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What is the Shape of the Universe

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Title: 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

2
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

3
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

5
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

6
Dimension Zero
  • Dimension zero consists of a single point
  • There is no notion of distance because the notion
    is vacuous in this dimension

7
Dimensions A 1st Attempt
D1
D3
D2
D4 ????
8
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

9
Dimension Two R2
  • Consists of two, mutually orthogonal infinite
    lines called axes

10
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

11
Dimension Three R3
  • Consists of 3 mutually orthogonal axes

12
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

13
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

14
Distance in R1
  • The distance between two points, x and y, is
    defined to be d (x , y)

15
Distance in R2
  • The distance between two points, x(x1,x2) and
  • y (y1,y2) is defined to be d (x , y)

16
Distance in R3
  • For two points, x and y, in R3 we define d (x ,
    y)

17
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)

18
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

19
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

20
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

21
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

22
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

23
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

24
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 )

25
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

26
The 2-Ball B2
  • B2 is just the familiar disk
  • The boundary of B2 is S1, the circle

27
The 3-Ball B3
  • B3 is just a solid 3-dimensional ball
  • Its boundary is the 2-sphere, S2

28
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

29
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)

30
S1

31
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

32
S2
33
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

34
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

35
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

36
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

37
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

38
The Hypersphere I
  • Using the first of these decompositions, we can
    view the hypersphere, S3, as infinitely many
    nested copies of S2

39
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

40
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

41
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

42
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)

43
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

44
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

45
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

46
Hubble III
  • Continuing in this way, we see that Hubbles Law
    implies that all galaxies were at the same point
    14 billion years ago

47
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!

48
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

49
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!
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