Transcript Physics Done Right - english.rejbrand.se

Physics Done Right
Diploma Work (45 p)
• Title
– A clear, concise, and rigorous treatment of
classical physics and relativity theory?
• Aim
– To write a book on classical physics and
relativity theory.
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This Presentation
• Part A: Motivation
• Part B: Preparation
• Part C: The Writing Process
• Part D: The End Result
• Part E: Future Work
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Part A
Motivation
Part A: Motivation
• Issues with Current Literature:
– General Issues
• Lack of coherency
• Mathematics as the language of physics
• Lack of rigor and clarity
– Survey of Existing Works
• Taylor, d’Inverno, Wald, etc.
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Part A: Motivation
• My motto:
“It can’t be that hard!”
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Part A: Motivation
• Two Additional Incentives:
– Responsibility to verify theories
– Learning by writing
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By the way…
An ‘unorthodox’ diploma work?!
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Part B
Preparation
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Part B: Preparation
•
Reference literature includes (but is not limited to):
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Berry, M.V. (1976). Principles of Cosmology and Gravitation. Institute of Physics Publishing, Cambridge (2009). ISBN 0-85274-037-9 (Paperback)
(179 pages)
Conlon, Lawrence (2001). Differentiable Manifolds (Second Edition). Birkhäuser, United States of America (2008). ISBN 978-0-8176-4766-7
(Paperback) (English) (418 pages)
Darling, R.W.R. (1994). Differential Forms and Connections. Cambridge University Press, United States of America (2009). ISBN 978-0-521-46800-8
(Paperback) (English) (250 pages)
D'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford University Press, United Kingdom. ISBN 978-0-19-859686-8 (Paperback) (English)
(383 pages)
Frankel, Theodore (1997, 2004). The Geometry of Physics—An Introduction (Second Edition). Cambridge University Press, United States of America
(2009). ISBN 978-0-521-53927-2 (Paperback) (English) (694 pages)
Ludvigsen, Malcolm (1999). General Relativity—A Geometric Approach. Cambridge University Press, United States of America (2000). ISBN 0-52163976-X (Paperback) (English) (217 pages)
Spivak, Michael (1965). Calculus on Manifolds—A Modern Approach to Classical Theorems of Advanced Calculus. Westview Press (1998). ISBN 9780-8053-9021-6 (Paperback) (English) (146 pages)
Synge, J.L.; Schild, A. (1949). Tensor Calculus. Dover Publications, United States of America. ISBN 978-0-486-63612-2 (Paperback) (English) (324
pages)
Taylor, John R. (2005). Classical Mechanics. University Science Books, United States of America. ISBN 1-891389-22-X (Hardcover) (English) (786
pages)
Wald, Robert M. (1984). General Relativity. The University of Chicago Press, United States of America. ISBN 978-0-266-87033-5 (Paperback)
(English) (491 pages)
Weinberg, Steven (2008). Cosmology. Oxford University Press, United Kingdom. ISBN 978-0-19-852682-7 (Hardcover) (English) (593 pages)
Weintraub, Steven H. (1997). Differential Forms—A Complement to Vector Calculus. Academic Press, United Kingdom (2011). ISBN 0-12-742510-1
(Hardcover) (English) (256 pages)
Einstein, A. (1905). Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? Annalen der Physik, 323(13), 639-641.
Hafele, J., & Keating, R. E. (den 14 Jul. 1972). Around-the-World Atomic Clocks: Predicted Relativistic Time Gains. Science, ss. 166-168.
Hafele, J., & Keating, R. E. (den 14 Jul. 1972). Around-the-World Atomic Clocks: Observed Relativistic Time Gains. Science, ss. 168-170.
Rossi, B., & Hall, D. B. (den 1941 Feb. 1941). Variation of the Rate of Decay of Mesotrons with Momentum. The Physical Review, 59(3), ss. 223-228.
In total more than 4 000 pages.
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Part B: Preparation
• Choice of Topics:
– Classical mechanics
– Electromagnetism
– Special relativity
– Classical differential geometry of curves and
surfaces in ℝ3
– Modern differential geometry of manifolds
– General relativity
– Cosmology
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Part B: Preparation
• Choice of Target Audience:
– Very well familiar with calculus (in one and several
variables), linear algebra, and vector calculus.
– Preferably familiar with abstract algebra,
differential equations (including PDEs), Fourier
analysis, functional analysis, and complex analysis.
– Familiar with the basics of classical mechanics and
electromagnetism (a couple of first-year university
courses will suffice).
• Mathematically inclined audience?
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Part B: Preparation
• Choice of Standards:
– Coherency
– Separation of mathematics from physics
– Rigor and clarity
– Style and clarity
– Basics rather than depth
– Many colourful (3D) illustrations!
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Part C
The Writing Process
Part C: The Writing Process
• Word-processing software:
– Microsoft Word 2010
• Exceedingly obnoxious piece of software:
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Instability
Data corruption
Sporadic loss of characters
Sporadic loss of formatting
Visual bugs
Equation numbering
Heading numbering
Problems with cross-references
A few other bugs:
» PDF export ‘bug’, continuous section break bug, …
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Part C: The Writing Process
• Word-processing software:
– There seems to be a large number of bugs in
Word 2010, but please notice that the fact
• “I experience a lot of problems with the Microsoft
Word 2010 program installed on my computer”
– does not imply
• “The Microsoft Word 2010 software is buggy”.
