PHYM432 Relativity and Cosmology
fall 2012
1. Introduction
Dr. David K. Sing
1
PHYM432 General Relativity and Cosmology
Fall 2012 - 2013
Instructor: Dr. David K Sing
Office:
Physics building room 514
Email
sing@astro.ex.ac.uk
Website:
http://www.astro.ex.ac.uk/people/sing
CATS Credit Value: 10
ECTS Credit Value: 5
Duration: T1 - T11
Newman D
26 Sept 2012 to 13 Dec 2012; Wednesday 10:00, Thursday 15:00
20 x 1 hour Lectures, 2 discussion and problems classes
Private Study Time: 78+ hours
Assessment: One 90-minute examination (100%)
T2:00 (week of 13 Jan 2013)
Notes: On ELE & on my personal website
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Aims
The module aims to introduce the student to the special and general theories of relativity,
emphasizing the geometrical interpretation of the theory. Application of the general theory to
the standard cosmological model is also included at the end of the course. The course aims
to develop the minimum necessary advanced mathematical topics needed to understand the
concepts behind the theory, and students will require a good level of mathematical fluency
and intuition in order to fully engage with the material.
Core Text
Lambourne, Robert J., Relativity, Gravitation, and Cosmology, Cambridge University Press,
(2010)
Supplementary Text
Hartle, J., Gravity: An Introduction to Einstein’s General Relativity, Addison-Wesley
Benjamin Cummings, 1st edition (2003)
Roos, Matts, Introduction to Cosmology, 3rd edition, Wiley
Kenyon, I. R., General Relativity, Oxford Science Publications
Assignments
The student is expected to work through a set problems.
BUY THE BOOK!!!
only 30 quid
3
I. Introduction
II. Key aspects of special relativity (2-5)
1. Galilean and Lorentz transformations
2. Length contraction and time dilation
3. Doppler effect
4. Relativistic mechanics
5. 4-Vectors
6. SR Mechanics
III. From SR to GR (6-10)
1. Differential Geometry
2. Accelerated reference frames
3. Spacetime tells matter how to move
4. Geodesics
IV. Curved spaces
1. Differential Geometry
2. Euclidean spaces
3. Curvature in one and two dimensions
4. Intrinsic and extrinsic curvature
5. Riemannian curvature
6. Introduction to tensors
V. GR (11-16)
1. Einstein Field Equations
2. Newtonian Limit
3. Schwarzschild Solution
4. Orbits
5. Black Holes
VI. Cosmology (17-21)
1. The cosmological principle (M10)
2. Robertson-Walker metric
3. Red-shift distance relation
4. The Friedmann equations (M11)
5. Cosmic microwave background
VII. Review class (22)
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BUY THE BOOK!!!
only 30 quid
http://www.amazon.co.uk/Relativity-Gravitation-Cosmology-Robert-Lambourne/dp/0521131383/ref=sr_1_1?ie=UTF8&s=books&qid=1284393064&sr=8-1-spell
Link for PDF of Chapter 1 and homework solutions on ELE & on my website
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BUY THE BOOK!!!
only 25 quid
Homework problems will be given primarily from
Lambourne (contains complete solutions)
I will also likely give problems from Hartle
two classes dedicated to discuss problems
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BUY THE BOOK!!!
only 25 quid
These are for your own benefit, to gain practice working
with the equations and gaining insight into GR
Assignment: Read Lambourne Chapter 1
do problems 1.1, 1.2, 1.3
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Many many good books on this subject
Hartle is highly recommended
Subject matter can be Difficult to understand
Different viewpoints can be Very useful to understand
the material
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Gravity and Modern Physics
• Modern Physics is essentially now two separate,
very different, fundamental theories, GR and QM
• GR is a classical theory (not quantized)
• The entire rest of Modern physics is currently
explained by Quantum Field Theories
(standard model)
Short Range (quantum)
Long Range
Gravity
?
