Announcements

• Grades from the fourth exam are available on WebCT.

• Observing for the rest of the semester, until December 9th, will count for the fourth exam.

Expansion of the Universe and Curved Spacetime

• Expansion of the Universe

• Geometry

• Non-Euclidean geometry

• Geometry of the Universe

Hubble expansion v = H0d

Expansion of the Universe

Motion at constant velocity

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6

Time [s]

Dis

tan

ce [

cm]

distance = velocity time velocity = 0.5 cm/s

time = distance / velocity = 3 cm/(0.5 cm/s) = 6 s

Receding galaxy

01

23

45

67

8

0 2 4 6 8 10 12 14

Time [Gyr]

Dis

tan

ce [

Mp

c]

When were galaxies in the same place?

time = distance / velocity = 7 Mpc/(0.508 Mpc/Gyr)

= 13.8 Gyr ago

Velocity =

= 500 km/s

= 0.508 Mpc/Gyr

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.