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

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Announcements
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 and Curved Spacetime

  • Expansion of the Universe

  • Geometry

  • Non-Euclidean geometry

  • Geometry of the Universe




Motion at constant velocity
Motion at constant velocity

distance = velocity  time velocity = 0.5 cm/s

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


Receding galaxy
Receding galaxy

Velocity =

= 500 km/s

= 0.508 Mpc/Gyr

When were galaxies in the same place?

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

= 13.8 Gyr ago


Hubble expansion v h 0 d1
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

  • most of the galaxies approaching her.

  • the same Hubble’s law that we see.

  • about equal numbers of red and blue shifted galaxies.

  • everything rushing away from a point near the Milky Way galaxy.


Two dimensional geometry
Two dimensional geometry looking in our general direction would see

  • 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
Geometry looking in our general direction would see

  • 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
Geometry in flat space looking in our general direction would see

  • Circumference = 2  radius

     = 3.1415926… 2 = 6.28…

  • The sum of the angles in a triangle is 180º


Non euclidean geometry
Non-Euclidean Geometry looking in our general direction would see

  • How about in a curved space?

  • Do demo 8C10.50


Geometry in positively curved space
Geometry in positively curved space looking in our general direction would see

  • Circumference < 2  radius

  • The sum of the angles in a triangle > 180º


Properties of positively curved space
Properties of positively curved space looking in our general direction would see

  • Finite

  • Unbounded

  • No center

How do the properties of positively curved space differ from flat space?


Geometry in negatively curved space
Geometry in negatively curved space looking in our general direction would see

  • Circumference > 2  radius

  • The sum of the angles in a triangle < 180º


Review questions
Review questions looking in our general direction would see

  • 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.


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