GR
E&M
QED
Maxwell
Weak
QFT
-
Strong
QFT
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Gravity
• Universal attraction between ALL mass and
all Energy (E=mc2)
• Only attractive, Gravity is Unscreened
• Long Range (1/r )
• Weakest of the 4 forces
2
Short Range (quantum)
Long Range
Gravity
?
GR
E&M
QED
Maxwell
Weak
QFT
-
Strong
QFT
10
Gravity
does GR theoretically break down?
• Where
r
r
G~
= 1.62 ⇥ 10
3
c
33
cm
G~
44
=
5.39
⇥
10
s
5
r c
5
c
~c5
19
3
=
1.22
⇥
10
g/cm
= 1.22 ⇥ 1019 GEV
~G2
G
Short Range (quantum)
Long Range
Gravity
? - Theory of Everything
GR
E&M
QED
Maxwell
Weak
QFT
-
Strong
QFT
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Gravity
Quantum
A fundamental Challenge to bring them together
Measurements: cosmological → mm
Requirements:
dynamic spacetime
no preferred reference frames
space-time
Time Problem:
time comes in only through metric,
describing curved spacetime
Technical Difficulties:
highly non-linear theory
mm → 10-19 meters
fixed background (flat spacetime geometry)
splitting of space & time
Not a predictive theory, obvious QG
wave functions are not normalizable
Quantization of space-time itself, what does
that even mean?
Non-linear Schrodinger’s equation, principle
of superposition of states breaks down.
Without Quantum Gravity (Does grav. wave lead to wavefunction collapse?)
Yes. Uncertainty principle violated
No. Gravitational interactions with
quantum (use gravitational measurements
quantized matter could be used for
to give exact time & space locations)
faster than light signals
Alternatively, instead of quantizing GR, Quantum Mechanics would have to change
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General Relativity works:
Einstein hard to prove wrong
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FLT - violation Lorentz invariance
• Neutrino result almost certainly wrong
•
worth checking (latest update: wrong, bad cables)
• Einstein formulated GR from fundamental
philosophical reasoning & deep insight into the laws
of Nature
• FTL would put causality under question
•
•
space-like world lines
•
“exotic” physics to explain
reference frames where they appear to move back in
time (effect before cause)
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dual views of Gravity
Gravity is a field
(force)
Gravity is a manifestation of spacetime curvature
(geometry)
Both views useful when solving problems
Geometric interpretation is stressed here
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General Relativity
Arguably, the geometric interpretation used to
explain gravitational forces is the most elegant,
most beautiful, most insightful view into the laws
of nature anyone has ever had.
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Gravity = Field
mass tells the gravity field what to be
g=
GM r̂
r2
the gravity field tells the mass how
to accelerate
d2 X
F = m 2 = mg
dt
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Gravity = Geometry
matter tell spacetime how to curve
Gµ
8 G
= 4 Tµ
c
curvature of spacetime (curve) = energy & momentum (matter)
spacetime tells matter how to move
d2 X ↵
m
=m
2
d
↵
dX dX
d
d
move = curvature (“straight” line in curved space)
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General Relativity
Consequences are Very rich in unique “weird”
physics
Black Holes
Worm Holes
Time warps
Space-time coordinates
“flip/switch roles” inside Blackhole
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Cosmology
• Mathematical model of the large-scale feature
of the Universe
• All you need is GR
•
•
E&M screened out, SF/WF to short range
Well... for all except the first tinniest fraction of a second (Inflation.. GUT scale)
Short Range (quantum)
Long Range
Gravity
?
GR
E&M
QED
Maxwell
Weak
QFT
-
Strong
QFT
20
Cosmology
• Currently in a Golden age of Cosmology
• Now have a “standard model”
• ΛCDM cosmology
• Basic parameters of the
universe are now fairly well
known.... how old, how much
matter, dark energy, dark
matter
13.75 ± 0.17 yr
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Burning Questions?
• List of questions we’ll try to address before
the end of class
- What happens when you fall into a black hole?
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