Seonglae ChoContinuous Symmetry of the Action of a Physical System with force has a corresponding Conservation law
Symmetry indicates same physical principles against action (Noether Current’s derivative is 0)
- Symmetry against Space transition (Space translation invariance) → Conservation of momentum → Inertia
- Symmetry against time transition (Time translation invariance) → Conservation of Energy
- Symmetry against rotation (Rotational invariance) - Conservation of Angular Momentum
- Symmetry against Global phase invariance (U(1)) → Charge conservation
This shows that the Conservation law is not absolute and inevitable, but rather a conditional property that is not absolute in all situations and depends on symmetry. In other words, it means that if symmetry is broken, conservation laws may not apply. This can be seen simply when interpreting Redshift, where the time translation symmetry is broken, and the energy conservation law does not seem to hold due to changes in wavelength.
Einstein explained the consistency of physical laws in General Relativity, clarified by Noether's work. This gives symmetry to spacetime transformations by fixing physical laws, including the speed of light, despite the foreign nature of spacetime. The conserved quantity for this symmetry of spacetime transformation is the Stress-energy tensor, and it is conserved without change rate as . In other words, in redshift, while time symmetry is broken, it is interpreted that energy is redistributed in the expanded spacetime that flexibly changes with spacetime symmetry.
Because the theory of relativity explains gravity as the curvature of spacetime itself, it requires modeling the entire curved spacetime gravitational field as a Stress–energy–momentum pseudotensor for the expression of gravitational energy.
Symmetry of physical laws under variation of the path in time
Principle of stationary action → Euler–Lagrange equation
Conservation of Energy
For a static empty universe
However, our universe is not globally symmetric with respect to time since spacetime itself evolves. Global energy conservation does not hold, even though local conservation () still applies.
Contracted Bianchi identities
Noether's theorem
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems published by mathematician Emmy Noether in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries of physical space.
https://en.wikipedia.org/wiki/Noether%27s_theorem
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Special thanks to Alessandro from ScienceClic for his help on animations and input on the script. Check out his excellent video on Noether’s theorem here: • The Symmetries of the universe
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0:00 대칭성이란 무엇인가?
4:25 에미 뇌터와 아인슈타인
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23:20 뇌터의 뒷이야기
24:49 표준 모형 - 힉스, 쿼크 등등
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References:
MathHistory Biography - Emmy Amalie Noether via mathshistory.st-andrews.ac.uk - https://ve42.co/BioNoether
Bartlett, J. G. (2021). The Standard Cosmological Model and CMB Anisotropies. New Astron.Rev. 43 (1999) 83-109 - https://ve42.co/CosmoCMB
Kimberling, C. (1982). Emmy Noether, Greatest Woman Mathematician. Mathematics Teacher, March 1982, Volume 84, Number 3, pp. 246–249. - https://ve42.co/NoetherGreat
Blunck, A., et al. (2016). Mathematik und Gender. Verlag Franzbecker - https://ve42.co/MathGender
Evans, R. J. (1976). The Feminist movement in Germany, 1894-1933. Sage Publications - https://ve42.co/FeministGer
Without Emmy Noether, there would be a huge gap in mathematics and its understanding via www.mpg.de - https://ve42.co/NoetherGap
Dihedral group of order 6 via Wikipedia - https://ve42.co/Dihedral6
Emmy Noether University of Erlangen via Wikipedia - https://ve42.co/Erlangen
Timeline of women in mathematics via Wikipedia - https://ve42.co/WomenMath
Elizaveta Litvinova via Wikipedia - https://ve42.co/LitvinovaE
Sofya Kovalevskaya via Wikipedia - https://ve42.co/KovalevskayaS
Charlotte Scott via Wikipedia - https://ve42.co/ScottC
Cornelia Fabri via Wikipedia - https://ve42.co/FabriC
Emmy Noether, the renowned mathematician who taught 20 years of unpaid classes just because she was a woman via english.elpais.com - https://ve42.co/NoetherPay
Noether, E. (1908). Über die Bildung des Formensystems der ternären biquadratischen Form. Reimer - https://ve42.co/NDissert
Noether, E. (1983). Invariante Variationsprobleme. Gott.Nachr.1918:235-257,1918; Transp.Theory Statist.Phys.1:186-207,1971 - https://ve42.co/InvarProb
Noether, E. (2018). M. A. Tavel’s English translation of “Invariante Variationsprobleme”. arXiv - https://ve42.co/InvarProbEn
Byers, N. (1998). E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws*. Mod.Phys.Lett. A14 (1999) 99-104 - https://ve42.co/SymCon
De Haro, S. (2021). Noether’s Theorems and Energy in General Relativity. Cambridge University Press, 2021 - https://ve42.co/EnergyGR
Rowe, D. E. (2019). Emmy Noether on Energy Conservation in General Relativity. arXiv - https://ve42.co/EnergyCon
Branding, K. (2005). A Note on General Relativity, Energy Conservation, and Noether’s Theorems. Springer - https://ve42.co/GRanEC
Additional References:
https://ve42.co/NoethersAdRefs
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