Transcript GTR Continued

GTR Continued
Sections 29.3 - 29.4
Professor David Marx
Reminders
• Lab this week LAB B1-WA Wave Addition: due
by 4 p.m. Friday
• Examination #3 on Thursday:
– Addresses Chapters 8, 9, 11-13 only
– 29 multiple-choice questions (one 2-pt question
– 30 points possible (120% maximum)
– Professor Khondkar Karim will proctor exam.
• Don’t forget Weekly Reflection #12 due 11/18
• Remember In-class Quiz #6 moved to 11/20
Einstein, GTR, and the Universe
• Einstein’s equation of general relativity:
1
Rmn - gmn R = 8p GTmn
2
• The equation required an expanding universe,
so Einstein arbitrarily added a “cosmological”
constant to avoid this conclusion…
• Recall that galactic nebulas had yet to be
discovered as galaxies, and neither had the
expanding universe!
Friedmann’s Universe 1
1 2 GMm
• Starts with total energy: E = mv 2
r
• Makes 3 simplifying assumptions:
– Universe is essentially empty (on a larger scale)
– Universe is isotropic (no preferred direction)
– Universe is homogeneous (no preferred point)
• and gets:
2E æ v ö 8p G
=ç ÷ r
2
3
mR è R ø
2
Friedmann’s Universe 2
• Inserting E=mc2 and taking into account GTR, the
prior equation results in:
2
2
æ v ö 8p G
kc
=
r
ç ÷
3
R2
è Rø
• where k is the curvature of the universe:
– If k < 0, then universe is closed and bounded
(positively curved).
– If k > 0, then the universe is open and unbounded
(negatively curved).
– If k = 0, then the universe is flat.
Friedmann’s Universe
• Solving this equation, Friedmann
predicted that the universe had to
be expanding if it were to exist.
• Einstein added the “cosmological
constant” to his own field equations
because he could not believe in an
expanding universe. Later Einstein
remarked, “That was the greatest
scientific blunder that I ever made.”
Hubble’s Relationship 1
• Following in the footsteps of others, Hubble
determines recessional velocities from
Doppler shifts.
Dl
v
=
l c
Hubble’s Relationship 2
• Hubble determines distances to “nebulas”
using a variety of means:
– Study of variable stars whose periods are related
to their intrinsic brightness
– Angular diameters by type
– Presence of brightest stars in a galaxy by type
– Presence of globular star clusters
– Presence of supernovas by type (I and II)
Hubble’s Law for Galaxies
• Hubble used Doppler shifts and distances to
study galaxies. He found a linear relationship.
v = HR
Abbé Georges Lamaître
• Introduces the “Big Bang”
• Concludes that time since
universal expansion
began could be found
from Hubble parameter.
• v = HR; now d/v = t;
• v/R = H and d/v = 1/H
• The current value of H
(21-23km/s/Mly) shows
the age of the universe to
around 13.8 billion years.
Hubble’s Law is consistent with the
expansion of the universe.