1. Announcements Grades from the fourth exam are available on WebCT. Observing for the rest of the semester, until December 9 th, will count for the fourth exam.

2. Expansion of the Universe and Curved Spacetime Expansion of the Universe Geometry Non-Euclidean geometry Geometry of the Universe

3. Hubble expansion v = H 0 d

4. Expansion of the Universe

5. Motion at constant velocity distance = velocity  time velocity = 0.5 cm/s time = distance / velocity = 3 cm/(0.5 cm/s) = 6 s

6. Receding galaxy 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

7. Hubble expansion v = H 0 d Time = distance/velocity = d/H 0 d = 1/H 0 = 1/(71 km/s/Mpc) = 13.8 Gyr

8. 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.

9. 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

10. Geometry How are the diameter and circumference of a circle related? What is the sum of all of the angles in a triangle?

11. Geometry in flat space Circumference = 2   radius  = 3.1415926… 2  = 6.28… The sum of the angles in a triangle is 180º

12. Non-Euclidean Geometry How about in a curved space? Do demo 8C10.50

13. Geometry in positively curved space Circumference < 2   radius The sum of the angles in a triangle > 180º

14. Properties of positively curved space Finite Unbounded No center How do the properties of positively curved space differ from flat space?

15. Geometry in negatively curved space Circumference > 2   radius The sum of the angles in a triangle < 180º

16. 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.