– However, I have seen similar bugs on other
systems with Word 2007 and 2010.
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Part C: The Writing Process
• Graphics Software:
– AlgoSim 2.0
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Part D
The End Result
Part D: The End Result
• Recall the choice of topics:
– Classical mechanics
– Electromagnetism
– Special relativity
– Classical differential geometry of curves and
surfaces in ℝ3
– Modern differential geometry of manifolds
– General relativity
– Cosmology
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Part D: The End Result
• Statistics:
– The end result is a book consisting of 307 pages
(A4 paper size) containing 83 000 words.
Chapter
Number of Pages
Classical Mechanics
68
Classical Electromagnetism
20
Special Relativity
65
Classical Differential Geometry
78
Manifolds and Tensors
56
Appendices
11
Sum
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Part D: The End Result
• The Title of the Book:
Physics Done Right – An Attempt
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Part D: The End Result
• Specific Benefits:
– Classical mechanics
• Elegant and efficient. Fun facts. All elementary
examples. Kepler’s laws. Gaussian formalism of
gravity.
– Electromagnetism
• Classical laws vs. Maxwell’s equations. EM radiation.
Incompatibility.
– Special relativity
• First naïve, then powerful. The ‘wow effect’. SR
dynamics. Magnetism.
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Part D: The End Result
• Specific Benefits:
– Differential Geometry
• Classical first, modern later.
• The covariant derivative made natural. Also serves as
a motivation for the summation notation.
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Part D: The End Result
Colourful Illustrations
made in AlgoSim
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Part D: The End Result
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Part D: The End Result
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Part D: The End Result
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Part D: The End Result
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Part D: The End Result
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Part D: The End Result
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Part D: The End Result
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Part D: The End Result
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Part D: The End Result
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Part D: The End Result
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Part D: The End Result
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Part D: The End Result
• The Standards:
– Coherency
• Classical mechanics as the foundation. Gaussian
formalism for gravity -> Maxwell’s equations.
• Electromagnetism -> Special relativity
• Preparing for GR
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Part D: The End Result
• The Standards:
– Separation of Mathematics from Physics
• Classical mechanics.
– Full vector formalism. Force fields. Conservative fields.
Gaussian formalism. Work-KE theorem:
1
𝑊≝
𝐅 𝐱 ⋅ 𝑑𝐥 =
Γ
1
𝐅 𝐱 𝑡
0
⋅ 𝐱 𝑡 𝑑𝑡 =
1
= 𝑚 𝑢𝑄2 − 𝑢𝑃2 ≝ Δ𝐸𝑘 .
2
1
𝑚𝐱 𝑡 ⋅ 𝐱 𝑡 𝑑𝑡 = 𝑚
0
0
𝑑 1
𝐱 𝑡
𝑑𝑡 2
2
𝑑𝑡 =
– First example on kinematics: two gravitationally-attracting
bodies.
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By the way…
• Two gravitationally attracting bodies
𝑟0
A
𝑑1 𝑡
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𝑚2
𝑚1
𝑟 𝑡
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B
𝑑2 𝑡
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By the way…
𝑘
𝑟 𝑡 +
𝑟 𝑡
2
= 0,
𝑟 0 = 𝑟0 > 0
where
𝑘 = 𝐺 𝑚1 + 𝑚2 > 0.
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By the way…
Have to invert
𝑝
𝑝 ↦ 𝐹 𝑝 = arctan 𝑝 + 2
,
𝑝 +1
Clearly 𝐹
−1
∃ and 𝐹
−1
:
𝜋
0,
2
𝑝 ≥ 0.
→ 0, ∞ . Define 𝒢 ≔
𝐹 −1 .
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By the way…
Then
𝑟 𝑡 =
𝑟0
𝒢 𝑟0−3
2
2𝑘 ⋅ 𝑡
2
+1
and
𝑡collision =
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𝜋
2 2𝑘
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3 2
𝑟0 .
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By the way…
r(t)
1.2
Distance r (metres)
1.0
0.8
0.6
0.4
0.2
0.0
0
10
20
30
40
50
60
70
80
× 103
Time t (seconds)
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Part D: The End Result
• Rigor and Clarity
– Example: curves
• Style and Clarity
– Examples: boxes, definitions
• Basics rather than Depth
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Part D: The End Result
• The English Language:
– British English
– Oxford (serial) comma
– Logical punctuation
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Part E
Future Work
Part E: Future Work
• The Obvious:
– Write the chapter introducing GR.
– Write the chapter introducing cosmology.
– Find a way to number theorems and equations.
– Proof-read the entire text.
– Consider how to publish it.
• Done.
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Part E: Future Work
Left
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Part E: Future Work
• Software:
– Word-Processing software
• Microsoft Word 2010
– Graphics software
• AlgoSim 2.0
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Thank you for your attention!
www.rejbrand.se
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