Updated slides

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Lecture 11

• Updated slides

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Expansion history S(t)
accelerating
1
Scale Factor S
decelerating
0
Big Bang
Now
Time t
3
An expanding sphere of rocks
Empty Space
Outermost rock
Outward velocity Hubble Law

Empty Space
Empty Space
4
Backwards Gravitational Force
F/9
Backwards Force
3R
F/4
2R
Rock distance
F
R

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Kinetic and Gravitational Energies
Kinetic Energy
V
KE 1/2 mV2
m
R
M
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Gravitational Energy is Negative!
m
Let go from rest
Kinetic energy increases (positive)
m
Gravitational Energy
R
gravitational energy gets more (negative)
R
M
M
G Mm R
GE
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Total Energy, TE, condition for re-collapse
Kinetic Gravitational Total Energy
G Mm R
1/2 mV2
TE
If KE lt GE then TE lt 0 ? re-collapse
If KE gt GE then TE gt 0 ? infinite
expansion
If KE GE then TE 0 ? marginal.
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The Escape Velocity TE 0
G Mm R
1/2 mV2
TE 0
G Mm R
1/2 mV2
2GM R
V2

2GM R
?? ?
Escape velocity
Vesc
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The total mass and density of the sphere
Outward velocity
Vo Ho ? Ro
Mass Volume
Ro
Density
Mass Volume ? Density

Mo 4/3 ? Ro3 ? ?o
Mo total mass in sphere
?o average density of sphere
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Todays Critical Density TE 0
KE GE
Mo 4/3?Ro3 ? ?o
Vo Ho Ro
G Mom Ro
1/2 mVo2
G 4/3 ? Ro3 ?o m Ro
1/2 m Ho2 Ro2
Ho2 ?o
8 ? G 3

3 Ho2 8 ? G
?o,crit
Critical density
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Total Energy Remains Constant
5Ro
KE decreasing (rock slows down)
4Ro
3Ro
GE increasing (less negative)
2Ro
Ro

Maintain colors KE yellow GE blue TE
red Velocity purple Size green
Total Energy KE GE TE
Remains Constant
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Energy vs size for TE 0
2
KE 1/2mV2
Velocity ? ?KE
1
TE GE KE
Energy
0

1
2
3
Size
1
G Mm R
GE
2
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Size Velocity History for TE 0
2
5
4
Size
3
Expansion Velocity V
1
Scale Factor S
2
Velocity
1
0
0
8
2
4
6
0
Big Bang
Time t
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Energy vs size for TE gt 0
KE 1/2mV2
2
Velocity ? ?KE
TE GE KE
1
Energy
0

3
1
2
Size
1
G Mm R
GE
2
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Size Velocity History for TE gt 0
2
5
4
Size
3
Expansion Velocity V
1
Scale Factor S
2
Velocity
1
0
0
8
2
4
6
0
Big Bang
Time t
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Energy vs size for TE lt 0
2
KE 1/2mV2
Velocity ? ?KE
1
Turnaround
Energy
0

3
1
2
Size
1
TE GE KE
G Mm R
GE
2
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Size Velocity History for TE lt 0
2
5
4
Turnaround
3
Size
Expansion Velocity V
1
Scale Factor S
2
Collapse Velocity
Expansion Velocity
1
0
0
14
2
4
6
8
10
12
0
Big Crunch
Big Bang
Time t
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The Three Matter Models
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TE gt 0
6
TE 0
Scale Factor S
4
TE lt 0
2
0
14
2
4
6
8
10
12
0
Big Bang
Time t
Big Crunch
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A Little Algebra
At the start
At any time
GE

TE
GEo

KE
KEo
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A little algebra
M 4/3 ? R3 ?
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A little algebra
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Final Equation

TE
KE
GE

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Final Equation
Density of sphere changes with expansion

TE
KE
GE

Solutions give Expansion History S(t)
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Einsteins Field Equations
8pG c4
gi,j ? Ti,j
x,y,z,t
Space-time curvature
Simple constant
Mass Energy distribution
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Friedmanns Equation
Geometry of Space
Cosmic density
Scale factor
Curvature sign 1 negative (open) 0 zero
(flat) 1 positive (closed)
Curvature radius
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Friedmanns Equation
V H R
R S Ro
1 S
dS dt
H
R S ? Ro
The Newtonian Equation!
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The GR - Newtonian Link
Arthur Milne 1896 - 1950
Georges Birkhoff 1884 - 1944
Bill McCrea 1904 - 1999