
• Thus, all stars 1 billion L.Y. from us now must
  have been where we were 14 billion years ago

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Hubble II

• Since the rate of recession is proportional to
  the distance, all stars 2 billion L.Y. away are
  receding at a rate of 1/7 of a L.Y. per year
• Thus, these stars, too, must have been where we
  were 14 billion years ago

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Hubble III

• Continuing in this way, we see that Hubbles Law
  implies that all galaxies were at the same point
  14 billion years ago

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The Universe is a Hypersphere

• As we look out into space, the sphere of stars
  starts off (relatively) small and gets larger and
  larger, the farther out we go
• At some point, these spheres stop growing and
  start getting smaller
• Nothing is farther away than 14 billion light
  years, so the sphere of stars with radius 14
  billion L.Y. is a point
• Inescapable conclusion the Universe is a
  Hypersphere!

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The Edge of the Universe

• Note that this description solves a major paradox
• The Big Bang together with Hubbles Law imply
  that the Universe is finite
• What, then, is outside it?
• As S3, like all spheres, has no boundary, there
  is nothing outside

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What Now?

• Either the Universe will go on expanding forever
  or, if gravitational pull becomes sufficient, it
  will eventually stop expanding and start
  contracting
• In the first case, the Universe will grow to be
  larger and larger hyperspheres
• In the second case, the Universe must be an S4
  i.e. a hyper-hypersphere!