Transcript Expansion of the Universe, Curved Spacetime

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Expansion of the Universe and
Curved Spacetime
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Expansion of the Universe
Geometry
Non-Euclidean geometry
Geometry of the Universe
Hubble expansion v = H0d
Expansion of the Universe
Motion at constant velocity
Distance [cm]
3.5
3
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
Time [s]
distance = velocity  time velocity = 0.5 cm/s
time = distance / velocity = 3 cm/(0.5 cm/s) = 6 s
Distance [Mpc]
Receding galaxy
8
7
6
5
4
3
2
1
0
Velocity =
= 500 km/s
= 0.508 Mpc/Gyr
0
2
4
6
8
10
12
14
Time [Gyr]
When were galaxies in the same place?
time = distance / velocity = 7 Mpc/(0.508 Mpc/Gyr)
= 13.8 Gyr ago
Hubble expansion v = H0d
Time =
distance/velocity
= d/H0d
= 1/H0
= 1/(71 km/s/Mpc)
= 13.8 Gyr
An observer at a distance of 3 billion light years
from us looking in our general direction would see
1. most of the galaxies approaching her.
2. the same Hubble’s law that we see.
3. about equal numbers of red and blue shifted
galaxies.
4. everything rushing away from a point near
the Milky Way galaxy.
Two dimensional geometry
• Only two directions: up/down and left/right
: north/south and east/west
• All motion of particles, light confined to two
dimensions
• Examples: black board, piece of paper, surface of
sphere, surface of donut, surface of saddle
Geometry
• How are the diameter and circumference of
a circle related?
• What is the sum of all of the angles in a
triangle?
Geometry in flat space
• Circumference = 2  radius
 = 3.1415926… 2 = 6.28…
• The sum of the angles in a triangle is 180º
Non-Euclidean Geometry
• How about in a curved space?
• Do demo 8C10.50
Geometry in positively curved space
• Circumference < 2  radius
• The sum of the angles in a triangle > 180º
Properties of positively curved space
• Finite
• Unbounded
• No center
How do the properties of positively curved space
differ from flat space?
Geometry in negatively curved space
• Circumference > 2  radius
• The sum of the angles in a triangle < 180º
Review questions
• Suppose it was discovered that the Hubble
constant is twice the value that it is currently
thought to be, how would this change our
estimate of the age of the Universe?
• Compare a flat universe with a positively curved
universe as to whether they are finite, bounded,
and have a center.