Download What is the Shape of the Universe?

By
D.N. Seppala-Holtzman
St. Joseph’s College
faculty.sjcny.edu/~holtzman
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
A Hypersphere??!!!
 Yes, a hypersphere
 Specifically, a 3-dimensional sphere in Euclidean 4-
dimensional space
 Don’t panic
Some Basic Background
 Higher dimensional space
 Points
 Distance
 Spheres
 Balls
Dimensions
 Most people feel comfortable with dimensions 1, 2 and
3
 Mathematicians are comfortable in higher dimensions
because they don’t try to visual them; they reason by
analogy
Dimension Zero
 Dimension zero consists of a single point
 There is no notion of distance because the notion is
vacuous in this dimension
Dimensions: A 1st Attempt
D=1
D=3
D=2
D=4 ????
Dimension One:
1
R
 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
Dimension Two:
2
R
 Consists of two, mutually orthogonal infinite lines
called axes:
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
Dimension Three:
3
R
 Consists of 3 mutually orthogonal axes
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
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
Distance in
1
R
 The distance between two points, x and y, is
defined to be d(x , y) =
| x  y | ( x  y )
2
Distance in
2
R
 The distance between two points, x=(x1,x2) and
y =(y1,y2) is defined to be d(x , y) =
( x1  y1 )  ( x2  y2 )
2
2
Distance in
3
R
 For two points, x and y, in R3 we define d(x , y) =
( x1  y1 )  ( x2  y2 )  ( x3  y3 )
2
2
2
Distance in
n
R
 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) =
( x1  y1 )  ....  ( xn  yn )
2
2
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
n-Spheres
 An n-sphere, denoted by Sn, is an n-dimensional
subset of Rn+1
 An n-sphere of radius r with center at a fixed point, c,
is just the set of points, x, in Rn+1 which are distance r
from c. Thus:
Sn = {x in Rn+1 | d(x , c) = r}
The 0-sphere:
0
S
 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
The 1-sphere:
1
S
 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:
The 2-sphere:
2
S
 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
The 3-Sphere:
3
S
 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
The n-Ball:
n
B
 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 Rn+1 | d(x , c) = r} )
The 1-Ball:
1
B
 B1 is just a line segment connecting the 2 points that
make up S0
 Thus, the boundary of B1 is S0
The 2-Ball:
2
B
 B2 is just the familiar disk
 The boundary of B2 is S1, the circle
The 3-Ball:
3
B
 B3 is just a solid 3-dimensional ball
 Its boundary is the 2-sphere, S2
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
Bn+1
 Bn+1 consists of Sn together with its interior
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)
1
S
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
2
S
Other Properties of
2
S
 The circumference of a circle with radius R on the
surface of S2 is less than 2πR
 The maximal circumference for the circle is reached
when its radius is ¼ of the circumference of the 2sphere (i.e. at the equator)
 When the radius of the circle reaches ½ the
circumference of S2, its circumference becomes zero
Circle in Space vs. Circle on Sphere
The Earth as an Example
Approx. Dist.
From N.P. (R)
70
1645
6300
10955
12,600
Lat.
Approx.
Circum.
89N
399.6
66.5N 10,016
Eqtr
25,120
66.5S
10,016
S.P.
0
2πR
440
10,330
39,564
68,797
79,128
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
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
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
The Hypersphere I
 Using the first of these decompositions, we can view
the hypersphere, S3, as infinitely many nested copies of
S2
The Hypersphere II
 Using the second of these decompositions, we can
view the hypersphere, S3, as two copies of B3 glued
together by identifying corresponding points on their
common boundary, S2
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
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
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)
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
Hubble’s 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
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
Hubble III
 Continuing in this way, we see that Hubble’s Law
implies that all galaxies were at the same point 14
billion years ago
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!
The Edge of the Universe
 Note that this description solves a major paradox
 The Big Bang together with Hubble’s Law imply that
the Universe is finite
 What, then, is outside it?
 As S3, like all spheres, has no boundary, there is
nothing outside
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!
Further Reading
 Poetry of the Universe by Robert Osserman
 The Shape of Space by Jeffrey Weeks